Prospects to study a long-lived charged next lightest supersymmetric particle at the LHC

TL;DR

This paper explores the potential to study long-lived scalar tau particles at the LHC by proposing additional stopper detectors, aiming to measure decay properties and distinguish decay channels involving gravitinos and axinos.

Contribution

It proposes a novel detector setup to study long-lived stau decays at the LHC and analyzes its sensitivity to decay parameters and different decay channels.

Findings

A maximum stopper detector can be added in the CMS cavern.

The sensitivity to stau lifetime and gravitino mass can be estimated.

Decays involving axinos can be distinguished from gravitino decays using the proposed method.

Abstract

If the scalar tau $\stau$ is the next lightest supersymmetric particle and decays into a gravitino ($\gravitino$) being the lightest supersymmetric particle, it will have generally a very long lifetime. In this paper, we investigate the possibility to study the decay of such a long lived scalar tau at the LHC. If we can add to the present LHC experiments additional detectors which are able to stop the stau particles and measure the produced decay products, the decay characteristics can be studied precisely at the LHC. We identify a maximum "stopper detector" that could be added in the CMS cavern, and estimate the sensitivity to the lifetime of the stau and to the mass of gravitino with this detector. The decay of the scalar tau may be significantly modified if the decay channel to the axino $\axino$ is open. We study the possibility to distinguish such decays from decays into gravitinos by measuring the process $\stau\to \axino (\gravitino) \tau \gamma$ using the stopper detector.

Tables

	diameter	weight of the detector	length
ATLAS	22m	7Kt	44m
CMS	15m	12.5Kt	21m

Table 2. Table 1: Some model points in mSUGRA model from albert . The mass spectrum and production cross section relevant to our study are shown.

Point	$ϵ$	$\zeta$	$\eta$
$M_{1/2}$	440	1000	1000
$m_0$	20	100	20
$\tan \beta$	15	21.5	23.7
$m_{\tilde{g}}$[GeV]	1025	2191	2190
$m_{{\tilde{\chi }}_1^0}$[GeV]	175	417	416
$m_{{\tilde{\tau }}_1}$[GeV]	154.2	343.5	324.3
$\sigma$(SUSY)[pb]	$3.03$	$2.27\times {10}^{-2}$	$2.34\times {10}^{-2}$
stopped in the stopper–detector per ${10}^5$ events	255	250	254
stopped for 300 fb-1	4636	34	36

Table 3. Table 2: Some model points in gauge mediation model. The production cross section and mass spectrum relevant in this study are also shown.

$\Lambda$[TeV]	40	50	60	70	80
$m_{\tilde{g}}$[TeV]	0.93	1.13	1.34	1.54	1.74
$m_{{\tilde{\chi }}_1^0}$[GeV]	161.7	205.3	248.7	292.1	335.4
$m_{{\tilde{\tau }}_1}$[GeV]	120.5	150.1	179.9	209.8	239.9
$\sigma$(SUSY)[pb]	5.24	1.68	0.64	0.28	0.13
stopped in the stopper–detector per ${10}^5$ events	282	274	274	294	302
stopped for 300fb-1	8830	2762	1052	494	236

Equations

$$m_{{\tilde{q}}_L}^2$$

$$m_{{\tilde{u}}_R}^2\simeq m_{{\tilde{d}}_R}^2$$

$$m_{{\tilde{\ell }}_L}^2$$

$$m_{{\tilde{e}}_R}^2$$

$$\frac{M_i}{g_i^2}$$

$$m_{\tilde{g}}$$

$$m_{{\tilde{q}}_L}^2$$

$$m_{{\tilde{u}}_R}^2$$

$$m_{{\tilde{d}}_R}^2$$

$$m_{{\tilde{\ell }}_L}^2$$

$$m_{{\tilde{e}}_R}^2$$

$${\cal M}^{2}=\left(\begin{array}[]{cc}m^{2}_{\tilde{\ell}_{L_{3}}}+m^{2}_{\tau}-\frac{1}{2}(2m_{W}^{2}-m_{Z}^{2})\cos 2\beta&-m_{\tau}(A_{\tau}+\mu\tan\beta)\cr-m_{\tau}(A_{\tau}+\mu\tan\beta)&m^{2}_{\tilde{\tau}_{R}}+m^{2}_{\tau}+(m_{W}^{2}-m_{Z}^{2})\cos 2\beta\end{array}\right).$$

$$\frac{2500g/c{\mathrm{m}}^2}{\sin \theta }<R<\frac{2500g/c{\mathrm{m}}^2}{\sin \theta }+l(\max )\times {\rho }_{\mathrm{stop}},$$

$${\Gamma }_{\tilde{\tau }}(\tilde{\tau }\to \tilde{G}\tau )$$

$$E_{\tau }=\frac{m_{\tilde{\tau }}^2+m_{\tau }^2-m_X^2}{2m_{\tilde{\tau }}}.$$

$$m_X$$

$$M_{\mathrm{pl}}^2(\mathrm{supergravity})$$

$$L_{\tilde{a}}=i\frac{\alpha C_{aYY}}{16\pi \cos {\theta }_W^2{f}_a}\overline{\stackrel{~}{a}}{\gamma }_5[{\gamma }_{\mu },{\gamma }_{\nu }]\tilde{B}(\cos {\theta }_W{F}_{\mu \nu }-\sin {\theta }_W{Z}_{\mu \nu }).$$

$$L\simeq -\xi C_{aYY}\frac{3\sqrt{2}{\alpha }^2}{8{\pi }^2{\cos }^4{\theta }_W^4}\frac{m_{\tilde{B}}}{f_a}\log \frac{f_a}{m}{\tilde{\tau }}_R\bar{\tau }{P}_L\tilde{a}+hc,$$

$$\Gamma (\tilde{\tau }\to \tilde{a}\tau )$$

$$=$$

$$\ln L(\tau )$$

$${\nu }_i(\tau )$$

$$E_{\mathrm{jet}}\mathrm{distribution}\to E_{\tau }\to m_X.$$

$$\frac{dN}{d{E}_{\mathrm{jet}}}(E_{\mathrm{jet}};E_{\tau })$$

$$\frac{dN}{d{E}_{\mathrm{jet}}}(E_{\mathrm{jet}};E_{\tau },P_{\tau },\Delta E_{\mathrm{jet}}).$$

$$\ln L(E_{\tau },P_{\tau })=\sum _{i=\mathrm{bins}}\ln f_{\mathrm{P}}\left(N_i^{\mathrm{low}};N_i^{\mathrm{high}}(E_{\tau },P_{\tau })\right)$$

$$\delta E_{\tau }/E_{\tau }$$

$$\widehat{m_X}$$

$$E_{\tau }-\delta E_{\tau }(E_{\tau })\le \widehat{E_{\tau }}\le E_{\tau }+\delta E_{\tau }(E_{\tau })$$

$${\widehat{M_{\mathrm{pl}}}}^2$$

$$L=\frac{m_{\tilde{\tau }}^2}{\sqrt{3}{m}_{\tilde{G}}{M}_{\mathrm{pl}}}\left(\overline{\stackrel{~}{\chi }}{\tau }_R{\tilde{\tau }}_R^{\ast }+H.c.\right)+\frac{-m_{\tilde{B}}}{4\sqrt{6}{m}_{3/2}{M}_{\mathrm{pl}}}\overline{\stackrel{~}{\chi }}[{\gamma }^{\mu },{\gamma }^{\nu }]\tilde{B}(\cos {\theta }_W{F}_{\mu \nu }-\sin {\theta }_W{Z}_{\mu \nu }).$$

$$R(X)=\Gamma (\tilde{\tau }\to X\gamma \tau ){|}_{\mathrm{after}\mathrm{cut}}/\Gamma (\tilde{\tau }\to X\tau )$$

Full text

††preprint: UT-06-23††preprint: KEK-TH-1119

I introduction

The minimal supersymmetric standard model (MSSM) is one of the most important candidates for the physics beyond the Standard Model. The strongly interacting superpartners, gluinos and squarks with mass lighter than 2.5 TeV can be discovered at the Large Hadron Collider at CERN (LHC) ATLASTDR . The physics run of LHC will start in 2008.

Among the supersymmetric (SUSY) particles, the lightest SUSY particle (LSP) plays a key role. In cosmology, it is a candidate for the cold dark matter in the universe. At the LHC, the signals of the supersymmetric particles depends on the nature of the LSP. It may be the lightest neutralino $\tilde{\chi}^{0}_{1}$, which escapes from detection and leads to missing $E_{T}$ in the event. Another possibility for the LSP is the gravitino $\tilde{G}$, the superpartner of the graviton. The gravitino coupling to other particles in the MSSM sector is extremely small, namely proportional to $1/M_{\mathrm{pl}}$. The next lightest SUSY particle (NLSP), whose decay products necessarily include the gravitino, is therefore long-lived.

