) and K · f1(1285)! a0(980)ˇ decay: formalism Vertices: f1 K (K 1) K (K).. it1 = igf1C1ϵ ϵ′...

The molecular K∗K̄ nature of the f1(1285) and itsdecays into πa0(980) (πf0(980)) and πKK̄

Francesca Aceti

Instituto de Física Corpuscular, Universidad de Valencia - CSIC

September 17, 2015

Co-authors:J. M. Dias, Ju-Jun Xie and E. Oset

Francesca Aceti Hadron2015The molecular K∗K̄ nature of the f1(1285) and its decays into πa0(980) (πf0(980)) and πKK̄ 1 / 29

Outline

F. A., J. M. Dias, E. Oset, Eur. Phys. J. A51 (2015) 48F. A., Ju-Jun Xie, E. Oset, arXiv:1505.06134 [hep-ph]

• introduction

• f1(1285) → πa0(980) (πf0(980)) and isospin breaking- formalism- results for the decay width- invariant mass distributions

• f1(1285) → πKK̄

- formalism- results for the decay width- invariant mass distributions

• conclusionsFrancesca Aceti Hadron2015The molecular K∗K̄ nature of the f1(1285) and its decays into πa0(980) (πf0(980)) and πKK̄ 2 / 29

Introduction

STARTING POINT: f1(1285), a0(980) and f0(980) are dynamicallygenerated by meson-meson interaction


Introduction


• f1(1285) [I G(J PC) = 0+(1++)]

• appears in the pseudoscalar-vector interaction from the single channel K∗K̄ − c.c.[1−3]

• mostly considered as a qq̄[4−6] state

• small width, Γ ≃ 24 MeV

• BR(f1(1285) → a0(980)π) = (36 ± 7)%[7] excluding a0(980) → KK̄

• BR(f1 → πKK̄)|exp = (9.0 ± 0.4)%[7]

[1] Roca et al., Phys. Rev. D 72; 014002 (2005)[2] Zhou et al.; Phys. Rev. D 90, 014020 (2014)[3] Geng et al., Phys. Rev. D 92, 014029 (2015)[4] Chen et al., Phys. Rev. D 91, 074025 (2015)[5] Klempt et al., Phys. Rept. 454, 1 (2007)[6] Gavillet et al., Z. Phys. C 16, 119 (1982)[7] Particle Data Group, Chin. Phys. C, 38, 090001 (2014)


Introduction


• a0(980) [I G(J PC) = 1−(0++)]

• appears in the pseudoscalar-pseudoscalar interaction• building blocks: πη, KK̄[1]

• f0(980) [I G(J PC) = 0+(0++)]

• appears in the pseudoscalar-pseudoscalar interaction• building blocks: ππ, KK̄[1]

• success in the study of Φ → π0π0γ, π0η[2], J/ψ → Φ(ω)f [3]0 ... gives support to this

assumption[1] Oller et al., Nucl. Phys. A 620, 438 (1998)[2] Palomar et al., Nucl. Phys. A 729, 743 (2003)[3] Meissner et al.,Nucl. Phys. A 679, 671 (2001)


....Part 1: f1(1285) → a0(980)π decay


f1(1285) → a0(980)π decay: formalism

in this picture the process is described by the triangular diagrams

f1

K∗+

K−

K+

π0

a0

π0

η

f1

K∗0

K̄0

K0

π0

a0

π0

η

f1

K̄∗0

K0

K̄0

π0

a0

π0

η

f1

K∗−

K+

K−

π0

a0

π0

η

A) B)

C) D)



Vertices:

f1K∗(K̄∗)

K̄ (K)

.

......−it1 = −igf1C1ϵµϵ′µ

• gf1 = 7555 MeV, evaluated as the residue at the pole of T = [1− VG]−1V for K∗K̄ − c.c. in I = 0 withV taken from [1]

• the loop function G is regularized by means of a cutoff qmax ≃ 1 GeV

• factors C1 due to the fact that the f1 couples to the I = 0, C = + and G = + combination of K∗K̄:

− 12 (K

∗+K− + K∗0K̄ − K∗−K+ − K̄∗0K0)

[1] Roca et al., Phys. Rev. D 72, 014002 (2005)]



Vertices:

π0

K∗(K̄∗)

K̄ (K)

.

