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Kaluza-Klein gravitino production with a single photon at e^+ e^- colliders

In a supersymmetric large extra dimension scenario, the production of Kaluza-Klein gravitinos accompanied by a photino at e^+ e^- colliders is studied. We assume that a bulk supersymmetry is softly broken on our brane such that the low-energy theory resembles the MSSM. Low energy supersymmetry breaking is further assumed as in GMSB, leading to sub-eV mass shift in each KK mode of the gravitino from the corresponding graviton KK mode. Since the photino decays within a detector due to its sufficiently large inclusive decay rate into a photon and a gravitino, the process e^+ e^- ->photino + gravitino yields single photon events with missing energy. Even if the total cross section can be substantial at sqrt(s)=500 GeV, the KK graviton background of e^+ e^- ->photon + graviton is kinematically advantageous and thus much larger. It is shown that the observable, sigma(e^-_L)-sigma(e^-_R), can completely eliminate the KK graviton background but retain most of the KK gravitino signal, which provides a unique and robust method to probe the supersymmetric bulk.

a detector due to the sufficiently large inclusive decay rate ofγ → γG, the process e + e − →γG yields single photon events with missing energy. Even if the total cross section can be substantial at √ s = 500 GeV, the KK graviton background of e + e − → γG is kinematically advantageous and thus much larger. It is shown that the observable ∆σ LR ≡ σ(e − L e + R ) − σ(e − R e + L ) can completely eliminate the KK graviton background but retain most of the KK gravitino signal, which provides a unique and robust method to probe the supersymmetric bulk.

I. INTRODUCTION
The standard model (SM) has been thoroughly tested in various experiments even if the Higgs boson remains as the only missing ingredient. In the theoretical view point, however, the SM has several unsatisfactory aspects such as the gauge hierarchy problem: The Higgs boson mass near the electroweak scale requires a fine tuning to eliminate quadratically divergent radiative corrections. Low-energy supersymmetry is known to cancel the quadratic divergence by introducing a supersymmetric partner for each SM particle [1]. Supersymmetry protects the electroweak scale Higgs mass from the Planck scale, as the chiral (gauge) of anti-deSitter space [3]: The hierarchy problem is explained by a geometrical exponential factor. Very interesting is that these extra dimensional models can lead to distinct and rich phenomenological signatures in the future colliders, characterized by low-energy gravity effects [4,5]. In ADD case, for example, the multiplicity of gravitons below an energy scale E is proportional to (ER) δ (= M 2 Pl E δ /M δ+2 S ), which is extremely large and compensates the small gravitational coupling.
An economical description of new physics to solve the gauge hierarchy problem would introduce either low-energy supersymmetry or extra dimensions. Nevertheless supersymmetric bulk is theoretically more plausible [6,7] since the most realistic framework of extra dimensional models, string/M theory [8], indeed possesses supersymmetry as a fundamental symmetry. Moreover, extra dimensions can play the role of supersymmetry breaking on a hidden brane [9] or in the bulk by Scherk-Schwarz compactification [10].
Obviously phenomenological signatures of supersymmetric bulk are crucially dependent on how many supersymmetries survive on our brane below the scale M S . One interesting pos-sibility is that a single supersymmetry is softly broken on our brane such that our low-energy effective theory yields supersymmetric spectra as in the minimal supersymmetric standard model (MSSM). One distinctive feature of this scenario is the presence of the gravitino, the superpartner of the graviton. Since this gravitino also propagates in the full dimensional space as it belongs to the same supermultiplet with the graviton, we have gravitino Kaluza-Klein modes on our brane. The soft and spontaneous breaking of a supersymmetry results in the mass shift between a graviton KK mode and the corresponding gravitino KK mode, Here the four-dimensional Planck mass M Pl scales the strength of gravitino coupling and Λ SUSY is the supersymmetry breaking scale. If low-energy supersymmetry breaking is assumed, e.g., in the gauge mediated supersymmetry breaking (GMSB) scenario, the resulting mass shift is very light: For Λ SUSY ∼ 100 TeV, Λ 2 SUSY /M Pl ∼ 1 eV. Restricting ourselves to the ADD scenario, we have almost continuous spectrum of KK gravitinos with the zero mode mass at sub-eV scale. In Ref. [11], the four-dimensional effective theory in a supersymmetric ADD scenario has been derived, including the couplings of the bulk gravitino KK states to a fermion and its superpartner. At e + e − colliders, the virtual exchange of KK gravitinos can occur only in the selectron pair production which was shown to substantially enhance the total cross section and change the kinematic distributions.
