Raxion at colliders
Abstract
We study the effective theory of a generic class of hidden sectors where supersymmetry is broken together with an approximate Rsymmetry at low energy. The light spectrum contains the gravitino and the pseudoNambuGoldstone boson of the Rsymmetry, the Raxion. We derive new modelindependent constraints on the Raxion decay constant for Raxion masses ranging from GeV to TeV, which are of relevance for hadron colliders, lepton colliders and Bfactories. The current bounds allow for the exciting possibility that the first sign of SUSY will be the Raxion. We point out its most distinctive signals, providing a new experimental handle on the properties of the hidden sector and a solid motivation for searches of axionlike particles.
pacs:
11.30.Pb (Supersymmetry), 14.80.Mz (Axions and other NambuGoldstone bosons)Saclayt17/014
CERNTH2017032
In this letter we argue that there are generic signs of supersymmetry (SUSY) to be looked for at colliders that have not yet been satisfactorily explored: those associated with the Raxion, the pseudo NambuGoldstone boson (PNGB) of a spontaneously broken Rsymmetry.
Although it is well known that supersymmetry must be broken in a “hidden sector”, its dynamics is left unspecified in the vast majority of phenomenological studies, which instead focus on the “visible sector”, e.g. the MSSM. Here we point out that in an extensive class of models where both SUSY and Rsymmetry are broken at low energy, the hidden sector leaves its footprints in observables accessible to the current experimental program. In particular, we perform a thorough phenomenological study of the Raxion at high and low energy hadron and lepton colliders.
The SUSY algebra contains a single (“Rsymmetry”) under which supercharges transform, , such that components of a given supermultiplet have Rcharges differing by one unit (e.g. gauge fields carry no Rcharge while gauginos have ). Rsymmetry plays a crucial role in models of low energy dynamical SUSY breaking. According to the general result of Nelson and Seiberg, an Rsymmetry must exist in any generic, calculable model which breaks SUSY with Fterms, and if the Rsymmetry is spontaneously broken then SUSY is also broken Nelson and Seiberg (1994). Spontaneous Rsymmetry breaking often occurs also in incalculable models like Affleck et al. (1984a, b). If the SUSY breaking vacuum is metastable, like in ISS constructions Intriligator et al. (2006), then an analogue of the Nelson Seiberg result holds for an approximate Rsymmetry Intriligator et al. (2007). When is explicitly broken by a suitable deformation of the hidden sector, the Raxion gets a mass in addition to the irreducible contribution from supergravity Bagger et al. (1994) but can remain naturally lighter than the other hidden sector resonances.
In light of the above observations, and contrary to what previously explored in the literature, in our phenomenological study we treat the Raxion mass as a free parameter, together with its decay constant. Our analysis shows that such a particle could well be the first sign of supersymmetry to show up at experiments. We also derive modelindependent bounds on the scale of spontaneous Rsymmetry breaking, opening a new observational window on the properties of the SUSYbreaking hidden sector. The intimate connection of the Raxion with the hidden sector dynamics is reflected in its sizeable decay into the Goldstone of spontaneously broken SUSY, the Goldstino. This decay mode provides a way to distinguish the Raxion from other axionlike particles.
Setup
The Raxion is a PNGB realizing nonlinearly the spontaneously broken approximate Rsymmetry of the hidden sector. Its decay constant and mass are a priori free parameters. If we parametrize with the SUSY mass gap of the hidden sector and with the coupling strength between hidden sector states at , the generic size of the SUSY breaking VEV is , an outcome of naive dimensional analysis (NDA) with a single scale and coupling Cohen et al. (1997); Luty (1998). The Raxion decay constant is , while the Raxion mass should satisfy in order for the Raxion to be a PNGB.
As a generic consequence of spontaneous SUSY breaking, a light gravitino is also present in the low energy spectrum. In the rigid limit (i.e. ) the transverse degrees of freedom of the gravitino decouple, leaving a massless Goldstino in the spectrum. The effective action of the Goldstino and the Raxion can be written using the nonlinear superfield formalism of Komargodski and Seiberg (2009) and reads
(1) 
where and carry Rcharge 2 and 1 respectively and satisfy the nonlinear constraints and . As a solution of the first constraint, the bottom component of is integrated out in terms of the Goldstino bilinear and its Fcomponent gets identified with (minus) the SUSYbreaking scale . Analogously, all the degrees of freedom of the chiral field become functions of the Goldstino and its real bottom component , which we identify as the Raxion (see the Appendix for details).
