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Angular analysis of X ➙ VV systems

LHCb-Deutschland 2026, Aachen · 2 October 2026

Ilya Segal, Mikhail Mikhasenko, Marian Stahl · Ruhr-Universität Bochum

Liupan An, Zifan Li · Peking University

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Motivation of the double J/ψ analysis

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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Observation of T_cc̄cc̄ states and their quantum numbers
Probe of quarkonium production mechanisms �single (SPS) vs double (DPS) parton scattering
Tests NRQCD and gluon dynamics in pp collisions
Previous measurements:
first observed by LHCb
confirmed by ATLAS (also J/ψψ(2S)) and CMS
last year CMS reported a family of tetraquarks, the first quantum-number measurement (2n⁺) and the J/ψψ(2S) spectrum
Analysis goals:
observe X(7100)
first LHCb measurement of J/ψψ(2S), including J/ψψ(2S) → 4μ 2π
reproduce the CMS results and test their hypothesis with our technique

Science Bulletin �65 (2020) 1983

Nature 648 (2025) �8092, 58-63

Phys.Rev.Lett.� 131 (2023) 15, 151902

~ 200 cands.

~ 2000 cands.

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Introduction

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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Process considered: production of the vector-vector system (e.g. J/ψJ/ψ, ϕϕ, ZZ etc.)
Considered final states: V(➙ℓ⁺ℓ^-)V(➙ℓ⁺ℓ^-) or V(➙S⁺S^-)V(➙S⁺S^-)
ℓ is a lepton, S is a scalar meson (spin J=0), V is a vector (J=1)
Angular observables:
θ₁, θ₂: polar angles between leptons (scalar mesons) and the direction of vector mesons
Δφ: is a sum of two azimuthal angles between leptons (scalar mesons) planes and vector mesons planes
All calculated from the four-momenta of the leptons (scalar mesons) candidates

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Amplitude model and helicity matrix

by Liupan, Ronan and Misha

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Decay amplitude built from parity conservation and permutation symmetry

H^J_λ1,λ2 is the helicity matrix
d^J_λ,ξ(θ) are Wigner d-functions
λ₁, λ₂ are the helicity states of the vector mesons
ν and ξ₁, ξ₂ are differences between the vector-meson and lepton helicities
Helicity matrix contains the helicity couplings a, b, c, d

Naturality:

Signum:

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Angular functions of the 3D intensity

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Using the amplitude, the intensity takes a simple form:

Helicity couplings to fit

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Angular functions: one-dimensional projections

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The intensity is a sum over helicity states:

Integrating over the other angles gives the one-dimensional distributions:

~β

~ζ

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Motivation

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Angular analysis of X → VV decays is a powerful tool to study:
exotic hadrons (tetraquarks, glueballs)
quantum numbers (spin J and parity P), partial waves
QCD production mechanisms
Examples:
J/ψJ/ψ production helps to study the T_cc̄cc̄ tetraquark states
φφ spectra might contain a glueball candidate
central exclusive production (CEP) of double vector systems probes multi-parton and double-pomeron exchange

Phys. Rev. Lett. 132 (2024)

LHCb-ANA-2020-029

PDG Phys. Rev. D 110, 030001

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J/ψJ/ψ data sample

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Run 1–3: 37 fb⁻¹
Run 1+2: 9 fb⁻¹, selection of LHCb-PAPER-2020-011
Jpsi1_L0Global/Hlt1Phys || Jpsi2_L0Global/Hlt1Phys
Run 3: ~28 fb⁻¹
Hlt2BandQ_DoubleDiMuon_Jpsi_Jpsi_DPS
Background subtraction
2D fit of the two dimuon masses in each m(J/ψJ/ψ) bin
per-event 2D sWeights

Candidate	Variable	Requirement
each muon	p	> 5 GeV/c
	p_T	> 0.4 GeV/c
	track χ²/ndf	< 5
	PIDmu	> −1
	ProbNNmu	> 0.8
	TRACK_GhostProb	< 0.4
each J/ψ	m(μ⁺μ⁻)	3000–3200 MeV/c²
	constrained-child DTF χ²/ndf	< 5
J/ψJ/ψ	m(J/ψJ/ψ)	6200–9000 MeV/c²
	parent DTF χ²/ndf	< 3
	p_T	> 5.2 GeV/c
same-sign muons	m(μ^±μ^±)	> 220 MeV/c²
different J/ψ	opening angle	> 0.001 rad

