# Light Sgoldstino: Precision Measurements versus Collider Searches

###### Abstract

We study sensitivity of low-energy experiments to the scale of supersymmetry breaking in models with light sgoldstinos — superpartners of goldstino. The limits on may be obtained from direct and indirect measurements of sgoldstino coupling to photons, leptons, mesons and nucleons. There are three sources of constraints: () astrophysics and cosmology; () precision laboratory experiments at low energies; () rare decays. We discuss only processes with real sgoldstinos. For sgoldstino lighter than a few MeV and superpartner masses of the order of electroweak scale, astrophysical and reactor bounds on are significantly stronger than limits which may be reached at future colliders. In models with heavier sgoldstino (up to 5 GeV), constraints from flavor conserving decays of mesons are complementary to ones coming from collider experiments. The most sensitive probes of sgoldstinos are flavor violating processes, provided that flavor is violated in squark and/or slepton sector. It is shown that sgoldstino contributions into FCNC and lepton flavor violation are strong enough to probe the supersymmetry breaking scale up to GeV, if off-diagonal entries in squark (slepton) mass matrices are close to the current limits in MSSM.

## 1 Introduction

Superpartners of goldstino — longitudinal component of gravitino — may be fairly light. In a variety of models with low energy supersymmetry they are lighter than a few GeV. Such pattern emerges in a number of non-minimal supergravity models [1, 2] and also in gauge mediation models if supersymmetry is broken via non-trivial superpotential (see, e.g., Ref. [3] and references therein). To understand that superpartners of goldstino may be light, it suffices to recall that in globally supersymmetric theories with canonical Kähler potential and in the absence of anomalous abelian gauge factors, the sum of scalar squared masses is equal to the sum of fermion squared masses in each separate sector of the spectrum. Since goldstino is massless, its spinless superpartners (scalar and pseudoscalar particles, and , hereafter, sgoldstinos) are massless too; they are associated with a non-compact flat direction of the scalar potential. Higher order terms from the Kähler potential contribute to sgoldstino masses. Provided these terms are sufficiently suppressed, sgoldstinos remain light. Of course, these arguments in no way guarantee that sgoldstinos are always light, but they do indicate that small sgoldstino masses are rather generic. The theoretical discussion of sgoldstino masses is contained, e.g., in Ref. [4]; here we merely assume that sgoldstinos are light and consider their phenomenology.

Sgoldstinos couple to MSSM fields in the same way as goldstino [5]; constraints on their couplings may be translated into the limits on the supersymmetry breaking parameter .

There are several papers devoted to astrophysical [6], cosmological [7] and collider [8, 9, 10] constraints on models with light sgoldstinos. However, the role of light sgoldstinos in low-energy laboratory measurements has not been studied in detail. To the best of author’s knowledge, the only paper discussing this issue, Ref. [11], concentrated on sgoldstino contribution (as well as the contribution from light gravitino) into anomalous magnetic moment of muon. Here we consider a variety of low energy experiments sensitive to light sgoldstinos.

In this paper we identify those experiments which are most sensitive to different sgoldstino vertices for various sgoldstino masses. These experiments provide constraints on the corresponding coupling constants. These constants are in fact proportional to the ratios of soft terms (squark and gaugino masses, trilinear coupling constants) and . The latter parameter is related to the gravitino mass in a simple way, ; small corresponds to light gravitino (). Hence, the constraints derived in this paper are of importance for models with light gravitino, whereas sgoldstino effectively decouple from the visible sector in models with heavy gravitino.

In principle, there are both flavor-conserving and flavor-violating sgoldstino couplings to fermions. We present our results in the form of bounds on setting soft flavor-conserving terms to be of the order of electroweak scale, as motivated by the supersymmetric solution to the gauge hierarchy problem. Flavor-violating couplings are governed by soft off-diagonal entries in squark (slepton) squared mass matrices. When evaluating bounds on we set these off-diagonal entries equal to their current limits derived from the absence of FCNC and lepton flavor violation in MSSM [12]. In this way we estimate the sensitivity of various experiments to the supersymmetry breaking scale.