Since the Rcharge of is 1, its effective action differs from the one of a SUSY axion in that a superpotential term is allowed. This is controlled by the dimension three parameter , which is related to the VEV of the superpotential and satisfies the inequality , under the assumption of no extra light degrees of freedom other than the Raxion and the Goldstino Dine et al. (2010); Bellazzini (2016). The superpotential term induces cubic interactions between the Raxion and two Goldstini, proportional to , that lead to an invisible decay channel for the Raxion. The corresponding decay rate of the Raxion into two Goldstini is
(2) 
and it is bounded from above as a consequence of the upper bound on , saturated only in free theories. Our power counting gives , making the width within an factor of the upper limit in Eq. (2). For ordinary axions would instead break explicitly the associated global symmetry, resulting in a suppression of the decay width into Goldstini by extra powers of . Hence, a sizeable invisible decay width is a distinctive feature of the Raxion compared to other axionlike particles.
The Raxion mass is generated by sources of explicit Rsymmetry breaking and can be parametrized as
(3) 
where is the Rcharge of the explicitbreaking spurion , with technically natural. Explicit examples of this mass hierarchy arise by adding suitable Rsymmetry breaking deformations in calculable models of dynamical SUSY breaking like the 32 model Affleck et al. (1985); Nelson and Seiberg (1994); Bellazzini et al. (to appear) or in SUSY QCD at large once the hidden gauginos and squarks get soft masses Martin and Wells (1998); Dine et al. (2016). Moreover, in SUSYbreaking models like the one in Intriligator et al. (2007), the explicit breaking of the Rsymmetry is generically bounded from above () by requiring the SUSYbreaking vacuum to be metastable.
The Rsymmetry breaking contribution (3) can well be expected to dominate over the unavoidable SUGRA contribution arising from the tuning of the cosmological constant Bagger et al. (1994)^{1}^{1}1 The latter arises from a constant term in the superpotential generated by a sequestered sector, which in the rigid limit is completely decoupled from the sector where the Raxion lives. Including gravitational interactions at the linearized level, the two sectors gets coupled via Di Pietro et al. (2014) and the resulting potential after integrating out the auxiliary fields is Since is bounded from above and , the flat space time is recovered by tuning the explicit Rsymmetry breaking parameter . The resulting mass for the Raxion is where in the second equality we used for the gravitino mass. , which gives rise to .
We are now ready to study the couplings of the Raxion with the visible sector fields, which we take to be the MSSM (with matter/Rparity). The superpartners get SUSYbreaking masses from their interactions with the hidden sector, which are controlled by a perturbative coupling . This coupling is a proxy for the SM gauge coupling constants in gauge mediation models Giudice and Rattazzi (1999); Meade et al. (2009) or for Yukawatype interactions in extended gauge mediation, see Evans and Shih (2013) for a review. The scaling of strongly depends on the type of mediation mechanism. We can estimate it as
(4) 
In this letter we assume that gauginos get a mass via their coupling to the hidden sector global current, so that . Notice that if we recover the ordinary gauge mediation scaling where is the number of messengers. Other scaling, e.g. the one of Gherghetta and Pomarol (2011) for Dirac gauginos () will be discussed elsewhere. Besides, whatever the scaling in eq. (4), there is always a large portion of parameter space where the Raxion is lighter than the superpartners, which correspond to . It would also be interesting to depart from the NDA expectation for the scales and and explore models where a large separation between the two is realized.
We consider in the following a small SUSY breaking scale in the range from 1 to a few 10’s of TeV. This regime is welcome for finetuning and Higgs mass considerations. The resulting gravitino mass lies in the window , where the upper limit comes from cosmological and astrophysical bounds on gravitino abundance Pierpaoli et al. (1998); Viel et al. (2005); Osato et al. (2016), while the lower limit comes from collider bounds on gravitino pair production in association with a photon or a jet at LEP Brignole et al. (1998a) and at the LHC Brignole et al. (1998b); Maltoni et al. (2015).
