Search for Supersymmetry with jets and missing transverse momentum and one lepton at √s = 7 TeV

(1)

Vasia Mitsou Elena Romero

Emma Torró

IFIC - València aTOPe

13.12. 2011

Search for Supersymmetry with jets and missing transverse momentum and one lepton at √ s = 7 TeV

Paper accepted for Physical Review D;

arXiv:1109.6606v1

(2)

RPV model presented

SUSY:

• one of the most popular extensions of the SM

• Assigns a superpartner to every SM particle: same quantum numbers except for spin (± 1/2)

• It answers many open questions in the SM:

- stabilises the Higgs mass

- allows unification of gauge couplings

- provides a dark matter candidate (if R-Parity is conserved)

But

• mechanism to spontaneously break SUSY unknown: several models

• the minimal effective theory including SUSY breaking contains 124 parameters complicated search strategy

• common signatures (

R-Parity conservation assumed)

:

- multiple high energetic jets

- missing transverse momentum (ETmiss)

(3)

W

RP

= ∑ ε

ⁱ

L

ⁱ

H

^u

+ ∑ λ

^ijk

L

ⁱ

L

^j

E

^k

+ ∑ λ’

^ijk

L

ⁱ

Q

^j

D

^k

+ ∑ λ’’

^ijk

U

ⁱ

D

^j

D

^k

- LSP: no need to be neutral nor stable.

- LSP decay: possibility to explore new signals, exploit LSP invariant mass and decay properties

- Single Sparticles production is possible - Not so large E

Tmiss

R-Parity

R = (-1)

^{3B + L +2s}

= +1 for SM particles

- 1 for SUSY particles

L-number violating terms

B-number violating terms

^ ^ ^ ^ ^ ^

^ ^ ^ C ^ C ^ C

RPConserving SUSY models: RPViolating SUSY models:

- Neutral Stable LSP

- Sparticles produced in pairs - Large E

Tmiss

- Standard SUSY searches involve R-Parity Conservation (RPC)

- Lepton and/or baryon number violation constrained by previous experiments but not forbidden.

i i,j,k i,j,k i,j,k

i,j,k = quark and lepton generations

(4)

• Bilinear R-Parity violating terms in superpotential introduce neutrino masses.

• χ₁ LSP not stable, connected to neutrino physics through:

• Theoretical framework:

- mSUGRA: tanβ = 10, A₀ = 0 GeV, sgn μ = +1 m0 = 100 - 800 GeV

m1/2 = 200 - 600 GeV

- bRPV: parameters chosen to be consistent with constraints from neutrino experiments:

Δm_atm², Δm_sol², tan²θ_atm, tan²θ_sol

Bilinear RPV: description

BR (χ₁ W μ) BR (χ₁ W τ)

~⁰

~⁰ ≈ tan² θ_atm

~⁰

(5)

~

1-lepton search: description

Interpretation of this analysis on bRPV.

- χ

₁

LSP not stable, prompt decaying to a muon and two jets

- Analysis optimised for 1 lepton RPC requiring exactly one muon and several jets.

- Analysis performed using 1.04 fb

^-1

of 2011 collision data

~₀

RPC bRPV

mSUGRA scenario assumed, trigger based on muons, object definition, GRL...

- Analysis optimised for RPC with one lepton searches.

long cascade decay long cascade decay, same as RPC but with neutralino decaying.

Kinematic selection: 1 lepton

and at least 3 jets Minimal Kinematic selection: at least 1 lepton and at least 2 jets MET due to (2) stable LSPs MET mainly due to LSP-decay

neutrinos

q q

q

τ ν τ

W μ q

χ₁ χ₁

~

p p

+ 0

(6)

1-lepton search

•1 lepton + n Jets + ETmiss searches

• Common Event Pre-Selection:

- Data quality flags - Single muon trigger - Good primary vertex - Cosmic muon veto

- exactly one isolated muon with pT > 20 GeV, used to trigger the events.

- veto for events with at least one electron with pT > 20 GeV, aimed to avoid overlap with other analyses.