We discuss the scenario where the LSP is the gravitino and the long-lived NLSP carries charge. A natural candidate for a charged NLSP (CNLSP) is the lightest scalar tau, $\tilde{\tau}_{1}$, which can be significantly lighter than the other sleptons due to the left-right mixing induced by the off diagonal matrix element $m^{2}_{LR}\sim m_{\tau}\mu\tan\beta$. The charged particle leaves tracks in the central detectors (ATLAS and CMS), giving additional information for the SUSY particle reconstructions. If the NLSP decays in the main detectors, a displaced vertex may be observed as well.

The expected lifetime of the CNLSP $\tilde{\tau}$ is unconstrained, because it is proportional to $(m_{\tilde{G}})^{2}$ the yet unknown gravitino mass squared. On the other hand, the gravitino mass is proportional to the total SUSY breaking scale in the hidden sector, therefore the determination of the lifetime is an important physics goal. The lifetime measurement gives direct information on the hidden sector.

A particle decays efficiently in the main detectors (CMS and ATLAS) if the decay length is sufficiently short, $c\tau\ll$ (10m)$\times N_{\rm produced}$, where $N_{\rm produced}$ is the number of produced SUSY particles. On the other hand, for typical SUSY production cross sections, a direct observation of the decay is very difficult for $\tau_{\mathrm{CNLSP}}>0.01$ sec. However, it has been pointed out that the CNLSP stopped by the ionization loss in the material may be studied in detail Hamaguchi:2004df ; Feng:2004yi . An idea for a stopper based on a water tank is presented in Ref.Feng:2004yi , where the water can be transported away from the detector site, for concentration and further study. In Ref. Hamaguchi:2004df , a detector consisting of a tracker and heavy stopping material is proposed, which can measure the arrival time and the location where the CNLSPs are stopped, in addition to the energy of the decay products. Another possibility, which requires minimal experimental modification, is to study the CNLSP which are stopped in the main detector or surrounding rock albert .

As pointed out in Ref. Buchmuller:2004rq , the study of the CNLSP decays can probe the underlying supergravity in nature. With the gravitino mass inferred from the kinematics, the additional CNLSP lifetime measurement will test an unequivocal prediction of supergravity. Moreover, the study of a rare 3–body decay $\tilde{\tau}\to\tau\gamma\tilde{G}$ can reveal the peculiar couplings of the gravitino and the gravitino spin $3/2$.

In this paper, we consider the physics of the CNLSP $\tilde{\tau}$ decays that can be done with the heavy material stopper–detector Hamaguchi:2004df , because only a detector of this type can cover a wide range of lifetime $O(10{\rm nsec})$ $<\tau_{\mathrm{CNLSP}}<\mathcal{O}(10$ years). We find that the mass of the gravitino can be measured if it is sufficiently heavy (roughly $m_{\tilde{G}}>0.2m_{\tilde{\tau}}$). In that case, one can check if the lifetime is consistent with the supergravity interpretation.

As we shall see, the LSP mass resolution is however poor if $m_{\tilde{G}}\buildrel<\over{{}_{\sim}}0.2m_{\tilde{\tau}}$ . In that case, it is very hard to prove that the decay $\tilde{\tau}\rightarrow\tau$ plus an invisible particle $X$ is indeed caused by the supergravity interaction involving $\tilde{G}$. We should note that, because of the extremely weak coupling of the gravitino, if there is any unknown supersymmetric particle $X$ which couples rather weakly to the MSSM particles, the lightest MSSM particle may decay into $X$ instead of $\tilde{G}$ even if the $\tilde{G}$ is the LSP. For example, the superpartner of the axion, the axino $\tilde{a}$, can be such a particle. However, we found that it may be possible to discriminate the case of $X=\tilde{G}$ and the case of $X=\tilde{a}$, by investigating the three body decay $\tilde{\tau}\rightarrow\tau\gamma X$ as suggested in Buchmuller:2004rq ; Brandenburg:2005he , with enough statistics.

This paper is organized as follows. In section II, we discuss a possible detector setup. We found an $\mathcal{O}(1)$ kton detector (up to 8 kton) may be placed next to the CMS detector without serious modification of the CMS experiment itself, but with non-negligible modifications to the CMS cavern side walls. In section III, we select several model points and estimate the expected number of the stopped particles for $\int{\cal L}=300$ fb*-1* and 3000 fb*-1* . In Section IV, we discuss the measurement of the two body decay $\tilde{\tau}\rightarrow\tau X$. The three body decay of the CNLSP is studied in Section V. Section VI contains the discussion and comments.

II Assumptions on the stoppers

In this section, we discuss the possibility to install massive stoppers next to the LHC detectors, CMS and ATLAS. It turns out that the CMS cavern may allow for an easier installation and more room for a massive stopper, compared to the ATLAS cavern. The parameters of the two detectors are listed in Table. II. The diameter of the CMS detector is smaller, therefore the massive stopper can be placed closer to the interaction point at CMS. The weight of the CMS detector is about twice as large as that of ATLAS. Because of the large weight, the cavern of CMS is designed so that it can sustain a massive object safely, which includes a reinforced floor to spread the detector pressure more equally. A potential massive stopper with a weight of around a few kton can be placed on both sides of the CMS detector, but it will need a reorganization of the scaffolding and gallery paths on the cavern walls, to make room for such an additional detector.

The assembly and construction of the two detectors is also very different. Most of the CMS detector components are assembled on the ground, and about 15 large detector units will be lowered in the cavern for final assembly works. Hence the installation is less integrated with the cavern, leaving relatively more freedom and thus changes needed in the cavern to install massive stoppers are somewhat simpler CMS_proposal . On the other hand ATLAS detector is assembled mostly in its cavern. The huge magnets toroids and outside muon system fill up the cavern. The cryogenic system in the ATLAS cavern is also taking space outside the detector ATLAS_proposal .

We assume two stoppers with the size 3.5m $\times$ 15m $\times$ 15m and the average density $\rho_{\rm stop}=5$g/cm3, hence, the total weight of the detector is 8 kton. This is maximum possible rectangular parallelepiped volume that can be placed in the cavern, with the given space to the cavern wall and with its long edge being the same as that of the CMS barrel part (see Fig. 1). The stoppers are thus placed 8.5 m away from the interaction point. We also assume that stopping power of the CMS detector is equivalent to $(2500/\sin\theta)$ g/cm2 iron, where $\theta$ is angle between the CNLSP direction and the beam direction. The number comes from the average density of CMS detector, 3.37g/cm3, which leads to the weight per cm2 for the radial direction of $2500{\rm g/cm}^{2}$.

As discussed in the previous paper Hamaguchi:2004df , the stopper can be a hadronic and electromagnetic calorimeter simultaneously, if the detector consists of layers of dense stopper and tracking devices. The measurement of the energy of the decay product of the CNLSP is the key ingredient to explore the CNLSP interactions to the $X$ particle. In this paper we assume that the CNLSP is the scalar tau lepton $\tilde{\tau}$, which decays mostly as $\tilde{\tau}\rightarrow\tau X$ where $X=\tilde{G}$ or $\tilde{a}$.

The $\tau$ decays into $l\bar{\nu}_{l}\nu_{\tau}$, or into $\pi^{\pm}$ and $\pi^{0}$’s. We do not consider the decays into $\mu$, because the muon energy cannot be measured unless the stopper contains a magnetic field. The energy of the leptons are much softer than the parent $\tau$ energy anyway, so that they are less useful for the study of the decay kinematics.

A large volume detector is advantageous to measure the energy of the $\tau$ decay products, because the detector must contain most of the energy of the showers from the $\tau$ decay products. To fully absorb the hadronic cascade one needs sufficient thickness of the calorimeter. The required thickness for an iron calorimeter (density is 7.87g/cm3) is listed in Ref. Eidelman:2004wy ; Bock:1980rs , and is about 170 (120) cm for 100 GeV single hadron energy for 99% (95%) containment respectively, equivalent to 1337.9g/cm2 (944.4g/cm2). To measure the energy with sufficient accuracy by a stopper with $\rho=$ 5g/cm3, the distance between the decay position and the end of the detector along the shower path must be at least 190cm. A simplified Monte Carlo simulation shows that 63% of the $\tilde{\tau}$ decays satisfy this condition. High energy photons and electrons initiate electromagnetic cascades, which are much easier to contain in a detector. The energy deposition along the axis of the cascade is well described by a gamma distribution, where the maximum occurs at $\sim 8X_{0}$, where $X_{0}$ is the radiation length. The energy deposition terminates around $15X_{0}$. (See Fig.27.17 of Eidelman:2004wy .) For the case of iron ($X_{0}=13.84{\rm g/cm}^{2}$), 200g/cm2 thickness of the material is needed to stop the electromagnetic cascade. The difference in the absorption length is used to discriminate isolated photons from hadron, which would be useful to study $\tilde{\tau}\rightarrow\tau\gamma X$ decay.