Another distinctive signature of KK gravitinos is their production at high-energy colliders. A superlight gravitino, which becomes the stable lightest supersymmetric particle (LSP), escapes any detector, leading to missing energy events. Moreover the decay modes of a supersymmetric particle X are now changed. Even if the X is the lightest among supersymmetric partners of the SM particles, e.g., the photino, a new decay mode ofγ → γG is opened and dominant. As shall be discussed later, this decay rate is large enough for the photino to decay within a detector. Therefore, the process e + e − →γG yields a typical signature of a single photon at large transverse momentum. And the summation over all possible extra-dimensional momenta yields a sizable production rate characterized by the M S scale. This process has kinematic advantages over the selectron pair production in case the selectron is too heavy to be pair-produced.
Of great significance is to signal not only the extra dimensions, but also the supersymmetric extra dimensions, i.e., KK gravitinos. Unfortunately, single photon events with missing energy in this scenario have two more sources, the SM process of e + e − → γνν and the KK graviton production of e + e − → γG. With an appropriate cut to reduce the Z-pole contributions of the SM, the KK graviton production can be compatible with the SM background at the future e + e − collider. However the KK gravitino production rate is smaller than the KK graviton case by an order of magnitude. This is due to the kinematic suppression by the production of the massive photino while the dependence of the M S and δ is the same for both the KK graviton and gravitino production. Total cross section alone cannot tell whether the bulk possesses supersymmetry or not. We shall show that the observable completely eliminates the KK graviton effects, but retains most of the KK gravitino effects. This is because in a supersymmetric model the sign of the coupling of a left-handed electron with a photino (and a selectron) is opposite to that of a right-handed electron (a left-handed anti-electron). The coupling with gravitino, which is gravitational, does not depend on the fermion chirality. Therefore, the scattering amplitudes of the t-and u-channel diagrams, where both couplings are involved, have opposite sign for e − L and e − R beams. As the t-and u-channel amplitudes are added to the s-channel one, mediated by a photon, the total scattering amplitudes are different for the left-and right-handed electron beam. For the KK graviton production accompanied by a single photon, all the involved interactions are chirality blind so that the corresponding ∆σ LR vanishes.
Our paper is organized as follows. In Sec. II, we review the four-dimensional effective Lagrangian in a supersymmetric ADD scenario. And analytic expressions for photino decay rate into a photon and KK gravitinos and for the process e + e − →γG are to be given. Section III devotes to the phenomenological discussions of this scenario, including total cross section, kinematic distributions, a specific observable by using the polarization of electron beam and so on. In Sec. IV we give our conclusions.

II. EFFECTIVE GRAVITINO LAGRANGIAN
In this paper, we assume that there are δ large and supersymmetric extra dimensions, and a single supersymmetry is softly broken on our brane such that our low-energy effective theory yields the MSSM spectra. The cases of more than three extra dimensions are to be considered since in the δ = 2 case astrophysical and cosmological constraints are too strong that the M S is pushed up to about 100 TeV, disfavored as a solution of the gauge hierarchy problem [12]. And the MSSM super-particles are assumed to be confined on our brane. New feature is then another KK tower of the gravitino. The compactification of the gravitino field in a supergravity theory leads to the four-dimensional effective action which is a sum of KK states of massive spin 3/2 gravitinos [11]. The free part of the effective Lagrangian gives the propagator of the n-the KK mode of the gravitino with momentum k and mass Here P n µν is where γ µ P n µν = 0 and k µ P n µν = 0. The effective interaction Lagrangian for the KK gravitino is obtained from the general Noether technique, irrespective to the detailed supersymmetry breaking mechanism. The coupling of each KK mode of graviton and gravitino is determined by the Planck constant Minimally coupled to gravity, the interactions of a KK gravitino with a fermion and photon field to leading order in κ are [16] For later discussion, we present the interaction Lagrangian for the electron-selectron-photino: Since each KK mode of gravitons and gravitinos escapes a detector, experimentally applicable are inclusive rates with all the kinematically allowed KK modes summed up. Due to the very small mass splitting among KK modes, the summation can be approximated by a continuous integration over the KK mode mass m such as [4] n where S δ−1 is the volume of the unit sphere in δ dimensions, given by S δ−1 = 2π Pl in the numerator, implying the tremendous number of accessible KK modes, compensates the gravitational coupling. In effect, the Planck scale is lowered to the M S of TeV scale.