• 4 signal regions defined

• Main background in signal regions: W + jets and top pairs

• Global variables used:

(7)

[GeV]

miss

ET

0 100 200 300 400 500 600

Data / SM

0 1 2

Events / 10 GeV

10-1

1 10 102

103

104

105

106

L dt = 1.04 fb-1

!

ATLAS

Muon Channel

)>0.2

1,2,3,4 miss,jet

(ET

"

# Data/MC overflow = 0/0.2

=7 TeV) s

Data 2011 ( Standard Model

multijets (data estimate) W+jets

Z+jets t

tsingle top Dibosons

1/2=330

=500 m MSUGRA m0

1-lepton search: signal regions

•

4 different Signal Regions analysed: 3-jet loose selection (3JL), 3-jet tight selection (3JT), 4-jet loose selection (4JL) and 4-jet tight selection (4JT).

•

Common Event Pre-Selection

Selection 3JL 3JT 4JL 4JT’

Number of jets ≥ 3 ≥ 4

Leading jet pT (GeV)

Subsequent jet pT (GeV) 60

25 80

25 60

40 ΔΦ(jet, E^missT) [>0.2(mod. π)]for all 3 (4) jets

mT (GeV) > 100

E^missT (GeV) > 125 > 240 > 140 > 200 E^missT / Meff > 0.25 > 0.15 > 0.30 > 0.15

Meff (GeV) > 500 > 600 > 300 > 500

(8)

1-lepton search: background studies

• Separate Control Regions (CR) for the 3-jet, 4-jet selections defined.

• Same Pre-Selection, lepton and jets requirements as in the corresponding SR.

• 3 types of main CR: W+jets (WR), Top-pairs (TR) and QCD (QR).

• Several additional Control Regions.

Control Region WR TR QR

ΔΦ(jet, E^missT) >0.2 (mod.π)

mT (GeV) 40 GeV < mT < 80 GeV no requirement

E^missT (GeV) 30 GeV < E^missT < 80 GeV > 30 GeV Meff (GeV) > 500 (3J) ; > 300 (4J) no requirement

Number of the 3 or 4 jets with higher pT tagged

as a b-jet

0 ≥ 1 no requirement

[GeV]

miss

ET

0 20 40 60 80 100 120 140 160 180

[GeV]

W TM

0 20 40 60 80 100 120 140 160 180

SR

WR no b−jet

1 b−jetTR

!

QR

eff/T

XR high m

XR W XR Top

XR QCD(1) XR QCD(2) XR WTop

XR Top dilep

(9)

1-lepton search: background studies

• W + jets and Top contributions:

- Normalize MC to data for each background and specific CR

- Assuming bkg MC shapes, extrapolate normalization to Signal Regions:

Npred,jSR = NdataiR x , i= W, T; j = W+jets, top

• QCD multijet treatment:

- Estimation of contamination of QCD background in each of the regions with data-driven methods (Matrix Method).

- Take into account contamination from QCD events with real leptons

misidentified leptons

(estimated using Z ee, tt and W+jets events).

NMC,jSR

NMC,jiR

• The assumption that the MC simulation is able to predict the backgrounds in the signal regions is validated by checking additional Control Regions.

• Possible contamination from atmospheric muons is studied and found to be negligible for | zμ ‐ zPV| > 

5 mm.

• Background from single top and dibosons studied, found to be small.

(10)

[GeV]

meff

0 200 400 600 800 1000 1200 1400 1600

Data / SM

0 1 2

Events / 50 GeV

10-1

1 10 102

103

104

105

106

L dt = 1.04 fb-1

!

ATLAS

Muon Channel

3J W+jets Control Region Data/MC overflow = 0/0.2

=7 TeV) s

Data 2011 ( Standard Model

multijets (data estimate) W+jets

Z+jets t

tsingle top Dibosons

1/2=330

=500 m MSUGRA m0

1-lepton search: background studies

•

Final determination of background is done performing a simultaneous likelihood fit of the different CR to account for cross contamination.

•

The determination of the QCD multijet contribution to the various regions is performed as part of the fit procedure.