LHCb-ANA-2019-055

SIGNAL

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Possible quantum numbers and the CMS benchmarks

CMS tests eight benchmark models; each is one point in our helicity-coupling space

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CMS model	Our group	Couplings	Fractions at 6.9 GeV	c₁, c₂, c₃, c₄, c₆	Meaning
0⁻	II, special	b	f_b = 1	0.25, 0, 0, −0.75, 2.25	pseudoscalar
0⁺_m	I, special	b, d	f_b = 0.48, f_d = 0.52	−0.12, 1.59, 0.76, −0.36, 1.08	SM-Higgs-like
0⁺_h	I, special	b, d	f_b = 0.81, f_d = 0.19	−0.20, 1.25, 0.60, −0.61, 1.82	higher-dim. scalar
1⁻	III	a	f_a = 1	0, 1.13, −0.50, 0.75, 0	vector
1⁺	IV, special	a	f_a = 1	0, −1.13, −0.50, 0.75, 0	axial vector
2⁻_m	II	b	f_b = 1	same as 0⁻	pseudotensor
2⁻_h	II	a	f_a = 1	same as 1⁻	higher-dim. pseudotensor
2⁺_m	I	a, b, c, d	0.47, 0.09, 0.23, 0.21	−0.02, −0.09, 0.08, 0.11, 0.72	graviton-like

Groups: I = (2n)⁺ with a, b, c, d · II = (2n)⁻ with a, b · III = (2n+1)⁻ with a · IV = (2n+1)⁺ with a, c
m = minimal (lowest-dimension) coupling, h = higher-dimension coupling, as in the Higgs anomalous-coupling analyses (JHUGen)
With three decay angles 0⁻ ≡ 2⁻_m and 1⁻ ≡ 2⁻_h; 1⁺ differs from 1⁻ only in the sign of c₂
We consider: 0⁺, 0⁻, 1⁺, 1⁻, 2⁺ or 2⁻ for each of the three structures

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Simulation

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SPS background: Pythia 6 gg → J/ψJ/ψ �(colour-singlet LO, LbOniaPairs) in 2024 conditions
full simulation and efficiency-weighted generator level
gives the SPS c_i background
LbOniaPairs in Pythia 8: ported to Gauss v60 (Sim11)
event-level Rivet validation against LHCb 13 TeV J/ψ-pair measurements, three shower tunes
J^P toys: LHCb Run 2 mass model �(three structures, threshold, NRSPS, DPS)
all 216 J^P assignments;
background c_i from the reweighted SPS simulation

Rivet plugin LHCB_2016_I1505592

Rivet plugin LHCB_2024_I2725992

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CMS strategy

Pairwise hypothesis tests in 2D space (m, D)

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Signal and alternative from the 3D intensity with the benchmark J^P

Per pair of hypotheses i, j: �one discriminant per event

Templates T_k,b in 36 × 5 bins of (m, D):

signal × acceptance, SPS, DPS, fake J/ψ

Extended binned likelihood

with all yields N_k free

Test statistic; its distribution from pseudo-experiments

CMS, 135 fb⁻¹ at 13 TeV

Nature 648 (2025) 58

	0⁻	0⁺_m	0⁺_h	1⁻	1⁺	2⁻_m	2⁻_h	2⁺_m
0⁻
0⁺_m
0⁺_h
1⁻
1⁺
2⁻_m
2⁻_h
2⁺_m

row preferred

row rejected

not published

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CMS strategy on LHCb Run 3 data

Same benchmarks, templates and test statistic; resonance poles fixed to the Run 2 values

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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LHCb Run 3, 28 fb⁻¹ at 13.6 TeV

	0⁻	0⁺_m	0⁺_h	1⁻	1⁺	2⁻_m	2⁻_h	2⁺_m
0⁻						≡
0⁺_m
0⁺_h
1⁻							≡
1⁺
2⁻_m	≡
2⁻_h				≡
2⁺_m

CMS, 135 fb⁻¹ at 13 TeV

	0⁻	0⁺_m	0⁺_h	1⁻	1⁺	2⁻_m	2⁻_h	2⁺_m
0⁻
0⁺_m
0⁺_h
1⁻
1⁺
2⁻_m
2⁻_h
2⁺_m

row preferred

row rejected

both rejected

neither rejected

identical in our angles (≡)

