Athanasios Dedes

University of Bonn, Germany

XIV Workshop, Beyond the Standard Model, March 11 - 14, 2002, Bad Honnef, Germany

B_{s} → μ^{+}μ^{−}

in the Minimal Supergravity Scenario

In collaboration with:

H. Dreiner and U. Nierste, Phys. Rev. Lett. 87, 251804 (2001),

                    hep-ph/0108037

1. mSUGRA scenario (definitions)

2. The light Higgs boson mass

3. Muon anomalous magnetic moment, (g − 2)_{μ}

4. The decay B_{s} → μ^{+}μ^{−}

5. Results

6. Work in progress.....

7. Conclusions

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  1

Supersymmetry

• Supersymmetry is a symmetry which relates particles of different spins. It has a virtue of allowing non trivial interactions between fermions and bosons.

• It is possible for the quadratic divergences to cancel between the bosonic and fermionic loops. To insure that this cancellation persists to all orders we need Supersymmetry.

—

• Supersymmetry is certainly not a symmetry in nature: for example, there is certainly no bosonic particle with the mass and charge of the electron. Supersymmetry must be broken at a scale of the electroweak symmetry breaking in order to keep in touch with the solution of the hierarchy problem.

Minimal SuperSymmetric Standard Model

Superfields SU(3)_{C} × SU(2)_{L} × U(1)_{Y}  Components
Q_{r}^{i,α}  (3, 2, \frac{1}{6})  ( ũ ; d̃)_{L} , ( \begin{matrix}u \\ d \end{matrix} )_{L}
D_{r,α}^{c}  (¯3, 1, \frac{1}{3})  d̃^{c} , d^{c}
U_{r,α}^{c}  (¯3, 1, −\frac{2}{3})  ũ^{c} , u^{c}
L_{r}^{i}  (1, 2, −\frac{1}{2})  ( ν̃_{e} ; ẽ )_{L} , ( \begin{matrix}ν_{e} \\ e \end{matrix} )_{L}
E_{r}^{c}  (1, 1, 1)  ẽ^{c} , e^{c}
H_{1}^{i}  (1, 2, −\frac{1}{2})  ( H̃_{1}^{0} ; H̃_{1}^{−} ) , ( \begin{matrix}H_{1}^{0} \\ H_{1}^{−} \end{matrix} )
H_{2}^{i}  (1, 2, \frac{1}{2})  ( H̃_{2}^{+} ; H̃_{2}^{0} ) , ( \begin{matrix}H_{2}^{+} \\ H_{2}^{0} \end{matrix} )

Superfields SU(3)_{C} × SU(2)_{L} × U(1)_{Y} Components
V_{3}^{(R)}  (8, 1, 0)  G̃^{(R)} , G_{μ}^{(R)}
V_{2}^{(Γ)}  (1, 3, 0)  W̃^{(Γ)} , W_{μ}^{(Γ)}
V_{1}  (1, 1, 0)  B̃ , B_{μ}

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  3

Superpotential

W = ϵ_{ab} [(Y_{E})_{ij}L_{i}^{a}H_{1}^{b}Ē_{j} + (Y_{D})_{ij}Q_{i}^{ax}H_{1}^{b}D̄_{jx} + (Y_{U})_{ij}Q_{i}^{ax}H_{2}^{b}Ū_{jx}]

+ ϵ_{ab} μH_{1}^{a}H_{2}^{b}

+ ϵ_{ab} [\frac{1}{2}λ_{ijk}L_{i}^{a}L_{j}^{b}Ē_{k} + λ′_{ijk}L_{i}^{a}Q_{j}^{xb}D̄_{kx}] →L̸ violation

+ \frac{1}{2}ϵ_{xyz}λijk^{′′}Ū_{i}^{x}D̄_{j}^{y}D̄_{k}^{z} →B̸ violation

+ ϵ_{ab} [κ^{i}L_{i}^{a}H_{2}^{b}] →L̸ violation.

⇓

Proton Decay!

• B number is violated by 9 λijk^{′′} couplings

• L bumber is violated by 9 λ_{ijk} + 27 λ′_{ijk} + 3 κ_{i} couplings

Minimal Supergravity Scenario

• Since the most general parametrization of SUSY breaking introduces more than 100 parameters a construction of a model with few parameters is needed : mSUGRA seems the most elegant!

• Supergravity → local SUSY transformations

Supergravity Scalar Potential :

H. P. Nilles ’82, A. Chamseddine, R. Arnowitt, P. Nath, ’82 R. Barbieri, S. Ferrara, C. Savoy, ’82 L. Hall, J. Lykken, S. Weinberg, ’83 S. K. Soni, H. A. Weldon, ’83

V = e^{(\frac{Z{† \atop A}Z_{A}}{M^{2}})} [F_{A}^{†}F_{A} − 3\frac{f^{†}f}{M^{2}} ] + \frac{1}{2}D^{α}D^{α} ,

√ V not positive definite! so it could be spontaneous SUSY breaking : < F_{A} >≠ 0 with zero vacuum energy, V = 0

→ New terms arive!