- Good agreement data / MC

Systematic uncertainties on bkg estimations:

• JES and JER measured from 2010 data + pile-up effects in 2011

• MC modeling uncertainties affecting transfer factors

• lepton energy scale and resolution

• lepton ID / miss ID

• heavy flavour tagging uncertainties

• luminosity

Systematic uncertainties on signal:

• dominated by theoretical uncertainties in scale, αS and PDF ( 20 to 30 % ), the rest below 10%

• lepton trigger ID

•^{JES, JER}

•^pile-up

• luminosity

(11)

1-lepton search: signal studies

• Due to

χ

₁decay modes to μ, better trigger efficiency than RPC.

• Cosmic muon veto lowers efficiency for bRPV:

discard every event with any cosmic muon candidate, defined as

| zμ ‐ zPV| > 5 mm

| d0μ | > 1 mm

• Global variables cuts optimised for RPC

bRPV(efficiency) ~10% * RPC(efficiency)

Decay length in the rest frame of the

~₀

(12)

[GeV]

m0

100 120 140 160 180 200 220 240 260 280 300 [GeV] 1/2m

240 250 260 270 280 290 300

= 3 mm c!

= 10 mm c!

(600 GeV) q~

(600 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

3 jets, loose SR 1 muon, #

ATLAS _{95% CL}

Observed CLS

Expected CLS

1$

S± Expected CL

0.013 0.0057 0.0049 0.0052 0.0033 0.0039 0.0027 0.0017 0.014 0.0077 0.0076 0.0064 0.0055 0.0045 0.005 0.0028 0.013 0.01 0.0064 0.0076 0.0094 0.0079 0.0064 0.0059 0.014 0.014 0.0099 0.011 0.0094 0.009 0.0096 0.0078 0.016 0.014 0.013 0.012 0.012 0.011 0.012 0.011 0.016 0.016 0.014 0.016 0.016 0.014 0.016 0.018 0.018 0.02 0.024 0.018 0.016 0.02 0.02 0.021 0.015 0.024 0.022 0.023 0.022 0.024 0.025 0.026 0.021 0.021 0.026 0.023 0.027 0.023 0.028 0.029

[GeV]

m0

100 200 300 400 500 600 700 800 [GeV] 1/2m

300 350 400 450 500 550 600 650

Acceptance

0.005 0.01 0.015 0.02 0.025 ATLAS 0.03

5 4.3 2.5 2.5 2.4 1.7 1.5 0.84

3.2 2.3 1.7 1.8 1.6 1.4 1.2 0.99

1.9 1.6 1.3 1.2 1.2 0.91 0.84 0.84

1.2 1.1 1.1 0.84 0.63 0.71 0.72 0.67

0.86 0.84 0.65 0.58 0.53 0.53 0.44 0.47

0.67 0.59 0.45 0.37 0.41 0.32 0.37 0.34

[GeV]1/2m

400 450 500 550 600

1 10

ATLAS Preliminary

-1) Expected number of events (1.04 fb

3Jet Loose Selection

( truth / initial)

1-lepton search: results

Muon channel

Observed Fitted sum of

•

No excess of events observed!

•

95% CL exclusion limits in the 3JL SR for mSUGRA bRPV

•

tan β = 10; A0 = 0; μ > 0

(13)

[GeV]

m0

100 150 200 250 300 350 400 450 500 550 600 [GeV] 1/2m

250 300 350 400 450 500

= 3 mm c!

= 10 mm c!

(600 GeV) q~

(700 GeV) q~

(800 GeV) q~

(600 GeV) g~

(700 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

3 jets, tight SR 1 muon, #

ATLAS _{95% CL}

Observed CLS

Expected CLS

1$

S± Expected CL

0.011 0.0054 0.003 0.0042 0.0039 0.0028 0.0025 0.0013 0.013 0.0086 0.0082 0.0061 0.006 0.0037 0.0042 0.0032 0.015 0.012 0.0076 0.0098 0.011 0.0078 0.0067 0.0055 0.018 0.018 0.014 0.014 0.012 0.01 0.0094 0.0065 0.02 0.019 0.017 0.017 0.015 0.014 0.01 0.0092 0.023 0.022 0.02 0.019 0.017 0.015 0.012 0.013 0.024 0.026 0.026 0.021 0.017 0.019 0.018 0.013 0.021 0.026 0.025 0.024 0.022 0.022 0.02 0.017 0.026 0.026 0.029 0.023 0.026 0.02 0.02 0.02