In a previous paper Hamaguchi:2004df , we discussed the possibility to re-use of the existing 1 kton detector, such as SOUDAN II Allison:1996wn as the CNLSP stopper. The SOUDAN II consists of the layers of $\mathcal{O}$(m) long drift tubes and thin iron plates. The physics goal of this detector is the search for proton decays. To be sensitive to the low energies involved in these decay processes, the average density of SOUDAN II is low, less than 2g/cm3. The size of the detector is probably enough to stop a certain amount of CNLSPs and measure the decay rate, and therefore it may well be appropriate for the a first stage of the CNLSP study. On the other hand, it is certainly not enough for a detailed energy measurement of the $\tilde{\tau}$ decay product. Most of the hadronic decay cascades are not fully contained for a detector with the geometry allowed by the space in the CMS cavern. A high density detector consisting of the layers of drift tubes/scintillators or RPCs to measure the charged particles between iron plates thicker than SOUDAN II will be more optimized for the CNLSP study.

In this paper we assume a conservative energy resolution for hadronic showers, which is around $150\%/\sqrt{E/{\rm GeV}}$. The value is not unrealistic for a simple massive and affordable detector, if a shower is sufficiently contained in the stopper. An additional complexity may occur for showers which develop parallel to the layers of the tracking devices. If the particles pass mostly through tracking devices, they feel a much lower average density, while the particles going mostly through the iron plates do not give detectable signature efficiently. We assume that the measured energy will be corrected depending on the shower directions. A detector uniform to all directions would be better to measure the energy of decay products of the stopped CNLSP. The calorimetery technologies studied for the ILC may satisfy such conditions; see Martyn:2006as for the CNLSP study at ILC.

III Supersymmetric Models with charged next lightest SUSY particles, and

expected number of stopped CNLSP

In this section, we briefly describe supersymmetric models with a long-lived charged next lightest SUSY particle. We also select some model points, and estimate at each model point the number of CNLSP that can be stopped at the stopper–detector.

In the minimal supergravity (mSUGRA) models, the scalar masses and gaugino masses, trilinear couplings are universal at the GUT scale $M_{\mathrm{GUT}}$ which are denoted by $m_{0}$, $M_{1/2}$, and $A_{0}$ respectively. The resulting mass spectrum is obtained by solving renormalization group equations (RGE’s). When the RGE is integrated up to $\mathcal{O}(m_{Z})$, slepton soft masses and gaugino masses are approximately given by the following convenient formulas

[TABLE]

where $g_{X}$ is the gauge couplings at the unification scale.111 The pole masses of strongly interacting SUSY particles receive a large corrections of $\mathcal{O}$(30%) if the mass scale is $\mathcal{O}$(1TeV). In this model, the mass of gravitino $m_{\tilde{G}}$ is order of $m_{0}$ or $M_{1/2}$, because both of them are proportional to the $F_{0}/M_{\mathrm{pl}}$ with $\mathcal{O}(1)$ coefficient, where $F_{0}$ is the fundamental SUSY breaking scale. Depending on the $\mathcal{O}(1)$ coefficients, the gravitino can be the lightest superparticle in general. Also in the gaugino mediation models gauginoMSB , the gravitino can be the LSP in a large domain of the parameter space Buchmuller:2005rt . In this case the scalar masses are very small at the boundary, i.e., $m_{0}\simeq 0$, and the stau naturally becomes the NLSP.

In the gauge mediation (GM) model Giudice:1998bp the supersymmetry is broken at lower energy scale, and the SUSY breaking in the hidden sector is mediated to the MSSM sector by gauge interactions. The simplest GM model is described by 6 parameters: $\Lambda=F/M_{\rm mes}$, $M_{\rm mes}$, $N_{5}$, $\tan\beta$, sgn$\mu=\pm 1$ and gravitino mass. Here $N_{5}$ is an effective number of vector-like heavy quarks and leptons in SU(5) representations, $\Phi$ and $\bar{\Phi}$. ($N_{5}=1$ for $5+\bar{5}$ quarks and leptons, and $N_{5}=3$ for $10+\overline{10}$.) A messenger field $Y$ couples to the vector-like fields as $W=\lambda Y\Phi\bar{\Phi}$ and develops a vacuum expectation value $\lambda\langle Y\rangle=M_{\rm mes}+\theta\theta F$. The gravitino mass is given by $m_{\tilde{G}}=F_{0}/(\sqrt{3}M_{\mathrm{pl}})$ where $F_{0}(\geq F)$ is the total SUSY breaking scale of the theory. We take $m_{\tilde{G}}$ as a free parameter in this model.

The masses of MSSM particles at $M_{\rm mes}$ are obtained by relatively simple formula. Gaugino masses satisfy the GUT relation, and

[TABLE]

Squark and slepton masses are given by

[TABLE]

All of the above models predict a large mass hierarchy between strongly interacting superpartners and weakly interacting superpartners. Heavy gluino and squarks are copiously produced at the LHC, and they decay into the light weakly interacting SUSY particles. The lightest SUSY particle in the MSSM sector is either the lighter stau $\tilde{\tau}_{1}$ or the lightest neutralino $\tilde{\chi}^{0}_{1}$.

The mass of the $\tilde{\tau}_{1}$ is the smaller eigenvalue of the mass matrix,

[TABLE]

Because of the off-diagonal elements of $\tilde{\tau}$ mass matrix, $\tilde{\tau}_{1}$ could be significantly lighter than the other sleptons. If the $\tilde{\tau}_{1}$ is the NLSP, the stopper–detector is useful to stop it and to study its decay. We therefore consider the phenomenology when $m_{\tilde{g}},m_{\tilde{q}}\gg m_{\tilde{\chi}^{0}_{1}}>m_{\tilde{\tau}_{1}}$ in this paper. In the following, we omit the subscript of $\tilde{\tau}_{1}$ and denote the NLSP stau as $\tilde{\tau}$.

In most part of this paper, we will discuss the CNLSP physics as model independent as possible. However, to give some numerical reference, we choose a few model points. For mSUGRA models, we take the points proposed in albert . The parameters are listed in Table. 1 and mass spectrum is given by the ISAJET ver. 7.69. In albert , the gravitino mass is taken to be $m_{\tilde{G}}=m_{0}$ in those model points.

For the gauge mediation model, we take the model points similar to that for the study in ATLASTDR . Namely, we fix $N_{5}=3$, $\tan\beta=15$ and $\Lambda/M_{\rm mes}=0.5$, where $\Lambda=40,50,60,70,80$ TeV. The mass spectrum and production cross sections are summarized in Table. 2. The gravitino mass for these model can be very small, whose minimum value is given by $\mathcal{O}(\Lambda^{2})/M_{\mathrm{pl}}$. If the messenger sector couples a fraction of total SUSY breaking, i.e. $F<F_{0}$, the gravitino mass, which is proportional total SUSY breaking $F_{0}$, can be large.

The pattern of the mass spectra are similar for both mSUGRA and GM sample points. The gaugino masses obey the GUT relation, and $\tilde{\tau}$ mass is lighter than $\tilde{B}$ mass because of the CNLSP assumption. For mSUGRA, this implies $m_{0}<M_{1/2}$, so that $\tilde{q}$ mass is about as large as $\tilde{g}$, as it is so in GM models. The relation $m_{\tilde{q}}\sim m_{\tilde{g}}\gg m_{\tilde{\tau}}$ is realized in our model points.

To estimate the number of stopped CNLSP, the production cross section of the SUSY particles and the velocity distribution of $\tilde{\tau}$ at LHC must be evaluated. They depend on the $\tilde{g}$ and $\tilde{q}$ masses, and the mass difference $m_{\tilde{g}(\tilde{q})}-m_{\tilde{\tau}}$. We estimate the production cross section and the decay distribution by using HERWIGHERWIG , where the mass spectrum and branching ratios are interfaced from ISAJETISAJET to HERWIG by using ISAWIGISAWIG . We generated $10^{5}$ SUSY events for each model point.

The flying range $R$ of the charged stable massive particle may be calculated by integrating the energy loss equation of heavy ionizing particle (Bethe-Bloch equation). The result is a function of $\beta=p/E$, with a linear dependence on the mass of the particle $M$. In this paper, the stopping power of the stopper–detector is calculated using the data in ICRU . See also the detailed discussion in Ref. Feng:2004yi .

