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1
LHCb analysis tools for finding resonances
@GreigCowan (Edinburgh)On behalf of the LHCb collaboration
PWA 9 / ATHOS 4Bad Honnef, 14th March 2017
nPVs ~ 2nTracks ~ 200
The LHCb detector
2
Promptbackground
[PRD 87, 112010 (2013)]
B decays withlifetime of ~1.5 ps
[EPJC 73 (2013) 2431]
σ(pp → bbX̅) = 515±2±53μbσcc ̅= 20xσbb ̅
Excellent PID
2 < η < 5
The LHCb detector
32 < η < 5
[PRL 111, 101805 (2013)]
[NPB 874 (2013) 663]
Excellent tracking
Calorimetryχc→J/ψγ
σ(pp → bbX̅) = 515±2±53μbσcc ̅= 20xσbb ̅
How do we find resonances?1. Isolate prompt or b/c-hadron signal
2. Estimate or subtract the background
3. Calculate the efficiency and resolution
4. Build amplitude model (using some assumptions about default resonances to include)
5. Perform a fit in one, two or more (typically kinematic) dimensions
4
See talks from Anton (Tue), Tomasz (Wed) and Jonas (Thurs)
for physics motivations and interpretations
~5 fb-1
• This talk: try and give specific examples of each of these stages
• Results from Run-1 only (many not covered)Run 1
[http://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/Summary_all.html]
Event selection• Selection of b-hadron decays based on track quality,
(transverse) momentum, impact parameter of the final state particles, flight distance and PID.
• Further reduce backgrounds using machine learning (neural nets, BDT…).
• Trained on data sidebands for the background and simulated signal (or the signal data itself).
• Uses many variables related to kinematics, vertexing and PID.
• Correct the simulation using weights to match the data (PID, trigger etc) using a control channel
5
[JHEP 06 (2015) 131]
B0B0s
Before neural net
e.g., B→J/ψKS neural net removes 99% of background and keeps 73% of signal.
After neural net
Dalitz plot analysis formalism
6
• LHCb analyses most often use the Isobar approach.
• Build amplitude from coherent sum of two-body resonances, described by Breit-Wigner, Flatté,… line shapes.
• Violates unitarity, so K-matrix formalism (for particular waves) being explored as alternative.
• Crucial to include efficiency and background effects.
Amplitude containingresonant dynamics
Complexcoefficients
Blatt-Weisskopf barrier factors Angular terms Line shapes[http://pdg.lbl.gov/2010/reviews/rpp2010-rev-dalitz-analysis-formalism.pdf]
Angular terms• Helicity formalism
• Angular distribution as function of helicity angle θ.
7
[Chung PRD 57 (1998) 431] [Fillipini et al PRD 51 (1995) 2247]
• Zemach tensors [Phys. Rev. 133, B1201 (1964)]
• Expressions involving 3-vectors
• Cross-check with covariant tensor form• Use 4-vectors and projection operators.
• Both approaches give same angular dependence (small modification of the line shape).
• Non-relativistic as defined in rest frame of resonance but can compute (process dependent) relativistic corrections
{B0→D0π+π−
[LHCb PRD 92 (2015) 032002]
[LHCb arXiv:1701.07873]
mB0X
Blatt-Weisskopf barrier factors
• BW factors for the production and decay of a resonance of spin-L
• z0 is z when the invariant mass is equal to the mass of the resonance
• rBW is the barrier radius, which may be different for the b-hadron and resonance decay (e.g., 5/GeV vs. 1.5/GeV)
• Altered as part of systematic uncertainties.
• Model uncertainty typically dominates (i.e., which (non-)reasonant components to include).
8[Blatt, Weisskopf, Theoretical Nuclear Physics, John Wiley & Sons (1952)]
Line shapes
• Relativistic Breit-Wigner used most often, even when resonance not isolated and far from threshold
9
• Flatté if second channel opens up near resonance (e.g., for f0(980)).
• Gounaris-Sakurai can be used for ρ(770), ρ(1700)… and Bugg for f0(500) (accounts for ππ, KK, ηη and 4π)
ρ-ω interference
phase space coupling constant
[PRL 21 (1968) 244] [JPG 34 (2007) 151]
Ks0
vetoρ-ω interference visible
f0(500)
[PLB 63, 224 (1976)]
[PLB 742 (2015) 38]
Line shapes
• “Non-resonant” terms• Exponential• LASS (K*0(1430) combined with a NR Kπ S-wave)
• Kappa (K*(800)), dabba (Dπ S-wave)…
10
• K-Matrix: ensure unitarity by construction
• used for spin-0 π+π− and spin-1/2 pπ− amplitudes
• Coherent description of the production, rescattering and decay of the partial wave.
• Parameters of K-matrix (pole couplings and scattering amplitudes) are taken from the global analysis of π+π− (pπ−) data.
