Supersymmetry, naturalness, and signatures at the CERN LHC

Weak scale supersymmetry is often said to be fine-tuned, especially if the matter content is minimal. This is not true if there is a large $A$ term for the top squarks. We present a systematic study on fine-tuning in minimal supersymmetric theories and identify low-energy spectra that do not lead to severe fine-tuning. Characteristic features of these spectra are: a large $A$ term for the top squarks, small top squark masses, moderately large $\tan \beta$, and a small $\mu$ parameter. There are classes of theories leading to these features, which are discussed. In one class, which allows a complete elimination of fine-tuning, the Higgsinos are the lightest among all the superpartners of the standard model particles, leading to three nearly degenerate neutralino/chargino states. This gives interesting signals at the LHC—the dilepton invariant mass distribution has a very small endpoint and shows a particular shape determined by the Higgsino nature of the two lightest neutralinos. We demonstrate that these signals are indeed useful in realistic analyses by performing Monte Carlo simulations, including detector simulations and background estimations. We also present a method that allows the determination of all the relevant superparticle masses without using input from particular models, despite the limited kinematical information due to short cascades. This allows us to test various possible models, which is demonstrated in the case of a model with mixed moduli-anomaly mediation. We also give a simple derivation of special renormalization group properties associated with moduli mediated supersymmetry-breaking, which are relevant in a model without fine-tuning.

Figure 1

Minimal values of $m_{\tilde{t}}\equiv (m_{Q_3}^2{)}^{1/2}=(m_{U_3}^2{)}^{1/2}$ giving $M_{\mathrm{Higgs}}\gtrsim 114.4\mathrm{GeV}$ as a function of $A_t/m_{\tilde{t}}$. The other parameters are fixed to be 500 GeV for the gaugino and sfermion masses other than $m_{Q_3}^2$ and $m_{U_3}^2$, $(500\mathrm{GeV})(A_t/m_{\tilde{t}})$ for the $A$ parameters other than $A_t$, $\tan \beta =15$, $\mu =+170\mathrm{GeV}$, and $m_A=250\mathrm{GeV}$.

Figure 2

Contours of ${\Delta }^{-1}$ on the $m_0-M_{1/2}$ plane for the constrained mSUGRA with $A_0=0$ (left) and $A_0=-3|m_0|$ (right). The sign of $\mu$ is chosen to be positive. The constraints from direct superparticle search, the Higgs boson mass bound, and the stau LSP are also shown.

Figure 3

Contours of ${\Delta }^{-1}$ on the $m_0-M_{1/2}$ plane in the mSUGRA model with $\mu =190\mathrm{GeV}$ and $m_A=250\mathrm{GeV}$ fixed at the weak scale and $A_0=-3|m_0|$ at $M_{\mathrm{unif}}$ (left). Contours of ${\Delta }^{-1}$ on the $m_{\tilde{q}}-M_{1/2}$ plane with $\mu >0$ and $m_{\tilde{l}}^2=(500\mathrm{GeV}{)}^2$, $A_0=-|m_{\tilde{q}}|$ and $m_H^2\equiv m_{H_u}^2=m_{H_d}^2=(100\mathrm{GeV}{)}^2$ at $M_{\mathrm{unif}}$, where $m_{\tilde{q}}^2$ and $m_{\tilde{l}}^2$ are the squark and slepton masses, respectively, (right). In both cases $\tan \beta =15$.

Figure 4

Contours of ${\Delta }^{-1}$ on the $M_{\mathrm{mess}}-F/M_{\mathrm{mess}}$ plane for the minimal gauge mediation models with $n_{\mathrm{mess}}=1$ (left) and $n_{\mathrm{mess}}=4$ (right) pairs of messenger fields in the $\mathbf{5}+{\mathbf{5}}^{*}$ representation. The instability bound implies the region in which the messenger fields are tachyonic, $F/|M_{\mathrm{mess}}{|}^2>1$. The Higgs sector parameters are fixed as $\tan \beta =15$ and $\mu >0$.

Figure 5

Characteristic spectra for the superparticles which give the correct scale for electroweak symmetry breaking without significant fine-tuning. The spectrum in (a) arises typically in a high scale supersymmetry-breaking scenario with ${\Delta }^{-1}\approx 10\%$, while that in (b) arises in a theory where supersymmetry is broken by boundary conditions, or moduli contributions, with small (effective) $M_{\mathrm{mess}}$.

