1 Introduction

LPT Orsay 16-07


A 750 GeV Diphoton Signal from a Very Light Pseudoscalar in the NMSSM

Ulrich Ellwanger and Cyril Hugonie

Laboratoire de Physique Théorique, UMR 8627, CNRS, Université de Paris-Sud,

Univ. Paris-Saclay, 91405 Orsay, France

School of Physics and Astronomy, University of Southampton,

Highfield, Southampton SO17 1BJ, UK

LUPM, UMR 5299, CNRS, Université de Montpellier, 34095 Montpellier, France

The excess of events in the diphoton final state near 750 GeV observed by ATLAS and CMS can be explained within the NMSSM near the -symmetry limit. Both scalars beyond the Standard Model Higgs boson have masses near 750 GeV, mix strongly, and share sizeable production cross sections in association with b-quarks as well as branching fractions into a pair of very light pseudoscalars. Pseudoscalars with a mass of  MeV decay into collimated diphotons, whereas pseudoscalars with a mass of  MeV can decay either into collimated diphotons or into three resulting in collimated photon jets. Various such scenarios are discussed; the dominant constraints on the latter scenario originate from bounds on radiative decays, but they allow for a signal cross section up to 6.7 fb times the acceptance for collimated multiphotons to pass as a single photon.

1 Introduction

In December 2015 the ATLAS and CMS collaborations have reported excesses in the search for resonances decaying into pairs of photons for diphoton invariant masses around 750 GeV [1, 2]. In ATLAS, excesses appeared in the two bins 710–750 GeV (14 events vs. 6.3 expected) and 750-790 GeV (9 events vs. 5.0 expected), with a local significance of (assuming a large width of  GeV; in the narrow width approximation). In CMS, excesses appear in the bin 750–770 GeV for photons in the EBEB category (5 events vs. 1.9 expected) and EBEE category (6 events vs. 3.5 expected), but less in the bin 730-750 GeV (4 events vs. 2.1 expected for photons in the EBEB category, 1 event vs. 4.0 expected for photons in the EBEE category, considered as less sensitive). The local significance of the excesses is for CMS in the narrow width approximation.

The global significances of the signals of are not overwhelming and compatible with statistical fluctuations. Still, the fact that the region of invariant diphoton masses is very similar for ATLAS and CMS has stirred quite some excitement resulting in a huge number of possible explanations. (The number of proposed models exceeds the number of observed signal events.)

Fits to the combined data should, in principle, also consider the informations from diphoton searches at 8 TeV [3, 4] where a mild excess was observed by CMS. However, the extrapolation of signal cross sections from 8 to 13 TeV depends on the assumed production mechanism [5, 6, 7, 8, 9]. Assuming the production of a resonance around 750 GeV by gluon fusion (ggF), combined fits to the signal cross sections at 13 TeV are in the range 2-10 fb [5, 6, 7, 9], with slightly better fits and a larger signal cross section assuming a larger width of 30-45 GeV [5, 6, 9].

It is notoriously difficult to construct a consistent model for such a resonance “”: Its production channel in proton proton collisions is typically assumed to be ggF through loops of colored particles. If these are the quarks of the Standard Model (SM), would decay into them leaving little branching fraction for -decays into , which has to be generated by loop diagrams as well.

Accordingly simple two Higgs doublet (or MSSM) extensions of the Standard Model, which could contain a resonance near 750 GeV [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], require additional scalars or vector-like fermions whose loops generate the coupling of to gluons and/or (unless -parity is broken [24, 25]). Large Yukawa couplings are required for a sufficiently large cross section, which risk to generate new hierarchy problems/Landau singularities (unless compositeness is invoked). Also in the Next-to-Minimal supersymmetric extension of the Standard Model (NMSSM) it has been argued [26, 27, 28] that additional vector-like quark superfields have to be introduced. In [29] a two-step decay cascade involving the two pseudoscalars of the NMSSM with masses of about 750 GeV and 850 GeV has been proposed which requires, however, to tune the corresponding mixing angle close to 0.

