In the present work, the strong decays of the discovered $ P_c(4380) $ , $ P_c(4440) $ , $ P_c(4457) $ and their possible isospin cousins are systematically studied via the assignment that they are meson-baryon molecular states. In detail, the strong decay constants and partial decay widths of their decay channels are calculated under the framework of QCD sum rules. The decay widths of the discovered $ P_c(4380) $ , $ P_c(4440) $ , and $ P_c(4457) $ are in good agreement with the experiments. The predictions of the decays of these three related possible isospin cousins are presented, which would shed light on their findings in experiments. In return, this may testify to the assignments of the discovered $ P_c $ states.

China

Xiu-Wu Wang

Xin Li

Zhi-Gang Wang

Systematic study of the strong decays of the $\textit{P}$ $\textit{c}$ states and their possible isospin cousins via the QCD sum rules * Supported by the National Natural Science Foundation of China (12575083) and the Natural Science Foundation of Hebei Province (A2024502002)

We discuss charm associated production of the pseudoscalar Higgs boson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>′′</mo></msup></math> in the Maximally CP symmetric Model’ (MCPM’). In our analysis we assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><msup><mi>h</mi><mo>′′</mo></msup></msub><mo>=</mo><mn>95.4</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, which corresponds to an enhancement observed by the CMS collaboration in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi><mi>γ</mi></math> channel. As discussed recently, the MCPM’ is consistent with the CMS enhancement. In this model the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>′′</mo></msup></math> Higgs boson of this mass decays dominantly into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> jets. The cross section for the associated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>′′</mo></msup></math> production in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi><mo stretchy="false">→</mo><msup><mi>h</mi><mo>′′</mo></msup><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> is found to be almost as big as that for the Drell-Yan (DY) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">→</mo><msup><mi>h</mi><mo>′′</mo></msup></math> production, much larger than that for two-gluon fusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mi>g</mi><mo stretchy="false">→</mo><msup><mi>h</mi><mo>′′</mo></msup></math>. Since in the MCPM’ the dominant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>′′</mo></msup></math> decay is into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> channel we have to deal with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi><mo stretchy="false">→</mo><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> process with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> jets. In addition to the signal processes, we have sizeable single parton scattering (SPS) and double parton scattering (DPS) nonresonant background. In our analysis we include diagrams of gluon-gluon fusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mi>g</mi><mo stretchy="false">→</mo><msup><mi>h</mi><mo>′′</mo></msup></math> which lead to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> final state. We discuss also the possible role of the reaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi><mo stretchy="false">→</mo><msup><mi>h</mi><mo>′′</mo></msup><msup><mi>h</mi><mo>′′</mo></msup></math>. Assuming 100% c and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> jet tagging we discuss in detail how to improve the signal-to-background ratio. Our analysis shows promising resonance-like enhancements and relatively large cross sections in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> channel within the MCPM’. In an Appendix we discuss the total cross section for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>′′</mo></msup></math> production at the LHC. We also compare distributions for the reactions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi><mo stretchy="false">→</mo><msup><mi>h</mi><mo>′′</mo></msup><mi>X</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi><mo stretchy="false">→</mo><mi>Z</mi><mi>X</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mo>′′</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Z</mi></math> decaying to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math>. We hope that the LHC collaborations will in the future perform the experimental studies corresponding to the theoretical studies discussed here.

Poland

R. Maciuła

Chile

M. Maniatis

Germany

O. Nachtmann

A. Szczurek

Charm associated production of the pseudoscalar Higgs boson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>h</mi><mrow><mo>′′</mo></mrow></msup></math> in a variant of the maximally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-symmetric model at the LHC

Four-dimensional chiral gauge theory can be formulated as the boundary theory on a five-dimensional manifold in a manner that may be realized on a finite lattice. There are interesting features of these theories that defy a purely four-dimensional conception of universality. We find that QCD when embedded in a chiral gauge theory (the Standard Model) and regulated this way can simultaneously avoid both the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>A</mi></msub></math> problem and the strong <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> problem, with a central role played by fermion zero modes localized far away in the fifth dimension. In this way it differs from conventional lattice QCD formulated as a stand-alone theory, universality being violated by inaccessible light modes in the five-dimensional bulk. Our analysis builds on recent work by others that highlights the role of global <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> symmetries in five-dimensional formulations of four-dimensional chiral gauge theories, and the generic appearance of fermion zero modes in the bulk.