CMS 2⁺_m preference �not reproduced
Data prefer 0⁺_h
Maybe hints at different J^P

not published

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Quantum-number hypotheses

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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CMS claims all three structures belong to group I (J^P = 2n⁺)
This can be tested against other hypotheses
Toy model with the Run 2 analysis model
216 hypotheses in total: �six effectively different J^P for three resonances
Background c_i from a linear fit of the reweighted SPS simulation
Test statistics to be used for the data

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Mass and angular fits

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CMS model:
Signal: three interfering Breit–Wigner functions convolved with a Gaussian
Background: Breit–Wigner + exponential (feed-down and combinatorial)
Three peaks are now visible in the Run 3 data
They can be fitted with the CMS model
Fractions are fixed from the mass fit
All resonances in group I (J^P = 2n⁺), all helicity couplings a, b, c, d = 1
Background c_i linear in m
Large interference and background bias

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CMS-like test with our c_i strategy

All 216 assignments, poles fixed to Run 2, free helicity couplings, six background models

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Fit of the acceptance-corrected c_i(m) for every J^P assignment, couplings free where J^P allows
Structure at 6.5 GeV: 2⁺ preferred for every background model
Structures at 6.9 and 7.2 GeV: not yet distinguished

Structure 1 (6.5 GeV)

	linear	quadr.	> 6.5 GeV	SPS gen.	SPS full	SPS + off.
0⁺
0⁻
1⁺
1⁻
2⁺
2⁻

Structure 2 (6.9 GeV)

	linear	quadr.	> 6.5 GeV	SPS gen.	SPS full	SPS + off.
0⁺
0⁻
1⁺
1⁻
2⁺
2⁻

Structure 3 (7.2 GeV)

	linear	quadr.	> 6.5 GeV	SPS gen.	SPS full	SPS + off.
0⁺
0⁻
1⁺
1⁻
2⁺
2⁻

best

compatible

disfavoured

strongly disfavoured

very preliminary

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Summary and next steps

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J/ψJ/ψ samples of Run 1–3 (37 fb⁻¹) are produced
SPS simulation is tested validated with Rivet and Pythia 8 LbOniaPairs
Three structures visible in the Run 3 data, fitted with the CMS model
CMS strategy on our data: �the 2⁺_m preference is not reproduced, the data prefer 0⁺_h
Hints that the three structures may have different J^P
c_i test of all 216 assignments: investigations are ongoing

Next steps:

finalise the J/ψJ/ψ mass fit
Try to improve statistics limitations
Tune the angular analysis
J/ψψ(2S) mass fit, including J/ψψ(2S) → 4μ 2π

Thank you for your attention!

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Backup

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CMS strategy: CMS and our implementation

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	CMS	Our cross-check
Data	135 fb⁻¹, 13 TeV	LHCb Run 3, 28fb^-1, 13.6 TeV
Signal angles	JHUGen \|M\|² in (cosθ₁, cosθ₂, Φ)	3D intensity I(Ω; c^H), fractions fixed at 6.9 GeV
Acceptance	JHUGen events with full CMS simulation	4D map from weighted Pythia 6 generator level
Mass model	CMS fit: three interfering BW + threshold	our mass fit with the CMS model
SPS / DPS angles	Pythia 8 simulation	Pythia 6 SPS simulation, mass dependent
Fake J/ψ	data driven	J/ψ mass sidebands
Threshold term	both hypotheses in both likelihoods	same
Templates and fit	36 × 5 (m, 𝒟) bins, extended likelihood	same
Test statistic	q = −2 ln(L_j/ L_i), pseudo-experiments	same, Poisson toys from the fits to the data

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CMS strategy on our data: p-values and test statistics

Nominal configuration, poles fixed to Run 2; each cell gives the p-value of the row hypothesis and its significance

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Robustness of the CMS-style result

Observed 0⁺_h vs 2⁺_m preference in the configuration variations (free-pole mass fit)

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Sensitivity at the Run 3 yield

Pseudo-experiments with the simulated backgrounds (free-pole mass fit)

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Our c_i test: profiled Δχ²

Minimum Δχ² over the other two structures, per J^P of each structure and background model