A different approach towards an explanation of the diphoton events is to consider that a single photon in the detector can represent a collimated bunch of photons (typically two of them) which originate from a single very light state, for instance a light pseudoscalar  [5, 30, 31, 32, 33, 34, 35, 36, 37]. Then the observed processes correspond to an initial resonance decaying into a pair , where must be well below 1 GeV for the resulting photons to be sufficiently collimated (see below). This scenario opens the possibility to explain the diphoton events in different models which can accomodate resonances and a light pseudoscalar . In this paper we show that the simple -invariant NMSSM belongs to this class of models. (This has also been observed in [38].)

In the NMSSM (see [39, 40] for reviews), two CP-even Higgs states beyond the Standard Model-like Higgs (subsequently denoted as ) can play the role of a resonance . In terms of weak eigenstates, a singlet-like state can have a large coupling to a pair of mostly singlet-like pseudoscalars , originating from a cubic singlet self coupling in the superpotential (see below). However, a coupling to quarks or gluons inside protons has to be induced by a mixing of with one of the two SU(2) doublet-like Higgs states. If this state is , the mixing reduces the couplings of to SM particles (notably and ) and is severely constrained [9] by the measured signal rates by ATLAS and CMS [41]. An alternative is that mixes strongly with the other “MSSM”-like CP-even state . Then the physical eigenstates – preferably both of them with masses near 750 GeV – can profit from an enhancement of the couplings of to -quarks by , leading to sufficiently large signal cross sections into the (and hence diphoton) final state via associated production with -quarks. Given the diphoton mass resolution of the detectors and the slightly preferred large width of the excess it is clear that two (narrow) CP-even states near 750 GeV, mixtures of and , can also provide a good fit to the data. (A similar scenario has been discussed in [42].) For one of the benchmark points presented below (BP1) the signal originates, however, from one CP-even state only, the other one being significantly heavier.

A light pseudoscalar can appear in the NMSSM in the form of a pseudo-Goldstone boson (PGB). A priori two global symmetries can lead to such PGBs: First, a Peccei-Quinn symmetry emerges in the limit  [40, 43, 44, 45]. However, is required for the couplings of the heavy Higgs states to . Second, the scalar potential of the NMSSM is invariant under an -symmetry [40, 46, 47, 48] if the soft supersymmetry breaking trilinear couplings and vanish, leading to a PGB due to its spontaneous breakdown by the phenomenologically required vacuum expectation values. We find indeed, that the interesting part of the parameter space of the NMSSM corresponds to small values of and . However, since the -symmetry is broken by radiative corrections to the scalar potential involving the necessarily non-vanishing gaugino masses and trilinear couplings and , it helps only partially to explain a very light pseudoscalar . Still, it represents a “go-theorem” showing that a standard supersymmetric extension of the SM – without additional vector-like quarks and/or leptons – could explain the observed diphoton excess.

Different assumptions on the mass of can be made. For one set of scenarios we assume  MeV, just below the threshold. These scenarios lead to visibly displaced vertices from the decays. For a large value of the NMSSM trilinear coupling , the signal can originate from a single Higgs state near 750 GeV. For smaller values of , the signal can originate from two Higgs states with masses near 750 GeV. For another set of scenarios we assume  MeV, not far from the mass. For near 550 MeV, mixes with the meson and inherits its decays into and ; the latter lead to photon-jets. The average separation in rapidity of the diphotons and the two leading photons from will be studied. For near 510 MeV, constraints from searches for radiative decays into by CLEO [49] are alleviated, but estimates of the decay widths are more uncertain. But in both cases the life time is short enough avoiding macroscopically displaced vertices, and two Higgs states near 750 GeV can generate a signal.