United States

David B. Kaplan

Srimoyee Sen

Regulated chiral gauge theory and the strong <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> problem

We present a determination of the scalar and tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>p</mi></math> transition form factors using lattice QCD. These form factors are relevant for semileptonic hyperon decays in the presence of extensions of the Standard Model that include scalar and tensor interactions. The calculation is carried out using a gauge ensemble of twisted mass fermions at the physical pion mass, following the same strategy as our recent study on vector and axial form factors for the same transition [S. Bacchio and A. Konstantinou, <a href="http://dx.doi.org/10.1103/bvs9-wcsj">Phys. Rev. Lett. 135, 231901 (2025)</a>]. We provide the complete set of form factors as functions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> employing a model-independent parametrization. We examine their impact on searches for nonstandard charged-current interactions via the muon-to-electron decay-rate ratio <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>μ</mi><mi>e</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>p</mi><mi>μ</mi><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mi>μ</mi></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>p</mi><mi>e</mi><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math>, where scalar and tensor contributions enter linearly and are helicity enhanced relative to the electron channel. We compare this first-principles prediction for the decay-rate ratio with recent experimental measurements, thereby enabling improved constraints on nonstandard charged-current interactions.

Cyprus

Constantia Alexandrou

Simone Bacchio

Andreas Konstantinou

Eleni Vakana

Scalar and tensor form factors for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">Λ</mi><mo stretchy="false">→</mo><mi>p</mi><mo>ℓ</mo><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> from lattice QCD

The time-dependent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetry in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">→</mo><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> decays is measured using proton-proton collision data corresponding to an integrated luminosity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>6</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, collected with the LHCb detector at a center-of-mass energy of 13 TeV during the years 2015–2018. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violation parameters for this process are determined to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><msubsup><mrow><mi>β</mi></mrow><mrow><mi>c</mi><mover accent="true"><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>d</mi></mrow><mrow><mi>eff</mi></mrow></msubsup><mo>=</mo><mn>0.710</mn><mo>±</mo><mn>0.084</mn><mo>±</mo><mn>0.051</mn><mtext> </mtext><mtext> </mtext><mi>rad</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">|</mo><mi>λ</mi><mo stretchy="false">|</mo><mo>=</mo><mn>1.019</mn><mo>±</mo><mn>0.034</mn><mo>±</mo><mn>0.024</mn></mrow></math>, where the first uncertainty is statistical and the second systematic. This constitutes the first observation of time-dependent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">→</mo><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> decays. Assuming approximate SU(3) flavor symmetry, these results are combined with the previous consistent LHCb measurement to set the most stringent constraint on the penguin contribution, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msub><mi>ϕ</mi><mi>s</mi></msub></math>, to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating phase <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ϕ</mi><mi>s</mi></msub></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>s</mi><mn>0</mn></msubsup><mo stretchy="false">→</mo><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mn>1020</mn><mo stretchy="false">)</mo></math> decays, yielding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msub><mi>ϕ</mi><mi>s</mi></msub><mo>=</mo><mn>5.0</mn><mo>±</mo><mn>4.6</mn><mtext> </mtext><mtext> </mtext><mi>mrad</mi></math>.