• m_{ij}^{2} = M_{0}^{2} δ_{ij} , • M_{1} = M_{2} = M_{3} = M_{1/2}

• A_{ijk} = A_{0}^{2} , • ϕ_{A_{0}}

• B_{ij} = B_{0} δ_{ij} , • ϕ_{B_{0}}

Gauge coupling Unification in the MSSM

• 2 loop + thresholds from SUSY particles

⇒ but α_{s}^{exp} = 0.118 ± 0.002 ⇒ We need a GUT model !!

• Minimal SU(5) model : Large α_{s} > 0.13 • Missing Doublet model based on SU(5) gauge group : Small α_{s} < 0.12 • Flipped SU(5) and SO(10) correct values of α_{s}

The Light Higgs Boson in mSUGRA

50 GeV ≤ M_{0} ≤ 1 TeV
50 GeV ≤ M_{1/2} ≤ 1 TeV
−3 TeV ≤ A_{0} ≤ 3 TeV
1.5 ≤ tan β ≤ 60
μ = ±1

S. Ambrosanio, A. Dedes, S. Heinemeyer, S. Su and G. Weiglein ’01

(g − 2)_{μ} : Before November 2001...

• Based on Davier-Höcker calculation for the Hadronic contributions to the muon g − 2, BNL reports a 2.6σ deviation from the SM

There are four possible interpretations of this discrepancy :

• a statistical fluctuation with 0.9% probability.

• something is “wrong” with the experiment

• something is “wrong” with the theoretical analysis of the hadronic and/or the light by light contribution to the muon anomalous magnetic moment.

• the by far most exciting possibility:

there is “New Physics” BSM

21.5 × 10^{−10} < ∼ δa_{μ} < ∼ 63.7 × 10^{−10} @90% CL

• δa_{μ} is positive

• δa_{μ} is of the same order of the EW contributions, δa_{μ}^{NP} ∝

\frac{G_{F} m_{μ}^{2}}{4π^{2}\sqrt{2}}

• mSUGRA
(δa_{μ})_{SUSY} ∝ tan β _{\frac{f(M_{0})}{M_{1/2}^{2}}}

invites large tan β > ∼ 10 and M_{1/2} < ∼ 750 GeV

(g − 2)_{μ} : After November 2001...

M. Knecht, A. Nyffeler, M. Perrottet and E. De Rafael Nov. ’01 M. Hayakawa and T. Kinoshita, Dec. ’01

• : A mistake on the light by light scattering calculation :

−4.2 × 10^{−10} < ∼ δa_{μ} < ∼ 41.3 × 10^{−10} @90% CL

→ only just above 1σ significance...

√ A more definite claim is waiting for a more precise a_{μ} measurement...

→ one ideally wishes to study other observables whose sensitivity to supersymmetric loop corrections is correlated with δa_{μ}.

B_{s} → μ^{+}μ^{−} vs. B → X_{s}γ

• SUSY modifies B-meson observables if tan β is large [⇒ Y_{b} large ]

• Especially sensitive observables : B → X_{s}γ, B → l^{+}l^{−}

• B → X_{s}γ : relies on a cancelation between H^{±} − t and χ̃ − t̃ loops.

• In mSUGRA the low energy value for the trilinear term A_{t} is dominated by M_{1/2}, A_{t} ≃ −2M_{1/2}. Then μ > 0 implies that the charged-Higgs-top loop and the chargino-stop loop tend to cancel in B(B → X_{s}γ), so that the sensitivity to mSUGRA corrections is weakened.

• B → l^{+}l^{−} : large for large tan β, almost independent from the sign of μ. No accidental cancelation required!

• Since B(B → ℓ^{+}ℓ^{−}) ∝ m_{ℓ}^{2}, the branching ratio is largest for ℓ = τ: hard to detect!

• branching ratios of B_{d} mesons are smaller by a factor of |V_{td}/V_{ts}|^{2} < ∼ 0.06.

• BaBar and BELLE could look for B_{d} → τ^{+}τ^{−} ?

K. S. Babu and C. Kolda, ’99, C. Huang, W. Liao and Q. Yan, ’99 P. H. Chankowski and L. Slawianowska, ’01, C. Bobeth, T. Ewerth, F. Kruger and J. Urban, ’01 G. Isidori, A. Retico, ’01.