In the next section we describe with the help of analytic approximations to the mass matrices (including only the dominant radiative corrections) which region in the parameter space of the NMSSM can generate the diphoton events. In section 3 we discuss various constraints from low energy physics on light pseudoscalars, and discuss separately the different scenarios. Benchmark points are presented with the help of the public Fortran code NMSSMTools [50, 51]. For the different masses we study the average separation in rapidity of the diphotons and the two leading photons from , which allows to estimate the corresponding acceptances. In the final section 4 we summarize and discuss possible alternative signatures, which could help to distinguish different scenarios if the excess survives the next runs of the LHC.

2 Parameter regions with diphoton-like events at 750 GeV in the NMSSM

We consider the CP-conserving -invariant NMSSM. The superpotential of the Higgs sector reads in terms of hatted superfields


Once the real component of the singlet superfield develops a vacuum expectation value (vev) , the first term in generates an effective term


The soft SUSY–breaking terms consist of mass terms for the gaugino, Higgs and sfermion fields


as well as trilinear interactions between the sfermion and the Higgs fields, including the singlet field


The tree level scalar potential can be found in [40], from which the mass matrices in the CP-even and CP-odd sectors can be obtained. Once the soft Higgs masses are expressed in terms of , and using the minimization equations of the potential, the mass matrices depend on the six parameters


Initially, the CP-even mass matrix is obtained in the basis of the real components of the complex scalars after expanding around the vevs and . It is convenient, however, to rotate by an angle in the doublet sector sector into in the basis :


The advantage of this basis is that only the component of the Higgs doublets acquires a vev and that, for typical parameter choices, it is nearly diagonal: has SM-like couplings to fermions and electroweak gauge bosons, the heavy doublet field is the CP-even partner of the MSSM-like CP-odd state , while remains a pure singlet. The mass matrix in the basis has the elements


where and


is the mass squared of the MSSM-like CP-odd state . denotes the dominant radiative corrections due to top/stop loops,


where and .

As discussed in the introduction, we intend to describe the diphoton signal at  GeV by a mixture of the two states and . Then, both diagonal matrix elements and should have values close to . Furthermore we will be interested in the -symmetry limit . This implies the relations (for )




The matrix element inducing mixing is given by


and the matrix element inducing mixing by


Next we turn to the CP-odd sector. The CP-odd mass matrix contains always a Goldstone boson which will be eaten by the boson. The remaining CP-odd states are a singlet , and the “MSSM”-like SU(2)-doublet . In the basis , in the -symmetry limit , the CP-odd mass matrix is given by


Obviously has a vanishing eigenvalue , and is diagonalised by an angle with (for )


An important quantity will be the (reduced) coupling of to down quarks and leptons, which is obtained through the mixing of with . Since the reduced coupling of the MSSM-like state is given by , one obtains


Radiative corrections to the tree level potential and hence to the CP-odd mass matrix include terms proportional to the electroweak gaugino masses and , and terms proportional to the soft SUSY breaking trilinear couplings and . These corrections break the -symmetry present for , which is expected since is not invariant under scale transformations. Hence, depending on the scale where is assumed to hold, is a pseudo-Goldstone boson with a mass of typically a few GeV. For small, but one can obtain  MeV or  MeV as it will be assumed in the next section.

Finally we note that, for the parameter region considered below, the dominant contribution to the coupling of to scalars originates from the quartic coupling . After shifting by its vev one obtains


Next we observe that eqs. (2.11) and (2.16) allow to express in terms of : From (2.11) one finds


where (2.16) was used in the last step. Inserting  GeV one obtains


In the next section, for the scenarios with  MeV, we will obtain upper bounds on from upper bounds for the from CLEO [49]. These will thus imply upper bounds on according to (2.19). On the other hand a large signal rate, generated by a mixture of the states and decaying into , requires to be as large as possible. Accordingly and should saturate corresponding upper bounds.

If the 750 GeV signal is generated by a superposition of signals of two nearby physical states formed by the system, their mass splitting should not be too large, preferably of . Then the matrix element given in (2.12) should be as small as possible. With already determined, this implies as small as possible, preferably close to the lower bound  GeV from the LEP lower bound on higgsino-like charginos. Then (2.16) requires that is relatively small. (Simultaneously, this avoids a strong push-down effect on the mass of the SM-like Higgs boson from mixing, which is induced by the matrix element given in (2.13).) Finally the condition (2.10) on fixes .