Netherlands

R. Aaij

United Kingdom

A.S.W. Abdelmotteleb

Switzerland

C. Abellan Beteta

F. Abudinén

T. Ackernley

A.A. Adefisoye

Spain

B. Adeva

M. Adinolfi

P. Adlarson

France

C. Agapopoulou

C.A. Aidala

Z. Ajaltouni

S. Akar

K. Akiba

M. Akthar

Italy

P. Albicocco

J. Albrecht

R. Aleksiejunas

CERN

F. Alessio

P. Alvarez Cartelle

R. Amalric

Brazil

S. Amato

J.L. Amey

Y. Amhis

L. An

L. Anderlini

M. Andersson

P. Andreola

M. Andreotti

S. Andres Estrada

A. Anelli

D. Ao

C. Arata

F. Archilli

Z. Areg

M. Argenton

Costa Rica

S. Arguedas Cuendis

L. Arnone

A. Artamonov

M. Artuso

E. Aslanides

R. Ataíde Da Silva

M. Atzeni

B. Audurier

J.A. Authier

D. Bacher

I. Bachiller Perea

S. Bachmann

M. Bachmayer

J.J. Back

Z.B. Bai

P. Baladron Rodriguez

V. Balagura

A. Balboni

W. Baldini

Z. Baldwin

L. Balzani

H. Bao

J. Baptista de Souza Leite

C. Barbero Pretel

M. Barbetti

I.R. Barbosa

R.J. Barlow

M. Barnyakov

S. Barsuk

W. Barter

J. Bartz

S. Bashir

B. Batsukh

P.B. Battista

A. Bavarchee

A. Bay

A. Beck

M. Becker

F. Bedeschi

I.B. Bediaga

N.A. Behling

S. Belin

A. Bellavista

K. Belous

I. Belov

I. Belyaev

G. Benane

G. Bencivenni

E. Ben-Haim

A. Berezhnoy

R. Bernet

A. Bertolin

F. Betti

J. Bex

O. Bezshyyko

S. Bhattacharya

M.S. Bieker

N.V. Biesuz

A. Biolchini

M. Birch

F.C.R. Bishop

A. Bitadze

A. Bizzeti

T. Blake

Observation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> Violation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> Decays

We report an experimental investigation of the reactions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>p</mi><mo stretchy="false">→</mo><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>p</mi><mo stretchy="false">→</mo><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>p</mi><mo stretchy="false">→</mo><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>10.087</mn><mo>±</mo><mn>0.044</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>9</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> events collected with the BESIII detector at the BEPCII storage ring. The antineutron (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>n</mi><mo stretchy="false">¯</mo></mover></math>) is produced in the decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo stretchy="false">→</mo><mi>p</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> with studied momentum from 200 to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1174</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>/</mo><mi>c</mi></mrow></math>, while the target proton originates from the hydrogen nuclei in the cooling oil of the beam pipe. This novel method pioneers the study of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>n</mi><mo stretchy="false">¯</mo></mover></math>-nucleon interactions at an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> collider, providing the first experimental data for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> momenta exceeding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>800</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>/</mo><mi>c</mi></mrow></math>.

M. Ablikim

Russia

M.N. Achasov

Sweden

X.C. Ai

R. Aliberti

A. Amoroso

Q. An

Y. Bai

JINR

O. Bakina

Y. Ban

H.R. Bao

X.L. Bao

V. Batozskaya

Mongolia

K. Begzsuren

N. Berger

M. Berlowski

M.B. Bertani

D. Bettoni

F. Bianchi

E. Bianco

A. Bortone

I. Boyko

R.A. Briere

A. Brueggemann

H. Cai

M.H. Cai

X. Cai

A. Calcaterra

G.F. Cao

N. Cao

Türkiye

S.A. Cetin

X.Y. Chai

J.F. Chang

T.T. Chang

G.R. Che

Y.Z. Che

C.H. Chen

Chao Chen

G. Chen

H.S. Chen

H.Y. Chen

M.L. Chen

S.J. Chen

S.M. Chen

T. Chen

W. Chen

X.R. Chen

X.T. Chen

X.Y. Chen

Y.B. Chen

Y.Q. Chen

Z.K. Chen

J. Cheng

L.N. Cheng

South Korea

S.K. Choi

X. Chu

G. Cibinetto

Study of the Reactions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>p</mi><mo stretchy="false">→</mo><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mn>2</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> Using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo stretchy="false">→</mo><mi>p</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mover accent="true"><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math>