Figure 6

The diagrams contributing to the ${\tilde{\chi }}_2^0\to {\tilde{\chi }}_1^0{l}^{+}{l}^{-}$ decay.

Figure 7

Dilepton invariant mass distribution from the ${\tilde{\chi }}_2^0\to {\tilde{\chi }}_1^0{l}^{+}{l}^{-}$ decay. Solid lines represent the distributions for the Higgsino LSP case (left panel) and the case of gauginolike ${\tilde{\chi }}_1^0$ and ${\tilde{\chi }}_2^0$ with $\mathrm{sgn}(M_1)=\mathrm{sgn}(M_2)$ (right panel). The curves with dashed and dotted lines represent the ones obtained from Eq. (20) with ${\eta }_{\chi }=-1$ and $+1$, respectively, (in both panels).

Figure 8

Decay cascades used in the analysis.

Figure 9

The distribution of the dilepton invariant mass, $M_{ll}$, for point I. Hatched histogram represents the standard model background, which is smeared over five bins in order not to magnify the statistical uncertainty due to the scaling of the events. The solid line is the best fit function for the signal plus background, obtained using the theoretical curve with ${\eta }_{\chi }=-1$. It can be clearly distinguished from the ${\eta }_{\chi }=+1$ case, drawn by the dashed line. The endpoint is extracted to be $15.30\pm 0.15\mathrm{GeV}$.

Figure 10

The distributions of $M_{llq}$ (left) and $M_{lq}$ (right) for point I. Hatched histogram represents the standard model background, which for $M_{llq}$ is smeared over five bins in order not to magnify the statistical uncertainty due to the scaling of the events. The solid line is the best fit function for the signal plus background near the endpoint, obtained by using a linear function with Gaussian smearing for the signal and a linear function for the background. The endpoint is extracted to be $179.6\pm 5.8\mathrm{GeV}$ for $M_{llq}$. The $M_{lq}$ endpoint cannot be extracted clearly using a simple linear function.

Figure 11

The $M_{T2}$ (defined in the text) distribution for the input value of the neutralino mass $m_{{\tilde{\chi }}_1^0}=200\mathrm{GeV}$ for point I. Hatched histogram is the standard model background. The endpoint is extracted by fitting the signal plus background histogram with a linear function, and the background near the endpoint by a linear function. The endpoint is obtained to be $505.3\pm 0.6\mathrm{GeV}$.

Figure 12

Two curves on the $m_{{\tilde{\chi }}_1^0}-m_{\tilde{q}}$ plane deduced from the cascade decay analysis, $M_{ll}^{\max }$ and $M_{llq}^{\max }$, and the squark pair production analysis, $M_{T2}^{\max }$, for point I. Both curves are obtained by inputting hypothetical values of $m_{{\tilde{\chi }}_1^0}$, which is taken as the horizontal axis. The intersection determines the real values of $m_{{\tilde{\chi }}_1^0}$ and $m_{\tilde{q}}$. The obtained masses with the $1\sigma$ statistical errors are shown by shaded bands.

Figure 13

The statistical errors of the measured mass parameters $m_{{\tilde{\chi }}_1^0}$, $m_{\tilde{q}}$ and $\mathit{\Delta }m$ for point I. Arrows indicate the input values used in the Monte Carlo event generation. The accuracy at the level of 10%, 2% and 1% is obtained for $m_{{\tilde{\chi }}_1^0}$, $m_{\tilde{q}}$ and $\mathit{\Delta }m$, respectively.

Figure 14

The $M_{jj}$ (left) and $M_{bb}$ (right) invariant mass distributions for point I. In the $M_{jj}$ analysis, we select events with three hard jets and take a combination that gives the smallest $M_{jj}$, out of the three combinations, in order to see the endpoint. A fit is performed with a linear function with Gaussian smearing together with a linear function for the background events. The endpoint is obtained as $377\pm 10\mathrm{GeV}$. In the $M_{bb}$ analysis, we select events with three hard jets including two hard $b$-jets. A similar endpoint, $370\pm 11\mathrm{GeV}$, is obtained using the same fitting function.

Figure 15

The statistical error of the gluino mass for point I. The arrow indicates the input value used in the Monte Carlo event generation. The accuracy at the level of 2% is obtained.