The remaining NMSSM parameters in (2.5) are and . Both -symmetry breaking parameters have an impact on the mass of the pseudo-Goldstone boson . We find that one can chose small values of and such that assumes the desired value; due to radiative corrections to the scalar potential the precise value of depends on the other -symmetry breaking parameters , , and . Herewith all NMSSM parameters are nearly uniquely determined.

3 Viable scenarios with a light NMSSM pseudoscalar

As discussed in the introduction we will study scenarios with different values of the mass of a light pseudoscalar, denoted subsequently by . Constraints on such a light NMSSM pseudoscalar with a mass below  GeV have been discussed previously in [54, 52, 55, 56, 57, 53, 58, 59]. Strong constraints originate from the mediation of FCNCs. Assuming minimal flavour violation, flavour violating couplings of still originate from SUSY loops involving stops, sbottoms and charginos and depend on the corresponding masses and trilinear couplings like . These contribute notably to -physics observables like , and . We have implemented the computation of these and many more -physics observables and some -physics observables in the code NMSSMTools [50, 51] following the update in [59] and checked that, for the scenario presented here, the constraints are satisfied due to the mostly singlet-like nature of and the relatively heavy SUSY spectrum.

For near 210 MeV, additional strong constraints originate from rare flavour changing processes . (In [53] it has been argued that the corresponding constraints exclude scenarios with  MeV, where the branching fraction of into is sizeable.) We have verified the assertion in [38] that, for suitable choices of soft SUSY breaking parameters, the coupling responsible for these processes (see [52]) can be arbitrarily small1. Light pseudoscalars have been searched for in radiative decays by CLEO in [60]; these are also verified by NMSSMTools_4.9.0 and satisfied by the benchmark points given below.

Due to the mostly singlet-like nature of , its contributions to the muon anomalous magnetic moment are negligibly small. However, for and assuming relatively light slepton masses of 300 GeV, the scenarios below can reduce the discrepancy between the measured value and the Standard Model to an acceptable level.

Further constraints stem from possible production in and decays. The relevance of bounds on light pseudoscalars (or axion-like particles) from searches for at LEP (where a photon can correspond to a bunch of collimated photons) has been investigated in [58]. These bounds constrain the loop-induced coupling . This coupling is also constrained by the upper bound on  [61]. We have checked that in our cases this coupling is about four orders of magnitude below the bounds derived from [58, 61]. Searches for have been undertaken by ATLAS using 4.9 fb of integrated luminosity at 7 TeV c.m. energy in [62] for  MeV. One can assume that the corresponding upper bound on applies to our scenario as well, which leads to . If imitates a single photon, bounds on should be respected. In our scenarios we require , hence these constraints are well satisfied. Notably this small branching fraction has no impact on the measured signal rates of into the other Standard Model channels, which agree well with the Standard Model predictions.

Additional constraints depending on will be discussed in the corresponding subsections below.

3.1 near 210 MeV

For a light , too light for hadronic final states ( below ), the possible decays are into , and the loop induced decay into . The couplings of to Standard Model fermions are obtained via mixing with as discussed in eqs. (2.14) and (2.15) in the previous section, and lead to a reduced coupling of to leptons , see (2.19). These couplings determine also the partial width into . For a sizeable branching fraction into , the decay into must be kinematically forbidden. On the other hand, for  MeV the remaining decays into and lead generically to a too small total width implying, for a boosted with an energy of about 375 GeV, a decay length larger than the size of the detectors (unless mixes strongly with as discussed in [38]). However, for very close to , the loop contribution of muons to the width reaches a maximum. It is given by (neglecting all other contributions; see, e.g., [63])


with and, for ,


accordingly it increases with (remaining finite for ). We find that, for near or slightly above 210 MeV, the partial width dominated by the muon contribution is large enough to dominate the width leading to a .