• Assumes two-body system does not interact with rest of final state in production process
complex free parameter
Production vectorfor process-dependent
contributions
LHCb [PRD 92, 032002 (2015)] [PRL 117 (2016) 082003]
[Anisovich and Sarantsev, EPJA 16, 229 (2003)]Matrix describing S-wave scattering
Dalitz plot likelihood fit• Extended likelihood:
11
signal background
=
number ofsignal/bkg
Efficiency overDalitz plot
Resonanceparameters
“the physics”
Numerical integration
Some analyses use sum over fullyreconstructed phase-space events
to automatically include the efficiency
Can constrain these fromthe fit to the B-hadron
invariant mass
[PRD 94, 072001 (2016)]
The square Dalitz plot
• Alternative parameterisation of the phase space
• Enlarge the most populated signal region.
• Reduce dependence on binning for combinatorial background and efficiency parameterisation.
12
B0s→DKπ[PRD 90, 072003 (2014)]
Background parameterisation
• Use sideband to model background in signal region.
• If necessary, add specific components from simulation.
• Weight the simulation to match the data (production kinematics, PID, trigger, tracking)
• Use histograms directly in the fit, or parameterise using polynomials, splines, kernel density estimation…
13
sideband B-→D(*)K−π−
[PRD 94, 072001 (2016)]
[PRD 94, 072001 (2016)] [PRD 92, 032002 (2015)]
Use control samples
Parameterisation becomes more difficult for fits in higher dimensions
All methodslead to systematic
sWeight0 0.5 10
20
40
60
80
100
120
140310×
sWeights for background subtraction
• Use the sPlot method to statistically subtract background
• From fit to “mass”, get per-event weight Wi• For subsequent amplitude analysis perform sWeighted log-likelihood fit using
signal-only PDF.
• Requires that control variable (“mass”) is uncorrelated with variables in PDF• Requires that you correct the likelihood to have correct statistical
uncertainties.
14
[NIM A555 (2005) 356]
background
signal
toyexample
Efficiency parameterisation
• Generate large sample of phase space events and pass through the full detector reconstruction.
• Correct simulation to “look” like the data for trigger, PID etc
• Parameterise the efficiency across the phase space.
• Fit using polynominals, cubic splines or compute coefficients from sum over accepted events
15
[PLB 747 (2015) 484][PRD 94, 072001 (2016)]
B-→D+π−π−
B → ψ(2S)K+π−
LHCb simulation
Fitting technologies
• Different amplitude fitters used to extract the physics:
• Laura++ (http://laura.hepforge.org)
• 2D Dalitz-plot fits.
• Model-dependent (isobar, K-matrix) and semi model-independent (splines) approaches.
• Mint
• 4 body final states
• GooFit
• Takes advantage of GPUs for significant speed-up of likelihood calculation
• Many custom codes…
16Sebastian collecting examples here: https://github.com/PHASE-network
Speed of execution becoming prohibitively large as datasets and
model complexity increase
Putting it all together: B0→D0π+π−
17
[PRD 92, 032002 (2015)]
Interferenceeffects visible
ρ(770)
f2(1270)D*(2460)
D*3(2760)
Goodness of fitevaluated using
χ2 evaluated in 256bins acrossDalitz plot
BuggFlatte
Hypothesis testing use generated pseudoexperiments to determine spin (> 11σ)
B0→D0π+π−: S-wave
• Small differences visible, but results compatible
• f0(1370) and f0(1500) not significant when added to isobar model
18
[PRD 92, 032002 (2015)]
Model independent: Legendre moments of B−→D+π−π−
• Use moments to motivate resonances to include in amplitude fit.
• Weight events by PL(cosθDπ) as function of m(Dπ).
• Partial waves with spins up to J give moments up to 2J.• S,P-wave and P,D-wave interference indicate broad spin-0 and spin-1
components.
• L=7,8 consistent with zero, indicating no higher partial waves.19
D∗2(2460)0
Another spin-2resonance?
A possible spin-3resonance?
S,P-wave andP,D-wave
interference
SymmetrisedDP since twoidentical pionsin final state
~28 k candidatesin signal region
(very pure)
[PRD 94, 072001 (2016)]
Model dependent amplitude analysis of B−→D+π−π−
• Dominant systematic from efficiency model across the Dalitz plot.• 42 free parameters: fit performed many times with randomised
initial values of the amplitudes to avoid local minima.
20
Firstobservation
From B mass sidebandsand simulation
[PRD 94, 072001 (2016)]
data - fit modelconsistent with Gaussian(0,1)
S-wave model in B−→(D+π−)S-waveπ−
• No D+π− scattering data to constrain S-wave to determine directly from data.
• Use cubic spline function to give smooth description of the magnitude and phase of D+π− S-wave as function of m(D+π−).
• The anticlockwise rotation of the phase at low m(D+π−) is as expected due to the presence of the D0
∗(2400)0 resonance.