Since the visible sector feels the SUSYbreaking only through effects, we can treat the MSSM superfields linearly and “dress” the Rcharged operators with appropriate powers of the Raxion. We also neglect subleading effects in the explicit Rbreaking, suppressed by powers of . The interactions of the Raxion with the MSSM gauge sector are then
(5) 
where is the field strength superfield carrying Rcharge 1 and labels the SM gauge group, where we defined and . The Majorana gaugino masses are of order by assumption. The coefficients encode the hidden sector contributions to the mixed anomalies of the with the SM gauge groups. For example, we get for , for messengers chiral under and in the of with zero Rcharge (in our NDA ). The contributions to the anomalies from the MSSM fields will be encoded in the full loop functions.
The interactions in the Higgs sector can be written as
(6) 
where are the Higgsino Weyl spinors and the complex Higgs scalar doublets. We have assumed the term to be generated by the hidden dynamics^{2}^{2}2A term in the superpotential would break explicitly for , yielding . , so that the total Rcharge of the Higgses depends on the charge assignments in the sector responsible for generating and . The charge assignment of the visible sector fields is modified by higher dimensional operators in the Kahler like , etc., which lead to suppressed effects that will be neglected in what follows. Notice also that the NDA size of and reflects the well known problem in low energy SUSYbreaking scenarios.
The coupling to the MSSM Higgses proportional to induces, after electroweak symmetry breaking, a small mixing between and the MSSM Higgs boson ^{3}^{3}3Eq. (7) accounts for both mass and kinetic mixing between and . In fact, in the limit , parametrizes the misidentification of as the Raxion after EWSB (given that for the Higgs is Rcharged, ).
(7) 
If we assume the Yukawa interactions in the superpotential to be allowed in the limit of exact (, etc.), the mixing is the only source of couplings between and the SM fermions and we get
(8) 
The same mixing induces
(9) 
where is the SMlike Higgs, as well as extra interactions with the MSSM Higgses whose phenomenological consequences we leave for future work Bellazzini et al. (to appear). Finally, the couplings to sfermions also arise from its mixing with and are proportional to the Aterms. Since we assume all the sfermions to be heavy and the Aterms to be small, these couplings do not play any role in the Raxion phenomenology discussed here.
Finally, we note that all the couplings derived via the spurion analysis can be thought as derivative couplings of the Raxion with the Rcurrent , up to anomalous and explicit breaking terms.
Phenomenology
We now discuss the phenomenological implications of the Raxion. We focus on Raxion masses in the range between and , and we refer to Goh and Ibe (2009) for an LHC study for masses of MeV.^{4}^{4}4A much lighter Raxion would be similar to traditional axionlike particles (see Kim and Carosi (2010) for a review): for the Raxion couplings to fermions are the ones of the DFSZ axion Dine et al. (1981); Zhitnitsky (1980), while for the couplings to fermions are zero at the tree level and the phenomenology is dominated by the couplings to gluons and photons like in the KSVZ model Kim (1979); Shifman et al. (1980). The phenomenological study of the low mass window is left for future work. For definiteness we fix , which allows for an Rsymmetric term and a term from spontaneous breaking. We will comment on the phenomenological differences of the case, where the role of and is reversed. The Majorana gaugino masses cannot be arbitrarily larger than the scale of spontaneous Rbreaking so we take and fix for illustrative purposes the gaugino masses to the GUT universal values TeV (different values do not change the Raxion phenomenology as long as ). For , obtaining such heavy gauginos present model building challenges which are beyond the scope of this paper.
We now discuss the different production modes of the Raxion at the LHC, at LEP and at Bfactories. For the purposes of this paper we ignore Raxion production from SUSY decay chains. As for any axionlike particle, the single production modes scale with and double production ones with :

At the LHC, the resonant (+ SM) production is dominated by gluon fusion. To determine we use the leading order prediction at 13 TeV (including the hidden sector anomaly and the full loop functions for gluino, top and bottom) multiplied by a constant factor of 2.4 Ahmed et al. (2016). For TeV, and we get while for .

Also at the LHC, we have double production from Higgs decays which is driven by the coupling in (9). The BR goes up to for and TeV.^{5}^{5}5Even if potentially important we do not discuss here other pair production mechanisms.

At LEP the Raxion can be produced via its coupling to the boson. At LEP I we consider onshell production which then decays to with BR for and . At LEP II we consider the associated production of or from an offshell and . These cross sections are around for .