The total width depends then essentially on its reduced coupling to muons related to via (2.19). First we consider a scenario with a total width of  GeV, leading to a decay length of for an energy of 375 GeV of about 2 m. Given that the distance of the EM calorimeter cells to the interaction point is larger than 1.3 m for the ATLAS and CMS detectors (depending on the angle ), one can estimate that somewhat more than 60% of all pseudoscalars decay before the EM calorimeter cells.

This scenario requires , in which case runs into a Landau singularity at about 400 TeV where the NMSSM would require a UV completion (e.g. GMSB). Then a single Higgs state near 750 GeV is able to generate a visible signal. (The second Higgs state is heavier near 1 TeV and has a significantly smaller production cross section. A scenario where a single Higgs state near 750 GeV is responsible for the signal and another Higgs state is far below 750 GeV is not possible: Then the lighter state would generate a larger signal, which is excluded.) For a large enough production cross section of the state near 750 GeV from its coupling to -quarks it must have a dominant (MSSM-like) component. Still, for a large enough branching fraction into , the mixing angle (2.12) in the heavy scalar Higgs sector must not be too small and, notably, the coupling in (2.17) must be large. Both of these conditions are satisfied for , which is required if a single state should generate a visible signal.

Suitable values of , and for the desired masses and mixings are given by a benchmark point BP1 in Table 1. (Since the mass of the second heavy Higgs state is near 1 TeV and not near 750 GeV, these values deviate somewhat from the ones obtained in the previous section. Radiative corrections of can require corresponding readjustments of these values.) Since is , the NMSSM-specific uplift of the Standard Model like Higgs mass at low is not available. Then the Standard Model like Higgs mass of  GeV requires large radiative corrections as in the MSSM.

As stated above and discussed in [38], the squark masses and can be chosen such that flavour violating couplings of are suppressed. In order to generate simultaneously large enough radiative corrections to the Standard Model like Higgs mass, both parameters have to be relatively large in the multi-TeV range. Possible numerical values are also indicated in Table 1. The remaining NMSSM specific parameters and are chosen small, such that the (depending somewhat on ) is below , and sufficiently close to such that the total width of is large enough, i.e. that its decay length at 375 GeV is small enough: For the BP1 in Table 1 with  MeV we get  GeV and  m, for which we estimate that of all decays take place before the EM calorimeters. ( denotes the average distance to the calorimeter cells of  m.) For the production cross sections of the Higgs state at 750 GeV we find from SuShi_1.5.0 [64] (at NNLO with MMHT2014 PDFs)  fb,  fb, and from NMSSMTools we find with a total width of of  GeV. Together with a we obtain a signal rate of  fb. This signal rate remains to be multiplied by an acceptance for the diphotons to simulate a single photon in the detector. This issue will be discussed for all scenarios in section 3.3; for the time being the signal rates appear with a factor in Table 2.

If we assume a slightly smaller value of  MeV, decreases to  GeV leading to  m, reducing the percentage of decays before the EM calorimeters to and hence the signal rate by .

Scenarios with smaller values of are also possible. Then, however, the reduced coupling of to leptons is smaller (see (2.19)), and the total width decreases. Hence the decay length increases, and a smaller fraction of ’s decay before 2 m. This loss can be compensated for if two states and with large production cross sections and branching fractions into contribute to the signal.

The benchmark point BP2 is of this type, where we take , nearly (but not quite) small enough for the absence of a Landau singularity below the GUT scale. For  MeV the total width is  GeV, leading to  m. We estimate that then only of all decays take place before the EM calorimeter cells. On the other hand, two Higgs states and with masses near 730 GeV and 762 GeV contribute to the signal. Both are strong mixtures of the pure MSSM-like and singlet-like states. For , the production cross section is  fb, and . For , the production cross section is  fb, and . Together with a we obtain a signal rate of  fb times , as shown in Table 2.