Figure 16

An empirical relation between the peak location of $M_{\mathrm{eff}}$ and a combination $m_{\tilde{q}}+m_{\tilde{g}}-m_{{\tilde{\chi }}_1^0}$. We have generated 50 000 events for each 30 sample points in the mixed moduli-anomaly mediation model. A very good linear relation is obtained. Note that this relation will be modified if different selection cuts are used. The standard model background has not been included in searching the peak location.

Figure 17

The distribution of $M_{\mathrm{eff}}$ for point I. The QCD background is smeared in order not to artificially magnify the statistical uncertainty due to the scaling of the events. The location of the peak is determined by fitting with a Gaussian function near the peak. It is given by $M_{\mathrm{eff}}^{\mathrm{peak}}=977\mathrm{GeV}$.

Figure 18

The distributions of $M_{ll}$ (left) and $M_{llq}$ (right) for point II. Hatched histogram represents the standard model background, which is smeared over five bins in order not to magnify the statistical uncertainty due to the scaling of the events. For $M_{ll}$, we have searched the endpoint by fitting with the theoretical curve assuming ${\eta }_{\chi }=-1$. The endpoint is obtained to be $9.48\pm 0.21\mathrm{GeV}$. For $M_{llq}$, we have used a linear function with Gaussian smearing for the signal and a linear function for the background. The endpoint is obtained to be $223\pm 12\mathrm{GeV}$.

Figure 19

The $M_{T2}$ distribution for the input value of the neutralino mass $m_{{\tilde{\chi }}_1^0}=200\mathrm{GeV}$ for point II. Hatched histogram is the standard model background. The endpoint is extracted by fitting the signal plus background histogram with a linear function, and the background near the endpoint by a linear function. The endpoint is obtained to be $716.6\pm 1.9\mathrm{GeV}$.

Figure 20

Two curves on the $m_{{\tilde{\chi }}_1^0}-m_{\tilde{q}}$ plane deduced from the cascade decay analysis, $M_{ll}^{\max }$ and $M_{llq}^{\max }$, and the squark pair production analysis, $M_{T2}^{\max }$, for point II. Both curves are obtained by inputting hypothetical values of $m_{{\tilde{\chi }}_1^0}$. The intersection determines the real values of $m_{{\tilde{\chi }}_1^0}$ and $m_{\tilde{q}}$. The obtained masses with the $1\sigma$ statistical errors are shown by shaded bands.

Figure 21

The $M_{jj}$ invariant mass distribution for point II. We select events with three hard jets and take a combination that gives the smallest $M_{jj}$, out of the three combinations, in order to see the endpoint. A fit is performed with a linear function with Gaussian smearing together with a linear function for the background events. The endpoint is obtained as $609\pm 13\mathrm{GeV}$.

Figure 22

The statistical errors of the measured mass parameters $m_{{\tilde{\chi }}_1^0}$, $m_{\tilde{q}}$, $\mathit{\Delta }m$ and $m_{\tilde{g}}$ for point II. Arrows indicate the input values used in the Monte Carlo event generation. The accuracy at the level of 15%, 2%, 2% and 2% is obtained for $m_{{\tilde{\chi }}_1^0}$, $m_{\tilde{q}}$, $\mathit{\Delta }m$ and $m_{\tilde{g}}$, respectively.

Figure 23

The $M_{\mathrm{eff}}$ distribution for point II. The QCD background is smeared in order not to artificially magnify the statistical uncertainty due to the scaling of the events. The peak location is determined by fitting with a Gaussian function near the peak. It is found to be $M_{\mathrm{eff}}^{\mathrm{peak}}=1306\mathrm{GeV}$.

Figure 24

A nontrivial test for the mixed moduli-anomaly mediation model in the case of $M_0=600\mathrm{GeV}$ (left) and $M_0=900\mathrm{GeV}$ (right). Three ways of calculating $M_0$, i.e. $m_Z^2/\mathit{\Delta }m$, $\sqrt{2}{m}_{\tilde{q}}$ and $m_{\tilde{g}}$, give the same value within $\approx 15\%$ theoretical uncertainties. For $m_{\tilde{g}}$, we have plotted the values obtained in two different ways, i.e. from $M_{jj}^{\max }$ and $M_{\mathrm{eff}}^{\mathrm{peak}}$.