We consider only low-energy processes with sgoldstinos on mass-shell. Processes with sgoldstino exchange deserve separate discussion, though we do not expect that the results obtained in this paper will be altered significantly. Also, behind the scope of this paper are loop processes with virtual sgoldstinos running in loops (for instance, -mixing, , etc.). These processes were analyzed in Ref. [13] in models with heavy sgoldstinos. Constraints on obtained in Ref. [13] are significantly weaker than ones presented in our paper, so the loop processes are less sensitive to in models with heavy sgoldstinos. However, models with light sgoldstinos have not been analyzed in detail yet, though it was pointed out in Ref. [13] that enhancement effects may appear if sgoldstinos are light. In view of the results obtained in this paper we also find it conceivable that light virtual sgoldstinos may give significant contributions into rare processes considered in Ref. [13].

Let us briefly review the current status of experimental limits on . If one ignores sgoldstino, then in models with light gravitino the strongest direct current bound on is obtained from Tevatron, GeV [14]. In models with light sgoldstinos, collider experiments become more sensitive to the scale of supersymmetry breaking. Namely, LEP and Tevatron provide constraints at the level of 1 TeV on the supersymmetry breaking scale in models with of order of 20 GeV [9, 10]. The most stringent cosmological constraint comes from Big Bang Nucleosynthesis [7]: models with light gravitino, eV, that corresponds to GeV, are disfavored if sgoldstinos decouple at temperature not less than MeV ( MeV). It has been argued in Ref. [6] that among the astrophysical constraints, the strongest one comes from SN1987A: the gravitino mass is excluded in the range eV for 1 keV MeV and in a wider range eV for keV. These excluded intervals correspond to GeV and GeV, respectively.

In this paper we consider various constraints on couplings of light ( GeV) (pseudo)scalars to SM fields coming mostly from astrophysics and direct precision measurements. So, we partially fill the gap between constraints coming from collider experiments and cosmology.

As there are flavor-conserving and flavor-violating interactions of sgoldstino fields, we have to consider both flavor-symmetric and flavor asymmetric processes. Let us outline our results referring to these two cases in turn.

We begin with constraints independent of assumptions concerning breaking of flavor symmetry. As expected, strongest bounds arise from astrophysics and cosmology, that is GeV, or eV, for models with keV and MSSM soft flavor-conserving terms being of the order of electroweak scale. For the intermediate sgoldstino masses (up to a few MeV) constraints from the study of SN explosion and reactor experiments lead to TeV. We will find that for heavier sgoldstinos, low energy processes (such as rare decays of mesons) provide limits comparable to ones from colliders but valid for different sgoldstino masses.

As concerns flavor-asymmetric processes, we find that these are generally very sensitive to light sgoldstino. Namely, with flavor-changing off-diagonal entries in squark (slepton) squared mass matrix close to the current bounds, direct measurements of decays of mesons (leptons) provide very strong bounds, up to TeV (valid at GeV), which is much higher than bounds expected from future colliders. If off-diagonal entries are small, the limits on become weaker: they scale as square root of the corresponding off-diagonal elements.

We will see that the rates of processes with one sgoldstino in final state are proportional to , whereas the rates of processes with two sgoldstinos in final state are proportional to . Hence, under similar assumptions about soft terms governing sgoldstino couplings, processes with one sgoldstino are more sensitive to the supersymmetry breaking scale. Nevertheless, the coupling constants entering one-sgoldstino and two-sgoldstino processes are generally determined by different parameters, so the study of two-sgoldstino processes is also important.