[GeV]

m0

100 200 300 400 500 600 700 800 [GeV] 1/2m

300 350 400 450 500 550 600 650

Acceptance

0.005 0.01 0.015 0.02 0.025 ATLAS 0.03

19 9.5 8.1 5.8 4.4 2.7 2.2 1.6

9.5 6.1 4.9 3.7 3.8 2.2 2 1.4

6.5 5.6 3.7 3.1 2.6 1.9 1.5 0.88

3.9 3 2.6 2.2 1.8 1.5 1 0.64

2.5 2.2 1.8 1.5 1.2 0.89 0.62 0.52

1.5 1.4 1.3 0.9 0.6 0.63 0.66 0.38

0.88 0.88 0.74 0.57 0.53 0.5 0.39 0.29

0.88 0.59 0.5 0.38 0.34 0.31 0.27 0.22

[GeV]1/2m

350 400 450 500 550 600

1 10

ATLAS Preliminary

3Jet Tigth Selection

( truth / initial)

Muon channel Observed

Events

Fitted sum of background

Events

•

tan β = 10; A0 = 0; μ > 0

1-lepton search: results

(14)

0.012 0.0072 0.0046 0.0059 0.0049 0.0049 0.004 0.002 0.014 0.011 0.0096 0.0073 0.0071 0.0061 0.0065 0.0046 0.016 0.015 0.0097 0.012 0.013 0.0094 0.0087 0.0084 0.02 0.02 0.016 0.016 0.014 0.012 0.012 0.0078 0.02 0.021 0.021 0.019 0.016 0.017 0.013 0.011 0.021 0.023 0.022 0.022 0.019 0.017 0.015 0.016 0.022 0.027 0.028 0.023 0.018 0.021 0.021 0.017 0.018 0.025 0.027 0.026 0.025 0.024 0.022 0.02 0.019 0.024 0.028 0.025 0.029 0.023 0.023 0.022

[GeV]

m0

100 200 300 400 500 600 700 800 [GeV] 1/2m

300 350 400 450 500 550 600 650

Acceptance

0.005 0.01 0.015 0.02 0.025 ATLAS 0.03

[GeV]

m0

100 150 200 250 300 350 400 450 500 550 600 [GeV] 1/2m

250 300 350 400 450 500

= 3 mm c!

= 7 mm c!

= 15 mm

! c (700 GeV)

q~

(900 GeV) q~

(700 GeV) g~

(900 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

ATLAS _{95% CL}

Observed CLS

Expected CLS

1$

S± Expected CL

6.9 6.1 4.4 3.7 2.9 2.3 1.8 0.99

4.1 3.6 3.1 2.4 2.2 1.7 1.3 0.86

2.5 2.3 2 1.7 1.3 1 0.85 0.75

1.4 1.5 1.3 1 0.7 0.72 0.77 0.57

0.79 0.9 0.81 0.65 0.6 0.57 0.43 0.37

0.61 0.55 0.49 0.43 0.39 0.34 0.32 0.28

[GeV]1/2m

400 450 500 550 600

1 10

ATLAS Preliminary

4Jet Tight Selection

( truth / initial)

Muon channel

Observed Fitted sum of

•

tan β = 10; A0 = 0; μ > 0

1-lepton search: results

(15)

Conclusions and outlook

• ATLAS results in searches for RPV SUSY presented.

‣ good understanding of detector performance & physics objects have been demonstrated, which are essential for these analysis

‣ background processes well under control

• Limit setting for within bRPV.

‣ no significant deviations from SM observed so far

‣ exclusion limits set have been set for the first time in this scenario

‣ limits presented are the most stringent to date

(16)

BACKUP

(17)

Bilinear RPV: background studies

•

Systematic uncertainties: dominated by theoretical uncertainties ( 20 to 30 % ), the rest below 10%.