21
knots in the spline
[PRD 94, 072001 (2016)]
m(D+π−)={2.01,2.10,2.20,2.30,2.40,2.50, 2.60,2.70,2.80,
2.90,3.10,4.10,5.14}
x
BR x 10-4
referencepoint
Fit fraction
Decays to non-scalar final states• Non-scalars adds more degrees of freedom to the amplitude analysis
22
e.g., B0(s)⟶J/ψh⁺h⁻4D phase space(helicity angles + m(h⁺h⁻)
e.g., B+⟶J/ψφK+
6D phase space
[PRD 89 (2014) 092006] [PLB 742 (2015) 38]
[PRD 87 (2013) 052001] [PLB 747 (2015) 484]…
[PRL 118 (2017) 022003][PRD 95 (2017) 012002]
Three interfering decay chains1. B+ → K*+J/ψ, K*+ → ϕK+
2. B+ → XK+, X → J/ψϕ3. B+ → Z+ϕ, Z+ → J/ψK+
mass windowfor analysis
~20% background
B+ → J/ψϕK+
sideband
Nsig = 4289 ± 151
Matrix element for K*+
→ ϕK+ and X → J/ψϕ contributions
23
Sum over K* resonancesHelicity couplings
for quasi-two-body decaySum over J/ψ andφ helictites
In-coherent sum over differencebetween muon helicities
Parity conservation in strong decay ofK* limits number of couplings
Wigner d-matrices
Angle to align coordinate axesin the X and K* decay chains
The X angles arenot independent
from the K* angles
Fit results with only K* components
• Use quark model as guide to building matrix element in K* sector.
• Float all masses/widths in the fit. Consider K* with sig > 2σ.
• m(ϕK+) fitted well, but not m(J/ψϕ).24
Wave Num in fitNR 1
2P1 2
1D2 2
13F3 1
13D1 1
33S1 1
31S0 1
23P2 1
13F2 1
13D3 1
13F4 1
104 free parameters in fit
p-value < 10-4
Boxes show±1σmass
Godfrey-Isgur predictionswell-established K*unconfirmed K*
Fit results including exotic components
• 7 K* states, 4 exotic X states and NR J/ψφ and φK* components.
• Inclusion of exotic Z states does not improve fit.• Significant enhancement near threshold.• X(4140) mass consistent with other expts, but width much larger. 25
98 free parameters in fitp-value = 22%
X(4140)8.4σ1++
X(4274)6.0σ1++
X(4500)6.1σ0++
X(4700)5.6σ0++
Baryon decays• Need to consider polarisation
• For Λb it is consistent with zero at LHCb: Pz = 0.06±0.07±0.02
• Use 6D phase space (angles and m(pK))
• Similar amplitude formalism as previously described, with 12-14 Λ* states from PDG
26
[PLB 724 (2013) 27]
ΛΛ
μ
μ
μ
μ
ψ
ppKKθ θ φ
θ
*
+
−
+
−K − −
ψ Λ
Λ
b
*ψ *
φ = 0
Λ
Λ
φ μψ*
Λ
b
lab frame
rest frame0
0
rest frame
∗
x
z
b
Λ
rest frame
[GeV]pψ/Jm4 4.2 4.4 4.6 4.8 5
Even
ts/(1
5 M
eV)
0
100
200
300
400
500
600
700
800
LHCb(b)
[PRL 115 (2015) 072001]
Now working with JPAC to better model Λ* states
Re A -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.1
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
LHCb
(4450)cP
(a)
15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
(4380)cP
(b)
Pc Re APc
Im A
P cm0
Γ0
180o
Quasi model-independent study• Replace BW with 6 independent complex numbers in 6 bins of m(J/ψp) in region of 𝑃𝑐+ mass peak.• Allows 𝑃𝑐+ shape to be constrained only by amplitudes in Kp sector.• Observe rapid change of phase near maximum of magnitude ⇒ resonance!
27
magnitu
de
BW amplitude
phase
Argand diagram
inconclusive
simulation
Pentaquark model-independent
• Λ* spectrum is largest systematic uncertainty in observation of Pc states.
• Model-independent approach: do not assume anything about Λ*, Σ* or NR composition, spin, masses, widths or mass-shape.
• Only restrict the maximal spin of allowed Λ* components at given m(𝐾p).
28
Only low-spinstates at low masses
Theory predictions for Λ*Well established Λ* states
[Extension of BaBar PRD 79 (2009) 112001]
Pentaquark model-independent• Can Λ* -only resonances describe the data?• Expand cosθΛ* distribution in Legendre polynomials.
29
filter out maximum spin for
each m(𝐾p)
[PRL 117, 082002 (2016)]
hypothesis test through likelihood ratio
using pseudoexperiments
Null hypothesis (Λ* only) rejected at 9σ
Summary
30
• LHCb is excellent laboratory for (light, heavy, exotic) hadron spectroscopy.
• Combination of model-independent and model-dependent Dalitz/amplitude analyses used to search for resonances.
• Combination of data-driven and simulation-based methods to control experimental effects (backgrounds and efficiencies).
• Systematic uncertainties tend to be dominated by model selection (i.e., which (non-)reasonant components to include).
(Probably incomplete) list of LHCb Dalitz/amplitude analyses
[http://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/Summary_all.html]