Further progress in the search for sgoldstino is expected in several directions. Among the laboratory experiments, the most sensitive to flavor-conserving sgoldstino coupling for sgoldstino lighter than a few MeV are experiments with laser photons propagating in magnetic fields and reactor experiments. For heavier sgoldstinos, measurements of partial widths exhibit the best discovery potential. If flavor violation in MSSM is sufficiently strong (say, at the level of current limits), the most promising is the study of charged kaon decays.

This paper is organized as follows. In section 2 the effective lagrangian for sgoldstinos is presented and sgoldstino decay modes are described. In section 3 we derive various constraints on the parameter of supersymmetry breaking by considering low energy processes. There we study separately processes with one and two sgoldstinos in final states (sections 3.1 and 3.2, respectively). First, we discuss astrophysical and cosmological limits on sgoldstino interactions (section 3.1.1). Then we present laboratory bounds coming from search for light (pseudo)scalars in electromagnetic and strong processes (section 3.1.2). In sections 3.1.3 and 3.1.4. we discuss rare decays with one sgoldstino in final state due to flavor-conserving and flavor-violating sgoldstino couplings to SM fermions, respectively. Sections 3.2.1 and 3.2.2 are devoted to rare meson decays with two sgoldstinos in final state. Our conclusions and comparison of the results with ones coming from collider experiments are presented in section 4.

## 2 Effective lagrangian

Let us introduce the effective lagrangian for light goldstino supermultiplet: scalar , pseudoscalar and goldstino . The free part reads

There exist two types of interactions in the low-energy effective theory involving sgoldstino fields: these are terms that couple one sgoldstino [5, 11, 9, 10] and two sgoldstinos [9], respectively, to SM gauge fields (photons, gluons) and matter fields (leptons , up- and down-quarks and ). Terms involving one sgoldstino are

(1) | |||

The direct coupling of two sgoldstinos is described by

(2) | |||

Here and are gaugino masses; for down-quarks , whereas for up-quarks ; , and are LR-, LL-, and RR-soft mass terms in squark squared mass matrix and for convenience we take them real. In what follows we do not discuss neutrino, so the corresponding couplings are omitted. Note that in MSSM the flavor-conserving one-sgoldstino coupling constants satisfy , where are fermion masses and are corresponding soft trilinear coupling constants. Off-diagonal soft terms , and are subject to constraints from the absence of FCNC and lepton flavor violation (see, e.g., Ref. [12]).

The first part of the effective lagrangian, Eq. (1), is suppressed by , whereas the second one, Eq. (2), is proportional to , so processes with two sgoldstinos are very rare. The most stringent bounds on come from processes with one sgoldstino in final state. Nevertheless, as we will see, the absence of processes with two sgoldstinos gives rise to constraints on supersymmetry breaking parameter comparable to bounds from high-energy experiments. The latter constraints are, strictly speaking, independent of the constraints coming from one-sgoldstino processes: one-sgoldstino and direct two-sgoldstino processes are governed by and , , respectively.

Let us discuss decay modes of light sgoldstino. First, sgoldstino decay into two photons is always open [9],

(3) |

Second, in models where sgoldstinos may decay into gravitino pairs; however, the corresponding rates are suppressed by squared ratio of sgoldstino mass and in comparison with the decay into two photons, hence this mode may be disregarded. Third, relatively heavy sgoldstinos () decay into gluons (light mesons) with larger width than into photons because of color enhancement and because the corresponding coupling is proportional to gluino mass which is usually the largest among the gaugino masses, i.e. . When analyzing hadronic decay modes of light sgoldstinos ( GeV), corresponding rates into quarks and gluons should be rewritten in terms of light mesons. This step will be presented below. Fourth, sgoldstino can decay also into light leptons if this process is allowed kinematically (). Since the corresponding coupling constants are proportional to fermion masses these rates are suppressed by a factor apart from the phase space volume [10],