......−it2 = −igC2(2k−P+q)µϵ′µ

• evaluated using the Hidden Gauge VPP Lagrangian LVPP = −ig⟨Vµ[P, ∂µP]⟩[1−4]

• gv =mV2f , mV ≃ mρ, f = 93 MeV

[1] Bando et al., Phys. Rev. Lett. 54, 1215 (1985)[2] Bando et al., Phys. Rept. 164, 217 (1988)[3] Meissner, Phys. Rept. 161, 213 (1988)[4] Harada et al., Phys. Rept 381, 1 (2003)

k

P

P − q

q

P − q − k

Minv



Vertices:

K (K̄)

K̄ (K)

π0 (π+)

η (π−)

.

......−it3 = −itif

• rescattering of the KK̄ pair

• tif is the element of the 5 × 5 scattering matrix for the channels K+K− (1), K0K̄0 (2), π0η (3), π+π−

(4), π0π0 (5) (see Ref. [1])

• loop function regularized with a cutoff q′max = 900 MeV

• i = 1 for A) and C), i = 2 for B) and D) while f = 3, 4

[1] Oller et al., Nucl. Phys. A 620, 438 (1998)



due to the physical masses of the kaons the rescattering of KK̄dynamically generates the a0(980) and f0(980) (they can mix!)[1]

leading to both π0η and π+π− in the final state

....isospin symmetry is broken

=⇒ simultaneous study of f1(1285) → f0(980)π to quantify this isospin breaking

[1] Achasov et al., Phys.Lett. B 88, 367 (1979)



Putting everything together we get, for each diagram:

.

......t = C gf1 g ϵ⃗ · k⃗ (2I1 + I2) tif = t̃ ϵ⃗ · k⃗ tif

C1 C2 CA) −1/2 1/

√2 −1/2

√2

B) −1/2 −1/√2 1/2

√2

C) 1/2 −1/√2 −1/2

√2

D) 1/2 1/√2 1/2

√2

I1 = −i∫ d4q

(2π)41

q2−m2K+iϵ

1(P−k−q)2−m2

K+iϵ1

2ω∗1

P0−q0−ω∗+iϵ

I2 = −i∫ d4q

(2π)41

q2−m2K+iϵ

1(P−k−q)2−m2

K+iϵk⃗·⃗q|⃗k|2

12ω∗

1P0−q0−ω∗+iϵ

• integration in dq0 done analytically• integration in d 3q → upper limit naturally provided by chiral unitary approach

→ same cutoff used in the KK̄ loop function



Peculiarities of the triangular mechanism:

• the cutoff qmax ≃ 900 MeV appears automatically as upper limit in I1 and I2

thanks to the KK̄ → PP potential used to generate the a0 and f0

V(⃗q, q⃗ ′) = v θ(qmax − |⃗q|) θ(qmax − |⃗q ′|)

t(⃗q, q⃗ ′) = t θ(qmax − |⃗q|) θ(qmax − |⃗q ′|)

• first applied in [1] to η(1405) → π0π+π−

• successfully used in [2] to evaluate Γ(η(1405)→π0f0(980))Γ(η(1405)→π0a0(980))

[1] Wu el al., Phys. Rev. Lett. 108, 081803(2012)

[2] F.A. et al., Phys.Rev. D 86, 114007 (2012)



TOTAL AMPLITUDE OF THE PROCESSES:.

......

Tπ0η = (2t̃(+) tK+K−→π0η + 2t̃(0) tK0K̄0→π0η) ϵ⃗ · k⃗ = T̃π0η ϵ⃗ · k⃗

Tπ+π− = (2t̃(+) tK+K−→π+π− +2t̃(0) tK0K̄0→π+π−) ϵ⃗ · k⃗ = T̃π+π− ϵ⃗ · k⃗

• t̃+ and t̃0 have opposite signs [see the table for the coefficients C]

• tK+K−→π0η = −tK0K̄0→π0η ; tK+K−→π+π− = tK0K̄0→π+π− [Oller et al., Nucl. Phys. A 620, 438 (1998)]

..

..if isospin symmetry were exact (mK0 = mK±) =⇒ Tπ+π− ≡ 0and the production of the the f0 would be forbidden

isospin symmetry is not exact −→ we can have both the π0η and π+π− pairs inthe final state


f1(1285) → a0(980)π decay: results

We plot the invariant mass distribution for both cases:.

......dΓ

dMinv= 1

(2π)3pπ |⃗k|4m2

f1

13|T̃|2




......dΓ

dMinv= 1

(2π)3pπ |⃗k|4m2

f1

13|T̃|2

0

0.004

0.008

0.012

0.016

0.02

920 960 1000 1040 1080

dΓ/d

mf [

a.u.