The maximum length of a particle track through the stopper–detector $l$(max), which depends on CNLSP direction, is calculated assuming that the track of the CNLSP goes straight from the production point. We regard the particle is stopped in the stopper–detector if the flying range $R$ satisfies:

[TABLE]

where $\rho_{\rm stop}=5$g$/$cm3 is the density of the stopper–detector. The number of stopped CNLSP in a stopper per $10^{5}$ events, and the number of stopped CNLSP for ${\cal L}=300{\rm fb}^{-1}$ in the two stoppers, are listed in Table 1 and 2. In Fig. 2 we also show the simulated stopped positions in a stopper–detector. The position distribution is uniform in the detector.

In Fig. 3, we show the $\beta\gamma$ distribution of the CNLSPs for a few model points. We find that CNLSP tends to be less relativistic when the SUSY scale is large, because momentum of parent squarks and gluino is reduced. In Fig. 3, the peak position of the $\beta\gamma$ is at $\sim 1.5(1.3)$ for $\Lambda=50(80)$ TeV. Because of that, the number of CNLSP in the smaller $\beta\gamma$ region is increased as the SUSY scale is increased. On the other hand, as $m_{\mathrm{CNSLP}}$ increases, CNLSPs with smaller $\beta\gamma$ are stopped in the detector while the number of events in the smaller $\beta\gamma$ bins are kinetically suppressed. For instance, for $\Lambda=50$ and 80 TeV, CNLSPs in the bins between $0.5<\beta\gamma<0.6$ and $0.45<\beta\gamma<0.55$ are stopped in the stopper–detector, respectively. Altogether, the number of stopped $\tilde{\tau}$ for $\Lambda=40$ to $80$ TeV for $10^{5}$ SUSY events is roughly constant as we can see in Table. 2.

The production cross section reduces when the gluino mass is increased because the parton distribution functions of gluon and quarks are quite small for $x\gg 0.1$. For 300fb*-1*, the number of events stopped at the assumed two 4 kton stoppers ranges from 8000 events to 30 events in the table. We will see in the next section that accumulation of $\mathcal{O}(100)$ CNLSPs are enough to measure the lifetime with $\mathcal{O}$(10)% accuracy. We will also estimate the resolution of $\tau$ lepton energy $E_{\tau}$ arising from the decay $\tilde{\tau}\to\tau X$ through the end point of the tau jet energy. Statistically $\mathcal{O}(1000)$ stopped CNLSPs are enough to measure the end point with a few GeV error. In the upgrade of LHC (SLHC), integrated luminosity of 3000fb*-1* is proposed, therefore the number of stopped events ranges from $\mathcal{O}(300)$ to $\mathcal{O}(80000)$ for the model points presented in Table 2.

IV Study of the $\tilde{\tau}$ two body decay in stopper–detector

In this section we study the two body decay of the CNLSP in the stopper–detector. Both in mSUGRA and GM models, the stau can be the CNLSP and decays into the gravitino $\tilde{G}$ and the $\tau$–lepton. The CNLSP decay width into a gravitino and a lepton is given by Buchmuller:2004rq

[TABLE]

We show the dependence of the stau lifetime on the gravitino mass in Fig. 4.

In general, the two–body decay $\tilde{\tau}\rightarrow X\tau$ can be triggered by a single tau jets initiating from the position where $\tilde{\tau}$ is stopped. The tau energy is monochromatic and expressed as

[TABLE]

Here $X$ is the invisible particle in the $\tilde{\tau}$ decay, in this case $X=\tilde{G}$. If one can measure both of the lifetime and the mass of the stau, the gravitino mass can be determined assuming that Eq.(14) is correct. Then the total SUSY breaking scale $F_{0}=\sqrt{3}m_{\tilde{G}}M_{\mathrm{pl}}$ is also determined, which is very important to understand the hidden sector physics.

The LHC main detectors can determine the mass of $\tilde{\tau}$ through the stau velocity measurement $\beta_{\tilde{\tau}}$, e.g., in the muon system of the CMS detector. However, the measurement of the lifetime may not be easy at the main detectors if the lifetime is too much longer than the detector size. The cross section is typically $\mathcal{O}(1)$ pb or less, therefore we have at most $\mathcal{O}(10^{5})$ event at hand for ${\cal L}=100{\rm fb}^{-1}$. Thus, only 10 events or less decay inside the detector if $c\tau>100$ km. Some of the CNLSP are stopped in the main detector, but measuring the decay precisely in the main detector would be challenging during the beam time. On the other hand a stopper–detector Hamaguchi:2004df can measure the position and the time where a CNLSP is stopped, and its decays without dead-time. The lifetime measurement will be discussed in Sec.IV.1. See albert also on the idea to measure the lifetime by triggering muons from the decays of the CNLSP stopped in the surrounding rock.

To predict the CNLSP lifetime, one has to determine the gravitino mass independently. This is possible through the extraction of $E_{\tau}$ from the energy distribution of the tau jet from the CNLSP decay, because $E_{\tau}$ is a function of $m_{\tilde{\tau}}$ and $m_{X}$, as can be seen in Eq. (15). $m_{X}$ is expressed as a function of $E_{\tau}$ as follows;

[TABLE]

Fig. 5 shows the dependence of the reconstructed LSP mass Eq.(16) on the tau energy $E_{\tau}$ for several values of $m_{\tilde{\tau}}$. As can be seen from the figure, it is crucial to measure the $E_{\tau}$ as precise as possible, especially for small $m_{X}$, in order to determine the mass $m_{X}$. Hence, the stopper–detector should offer a reliable measurement of $E_{\tau}$. We will discuss the $E_{\tau}$ measurement and the $m_{X}$ reconstruction in Sec. IV.2 and Sec. IV.3, respectively.

One can study the supersymmetric version of gravity interaction by studying the consistency between the observation and prediction of the decay rate. If the gravitino is the LSP, the decay width is given by Eq.(14). Now if one can independently determine the gravitino mass by means of kinematics, as described above, Eq.(14) can be used in the other way round, which leads to the measurement of the ’supergravity Planck scale’ Buchmuller:2004rq

[TABLE]

Comparison of the obtained $M_{\mathrm{pl}}({\rm supergravity})$ with the Planck scale of the Einstein gravity $M_{\mathrm{pl}}({\rm gravity})=(8\pi G)^{-1/2}=2.43534(18)\times 10^{18}~{}\mathrm{GeV}$ Eidelman:2004wy would be a crucial test of the supergravity. Prospects of the “Planck scale” measurement will be discussed in Sec.IV.4.

It should be noted that the undetectable particle $X$ may not be the gravitino $\tilde{G}$. Any particle which couples weakly to $\tilde{\tau}$ can be particle $X$. If the decay width into $X\tau$ is larger than the decay width into $\tilde{G}\tau$, i.e. $\Gamma(\tilde{\tau}\rightarrow\tau X)>\Gamma(\tilde{\tau}\rightarrow\tau\tilde{G})$,222This includes the trivial case where the decay into the gravitino is kinetically forbidden, $\Gamma(\tilde{\tau}\to\tau\tilde{G})=0$, i.e., $m_{\tilde{G}}>m_{\tilde{\tau}}>m_{X}$. the CNLSP lifetime may be different from the supergravity prediction obtained from measured $m_{\tilde{\tau}}$ and tau jet energy distribution. Inconsistency between the measured and predicted lifetime immediately means a discovery of a new sector that may not be accessible otherwise.

One of well motivated examples of such a non-SUGRA decay is $\tilde{\tau}\rightarrow\tilde{a}\tau$ where $\tilde{a}$ is the axino, superpartner of the axion. The CNLSP $\tilde{\tau}$ decay into axino is studied in Ref. Brandenburg:2005he for hadronic, or KSVZ axion models Kim:1979if+X . In this paper we adopt the set–up in Ref. Brandenburg:2005he for the axino interaction, which we briefly describe here.