A. Decay rate of a photino
It has been known that the presence of a light gravitino alters the decay modes of supersymmetric particles as the gravitino becomes the LSP; the decay mode of X → XG becomes dominant [16,17]. Even though the coupling strength of the gravitino is Planck-suppressed, the wave function of a light gravitino with mass m 3/2 , momentum k µ and helicity ±1/2 is an ordinary spin 1/2 wave function multiplied by the large factor 2/3 k µ /m 3/2 [16]. The gravitino mass m 3/2 , e.g., in the gauge mediation supersymmetry breaking (GMSB) where the supersymmetry breaking scale is generically low, is Thus the M Pl term in the m 3/2 cancels the gravitational coupling M Pl , so that the characteristic scale of the decay rate becomes the supersymmetry breaking scale Λ SUSY . The photino decay rate is known to be [17] Γ(γ → γG) = 1 48π where Mγ is the photino mass. For Λ SUSY < ∼ 10 3 TeV with Mγ ∼ 100 GeV, the photino decays within a CDF-type detector.
In a supersymmetric ADD scenario, a photino can decay into a photon and a KK mode of a gravitino, if kinematically allowed. The decay rate for the n-th KK gravitino is The inclusive decay rate of a photino is obtained by the sum in Eq. (7): Then, the average distance travelled by an photino with energy E in the laboratory frame . (13) Thus the photino decays within a detector, leaving a detectable photon signal. In the following, we investigate at e + e − collisions the production of a KK gravitino and a photino, which generates single photon events with missing energy.
The differential cross section is then where xγ ≡ 2Eγ/ √ s and λγ ≡ λ(1, M 2 γ /s, The range of xγ is [2Mγ/ √ s, 1 + M 2 γ /s]. The amplitudes squared are summarized in the Appendix. It is to be compared to the KK graviton production process: where x γ ≡ 2E γ / √ s and for f G we refer to Ref. [4]. Equations (16) and (18) show that both the differential cross sections have the same M S -dependence. The gravitino-production accompanied by a massive photino is at a kinematically disadvantage, relative to the graviton-production with a massless photon. The measurement of total cross section alone is not enough to probe supersymmetric bulk effects. Some kinematic distributions and other observables are needed.
We notice that there is one crucial characteristic for the gravitino production accompanied by a photino. As explicitly shown in Eq. (6), the coupling sign of a left-handed electron with a photino and a selectron is opposite to that of a right-handed electron: The holomorphy of the super-potential requires that a fermion should belong to a (left-handed) chiral superfield; the right-handed electron is to be described by a left-handed anti-electron, which possesses positive charge. The interaction with a gravitino, which is gravitational, does not distinguish the chirality of the involved fermion. Therefore, the scattering amplitudes of the t-and uchannel diagrams, which include one e− e−γ and one e− e−G coupling, have opposite sign for the left-and right-handed electron beam. In the s-channel diagram, the electron is coupled with the ordinary QED photon. Since two kinds of amplitudes (one changes the sign under the helicity flip of the electron beam, whereas the other does not) are added, we are ended up with chirality-sensitive total cross section. Note that without the s-channel diagram, the sign-change in the amplitudes alone cannot yield any observable effect, as clearly shown in Eq. (A1). It is to be emphasized that this feature is generic in any supersymmetric model which ensures a light gravitino. In the ordinary MSSM, this point is hard to probe. For example, in the photino pair production, double vertices of e − e −γ in the t-and u-channel Feynman diagrams eliminate the difference.