(4) |

Consequently, depending on MSSM mass spectrum, sgoldstino masses and the value of the supersymmetry breaking parameter , there are three possible situations in experiments where light (pseudo)scalar particle appears. This particle may live long enough to escape from a detector. For instance, in the theory with the superpartner scale of order 100 GeV and TeV this behavior would be exhibited by (pseudo)scalar particle with mass less than 10 MeV, at which sgoldstino width is saturated by two-photon mode. Another case is when (pseudo)scalar particle decays within detector into two photons or leptons. Apart from these cases, there is also a possibility of the decay into two gluons (quarks). For relatively light sgoldstinos (but with masses exceeding 270 MeV), the dominant hadronic decay is into two pions, while for heavier sgoldstinos and channels become available. Furthermore, there would be effects emerging due to mixing.

Let us estimate branching ratios of hadronic and photonic decay channels neglecting threshold factor. In order to estimate sgoldstino coupling to hadrons we make use of chiral theory of light hadrons. There are two different sources of sgoldstino-meson couplings in the effective lagrangian (1): interaction terms with gluons and coupling to quarks. We evaluate contributions from these two sources into meson-sgoldstino interactions separately.

First, we have to relate gluonic operators entering Eq. (1) to meson fields. We make use of the correspondence

(5) |

derived in Ref. [15]. Here is momentum of pion pair created with zero total angular momentum, ; is the -function of QCD, is the pion isotopic amplitude,

and quarks and mesons are considered massless. At higher energies also and pairs may be created by gluonic operator.

There is one more relation [16],

(6) |

where is a tensor dual to gluonic one, is a neutral pseudoscalar meson (, ) and is a normalization factor; MeV and is a parameter responsible for flavor symmetry breaking ( for , for ).

In fact, the lagrangian (1) describes sgoldstino interactions at the superpartner scale. Sgoldstino coupling constants at low energies may be obtained by making use of renormalization group evolution. Thus for the gluonic operator one has

Hence, we estimate the matrix element of the gluonic operator between the scalar and meson pair as

(7) |

and in a similar way we estimate the matrix element of another gluonic operator between the pseudoscalar and meson

(8) |

Note, that these matrix elements are highly suppressed by squared sgoldstino or meson masses.

Since direct sgoldstino coupling to quarks contributes also to meson production, we remind basic relations of chiral theory

(9) |

where

(10) |

If we parameterize sgoldstino couplings to the triplet of light quarks as

with and being matrices of the corresponding coupling constants (which are read off from Eq. (1)), then the standard procedure (see, e.g., Ref. [17]) gives the following low-energy effective lagrangian

(11) |

to the leading order in mesonic fields included in matrix . The constant is related to quark condensate as and may be evaluated from the masses of kaon and quarks, . We account only for one-sgoldstino terms since others are suppressed by sgoldstino masses and additional inverse power of .

The lagrangian (11) consists of two parts. The first one,

(12) |

is pseudoscalar sgoldstino mixing with , , and mesons, while the second one,

(13) | |||

describes scalar sgoldstino decays into mesons. Note that sgoldstino couplings with two different mesons is suppressed by off-diagonal term in squark mass matrix. In what follows we will not consider processes where real sgoldstino decays into such modes.

Now let us estimate matrix elements between sgoldstino and meson (i.e., sgoldstino-meson mixing) as a sum of two quantities, Eq. (8) and Eq. (12), while the amplitude of the scalar sgoldstino decay into pairs of light mesons is evaluated as a sum of Eq. (7) and Eq. (13). Let us compare contributions of gluon and quark operators into sgoldstino couplings to mesons. As an example, for the ratio of the corresponding contributions into coupling of the scalar to neutral pions and into pion-pseudoscalar mixing we obtain

These ratios are larger than 10 for . Hence, gluonic operators give rise to stronger coupling of light sgoldstinos to light mesons, as compared to sgoldstino-quark interactions.