In KSVZ axion model, the Peccei-Quinn (PQ) mechanism Peccei:1977hh+X is realized in an action with new heavy quarks. When these heavy quarks are integrated out, anomalous terms involving the axion and gauge bosons are generated at low energy effective action. When the axion interaction is supersymmetrized, its fermionic superpartner, the axino $\tilde{a}$, must be introduced. The axino mass $m_{\tilde{a}}$ can range between the eV and GeV scale depending on the model and SUSY breaking scheme axinomass , and we regard it as a free parameter in this paper. The coupling of the axino to the bino and the photon/$Z$-boson at the scale below the Peccei-Quinn scale $f_{a}$ is given by the Lagrangian

[TABLE]

The action does not contain direct $\tilde{\tau}\tau\tilde{a}$ coupling and also strongly suppressed by the PQ scale 10^{9}{\rm GeV}\buildrel<\over{{}_{\sim}}f_{a}\buildrel<\over{{}_{\sim}}10^{11}{\rm GeV}. The two body decay $\tilde{\tau}\rightarrow\tilde{a}\tau$ is induced by the one loop diagram shown in Fig.6. The loop integral has a logarithmic divergence. This is because the effective vertex (18) is applicable only if the momentum is smaller than the heavy (s)quark masses, whereas the loop momentum exceeds that scale. In the full theory, one calculates two–loop diagrams with the heavy (s)quarks, which leads to a finite result axino2loop . Here we regulate the logarithmic divergence with the cut–off $\sim f_{a}$ Brandenburg:2005he ; Covi:2002vw , and the effective $\tilde{a}\tau\tilde{\tau}_{R}$ coupling is parameterized as

[TABLE]

Here, $m\simeq m_{\tilde{\tau},\tilde{B}}\simeq\mathcal{O}(100~{}\mathrm{GeV})$ and we take $\log(f_{a}/m)=20.7$ hereafter. The parameter $\xi$ is an order one parameter to represent the uncertainty coming from the cut–off procedure mentioned above. In this paper, we regard this as a free parameter. The two body decay width is given as Brandenburg:2005he

[TABLE]

IV.1 Lifetime measurement

In this subsection, we estimate the statistical error in the CNLSP lifetime measured by the stopper–detector. The analysis is model–independent and applicable to any long-lived charged particle stopped in the stopper.

For each CNLSP stopped in the stopper–detector, the stopping time $t_{\mathrm{stop}}$ and the decaying time $t_{\mathrm{decay}}$ will be recorded. The lifetime is then measured by fitting the temporal distribution of the decaying events $N(t)$, where $t=t_{\mathrm{decay}}-t_{\mathrm{stop}}$. Here, we use a maximum–likelihood fitting and adopt the following procedure: First we define a time $t_{e}$ such that $N(t<t_{e})=(1-e^{-1})N_{\mathrm{total}}\simeq 0.632N_{\mathrm{total}}$ and $N(t>t_{e})=e^{-1}N_{\mathrm{total}}\simeq 0.368N_{\mathrm{total}}$, where $N_{\mathrm{total}}$ is total number of stopped event.333More precisely, $t_{e}$ is defined by $t_{e}=(t_{j}+t_{j+1})/2$, where $j<(1-e^{-1})N_{\mathrm{total}}<j+1$ and $t_{j}$ is the decay time of the $j$-th event: $t_{1}<t_{2}<\cdots<t_{j}<\cdots<t_{N_{\mathrm{total}}}$. (For large $N_{\mathrm{total}}$, this $t_{e}$ is already a good estimator of the lifetime.) We then calculate the $\ln L$ distribution as a function of lifetime $\tau$:

[TABLE]

where $f_{\mathrm{P}}(n;\nu)=\nu^{n}e^{-\nu}/n!$ is the Poisson distribution, $n_{i}$ is the number of events in the $i$-th bin, and $\nu_{i}(\tau)$ is the predicted average number of events in that bin:

[TABLE]

Here, we take 1 bin $=\Delta t=t_{e}/5$.

An example of the time distribution and the best fit curve for MC distribution are shown in Fig. 7 for $N_{\mathrm{total}}=100$.

The $n\sigma$ confidence interval can be estimated by the range $2\Delta\ln L=2(\ln L_{\mathrm{max}}-\ln L)\leq n^{2}$. Fig. 7 shows that the error of the lifetime is $\Delta\tau/\tau=(10-15)\%$ for $N_{\mathrm{total}}=100$. We have done the same analysis for $N_{\mathrm{total}}=1000$ and found $\Delta\tau/\tau=(3-4)\%$.

So far, we assumed that most of the stopped CNLSPs decay within the experimental time scale. We now consider the case where the lifetime is longer. Suppose that one observes $N_{\mathrm{total}}=1000$ stopped CNLSPs and only 10 events decaying within 1 year. In such a case, the lifetime is estimated from the number of the decaying events. For instance, 95% interval of the mean $\nu$ of Poisson variable for $n=10$ is $\nu=[5.4,17.0]$ Eidelman:2004wy . Using $\nu=N_{\mathrm{total}}(1-e^{-1~{}\mathrm{year}/\tau})$, a 95% interval $58<\tau<184$ years is obtained. For much longer lifetime or smaller $N_{\mathrm{total}}$, only a lower bound of the lifetime is obtained.

We have assumed that the background is negligible. The background from cosmic neutrino and hard neutrino produced from the main detector interaction point is small (cf. Hamaguchi:2004df ), however careful study on the accidental background is necessary when statistics is low. This is beyond of the scope of this paper.

IV.2 Measurement of the $\tau$ energy

from distribution of the $\tau$ jet energies

In this subsection, we estimate the uncertainty of the tau energy determination. Then, in Sec.IV.3 we discuss the kinematical reconstruction of the LSP mass. Schematically, the procedure is as follows:

[TABLE]

When $\tilde{\tau}$ decays into $\tau$ and invisible particle $X$, the tau energy $E_{\tau}$ is monochromatic (see Eq.(15)). $E_{\tau}$ can be obtained by fitting the $\tau$ jet energy distribution :

[TABLE]

Among the decay products of the $\tau$ lepton we omit the neutrinos and muons. In order to see the prospects of $E_{\tau}$ measurement with a finite number of events, we generate the events from $\tau$ decay by using the TAUOLA TAUOLA , and we perform a maximum–likelihood fitting of low statistics (“experimental”) event sets by high statistics (“theoretical”) distributions.

In practice, the observed jet energy distribution depends not only on the $E_{\tau}$, but also on the detector resolution $\Delta E_{\mathrm{jet}}$ and the tau polarization $P_{\tau}$:

[TABLE]

To obtain theoretical predictions for $dN/dE_{\mathrm{jet}}(E_{\mathrm{jet}};\,E_{\tau},P_{\tau},\Delta E_{\mathrm{jet}})$, we have generated tau decay events with high statistics run of TAUOLA for the parameter space $E_{\tau}=$ 30 — 125 GeV and $-1\leq P_{\tau}\leq 1$, while fixing $\Delta E_{\mathrm{jet}}/E_{\mathrm{jet}}=$ $150\%$ $/\sqrt{E_{\mathrm{jet}}/{\rm GeV}}$.444We have generated $10^{6}$ events for each of the parameter sets $E_{\tau}=$ 30, 31, 32,… 109, 125 GeV and $P_{\tau}=\pm 1$ (i.e., $96\times 2$ parameter sets $\times 10^{6}$ events), and interpolated the distribution between those parameter points. For each single $\tau$ decay generated by TAUOLA, the jet energy $E_{\mathrm{jet}}$ is calculated, and then smeared by a Gaussian fluctuation with a variance $\sigma^{2}=(\Delta E_{\mathrm{jet}})^{2}$.

Fig 8 shows examples of the jet energy distribution.

In order to see a realistic situation, we generate a small number of tau lepton decays for a fixed parameter set of ($E_{\tau}$, $P_{\tau}$, $\Delta E_{\mathrm{jet}}$), and then fit the result by the “theoretical” distribution obtained above. Here and hereafter, we assume that the energy resolution will be known in advance. Examples are shown in Fig. 9 and Fig. 10 for total number of decaying tau leptons $N_{\tau}=1000$ and $N_{\tau}=100$, respectively. The energy of $\tau$ jets which are not contained in the detector may not be measured precisely. The $N_{\tau}$ corresponds to the number of well contained events.

Some comments are in order here. (i) We take 1 bin = 10 GeV. (ii) For the maximum–likelihood fitting we take only the bins with number of events $\geq 1$, and the bins above the peak energy, which in the examples of Fig. 9(a) and 10(a) correspond to the bins between the two vertical lines. (iii) We then calculate the $\ln L$ distribution in the parameter space of ($E_{\tau}$, $P_{\tau}$) as follows

[TABLE]

where $f_{\mathrm{P}}(n;\nu)=\nu^{n}e^{-\nu}/n!$ is the Poisson distribution, $N^{\mathrm{low}}_{i}$ is the number of events in the $i$–th bin for the low statistics run, and $N^{\mathrm{high}}_{i}(E_{\tau},P_{\tau})$ is the predicted number of events in the $i$–th bin as a function of the parameter set $(E_{\tau},P_{\tau})$, normalized by the total number of decaying tau leptons. Here, we assume that the total number of stopped $\tilde{\tau}$ is known, i.e., the total number of events is not taken as a free parameter for the fit.