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Structure 1	linear	quadr.	> 6.5 GeV	SPS gen.	SPS full	SPS + off.
0⁺	1.8	0.8	0.7	2.0	8.1	1.1
0⁻	18.8	19.4	11.0	59.1	37.9	23.0
1⁺	11.6	7.1	8.2	30.5	19.9	47.9
1⁻	19.2	13.7	13.6	21.4	11.8	43.8
2⁺	0.0	0.0	0.0	0.0	0.0	0.0
2⁻	8.3	7.6	5.1	13.5	0.7	13.8

Structure 2	linear	quadr.	> 6.5 GeV	SPS gen.	SPS full	SPS + off.
0⁺	1.4	1.2	1.1	2.0	4.2	0.4
0⁻	4.6	4.7	3.8	15.4	10.8	2.4
1⁺	1.9	1.5	2.5	3.8	3.4	7.3
1⁻	4.0	3.7	4.3	1.8	1.2	6.8
2⁺	0.0	0.0	0.0	0.0	0.6	0.0
2⁻	1.0	0.9	1.2	1.6	0.0	0.4

Structure 3	linear	quadr.	> 6.5 GeV	SPS gen.	SPS full	SPS + off.
0⁺	0.3	0.2	0.0	2.0	3.3	0.0
0⁻	3.2	2.9	2.9	10.1	6.4	3.2
1⁺	1.0	1.1	0.8	5.1	2.4	2.0
1⁻	1.5	1.6	1.3	5.1	2.5	2.2
2⁺	0.0	0.0	0.1	0.0	0.0	1.1
2⁻	1.6	1.6	1.3	5.1	0.7	2.2

Hypotheses within Δχ² < 4 of the best: 9–28 depending on the background model
Best per background: 2⁺,2⁺,2⁺ (linear, quadratic, SPS generator level) · 2⁺,2⁻,2⁺ (SPS full simulation) · 2⁺,2⁺,0⁺ (above 6.5 GeV, SPS + offset)

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J/ψJ/ψ data selection: trigger and online cuts

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Same selection as LHCb-PAPER-2020-011 for Run 1 and Run 2:
full Run 1+2 data (9 fb⁻¹)
Jpsi1_L0Global/Hlt1Phys || Jpsi2_L0Global/Hlt1Phys
Full Run 3 data (~28 fb⁻¹)
Hlt2BandQ_DoubleDiMuon_Jpsi_Jpsi_DPS line

Candidate	Online requirement
all μ^±	isMuon == 1
	p_T > 650 MeV/c
	track χ²/ndf < 5
both J/ψ	vertex χ² < 20
	mass window ±125 MeV/c²

From the WG sign-off

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Background subtraction

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2D model of the two dimuon masses:
signal: double-sided Crystal Ball + Gaussian
background: exponential
m(J/ψJ/ψ): 100 bins of 28 MeV/c²
Fit parameters: yields, J/ψ mass, resolution σ, background slopes
Per-event 2D sWeights

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β and ζ definition (four-meson final state)

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β and ζ definition (four-meson final state)

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β and ζ definition (four-meson final state)

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Symmetry constraints and partial waves

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s₁	s₂	S=s₁-s₂	S=(s₁-s₂)+1	S=(s₁+s₂)	L
1^-	1^-

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1	S=2	L
1^-	1^-

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1	S=2	L
1^-	1^-

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0			1					2	0	s-wave
		1	0	1	2			1	2	3	1	p-wave
		2	1	2	3	0	1	2	3	4	2	d-wave
		3	2	3	4	1	2	3	4	5	3	f-wave

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺			1⁺					2⁺	0	s-wave
		1^-	0^-	1^-	2^-			1^-	2^-	3^-	1	p-wave
		2⁺	1⁺	2⁺	3⁺	0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		3^-	2^-	3^-	4^-	1^-	2^-	3^-	4^-	5^-	3	f-wave

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺			1⁺					2⁺	0	s-wave
		1^-	0^-	1^-	2^-			1^-	2^-	3^-	1	p-wave
		2⁺	1⁺	2⁺	3⁺	0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		3^-	2^-	3^-	4^-	1^-	2^-	3^-	4^-	5^-	3	f-wave
		2n⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n	n=2,3,4, ...
		(2n+1)^-	2n^-	(2n+1)^-	(2n+2)^-	(2n-1)^-	2n^-	(2n+1)^-	(2n+2)^-	(2n+3)^-	2n+1	n=2,3,4, ...