Couplings and mass parameters BP1 BP2 BP3 BP4
0.528 0.212 0.0332 0.0644
1.65 0.75 0.121 0.215
9.57 16.8 15.5 14.5
(GeV) 138.5 101.1 102.3 111.3
(GeV) 32.2 15.6 0.0 0.0
(GeV) 1.16
(TeV) 6 7.5 2 3
(TeV) -3.48 -3.95 3 3
Table 1: Parameters for the four benchmark points. The soft Susy breaking gaugino masses are  GeV (500 GeV for BP1),  TeV,  TeV, all squarks are assumed degenerate, and all slepton masses are 300 GeV (with vanishing trilinear couplings). More digits (for all parameters) than shown here are necessary in order to reproduce given in the Table 2 below, and in order to obtain suffiently small for BP1 and BP2, see the text.
(GeV) 122.1 124.3 123.7 122.2
(GeV) 750 730 744 740
(GeV) 1003 762 750 750
(MeV) 211.3 211.1 548.7 510.3
(GeV) 763 747 748 745
(GeV) 757 749 752 749
(fb) 4.8 2.2 1.7 1.9
(fb) 36.8 67.2 44.7 44.9
(fb) 0.1 1.8 2.0 1.9
(fb) 0.2 52.5 54.3 44.3
0.51 0.66 0.082 0.21
0.72 0.53 0.048 0.16
0.74 0.73 0.72 0.66
( GeV) 1.74 0.65 7500 19000
at 375 GeV 2 m 5.5 m 0.18 mm 0.08 mm
Signal cross section (fb) 4.6 3.7 3.4 6.7
Table 2: Higgs masses, production cross sections and branching fractions for the 4 benchmark points. For the points BP1 and BP2, the signal cross section takes into account losses from decays beyond 2 m according to a factor with in m. For the points BP3 and BP4 the includes the . denotes the acceptance for the di- or multiphotons from two pseudoscalars to simulate two single photons in the detector as discussed in section 3.3. allows for a theoretical error of  GeV. For BP4, and are the unreliable parton level results (at NLO).

3.2 at 510-550 MeV

The partial widths of a light pseudoscalar in this mass range can be estimated employing two complementary approaches. To begin with one can ask what one would obtain within the parton model, extrapolated into the nonperturbative domain of QCD. First, for a reduced coupling of to leptons as considered below, the partial width of into muons can still be computed reliably and is


The loop induced partial width of into is  GeV and hence negligibly small. At NLO QCD the partial width of into strange quarks is about  GeV and the loop induced width into gluons of the same order as the width into . These widths can only be rough estimates, however.

An alternative approach is to consider the case  MeV, where one can expect that mixes with the meson with a mass of 547.85 MeV. (The possible rôle of for the decays of a light pseudoscalar has been indicated earlier in [54] without quantitative statements, however.) Mixing with the meson of a lighter with  MeV has been considered in [38], where Partial Conservation of Axial Currents (PCAC) or the sigma model for light mesons is employed in order to determine the off-diagonal element of the -meson mass matrix; the same formalism will be used here for mixing for  MeV.

First we discuss this latter case, where the results can be considered as more reliable. Only subsequently we turn to the case  MeV, motivated by the alleviation of constraints from radiative decays in this mass range, see below. There, however, estimates of partial widths of are more speculative.

For  MeV, the relevant mass matrix of the system reads in the basis


For a small mixing angle ,


the eigenstate contains a small component:   . For the partial widths of one obtains then


The dominant decays are [61]


Next we require that the -induced decays into or of the eigenstate dominate its width into , since we ignore the unreliable widths of into strange quarks or gluons in this subsection. (Since the latter decays can also generate or final states, this assumption is conservative.) This leads to


In order to estimate the mixing matrix element above we use, following [38], PCAC. There one introduces the SU(3) flavour currents where denote the SU(3) generators. Assuming that is a pure octet, satisfies


with  MeV. At the quark level one has


where is the (anomalous) U(1) current whose divergence involves the meson. Using these relations, one can re-write the coupling of to strange quarks in the Lagrangian (proportional to the corresponding Yukawa coupling )


where we have dropped the terms . From (3.11) one can read off


Then the request (3.8) becomes, again for and using (3.5),


This estimate can be refined by including mixing with the meson, the anomalous U(1) current and the loop-induced coupling of to , where is the QCD field strength2. The additional contribution to leads to a replacement of the right hand side of (3.13) by  MeV.