The $n\sigma$ confidence interval can be estimated by the range $2\Delta\ln L=2(\ln L_{\mathrm{max}}-\ln L)\leq n^{2}$ projected onto the $E_{\tau}$ axis [see Figs. 9(b) and 10(b)]. One can see that the primary tau energy can be determined within an error of a few GeV. From Fig. 9(c) one can also see that the polarization is hardly determined by the energy distribution analysis even with $N_{\tau}=1000$. This is because the sensitivity to the polarization becomes very weak once the finite detector resolution is taken into account (cf. Fig. 8).

To estimate the statistical error in $E_{\tau}$ measurement we have generated the event sets with same statistics $N_{\tau}=100$ $(1000)$ and repeat the fit on $E_{\tau}$ and $P_{\tau}$ to obtain the best fit value of the $E_{\tau}$, $E^{\rm best}_{\tau}$. In figure 11, we show the distribution of $E^{\rm best}_{\tau}$ for $E_{\tau}=50,70,90$ GeV, $P_{\tau}=1$, and $\Delta E_{\mathrm{jet}}/E_{\mathrm{jet}}=150\%/\sqrt{E/{\rm GeV}}$.

From the variances of these distributions, we estimate the $1\sigma$ uncertainty of $E_{\tau}$ as

[TABLE]

IV.3 Determination of the LSP mass

Now we can estimate the uncertainty of the LSP mass. In Fig. 12 we plot the range of reconstructed LSP mass

[TABLE]

where

[TABLE]

and $E_{\tau}=(m_{\tilde{\tau}}^{2}+m_{\tau}^{2}-m_{X}^{2})/2m_{\tilde{\tau}}$.

Note that the reconstructed LSP mass depends on not only $E_{\tau}$ but also the measured CNLSP mass $\widehat{m_{\tilde{\tau}}}$. In Fig. 12 we show the range of LSP mass for $\widehat{m_{\tilde{\tau}}}=m_{\tilde{\tau}}$ (solid lines) and $0.99m_{\tilde{\tau}}\leq\widehat{m_{\tilde{\tau}}}\leq 1.01m_{\tilde{\tau}}$ (dashed lines), the latter corresponding to 1% uncertainty of the CNLSP mass. The stau mass determination from time of flight was discussed in ATLASTDR ; Hinchliffe:1998ys ; albert . For CMS detector the mass resolution is estimated as 10–20% in each event and less than 1% for $\sim 1000$ events.

As can be seen in Fig. 12, the kinematical reconstruction of the LSP mass is possible if $m_{X}$ is sufficiently large, $m_{X}\buildrel>\over{{}_{\sim}}0.15m_{\tilde{\tau}}$ for $N_{\tilde{\tau}}=1000$ and $m_{X}\buildrel>\over{{}_{\sim}}0.25m_{\tilde{\tau}}$ for $N_{\tilde{\tau}}=100$. Otherwise one can get only an upper bound on the mass $m_{X}$.

IV.4 measurement of the ”Planck scale”

Finally, if the LSP is the gravitino ($X=\tilde{G}$), the uncertainty of the reconstructed gravitino mass $\widehat{m_{\tilde{G}}}$ translates into an uncertainty of the supergravity Planck scale, which is obtained by substituting $\widehat{m_{\tilde{G}}}$ for $m_{\tilde{G}}$ in Eq.(17). To take into account the error of $m_{\tilde{\tau}}$, we also substitute $\widehat{m_{\tilde{\tau}}}$ for $m_{\tilde{\tau}}$. Eq.(17) then becomes

[TABLE]

which is shown in Fig. 13 for $\widehat{m_{\tilde{\tau}}}=m_{\tilde{\tau}}$ (solid lines) and $0.99m_{\tilde{\tau}}\leq\widehat{m_{\tilde{\tau}}}\leq 1.01m_{\tilde{\tau}}$ (dashed lines). Here we do not include the uncertainty of the lifetime measurement, which simply affects the measured Planck scale as $M_{\mathrm{pl}}\propto(1/\Gamma_{\tilde{\tau}})^{1/2}$. Neglecting the $\tau$–lepton mass, Eq. (32) is simplified as

[TABLE]

As discussed in the previous section, the kinematical reconstruction of the gravitino mass is possible only if $m_{\tilde{G}}$ is sufficiently large, $m_{\tilde{G}}\buildrel>\over{{}_{\sim}}0.15m_{\tilde{\tau}}$ for $N_{\tilde{\tau}}=1000$ and $m_{\tilde{G}}\buildrel>\over{{}_{\sim}}0.25m_{\tilde{\tau}}$ for $N_{\tilde{\tau}}=100$. For smaller values of the gravitino mass, one can get only a lower bound on the Planck scale $M_{\mathrm{pl}}$.

One can see whether the determined ‘Planck scale’ is inconsistent with the Planck scale of the Einstein gravity, $M_{\mathrm{pl}}=2.4\times 10^{18}$ GeV. In other words, one can test the assumption of the decay $\tilde{\tau}\to\tau\tilde{G}$ by comparing the observed lifetime with the predicted lifetime. For example, if the NLSP dominantly decays into axino, the lifetime would be far shorter compared with gravitino assumption for a fixed mass $m_{X}$, leading to a smaller value of fitted $M_{\mathrm{pl}}$. Suppose for instance one measures $m_{\tilde{\tau}}=150$ GeV and $m_{X}=30$ GeV. When $X=\tilde{a}$, the lifetime becomes $\mathcal{O}(10~{}\mathrm{sec})$ for $f_{a}\simeq 10^{11}~{}\mathrm{GeV}$ and $m_{\tilde{B}}\simeq m_{\tilde{\tau}}$ [cf. Eq. (21)]. If one uses Eq. (32), a “Planck scale” $\widehat{M_{\mathrm{pl}}}=\mathcal{O}(10^{15}~{}\mathrm{GeV})$ would be obtained, thereby falsifying the gravitino assumption.

IV.5 model points and cosmological constraints

We now discuss mSUGRA model points discussed in section III, together with cosmological constraints. In the early universe, the $\tilde{\tau}$ CNLSP has been in thermal equilibrium until its decoupling, $T_{d}\sim 0.04m_{\tilde{\tau}}$. If such a particle has a very long lifetime, as discussed in this paper, its decay during or after the big-bang nucleosynthesis (BBN), $T_{\mathrm{BBN}}\sim 1$ MeV, may spoil the successful prediction of BBN BBNearly ; Kawasaki:2004qu+Jedamzik:2006xz . In the models with $\tilde{\tau}$ NLSP and $\tilde{G}$ LSP, this leads to severe constraints on the parameter space of ($m_{\tilde{\tau}}$, $m_{\tilde{G}}$), in particular to upper bounds on the gravitino mass for a given stau mass Asaka:2000zh+Fujii:2003nr ; Feng:2004mt+Steffen:2006hw .

Furthermore, it has been recently pointed out that a heavy charged particle can form a bound state with light elements during the BBN, which can lead to new effects and/or severer constraints Pospelov:2006sc+Kohri:2006cn+Kaplinghat:2006qr . In this paper we do not discuss these effects because it is difficult to evaluate the net effect quantitatively and it still awaits detailed analysis.

We should also mention that those BBN constraints may disappear if there is entropy production between the stau decoupling ($T_{d}\sim m_{\tilde{\tau}}/20$) and the BBN ($T_{\mathrm{BBN}}\sim 1$ MeV), because the stau abundance is diluted before its decay Buchmuller:2006tt .

Keeping in mind the possibilities of severer bounds and also a possible loophole, let us discuss the cases of $m_{\tilde{\tau}}=150$ GeV and $m_{\tilde{\tau}}=340$ GeV, corresponding to the mSUGRA model points $\epsilon$ and $\zeta$. According to the latest analyses Feng:2004mt+Steffen:2006hw including the effects of the hadronic decay Kawasaki:2004qu+Jedamzik:2006xz , the bounds on the gravitino are $m_{\tilde{G}}\buildrel<\over{{}_{\sim}}(20$–$80)~{}\mathrm{GeV}$ for $m_{\tilde{\tau}}\simeq 150~{}\mathrm{GeV}$ and $m_{\tilde{G}}\buildrel<\over{{}_{\sim}}(40$–$200)~{}\mathrm{GeV}$ for $m_{\tilde{\tau}}\simeq 340~{}\mathrm{GeV}$.555These constraints were derived without using the bound on the 3He. If one adopts it, the constraints become severer (cf. Kawasaki:2004qu+Jedamzik:2006xz ). The ranges of upper bounds correspond to the uncertainties of various bounds from primordial light elements.