It is known that the availability of polarized electron and positron beams is highly expected at future linear collider [18]: The current LC performance goal is above 80% of electron polarization and 60% of positron polarization. We propose, therefore, that the effects of KK gravitinos can be most sensitively measured by For the graviton production with a photon, all the involved couplings are completely blind to the helicity of the electron beam; the ∆σ LR vanishes. Moreover, in the SM, the main contribution from the Z-pole to the ∆σ LR is proportional to where g V = −1/2 + 2 sin 2 θ W and g A = −1/2 [19]. The smallness of g V suppresses the SM Z-pole background also.

III. NUMERICAL RESULTS
The cross section of the process e + e − →γG obviously depends sensitively on the mass spectrum of the involved supersymmetric particles, a photino, and the left-and right-handed selectron. Since the contribution of the m e R to the total cross section is very small, two mass scales (Mγ and m e L ) effectively determine the production rate. One can obtain the mass spectrum of superparticles by specifying a concrete supersymmetry breaking model, such as the GMSB model which guarantees a light gravitino. Instead we rather consider the experimental mass bounds when the decay mode of X → XG is open: At the LEP the negative results of the γ / E T event search from e + e − →G χ 0 1 ( χ 0 1 → γG) lead to Mγ > ∼ 82.5 GeV, and those of the γγ / E T from e + e − → χ 0 1 χ 0 1 ( χ 0 1 → γG) to Mγ > ∼ 86.5 GeV [13]. Similarly, the LEP bound with a light gravitino is m e > ∼ 77 GeV. In the following numerical analysis, we adopt the lower mass bounds of a photino and a left-handed selectron as Mγ ≥ 90 GeV and m e L ≥ 80 GeV. The m e R is set to be 200 GeV, which affects little the total cross   [14,15], even the δ = 6 case with Mγ < ∼ 300 GeV can produce substantial events. Being conservative, we present the parameter space of (Mγ, m e L ) for σ tot > 0.1 fb. It can been seen that in the e L decoupling range with m e L > ∼ 500 GeV, the s-channel diagram alone can produce sizable cross section. As expected from the presence of light KK gravitinos, this single photino production mode can probe the photino mass much higher than √ s/2, the kinematic maximum for the photino pair production: For δ = 3, the photino with Mγ < ∼ 460 GeV can be sufficiently produced; for δ = 6, that with Mγ < ∼ 260 GeV. In the followings, we set Mγ = 90 GeV and m e L = 80 GeV. In Fig. 3, we compare the polarized cross sections of the single photon production at e + e − collisions, with the neutrino pair in the SM ( e + e − → γνν) denoted by σ ± SM , with the KK gravitons (e + e − → γG) by σ ± (G), and with the KK gravitinos (e + e − →γG) by σ ± (G). Here superscript ± denotes the chirality of the electron beam. To eliminate the SM Z-pole contribution as much as possible, we employ the following kinematic cuts: Since the σ SM with the above cuts are mainly through the t-and u-channel diagrams mediated by the W boson, the σ + SM is much smaller than the σ − SM . For the KK graviton production, the blindness of the interactions of the graviton and the photon to the fermion chirality guarantees the equality of σ − (G) and σ + (G). For the KK gravitino production, there are several interesting points. First its cross section is only a few tens percents of that for the KK graviton production. This is due to the kinematic suppression by the massive photino. Second, the behavior of the cross section with respect to √ s is the same as the KK graviton case, which increases due to the use of four-dimensional effective Lagrangian.
Finally the opposite sign of the photino coupling with the left-and right-handed electron leads to the domination of the σ − (G) over the σ + (G). In Fig. 4, we present the ratio of ∆σ LR (γG) to ∆σ LR (SM). As discussed before, the ∆σ LR vanishes for the KK graviton production. Therefore, any deviation of the ∆σ LR from the SM background hints the presence of supersymmetric extra dimensions. And this deviation increases with the beam energy. KK gravitinos are produced, where the differential cross section behaves like which the m 2 in the denominator comes from the amplitude squared in Eq. (A1). The different behavior of the δ < 4 case is explained. The measurement of this differential cross section can tell whether the number of extra dimensions is three or more.
In the δ = 3 case, the scattering angle of the photino can be well approximated by that of the photon decayed from the energetic photino. Figure 6   distribution shape is rather flat.