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  10

B_{s} → μ^{+}μ^{−}

H_{eff} = −\frac{4G_{F}}{\sqrt{2}} V_{tb}V_{ts}^{∗}[c_{10}O_{10} + c_{S}O_{S} + c_{P}O_{P}]

O_{10} = \frac{e^{2}}{16π^{2}}(s̄γ^{μ}P_{L}b)(l̄γ_{μ}γ_{5}l)
O_{S} = \frac{e^{2}}{16π^{2}}m_{b}(s̄P_{R}b)(l̄l)
O_{P} = \frac{e^{2}}{16π^{2}}m_{b}(s̄P_{R}b)(l̄γ_{5}l)

• SM → O_{10} • MSSM → O_{S} • MSSM → O_{P}

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  13

• Branching Ratio :

B(B_{s} → μ^{+}μ^{−}) = \frac{G_{F}^{2}α^{2}M_{B_{s}}^{3}τ_{B_{s}}f_{B_{s}}^{2}}{64π^{3}}|V_{tb}^{∗}V_{ts}|^{2}\sqrt{1 − \frac{4m_{μ}^{2}}{M_{B_{s}}^{2}}}

× \{(1 − \frac{4m_{μ}^{2}}{M_{B_{s}}^{2}})|\frac{M_{B_{s}}}{m_{b} + m_{s}}C_{S}|^{2} + |\frac{2m_{μ}}{M_{B_{s}}}C_{10} − \frac{M_{B_{s}}}{m_{b} + m_{s}}C_{P} |^{2}\}

• Standard Model :

G. Buchalla and A. J. Buras, ’94 M. Misiak and J. Urban, ’99

B(B_{s} → μ^{+}μ^{−}) = (3.7 ± 1.2) × 10^{−9}

• MSSM : Large tan β > ∼ 40

C_{S,P} ≃ ±\frac{m_{μ}}{4s_{W}^{2}M_{W}^{2}} × tan^{3} β × \frac{μA_{t}}{M_{A}^{2}} × C_{0}(μ, mt̃_{1}, mt̃_{2})

• MSUGRA : Large tan β > ∼ 40

B(B_{s} → μ^{+}μ^{−}) ∝ tan^{6} β/M_{1/2}^{4} for M_{1/2} ≫ M_{0}

• For large tan β, higher order corrections can be large

University of Bonn  13  Athanasios Dedes

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  14

• tan β-enhanced supersymmetric QCD corrections have been summed to all orders in perturbation theory.

M. Carena, D. Garcia, U. Nierste and C. E. Wagner, ’00

• We have incorporated these dominant higher order corrections by simply replacing the b Yukawa coupling h_{b} ∝ m_{b} tan β with h_{b}^{eff} = h_{b}/(1 + Δm_{b})

• QCD corrections to B_{s} → μ^{+}μ^{−} in the MSSM have been recently calculated.

C. Bobeth, A. J. Buras, F. Kruger and J. Urban, Dec. ’01

• \underline{Experiment}

• Only an experimental bound from CDF Run I

B(B_{s} → μ^{+}μ^{−}) < 2.6 × 10^{−6}, at 95% C.L.

• Near Future experiment: CDF Run IIa and IIb → Single event sensitivity (S) : Branching ratio for which we would expect to observe one event in 2 fb^{−1}

S(B_{s} → μ^{+}μ^{−}) = 1.0 × 10^{−8} × \frac{2 fb^{−1}}{∫ L}

→ 5 × 10^{−7} for 50 events with luminosity 2 fb^{−1} → 5 × 10^{−7} for 250-500 events with luminosity 10-20 fb^{−1}

K. Anikeev et al., “B physics at the Tevatron: Run II and beyond,” arXiv:hep-ph/0201071.

Results

• M_{1/2} = 450, M_{0} = 350, A_{0} = 0, μ > 0, m_{t} = 175 GeV √ Fine with b → sγ and direct searches...

A. Dedes, H. K. Dreiner and U. Nierste, ’01 A. Dedes, H. K. Dreiner and U. Nierste, ’01 A. Dedes, H. K. Dreiner and U. Nierste, ’01

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  18

Trileptons and B_{s} → μ^{+}μ^{−}at Tevatron

• The Mode

• Final State : lll, l^{±}l^{±}, l^{±}l^{±}τ, l^{±}ττ, τττ, τ^{±}τ^{±}

University of Bonn  18  Athanasios Dedes

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  19

Work in progress..........

A. Dedes, H. Dreiner, U. Nierste, P. Richardson

B_{s} → μ^{+}μ^{−} in the Minimal Supergravity Scenario,  hep-ph/0108037  20

SO(10) Unification

T. Blazek, R. Dermisek, S. Raby ’02

→ B(B_{s} → μ^{+}μ^{−}) > ∼ 10^{−5} already excluded by CDF

University of Bonn  20  Athanasios Dedes

T. Blazek, R. Dermisek, S. Raby ’02

!!!! Detailed analysis for B_{s} → μ^{+}μ^{−} is not done.. !!!! SO(10) predicts B_{s} → μ^{+}μ^{−} just around the corner...

Conclusions

√ Striking correlation between the muon anomalous magnetic moment a_{μ} and the branching ratio B(B_{s} → μ^{+}μ^{−}) in mSUGRA scenarios.

√ If a reported excess on a_{μ} is caused by mSUGRA corrections with large tan β, one faces more than an order of magnitude enhancement of B(B_{s} → μ^{+}μ^{−}) over its SM value.

√ Well within reach of Run-II of the Tevatron.

√ The two combined measurements significantly constrain the mSUGRA parameters, allowing a determination of tan β and M_{1/2}.

√ A measurement of B(B_{s} → μ^{+}μ^{−}) will further constrain the mass of the lightest Higgs boson.