For $m_{\tilde{\tau}}=150$ GeV (model point $\epsilon$), as can be seen in Table 1, one could collect more than 1000 staus for 300 fb*-1*. The bound $m_{\tilde{G}}\buildrel<\over{{}_{\sim}}(20$–$80)$ GeV then suggests that the measurement of the gravitino mass and the “Planck scale” may become possible if one assumes conservative BBN bounds and if the gravitino mass is sufficiently large (cf. Figs. 12 and 13). For $m_{\tilde{\tau}}=340$ GeV (model point $\zeta$), the measurement would become easier if one could collect the same number of staus (cf. Figs. 12 and 13). However, from Table 1 we find that expected number of stopped CNLSP is around 30 for 300 fb*-1*. This is because gluino mass is above 2 TeV for this point and production cross section is small. One needs SLHC ($\int{\cal L}=1000$fb*-1*) to collect $\mathcal{O}(100)$ events.

In the case of axino LSP, the BBN bound is much weaker because the lifetime of the CNLSP stau becomes much shorter [see Eq.(21)]. Hence, the axino mass measurement is plausible for sufficiently large $m_{\tilde{a}}/m_{\tilde{\tau}}$.

V Light Axino vs gravitino: The rare decay of the CNLSP

V.1 Low energy effective action of the axino and gravitino involving photon

When $m_{\tilde{G}}\buildrel<\over{{}_{\sim}}0.2m_{\tilde{\tau}}$, it is difficult to determine the gravitino mass from $\tau$ energy measurement at stopper–detector. Axino is a motivated candidate which couples weakly to the MSSM particle with comparable strength to the gravitino. In this section we therefore compare the decay $\tilde{\tau}\rightarrow\tilde{a}\tau\gamma$ with the decay $\tilde{\tau}\rightarrow\tilde{G}\tau\gamma$. For simplicity, we will assume the NLSP is pure ‘right-handed’ stau, $\tilde{\tau}=\tilde{\tau}_{R}$ throughout this section. Extension to the case with mixing with $\tilde{\tau}_{L}$ is straightforward, and the mixing angle dependence is expected to be small.

The gravitino is a spin-3/2 particle. However, in the limit where $m_{\tilde{G}}\ll m_{\tilde{\tau}}$, the effective interaction to MSSM particles would be reduced to that of spin 1/2 particle, goldstino $\tilde{\chi}$. The effective action relevant to the $\tilde{\tau}_{R}$ decay is given as follows,

[TABLE]

The action is similar to that of axino given in Eqs.(18) and (19) except the coupling coefficients. The relative weight of the two terms in Eq.(34) are fixed by the supergravity, while for the axino the coefficient in Eq.(19) is induced from Eq.(18) by the radiative corrections. Note that the term proportional to $X[\gamma^{\mu},\gamma^{\nu}]\tilde{B}F_{\mu\nu}$ is a non renormalizable coupling of the photon to gravitino or axino and induces significantly different $\gamma$, $\tau$ distribution.

The axino three body decay $\tilde{\tau}\rightarrow\gamma\tau\tilde{a}$ proceeds through the diagrams shown in Fig.14, where the hatched triangle express the effective vertex shown in Eq. (19). On the other hand, the relevant diagrams for the three body decay into goldstino $\tilde{\tau}\rightarrow\gamma\tau\tilde{\chi}$, are given in Fig.15. The diagram corresponding to the top right of Fig. 14 does not exist for the goldstino case. The difference of the actions and the relevant diagrams will appear as the deviation of the decay distributions. In the Appendix, we list the three body decay differential width into gravitino/axino in the limit where the gravitino/axino mass can be neglected compared to $m_{\tilde{\tau}}$. The formula for the massive gravitino and axino are given in Ref. Brandenburg:2005he .

V.2 Numerical results

The three body decay $\tilde{\tau}\to\tau\gamma X$ should be visible in the stopper–detector if it has an ability to measure charged tracks, and also segmented into small units. The position where $\tilde{\tau}$ decays in the detector is the position where the charged track by the $\pi^{+}$, $\mu$ and $e$ from $\tau$ is initiated. For hadronic tau decays, a $\pi^{\pm}$ is always in the decay products, sometimes with photons coming from $\pi^{0}$ decays. Photons are converted into electron after passing $\sim$ one radiation length $X_{0}$. Therefore for the iron based detector with $\rho=5(2)$g/cm3, the photon shower starts $2.8(7)$cm from the decay point. In summary, the three body decay of $\tilde{\tau}\rightarrow\tau\gamma X$ is identified as a charged track (which might be associated with collinear EM showers) + an isolated hard EM shower pointing back to the point where charged track is started. If the segmentation is not good enough, the efficiency to discriminate EM showers from $\pi^{\pm}$ would be reduced.

As can be seen in the appendix, the three body decay amplitude can be written as a function of the angle between photon and $\tau$, $\theta$, and $E_{\gamma}$. To be conservative we assume the energy resolution for the isolated photon shower is $\Delta E_{\gamma}/E_{\gamma}=100\%/\sqrt{E_{\gamma}/{\rm GeV}}$ and ignore the angular resolution of the photon momentum. (In the following $E_{\gamma}$ denotes the photon energy after taking into account of this finite resolution effect.) Note that the shortest length of the detector is 3.5 m, which corresponds to 1750g/cm2 for $\rho=5$g/cm3. The EM showers are likely contained in the stopper because we only need 200g/cm2 to fully absorb them.

We need to require several cuts for the accepted events.

• •

Experimentally, the angle between $\tau$ and $\gamma$ must be large enough to avoid the overlap between $\tau$ decay products and prompt $\gamma$. We only use the events where $\cos\theta<0.866$. We do not lose sensitivity to the differece between the two decays $\tilde{\tau}\rightarrow\tau\gamma\tilde{a}/\tilde{G}$ by cutting these events. In the collinear region, the amplitude is dominated by the contributions from QED vertex, which is common for both of the decays.

• •

The three body decay amplitude suffers soft and collinear singularity. Because we only adopt simple leading order calculation, we require $E_{\gamma}>$ 10 GeV and $E_{\tau}>10$ GeV.

We define the ratio

[TABLE]

The dependence of $R$ on the MSSM parameter is quite different between $X=\tilde{a}$ and $X=\tilde{G}$. While $\tilde{\tau}\rightarrow\tilde{a}\tau$ is one loop process controlled by the parameter $\xi$, the three body decay contains a tree level contribution which depends on non-renormalizable axino-$\tilde{B}$-gauge coupling (top left of Fig. 14).

When $\xi$ is small, the tree level contribution plays a dominant role in the three body decay into axino. This can be seen in Fig. 16, where the ratio $R(\tilde{a})$ is plotted as a function of the mass difference $m_{\tilde{B}}-m_{\tilde{\tau}}$. We also fix $m_{X}=1$ GeV but $R(X)$ is insensitive to $m_{X}$. $R(\tilde{a})$ is enhanced when the mass difference between $m_{\tilde{B}}$ and $m_{\tilde{\tau}}$ is small relative to the $\tilde{\tau}$ mass. This is because the two body decay of axino is suppressed by $m_{\tilde{B}}^{2}$. The branching ratio ranges from 8% to 0.5% for the model parameters given in the figure.

On the other hand, the branching ratio $Br(\tilde{\tau}\rightarrow\tau\gamma\tilde{G})$ is well below 1% for the parameter given in Fig.16. $Br(\tilde{\tau}\rightarrow\tau\gamma\tilde{G})/Br(\tilde{\tau}\rightarrow\tau\tilde{G})\sim 0.56\%$ $(0.84\%)$ for $m_{\tilde{\tau}}=100(190)$ GeV respectively.

When the three body decay branching ratio turns out be above 5%, 200 stopped $\tilde{\tau}$ is enough to see the 3 $\sigma$ deviation from the gravitino assumption. We estimate the number of events $N_{\rm event}$ stopped in the detector, which is required to find 3$\sigma$ deviation from the gravitino scenario as follows;

[TABLE]

The $N_{\rm event}$ as a function of $m_{\tilde{B}}-m_{\tilde{\tau}}$ is given in Fig.17. Each curve increases as $m_{\tilde{B}}-m_{\tilde{\tau}}$ is increased up to the value where $Br(\tilde{\tau}\rightarrow\tau\gamma\tilde{a})$ coincides with $Br(\tilde{\tau}\rightarrow\tau\gamma\tilde{G})$. When $m_{\tilde{B}}-m_{\tilde{\tau}}<20$ GeV, $\mathcal{O}(1000)$ ( $\mathcal{O}(10000)$ ) stopped $\tilde{\tau}$ are enough to see the deviation from the measurement of $R$ for $\xi=0.5(1)$ respectively.