Flavor experiments can constrain the Raxion parameter space for . In particular, for we consider Raxion emission in transitions and decays. The are computed from the general result of Hall and Wise (1981), accounting for the mixing of the Raxion with the CPodd Higgs (7) Freytsis et al. (2010), and choosing for reference TeV (we take the formfactor relevant for from Ball and Zwicky (2005)). This yields, for both and final states, for and . The is computed using the standard Wilczeck formula Wilczek (1977). This simple estimate neglects the mixing of the Raxion with mesons and it is reliable for . For the is around .
In Fig. 1 we compare the branching ratios of the Raxion in two extreme cases for the GUT anomaly coefficients . These are modified by the loop functions of the MSSM fields, which in the limit of a light Raxion are encoded in a shift of the anomaly coefficients, e.g. where stands for the number of SM families heavier than . Notice that the contribution from gaugino loops partially cancels the negative one from . For and within our range of Raxion masses the decay widths into gluons, dibosons and diphotons for are suppressed with respect to the case with . In particular the reduced color anomaly explains the smaller crosssection from gluon fusion for .
In Fig. 2 we summarize the present constraints on the Raxion in the plane as well as the most promising processes to search for it at future experiments. For , the most important bounds come from resonant production at the LHC. The most distinctive feature of the Raxion is the large invisible signal strength, which is enhanced at large because of the enhanced BR into Goldstini (see Fig. 1 and Eq. (2)). As a consequence current monojet searches from 8 and 13 TeV data Aad et al. (2015a); Aaboud et al. (2016) constrain the Raxion parameter space. To draw the monojet exclusions we have determined the ratio of the and + jet(s) production cross sections via a MadGraph Alwall et al. (2011, 2014) simulation, for the different missing energy cuts for which the bounds are given in Aad et al. (2015a); Aaboud et al. (2016) (we believe this approximation to be sufficient for our purposes).
At , we show how the decay into bino pairs opens up. Given the light gravitino, the bino promptly decays to , resulting in a final state which is constrained by inclusive searches Aad et al. (2015b). We translate such searches in a bound fb, which results in for and for , because of the reduced production cross section.
For large anomalies the dominant decay mode is in dijets for the full Raxion mass range (we included a constant factor of 1.5 Djouadi (2008)). The branching ratio into diphotons is always around and the diphoton resonant searches Aad et al. (2014); ATL (2016a); Khachatryan et al. (2016a) at 8 and 13 TeV dominate the collider phenomenology for ^{6}^{6}6We checked that the decays into dibosons and can be neglected for our choice of the anomalies.. The net result is a lower bound on around 10 TeV over the full mass range for . Searches for a resonance decaying into dijet Khachatryan et al. (2016b); Sirunyan et al. (2016); ATL (2016b), into at 8 TeV Aad et al. (2015c); Chatrchyan et al. (2013) and into ditau at 13 TeV ATL (2016c); Collaboration (2016) give complementary bounds for , , respectively. Notice that the axion is typically narrow, e.g. for TeV, TeV and .
For small anomalies the LHC constraints are sensibly weaken by the reduced production cross section and by the suppressed branching ratio in diphotons. A lower bound on can still be derived from a combination of , ditau and monojet searches. In particular the monojet searches give the dominant constraints for and are competitive with ditaus searches for .
For , the major constraint comes from the upper bound on Aad et al. (2016), which can be translated in a lower bound on . This bound depends only on the mixing in Eq. (7) and applies to both cases with and . We also include constraints arising from exclusive Higgs decays (e.g. or ) Khachatryan et al. (2017); Aad et al. (2015d). Finally, LEP constraints Abreu et al. (1994); Acciarri et al. (1995a); Rupak and Simmons (1995); Adriani et al. (1992); Acciarri et al. (1995b); Anashkin et al. (1999) are not relevant for the considered in this study.
For , stringent constraints on come from Babar searches on with decaying into tau or muon pairs Lees et al. (2013a, b) or hadrons Lees et al. (2011). These give a bound which goes up to 3 TeV for both and . For , the stronger bound on is given by the LHCb latest result Aaij et al. (2015), but we also considered bounds from Belle and older LHCb data Hyun et al. (2010); Aaij et al. (2013). This results in for which goes up to TeV for , where the is enhanced.