(18)

[GeV]

m0

200 400 600 800 1000 1200 1400

[GeV] 1/2m

150 200 250 300 350 400 450 500

(400 GeV) q~

(500 GeV)

~q

(700 GeV) q~

(900 GeV) q~

(400 GeV) g~

(500 GeV) g~

(600 GeV) g~

(900 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

MSUGRA/CMSSM: tan!

=7 TeV s

-1, = 1.04 fb Lint

1 lepton, combination

ATLAS Observed CLS^{95% CL}

Expected CLS

"

±1 Expected CLS

1

#±

LEP2 $

<0, 2.1 fb-1

=3, µ , tan! q~ ,

~g D0

<0, 2 fb-1

=5, µ , tan!

~q ,

~g CDF

RPC vs bRPV: results

Events

3JL SR 58 63 ± 19

3JT SR 11 13.9 ± 4.3

•

95% CL exclusion limits in the combined SRs for mSUGRA

•

tan β = 10; A₀ = 0; μ > 0

[GeV]

m0

100 150 200 250 300 350 400 450 500 550 600 [GeV] 1/2m

250 300 350 400 450 500

= 3 mm c!

= 7 mm c!

= 15 mm c!

(700 GeV) q~

(900 GeV) q~

(700 GeV) g~

(900 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

ATLAS _{95% CL}

Observed CLS

Expected CLS

1$

S± Expected CL

(19)

[GeV]

m0

100 120 140 160 180 200 220 240 260 280 300 [GeV] 1/2m

240 250 260 270 280 290 300

= 3 mm

! c

= 10 mm c!

(600 GeV) q~

(600 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

3 jets, loose SR 1 muon, #

ATLAS Observed CL_S_{95% CL} Expected CLS

1$

S± Expected CL

( truth / initial)

Events

•

tan β = 10; A0 = 0; μ > 0

[GeV]

m0

200 400 600 800 1000 1200 1400

[GeV] 1/2m

150 200 250 300 350 400 450 500

(400 GeV) q~

(500 GeV)

~q

(700 GeV) q~

(900 GeV) q~

(400 GeV) g~

(500 GeV) g~

(600 GeV) g~

(900 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

MSUGRA/CMSSM: tan!

=7 TeV s

-1, = 1.04 fb Lint

3 jets, loose SR

"

1 lepton,

ATLAS Observed CL_S^{95% CL} Expected CLS

1#

S± Expected CL

1

$±

LEP2 %

<0, 2.1 fb-1

=3, µ , tan! q~ ,

~g D0

<0, 2 fb-1

=5, µ , tan!

~q ,

~g CDF

RPC vs bRPV: results

(20)

[GeV]

m0

100 150 200 250 300 350 400 450 500 550 600 [GeV] 1/2m

250 300 350 400 450 500

= 3 mm

! c

= 10 mm c!

(600 GeV) q~

(700 GeV) q~

(800 GeV) q~

(600 GeV) g~

(700 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

ATLAS Observed CL_S_{95% CL} Expected CLS

1$

S± Expected CL

Muon channel

Fitted sum of

[GeV]

m0

200 400 600 800 1000 1200 1400

[GeV] 1/2m

150 200 250 300 350 400 450 500

(400 GeV) q~

(500 GeV)

~q

(700 GeV) q~

(900 GeV) q~

(400 GeV) g~

(500 GeV) g~

(600 GeV) g~

(900 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

MSUGRA/CMSSM: tan!

=7 TeV s

-1, = 1.04 fb Lint

3 jets, tight SR

"

1 lepton,

1#

S± Expected CL

1

$±

LEP2 %

<0, 2.1 fb-1

=3, µ , tan! q~ ,

~g D0

<0, 2 fb-1

=5, µ , tan!