Let us evaluate the rate of the scalar sgoldstino decay into light mesons, assuming that this decay is allowed kinematically. As an example, for the neutral pion mode we obtain

Taking into account only the largest contribution from the gluon operator and neglecting the threshold factor we estimate the ratio of rates of sgoldstino decays into photons and mesons,

We see that this ratio is smaller than 1 at . Since in most models gluino is several times heavier than photino, for sufficiently heavy sgoldstinos hadronic modes usually dominate over photonic one.

Let us estimate now the contribution of pion-sgoldstino mixing into pseudoscalar sgoldstino width. Recall that the pion width is almost saturated by the two-photon decay mode. Then

(14) |

where the two-photon width of virtual pion is taken at and may be approximated as

With account of only leading contributions from gluonic operator we obtain

(15) |

As discussed above, , so at and light () we obtain that the ratio (15) is numerically . In the opposite case of heavy () the ratio becomes even larger. Hence mixing with pions gives negligible contribution to sgoldstino decay into photons (unless ; we do not consider this case). The only exception is the degenerate case, when sgoldstino and pion masses are close and this branching becomes of order 1. (In the case of strong degeneracy there is also a correction to pion life-time which may give rise to a constraint on ). We do not consider this unrealistic situation. The interference with -meson gives nothing new. Indeed, Eq. (15) scales as which is invariant under the variation of meson mass, if the meson width is (almost) saturated by anomalous decay into two photons. Decay via neutral kaon is also negligible because of large kaon life-time.

To conclude this section we summarize the situation with sgoldstino branching ratios. Let us begin with scalar sgoldstino. In Figure 1

we present scalar sgoldstino branching ratios into photons, leptons and neutral pions evaluated for four different sets of supersymmetry breaking soft terms, , , . To determine photonic and leptonic widths we make use of Eqs. (3) and (4), while the width into two neutral pions is calculated according to Eq. (5) generalized to non-zero pion masses. Estimating hadronic sgoldstino partial width we account only for and decay modes. Other hadronic modes may be considered in the same way. Ratios between different hadronic channels are determined by chiral theory.

Scalar sgoldstino lighter than 270 MeV almost always predominantly decays into two photons. At sgoldstino mass close to or , rates of the decays into pairs of corresponding leptons become comparable to the two-photon rate and even exceed the latter in models with large trilinear soft terms. Far from the lepton mass, the corresponding lepton branching ratio decreases as . At sgoldstino masses exceeding 270 MeV hadronic modes emerge. Their rates are somewhat higher than the rate of the two-photon decay except for models with large , in which the photonic mode always dominates.

As regards pseudoscalar sgoldstino, it does not have the decay mode into two pseudoscalar mesons to the zero order in . Hence at its hadronic decay modes are suppressed unless is quite large (well above 1 GeV). In what follows we consider photonic and leptonic decay channels of the pseudoscalar sgoldstino only.

## 3 Searches for light sgoldstino

In accordance with the discussion of sgoldstino effective lagrangian presented in the previous section, there are two types of processes we are interested in. In the processes of the first type only one sgoldstino emerges while in the processes of the second type a pair of sgoldstinos appears in the final state. These processes are governed by different coupling constants and will be considered in turn.

### 3.1 Processes with one sgoldstino in the final state

#### 3.1.1 Bounds from astrophysics and cosmology

In subsections 3.1.1 and 3.1.2 we consider mainly pseudoscalar sgoldstino, though almost all constraints are valid for the scalar sgoldstino as well; a few exceptions will be pointed out.

Light pseudoscalar particles appear in particle physics models in various contexts; a well known example is an axion. There are numerous cosmological, astrophysical and laboratory bounds on interactions of light pseudoscalars which apply to sgoldstino. For completeness we collect in sections 3.1.1 and 3.1.2 the most stringent of these bounds and translate the bounds on sgoldstino coupling constants into bounds on supersymmetry breaking parameters and . Let us write the interactions of sgoldstino with photons and fermions as follows

(16) |

Then the limits on and imply limits on .

In what follows we set GeV in our quantitative estimates, since superpartner scale is expected to be close to the electroweak scale as motivated by supersymmetric solution to the hierarchy problem in SM.