Notice that different values of modify the value of the mixing in Eq. (7). In particular becomes smaller at large reducing the bounds from Higgs branching ratios measurements and transitions. The couplings to quarks and leptons are also dependent, most importantly the signal strength is reduced at large . For (and within our previous assumptions) the Raxion does not couple to SM fermions at the linear level. This makes it generically very difficult to be constrained for . For larger values of and irrespectively of the values of and , diphoton constraints give for large anomalies, while for small anomalies a milder bound on is anyway given by monojet and dijet searches.
We now discuss the relevant signatures for future discovery of the Raxion. The most distinctive one is the decay into two Goldstini which gives a large invisible signal strength (dashed grey lines in Fig 2). This will be probed by monojet searches at the LHC (multijet+MET searches could also be relevant Buchmueller et al. (2015)) and constitutes a very good motivation for the highluminosity LHC program.
Other promising signatures for the future experimental programs are shown in Fig. 2. For large anomalies diphoton will be the most promising final state at the LHC, while for small anomalies ditaus and will be more important. For we show how an improvement of the Higgs coupling measurements down to (which is within the reach of ILC Dawson et al. (2013)) would probe up to 1.5 TeV. Even bigger values of are within the reach of machines like CLIC, CEPC and FCCee which plan to probe Higgs coupling with a precision of roughly Dawson et al. (2013). For large anomalies, an important probe of a light Raxion would be measurements at future lepton colliders. A naive rescaling of the LEP I analysis Adriani et al. (1992), for example, indicates that BRs in the ballpark of could be probed at the FCCee, if ’s will be produced. We notice that the mass window GeV is less constrained by the searches we considered. This could be improved by extending the coverage of resonance searches, in particular , to lower invariant masses.
To distinguish the Raxion from other scalar resonances, a signal would certainly help in combination with a pattern along the lines discussed above. Of course, to reinforce the Raxion interpretation of a possible signal, one would eventually need to find evidence for superpartners.
Conclusions
The possibility that the Raxion could be the first sign of SUSY at colliders is well motivated from theoretical as well as phenomenological considerations.
In this letter we have investigated the low energy dynamics of SUSY breaking sectors with a light Raxion (and gravitino) coupled to the MSSM. Our results are summarised in Fig. 2, where we show how current and future colliders probe the space of Raxion masses and decay constants. We have also identified some promising signatures to cover the currently unconstrained part of the parameter space.
The Raxion constitutes a very interesting prototype of axionlike particles, with couplings that follow from well defined selection rules of the theory, and whose mass can be safely considered a free parameter.
The rich phenomenology of the Raxion certainly deserves further investigation. The Raxion can give rise to nonstandard heavy Higgs decays or SUSY decay chains, it can be a further motivation for high intensity experiments (in its light mass window), and could impact cosmological and astrophysical processes.
Finally, we wish to point out that other appealing features of SUSY, such as unification and dark matter, might find an interesting interplay with a light Raxion, opening new model building avenues. We leave the exploration of this exciting physics for the future.
.1 Acknowledgements
We thank Lorenzo Di Pietro, Zohar Komargodski, David Shih, Riccardo Torre and Lorenzo Ubaldi for useful discussions. The authors thank CERN and the LPTHE for kind hospitality during the completion of this work. F.S. is grateful to the Institut d’Astrophysique de Paris (Iap) for hospitality.
Funding and research infrastructure acknowledgements:

B.B. is supported in part by the MIURFIRB grant RBFR12H1MW “A New Strong Force, the origin of masses and the LHC”;

A.M. is supported by the Strategic Research Program High Energy Physics and the Research Council of the Vrije Universiteit Brussel;

F.S is supported by the European Research Council (Erc) under the EU Seventh Framework Programme (FP7/20072013)/Erc Starting Grant (agreement n. 278234 — ‘NewDark’ project).
Appendix
For convenience of the reader, we report here the explicit expressions of the constrained superfields and , that satisfy and Komargodski and Seiberg (2009); Dine et al. (2010)
(10) 
(11) 
(12) 
(13) 
(14) 
From Eq. (1) one has , where the dots here and in Eq. (12) stand for terms with more derivatives and fermions. Notice that our convention for the metric tensor is , and the one for , is defined by and , where are the Pauli matrices and (these conventions imply ).
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