~q ,

~g CDF

3Jet Tigth Selection

( truth / initial)

•

95% CL exclusion limits in the 3JT SR for mSUGRA bRPV

•

tan β = 10; A0 = 0; μ > 0

RPC vs bRPV: results

(21)

( truth / initial)

22 12 10 7.1 5.8 4.2 3.3 1.8

12 8.2 5.7 5.6 5 3.1 2.6 1.8

6.9 6.1 4.4 3.7 2.9 2.3 1.8 0.99

4.1 3.6 3.1 2.4 2.2 1.7 1.3 0.86

2.5 2.3 2 1.7 1.3 1 0.85 0.75

1.4 1.5 1.3 1 0.7 0.72 0.77 0.57

0.79 0.9 0.81 0.65 0.6 0.57 0.43 0.37

0.61 0.55 0.49 0.43 0.39 0.34 0.32 0.28

[GeV]1/2m

300 350 400 450 500 550 600

1 10

ATLAS Preliminary

0.0069 0.0031 0.0021 0.0035 0.0021 0.0022 0.00097 0.00071 0.0073 0.0048 0.0048 0.0034 0.004 0.0029 0.0033 0.0027 0.0078 0.0056 0.0038 0.0046 0.0055 0.0051 0.0047 0.0044 0.0084 0.0081 0.0051 0.007 0.0067 0.0066 0.0054 0.0067 0.012 0.0089 0.007 0.0088 0.0093 0.0086 0.0088 0.01

0.011 0.0094 0.009 0.01 0.011 0.0091 0.011 0.013 0.01 0.012 0.015 0.011 0.0096 0.013 0.015 0.016 0.0078 0.014 0.013 0.015 0.013 0.014 0.016 0.017 0.012 0.013 0.016 0.014 0.018 0.013 0.019 0.019

[GeV]

m0

100 200 300 400 500 600 700 800

[GeV]1/2m

300 350 400 450 500 550 600

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

ATLAS Preliminary

Acceptance

[GeV]

m0

200 400 600 800 1000 1200 1400

[GeV] 1/2m

150 200 250 300 350 400 450 500

(400 GeV) q~

(500 GeV)

~q

(700 GeV) q~

(900 GeV) q~

(400 GeV) g~

(500 GeV) g~

(600 GeV) g~

(900 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

MSUGRA/CMSSM: tan!

=7 TeV s

-1, = 1.04 fb Lint

4 jets, loose SR

"

1 lepton,

ATLAS Observed CLS^{95% CL}

Expected CLS

1#

S± Expected CL

1

$±

LEP2 %

<0, 2.1 fb-1

=3, µ , tan! q~ ,

~g D0

<0, 2 fb-1

=5, µ , tan!

~q ,

~g CDF

RPC vs bRPV: results

•

tan β = 10; A0 = 0; μ > 0

(22)

Muon channel

Fitted sum of

[GeV]

m0

200 400 600 800 1000 1200 1400

[GeV] 1/2m

150 200 250 300 350 400 450 500

(400 GeV) q~

(500 GeV)

~q

(700 GeV) q~

(900 GeV) q~

(400 GeV) g~

(500 GeV) g~

(600 GeV) g~

(900 GeV) g~

(800 GeV) g~

>0

= 0, µ = 10, A0

MSUGRA/CMSSM: tan!

=7 TeV s

-1, = 1.04 fb Lint

4 jets, tight SR

"

1 lepton,

1#

S± Expected CL

1

$±

LEP2 %

<0, 2.1 fb-1

=3, µ , tan! q~ ,

~g D0

<0, 2 fb-1

=5, µ , tan!

~q ,

~g CDF

[GeV]

m0

100 150 200 250 300 350 400 450 500 550 600 [GeV] 1/2m

250 300 350 400 450 500

= 3 mm c!

= 7 mm c!

= 15 mm c!

(700 GeV) q~

(900 GeV) q~

(700 GeV) g~

(900 GeV) g~

>0

= 0, µ = 10, A0

"

bRPV MSUGRA: tan

=7 TeV s

-1, = 1.04 fb Lint

ATLAS _{95% CL}

Observed CLS

Expected CLS

1$

S± Expected CL

4Jet Tight Selection

( truth / initial)

•

95% CL exclusion limits in the 4JT SR for mSUGRA bRPV

•

tan β = 10; A0 = 0; μ > 0