One of the sources of pseudoscalars are stars: light pseudoscalars are produced there by Primakoff process, that is conversion in external electromagnetic field. Another place of sgoldstino creation is galactic space where magnetic fields produce pseudoscalars from propagating photons.

In “helioscope” method, a dipole magnet directed towards the Sun is used. Inside the volume with strong magnetic field, solar pseudoscalars can transform into X-rays by inverse Primakoff process. An alternative method, “Bragg diffraction”, was applied in SOLAX experiment to detect solar pseudoscalars. The absence of anomalous X-ray fluxes from SN1987A related to possible pseudoscalar conversion into photons in galactic magnetic field gives the strongest constraint on . Since this limit is valid only for unrealistically light pseudoscalar ( eV), we consider the helium-burning life-time of Horizontal Branch Stars (HBS) in globular clusters as the most sensitive probe of at very small .

There are two more constraints on coming from cosmology and astrophysics. Light sgoldstinos are thermally produced in the early Universe via Compton process . Photons from sgoldstino decays contribute to the photon extragalactic background, if sgoldstinos outlive matter-radiation decoupling. If, on the other hand, sgoldstinos decay before matter-radiation decoupling, produced photons may heat electrons leading to distortion of CMBR spectrum, which is experimentally studied well enough to exclude wide range of at corresponding sgoldstino masses. The experiments on photon background and cosmic microwave background radiation, being combined, exclude a strip in plane (see Ref. [22]). In Table 1 we present the corresponding limits for two typical values of .

All these
constraints ^{1}^{1}1See also Ref. [22] for
constraints on coming from
Deuterium fission by scalars decaying into two photons.
on are collected in
Table 1. The
limits on
are obtained at GeV and scale as square
root of . For completeness, we included in
Table 1 also the limits obtained in
Ref. [6] by considering SN1987A.

Let us proceed with sgoldstino coupling to electrons. Restrictive limits come from delay of helium ignition in low-mass red giants. There are also two limits on coupling to electrons from bremsstrahlung process and Compton process in stars: these processes lead to energy loss of stars and are constrained by helium-burning life-time of Horizontal Branch Stars. Note that the life-time of HBS gives stronger constraints on electron coupling to scalar than to pseudoscalar.

Let us turn to (pseudo)scalar coupling to nucleons. In order to relate the corresponding constant to we make use of the analogy to axion. Then effective lagrangian reads

(17) |

where denotes the nucleon dublet and

(18) |

The energy loss of Horizontal Branch Stars via Compton process . Also, nucleon-sgoldstino coupling leads to shortening of SN1987A neutrino burst. gives rise to a bound on

Astrophysical constraints on sgoldstino-fermion interactions are presented in Table 2. Bounds on are obtained at GeV and scale as , . Note that the region GeV allowed by SN explosion [26] is not ruled out by astrophysical arguments or direct measurements if sgoldstino is relatively heavy (10 keV10 MeV) and its interactions conserve flavor (see below).

For constraints coming from Big Bang Nucleosynthesis see Ref. [7].

#### 3.1.2 Laboratory bounds on very light sgoldstinos

Let us now consider direct laboratory limits on couplings of very light sgoldstinos.

The first set of bounds on comes from the study of laser beam propagation through transverse magnetic field. The production of real sgoldstinos would induce the rotation of the beam polarization, while the emission and absorption of virtual sgoldstinos would contribute to the ellipticity of the laser beam. Such effects have not been observed and their absence implies a constraint on pseudoscalar-photon coupling. There is also a constraint on the interaction of a pseudoscalar particle with photons coming from experiments on photon regeneration. In these experiments, light pseudoscalars produced via Primakoff effect penetrate through optic shield and then transform back into photons (“invisible light shining through walls”). Similar scheme is applied in NOMAD experiment. The results are presented in Table 3 at GeV; limits on