]

mf [MeV]

f1(1285) → π0π0η

• a0(980) produced with its normal widthΓ ≃ 50 MeV

• high strength at the peak




......dΓ

dMinv= 1

(2π)3pπ |⃗k|4m2

f1

13|T̃|2

f1(1285) → π0π+π−

• narrow peak, Γ ≃ 10 MeV

• the peak is between the two K+K− andK0K̄0 thresholds

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

920 960 1000 1040 1080

dΓ/d

mf [

a.u.

]

mf [MeV]



due to the the difference between |̃t+| and |̃t0|−→ significant only between the two thresholds

960 980 1000 1020 1040

-0.0012

-0.0008

-0.0004

0

mf [MeV]

t˜⁽⁺⁾t˜⁽⁰⁾

0

0.01

0.02

0.03

0.04

0.05

0.06

920 960 1000 1040 1080

R

mf [MeV]

ratio of strength at the peak ≃ 6%



• f1(1285) → π0π+π− mass distribution → in agreement with what found inother reactions (like η(1405) → π0π+π− [1])

• f1(1285) → π0π+π− mass distribution → in agreement with BES recentobservations[2]

• ratio of strength at the peak ≃ 6% → ten times smaller than what we got in thecase of the η′ → π0π0η, π0π+π− [3]

[1] BESIII, Phys. Rev. Lett. 108,182001 (2012)[2] BESIII, Phys. Rev. D 92, 012007 (2015)[3] Aceti et al., Phys. Rev. D 86, 114007



PARTIAL WIDTH FOR THE DECAY TO a0π:.

......Γa0π = 3

∫dMinv

(dΓ

dMinv

)π0η

= 4.5 MeV

Two sources of uncertainties:• cutoff used for the generation of the f1(1285): we change it of ±20 MeV• cutoff used in I1 and I2: we change it of ±20 MeV

....BR(f1 → a0π)|th = (19 ± 2)%

We also get: Γ(π0π+π−)

Γ(π0π0η)= 0.82 × 10−2



BR(f1 → a0π)|exp = (36 ± 7)% BR(f1 → a0π)|th ≃ 19%

• qualitative agreement between the experimental and the theoretical value of thebranching ratio

• amount of isospin breaking → much smaller than

Γ(η(1405)→π0π+π−)

Γ(η(1405)→π0π0η)= 18× 10−2 [1]

and twice as big as

Γ(J/ψ→ϕf0(980))Γ(J/ψ→ϕa0(980)) = 0.6× 10−2 [2]

but in agreement with BES recent measurment of (3.6 ± 1.4)%[3]

[1] BESIII, Phys. Rev. Lett. 108,182001 (2012)[2] BESIII, Phys. Rev D, 83, 032003 (2011)[3] BESIII, Phys. Rev. D 92, 012007 (2015)


....Part 2: f1(1285) → πKK̄ decay


f1(1285) → πKK̄ decay: formalism at tree level

f1

K−

K∗+

K+

π0

A)

f1

K̄0

K∗0

K+

π−

C)

f1

K−

K∗+

K0

π+

B)

K0

f1

K̄0

K∗0

π0

D)

f1

K+

K∗−

K−

π0

E)

f1

K+

K∗−

K̄0

π−

F )

f1

K0

K̄∗0

K̄0

π0

H)

f1

K0

K̄∗0

K−

π+

G)



Vertices:

f1K∗(K̄∗)

K̄ (K)

.

......−it1 = −igf1C1ϵµϵ′µ

C1

A) B) −1/2C) D) 1/2E) F) −1/2G) H) 1/2

π0

K∗(K̄∗)

K̄ (K)

.

......−it2 = −igC2(k − p)µϵ′µ

C2

A) H) −1/√2

B) C) −1

D) E) 1/√2

F) G) 1



• we sum the amplitudes for the diagrams with same final state:

MA+Etree = MD+H

tree = Mtree MB+Gtree = MC+F

tree =√2Mtree

• we take diagrams A) and E) as reference −→ π0K+K− final state

.