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺			1⁺					2⁺	0	s-wave
		1^-	0^-	1^-	2^-			1^-	2^-	3^-	1	p-wave
		2⁺	1⁺	2⁺	3⁺	0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		3^-	2^-	3^-	4^-	1^-	2^-	3^-	4^-	5^-	3	f-wave
		2n⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n	n=2,3,4, ...
		(2n+1)^-	2n^-	(2n+1)^-	(2n+2)^-	(2n-1)^-	2n^-	(2n+1)^-	(2n+2)^-	(2n+3)^-	2n+1	n=2,3,4, ...

X

V

Identical vector mesons: only symmetric (even) or antisymmetric (odd) LS allowed

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺			1⁺					2⁺	0	s-wave
		1^-	0^-	1^-	2^-			1^-	2^-	3^-	1	p-wave
		2⁺	1⁺	2⁺	3⁺	0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		3^-	2^-	3^-	4^-	1^-	2^-	3^-	4^-	5^-	3	f-wave
		2n⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n	n=2,3,4, ...
		(2n+1)^-	2n^-	(2n+1)^-	(2n+2)^-	(2n-1)^-	2n^-	(2n+1)^-	(2n+2)^-	(2n+3)^-	2n+1	n=2,3,4, ...

Identical vector mesons: only symmetric (even) or antisymmetric (odd) LS allowed

V

X

V

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Symmetry constraints and partial waves

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Identical vector mesons: only symmetric (even) or antisymmetric (odd) LS allowed

s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
			2^-	3^-	4^-						3	f-wave
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n	n=2,3,4, ...
			2n^-	(2n+1)^-	(2n+2)^-						2n+1	n=2,3,4, ...

V

X

V

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

All possible appearances of the certain J^P happens over 5 different waves and divided into 4 groups

n=3,4,5, ...

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Symmetry constraints and partial waves

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

Group I (even and natural): (L,S)=(2n-2,2),(2n,0),(2n,2),(2n+2,2) ➙ 4 helicity couplings:

n=3,4,5, ...

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Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

Group II (even and unnatural): (L,S)=(2n-1,1),(2n+1,1) ➙ 2 helicity couplings:

n=3,4,5, ...

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Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

Group III (odd and natural): (L,S)=(2n+1,1) ➙ 1 helicity coupling:

n=3,4,5, ...

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Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

Group IV (odd and unnatural): (L,S)=(2n,2),(2n+2,2) ➙ 2 helicity couplings:

n=3,4,5, ...

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Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

0⁺ is a special case in Group I: (L,S)=(-2,2),(0,0),(0,2),(2,2) ➙ 2 helicity couplings: a=c=0,

n=3,4,5, ...

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Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

0^- is a special case in Group II: (L,S)=(-1,1),(1,1) ➙ 1 helicity couplings: a=0,

n=3,4,5, ...

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Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

1⁺ is a special case in Group IV: (L,S)=(0,2),(2,2) ➙ 1 helicity couplings: c=0,

n=3,4,5, ...

45 of 45

Symmetry constraints and partial waves

Ilya Segal · Angular analysis of X → VV systems · LHCb-Deutschland 2026

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s₁	s₂	S=0	S=1			S=2					L
1^-	1^-	0⁺								2⁺	0	s-wave
			0^-	1^-	2^-						1	p-wave
		2⁺				0⁺	1⁺	2⁺	3⁺	4⁺	2	d-wave
		(2n-2)⁺				(2n-4)⁺	(2n-3)⁺	(2n-2)⁺	(2n-1)⁺	2n⁺	2n-2
			(2n-2)^-	(2n-1)^-	2n^-						2n-1
		2n⁺				(2n-2)⁺	(2n-1)⁺	2n⁺	(2n+1)⁺	(2n+2)⁺	2n
			2n^-	(2n+1)^-	(2n+2)^-						2n+1
		(2n+2)⁺				2n⁺	(2n+1)⁺	(2n+2)⁺	(2n+3)⁺	(2n+4)⁺	2n+2

All other J^Ps can be divided into groups accordingly

n=3,4,5, ...