Assuming such a small mass difference, the decay length of is below a mm, and its branching fractions are the ones of given in (3.2) above. For , constraints from rare  decays are no longer relevant. However, since has couplings to -quarks , constraints from the search for the radiative decays by CLEO in [49] apply (and are more relevant than searches for ).

The can be obtained from the Wilczek formula [65, 66]


where and is a correction factor . The upper bound of CLEO [49] on is at the 90% CL level, or at the 95% CL level. Applying this bound to the , (3.2) gives


as used above. From (2.19) one finds that must then also be quite small, leading to relatively small branching fractions of the heavy Higgs states and into . Hence both of these states should contribute to the signal.

Next we consider the decays induced by its mixing with where decays as in (3.2). The decays into give diphotons plus muons, but due to the escaping neutrinos this final state will not allow to reconstruct the masses of the original resonances near 750 GeV. In addition to the mode, the mode leads to photon jets. The compatibility of such photon jets with a single photon signature in the detectors has been discussed in detail in [35]. In particular, due to the enhanced probability for photon conversions into in the inner parts of the detectors, such scenarios can be distinguished from single photons (or even diphotons) once more events are available. Adding both modes, about of all decays lead to di- or multi-photons. The resulting signal cross section remains to be multiplied by the acceptamce for the di- or multi-photons to fake a single photon discussed in section 3.3.

The parameters, masses, branching fractions, production and signal cross sections of a corresponding benchmark point BP3 are shown in Tables 1 and 2.

If differs by a few tens of MeV from the mass it becomes more difficult to estimate its decays; its mixing angle with the on-shell meson using PCAC as above becomes tiny. Its Yukawa couplings to Standard Model fermions are obtained through mixing with the (heavy) MSSM-like pseudoscalar . At the parton level and for , the relative couplings squared and hence the corresponding widths of are dominantly into (), into via top quark loops (), and into (). The hadronic or decays of can then be considered as being mediated by the CP-odd isospin and color singlet interpolating composite fields and . Both are known components of the wave function in Fock space, and the most reasonable assumption is that their hadronisation (decays into physical hadrons and ) proceeds again with branching fractions similar to the ones of .

The partial width for the sum of these decays of is less clear, however. It is relevant, since it competes with the width of into and determines consequently the branching fraction via the above interpolating fields relative to the . Since the widths for the above mentioned decays of into or pions are small (being electromagnetic or suppressed by isospin), one must assume that the widths for the decays of via the above interpolating fields are also smaller than estimated from the couplings squared at the parton level as at the beginning of this section. A quantitative statement is difficult, however, without a nonperturbative evaluation of the relevant matrix elements between physical states.

On the other hand, the sum of the couplings squared of to or and hence the sum of the partial widths of into or (for both of which -like branching fractions are assumed) is considerably larger than into : At NLO one has . Hence, reducing the sum of the partial widths of into or by a factor 1/10 leaves us still with a dominant .

In the scenario where differs by a few tens of MeV from the mass we will make the assumption that the reduction of the width of the decays is not too dramatic, i.e. the relevant branching fractions of can be parametrized as


where the factor is not too small .

Let us have another look at the searches by CLEO which were performed separately for the , and final states. The windows for the invariant masses were chosen differently for different final states,  MeV and  MeV. ( is fitted to a double Gaussian function centered at .) No candidates were found in the (background free) and final states, but two events in with  MeV just below the window. Also a mild excess of events for  MeV is observed. These events are not numerous enough to allow for the claim of a signal, but we conclude that the and final states do not lead to stronger upper limits on for  MeV than the limit on from the remaining final state. After translating the 90% CL upper limit from the latter final state into a 95% CL upper limit, we find from only the final state