Not only the branching ratio, but also the decay distribution contain the information of the invisible particle. The axino decay distribution can be enhanced at the region where $x_{\gamma}=2E_{\gamma}/m_{\tilde{\tau}}$ is large and $\cos\theta$ is small, namely hard photon and $\tau$ is back to back  Brandenburg:2005he . This occurs when $\xi$ is small and relative importance of the direct axino-bino-gauge boson coupling is enhanced.

In Fig. 18, left panel shows the $E_{\gamma}$ distribution for different $\cos\theta$ intervals. Here we fix $\xi=1$, $m_{\tilde{B}}=130$  GeV, and $m_{\tilde{\tau}}=100$ GeV. The distribution has significantly hard component for $\cos\theta<0$ when compared with that of gravitino in the right panel. The enhancement of back-to-back events is a signature of an axino, and clear $\tau$ and $\gamma$ separation is not required to distinguish them.

We now estimate the the number of CNLSP decays in the stopper to see the 3 $\sigma$ deviation between $\tilde{\tau}\rightarrow\tilde{a}\gamma\tau$ and $\tilde{G}\gamma\tau$ from the decay distribution. We define $\Delta\chi^{2}$ like function from the differential decay width;

[TABLE]

Here, $n_{i}(\tilde{a})$ is the number of event in a $i$-th bin for $\tilde{\tau}\rightarrow\tau\gamma\tilde{a}$ when the number of stopped $\tilde{\tau}$ is $N_{\rm gen}=10^{5}$. $\Delta E_{\gamma}$ and $\Delta\theta$ is the bin size in $E_{\gamma}$ and $\theta$. We divide the $E_{\gamma}$ and $\theta$ into 8 bins and 3 bins respectively, for the range $0<E_{\gamma}<m_{\tilde{\tau}}$ and $0<\theta<\pi$;

[TABLE]

The photon energy resolution of the detector is assumed $\Delta E_{\gamma}/E_{\gamma}=100\%/\sqrt{E_{\gamma}/{\rm GeV}}$. We apply the cut $\cos\theta<0.866$, $E_{\gamma}>10$ GeV and $E_{\tau}>10$ GeV. On the other hand, $\bar{n}_{i}(\tilde{G})$ is the number of event in a $i$-th bin for $\tilde{\tau}\rightarrow\tau\gamma\tilde{G}$ with $m_{\tilde{G}}=m_{\tilde{a}}$ normalized so that the total number is same to that of axino three body decay. Namely, we do not use the information for $R(x)$ in our fit.

From $\Delta\chi^{2}_{\rm dist}$, we define $N_{\rm dist}(3\sigma)$, the number of stopped $\tilde{\tau}$ required to see the 3$\sigma$ deviation between $\tilde{\tau}\rightarrow\tau\gamma\tilde{a}$ and $\tilde{\tau}\rightarrow\tau\gamma\tilde{G}$ as follows,

[TABLE]

We show the $N_{\rm dist}(\sigma)$ as a function of $m_{\tilde{B}}-m_{\tilde{\tau}}$ in Fig. 19. The sensitivity is significantly increased from the estimate using the branching ratio only. For $m_{\tilde{B}}-m_{\tilde{\tau}}=40$ GeV, the deviation may be visible for $\mathcal{O}(1000)$ events for $\xi=0.5$ ($\mathcal{O}(10000)$ events for $\xi=1$).

Finally, we estimate sensitivity at our model points in section 3. For simplicity we assume $\tilde{\tau}=\tilde{\tau}_{R}$. In general $\tilde{\tau}_{R}$ is mixed with $\tilde{\tau}_{L}$. The mixing angle is defined as

[TABLE]

For model points discussed in Table 2, the angle is $\sin\theta_{\tilde{\tau}}\sim 0.9$. The effect of the mixing angle in the axino decay is small because the amplitude of $\tau_{L}$ is suppressed by both by the small $\cos^{2}\theta_{\tilde{\tau}}$ factor and smaller hypercharge, and can be safely ignored.

In Fig.20 the expected sensitivity at the stopper–detector is shown. Here, long-dashed (long-short-dashed, dashed) line corresponds to the required statistics for $\xi=0.5$ (0.75, 1) for different $\Lambda$, while the upper and lower solid lines correspond to the number of stopped CNLSP for 300 fb*-1* and 3000 fb*-1*. One can address the difference between axino and gravitino for $\Lambda\sim 55(65)$TeV or less for ${\cal L}=3000$ fb*-1* and $\xi=1$ (0.75). Note that the expected reach in $\Lambda$ is essentially determined by the SUSY production cross section, as they decrease steeply with increasing $\Lambda$. We therefore show the gluino masses scale on the top of the figure.

VI Discussion

In this paper we investigate the physics of the long-lived charged next lightest SUSY particle (CNLSP), which may be explored at a massive stopper–detector placed next to the CMS detector at the LHC. We assume the CNLSP is the lighter scalar tau, $\tilde{\tau}$, which decay into $\tau X$ where $X$ is a invisible particle. A natural candidate of the particle $X$ is a gravitino but we also consider the case where $X$ is axino $\tilde{a}$.

In this paper, we assume very large stopper–detector next to the CMS detector, with total mass of 8 kton. The stopper must have a capability to measure the position where the CNLSP stopped, and also the energy of the $\tau$ decay products. If the detector can be highly segmented, it is also possible to identify the $\tau$ decay products separately. The number of stopped NLSP ranges from $\mathcal{O}(10^{4})$ to $\mathcal{O}(100)$ for gluino and squark with mass below 2 TeV. If the size of the detector should be smaller, the number of the events must be scaled down linearly.

We estimate the statistical error for $E_{\tau}$, which can be determined from the end point of the tau jet energy. We assume that the energy resolution of the detector is 150%/$\sqrt{E/{\rm GeV}}$. $\delta E_{\tau}/E_{\tau}\simeq 60\%/\sqrt{E_{\tau}/{\rm GeV}}$ ($15\%/\sqrt{E_{\tau}/{\rm GeV}}$) can be achieved for the statistics of the order of $N_{\rm stopped}=100(1000)$. When LSP mass is above $0.2m_{\tilde{\tau}}$, one can constrain the gravitino mass both from above and below. From the lifetime and gravitino mass measurement, the supergravity Planck scale $M_{\mathrm{pl}}$ can be measured. If many CNLSP can be accumulated, one can even study the three body decay of the CNLP $\tilde{\tau}\rightarrow\tau\gamma X$. When the dominant decay mode of the $\tilde{\tau}$ is $\tilde{\tau}\rightarrow\tilde{a}\tau$, we may be able to see the deviation of the decay branching ratio and the distribution from the $X=\tilde{G}$ case.

Finally we comment on the strategy to proceed this experiment. SUSY particles will be found in the early stage of the LHC experiment if SUSY scale is $\mathcal{O}(1~{}\mathrm{TeV})$. If LSP is gravitino or axino and the NLSP is charged and long-lived, it would also be recognized easily. The detector proposed in this paper may be placed after the existence of long-lived CNLSP is observed, roughly at the same time to the high luminosity run of the LHC, or the proposed super LHC run.

LHC experiment has a great potential to explore new physics in TeV regions. It is important to explore new possibilities that can be done with LHC. In this paper we have proposed a large additional detector in the CMS cavern when the long-lived CNLSP is found. This requires a significant modification of the CMS experiment. The reward is low systematics study of the CNLSP decay which primarily serve for the study of the gravitino sector. The determination of gravitino mass either from the lifetime or (independently) from $E_{\tau}$ measurement would give us a direct information of the the total SUSY breaking scale. Together with high precision determination of the MSSM sector expected with the CNSLP momentum information, the nature of the interaction of the MSSM sector and hidden sector can be studied in detail. We hope this paper is useful for further, and more realistic studies.

Acknowledgement

We thank J. Ellis, H. U. Martyn and F. D. Steffen for discussions. This work is supported in part by the Grant-in-Aid for Science Research, Ministry of Education, Science and Culture, Japan (No.16081207, 18340060) for MMN.

VII appendix

In this appendix we list the differential decay width of $\tilde{\tau}_{R}$ decay into $\gamma\tau X$ where $X$ is either gravitino or axino in limit where $m_{X}\ll m_{\tilde{\tau}_{R}}$. Formulas in this appendix are obtained by taking massless limits $m_{\tilde{G}}/m_{\tilde{a}}\to 0$ of the formulas in Ref. Brandenburg:2005he . The $\tilde{\tau}_{R}$ decay width to $\tau\gamma\tilde{G}$ is given as follows;

[TABLE]

where

[TABLE]

and

[TABLE]

Here

[TABLE]

and $\theta$ is the angle between $\tau$ and $\gamma$.

For the case of massless axino, we find

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]