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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

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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

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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

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[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

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• 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.

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[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

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• 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)…

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• 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:

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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.

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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…

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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.

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[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

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[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−

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[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

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[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.

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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.

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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

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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

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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

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[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!

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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).

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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.

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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

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• 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]