In Tables I and II, we summarize the sensitivity to the M S at 95% CL in two cases, when only the KK gravitons are produced and when the KK gravitinos are also produced. Table   I is for √ s = 183 GeV with the luminosity of 55.3 pb −1 , and Table II for √ s = 500 GeV with the luminosity of 100 fb −1 . We have applied the kinematic cuts in Eq. (20). With the KK gravitinos, the increased cross section generally raises the sensitivity bound on the M S .
Unfortunately, the resulting change is practically negligible. Originally, extra dimensional models have been introduced to solve the gauge hierarchy problem without resort to supersymmetry. However if the ultimate theory is string theory, we live in higher dimensional spacetime which has supersymmetry as a fundamental symmetry.
And branes tend to break supersymmetry. An interesting scenario is that there are large and supersymmetric extra dimensions and at least one supersymmetry survives on our brane below the scale M S so that the low-energy effective theory on our brane resembles the MSSM.
The gravity supermultiplet resides in the bulk, which includes the graviton and its superpartner, the gravitino. On our brane, we have Kaluza-Klein towers of the graviton and gravitino. If supersymmetry is not broken, KK modes of the graviton would have the same mass spectrum as those of the gravitino; the zero mode of gravitino remains massless. As the supersymmetry is broken by an expectation value of order Λ SUSY , each gravitino KK mode acquires additional mass of Λ 2 SUSY /M Pl . Under the assumption of low-energy Λ SUSY , this mass shift is sub-eV scale. In practice, KK gravitinos exist with almost continuous mass spectrum from zero.
In this scenario, we have studied the KK gravitino production at e + e − collisions. With R-parity conservation, the KK gravitino is produced with a supersymmetric particle, e.g., the photino. Since light KK gravitinos become the LSP, the photino decays into a photon and a KK gravitino (missing energy). It has been shown that the inclusive decay rate ofγ → γG n is large enough for the photino to decay within a detector. Therefore, the process e + e − →γG yields a typical signature of a single photon with missing energy. In the phenomenological allowed parameter space of (Mγ, m e L ), we have shown that the total cross section can be substantial: At √ s = 500 GeV, σ tot > 0.1 fb for Mγ < ∼ 460 GeV in the δ = 3 case and for Mγ < ∼ 260 GeV in the δ = 6 case. The dependence of m e L is rather weak; even in the range of m e L > ∼ 500 GeV, we have sizable cross section. Unfortunately, the background processes (the SM reaction of e + e − → γνν and the KK graviton production of e + e − → γG) have much larger cross sections. With the M S of TeV, the KK graviton production becomes compatible with the SM background around √ s = 500 GeV. However the production of a massive photino kinematically suppresses the KK gravitino production rate compared to the KK graviton case by an order of magnitude, since the M S -dependence is the same. To single out the effect of KK gravitinos, total cross section is not enough.
We have noticed that the observable ∆σ LR ≡ σ(e − L e + R )−σ(e − R e + L ) can completely eliminate the KK graviton background. This is because both the gravitational and QED interactions, which are involved in the KK graviton production, do not distinguish the electron beam chirality; ∆σ LR (γ G) vanishes. For the KK gravitino production accompanied by a photino, the electron chirality becomes important since the interaction of e − L − e L −γ has opposite sign to that of e − R − e R −γ, such that σ(e − L e + R ) ≫ σ(e − L e + R ). The ratio of ∆σ LR (SM) to ∆σ LR (γG) is demonstrated to increase with the beam energy, implying that the observable ∆σ LR is unique and robust to probe the supersymmetric bulk.
We also found that the differential cross section with respect to the photino energy fraction xγ can tell whether the number of extra dimensions is three or more: In the δ = 3 case, the dσ/dxγ increases rapidly as xγ approaches its maximum; energetic photinos are more likely produced. And the angular distribution shapes, e.g., for the δ = 3 case, are presented: For the KK gravitino it is more or less flat, while for the SM and the KK graviton they rapidly increase toward the beam line. The sensitivity bound of the M S at 95% CL does not practically change by taking into account of KK gravitino effects due to the kinematic suppression of the KK gravitino production cross section.