......Mtree =

ggf12√

2

([(⃗k− p⃗)− m2

π−m2K

m2K∗

(⃗k+ p⃗)]D1+[(⃗k− p⃗′)+m2

π−m2K

m2K∗

(⃗k+ p⃗′)]D2

)· ϵ⃗

• p and p′ are the momenta of K+ and K− respectively, k is the momentum of the π0

• D1 = 1

(k+p)2−m2K∗+imK∗ΓK∗

and D2 = 1

(k+p′)2−m2K∗+imK∗ΓK∗

are the propagators of the K∗

• ΓK∗ = Γon( qoff

qon

)3is the total width of the K∗


f1(1285) → πKK̄ decay: FSI

now we study the final state interactions (FSI) by means of the triangular mechanism

A) KK̄ interaction

B) πK interaction



A) KK̄ interaction:

f1

K∗

K

K̄

π0

K+

K−

f1

K∗

K

K̄

π−

K+

K̄0

f1

K∗

K

K̄

π0

K0

K̄0

f1

K∗

K

K̄

π+

K0

K−

• rescattering → generates the a0(980) (I = 1)

• scattering amplitude tI=1KK̄→KK̄(MKK̄): obtained using T = [1 − VG]−1V with V taken from [1]

−→ good description of the a0(980) using qmax = 900 MeV

• we sum the four diagrams with final state π0K+K−

[1] Oller et al., Nucl. Phys. A 620, 438 (1998)



A) KK̄ interaction:

f1

K∗

K

K̄

π0

K+

K−

f1

K∗

K

K̄

π−

K+

K̄0

f1

K∗

K

K̄

π0

K0

K̄0

f1

K∗

K

K̄

π+

K0

K−

.

......MKK̄FSI = − g gf1

2√2(2I1 + I2) 2 tI=1

KK̄→KK̄(MKK̄) ϵ⃗ · k⃗

M2KK̄ = (p + p′)2 = M2

f1 + m2π − 2Mf1

√|⃗k|2 + m2

π

I1 = −i∫ d4q

(2π)41

q2−m2K+iϵ

1

(P−k−q)2−m2K+iϵ

12ω∗


I2 = −i∫ d4q

(2π)41

q2−m2K+iϵ

1

(P−k−q)2−m2K+iϵ

k⃗·⃗q|⃗k|2

12ω∗


• integration in dq0 done analytically• integration in d 3q → upper limit naturally provided by chiral unitary approach



B) πK interaction:

f1

K̄∗

π

K

K̄0

π−, π0

K+, K0

f1

K̄∗

π

K

K−

π+, π0

K0, K+

f1

K∗

π

K̄

K+

π0, π−

K−, K̄0

f1

K∗

π

K̄

K0

π0, π+

K̄0, K−

• rescattering → generates the κ(800) (I = 1/2)

• scattering amplitude: tI=1/2πK→πK(MπK), with the potential V taken from [1]

−→ good description of the κ(800) using qmax = 900 MeV

• we sum the four diagrams with final state π0K+K−

[1] Bayar et al., Phys. Rev. D 90, 114004 (2014)



B) πK interaction:

f1

K̄∗

π

K

K̄0

π−, π0

K+, K0

f1

K̄∗

π

K

K−

π+, π0

K0, K+

f1

K∗

π

K̄

K+

π0, π−

K−, K̄0

f1

K∗

π

K̄

K0

π0, π+

K̄0, K−

.

......MπKFSI =

g gf12√

2(2I′

1 +I′2) tI=1/2

πK→πK(M(1)πK) ϵ⃗ · p⃗+

g gf12√2(2I′′

1 +I′′1 ) tI=1/2

πK→πK(M(2)πK) ϵ⃗ · p⃗′

M(1)πK =

√(k + p′)2

M(2)πK =

√(k + p)2

I′1, I′

2, I′′1 and I′′

2 obtained from I1 and I1 with suitable substitution of kinematic variables



=⇒ full amplitude for the π0K+K− final state:

M = Mtree + MKK̄FSI + MπK

FSI

=⇒ total decay width.

......ΓKK̄π = 6 1

64π3Mf1

∫ ∫dωK+dωK−

∑|M|2 θ(1− cos2θKK̄)θ(Mf1 −ωK+ −ωK− −mπ)

• factor 6 −→ accounts for all the final charges: π0K+K−, π+K0K−, π−K+K̄0 and π0K0K̄0, havingweights 1, 2, 2, and 1

• ωK+ =√

m2K + p⃗ 2 and ωK− =

√m2

K + p⃗′ 2, energies of the K+ and K−

• cosθKK̄ given by energy conservation


f1(1285) → πKK̄ decay: results

gf1 MeV ΓKK̄π MeV B.R.

7555 (1) 1.90 7.8%

7230 (2) 1.74 7.2%

(1) Aceti et al., Eur. Phys. J. A 51, 4, 48 (2015)

(2) Roca et al., Phys. Rev. D 72, 014002 (2004)

Two sources of uncertainties:

• cutoff used for the generation of the f1(1285):→ we change it of ±40 MeV

• cutoff used in I1 and I2:→ we change it of ±40 MeV

BR(f1 → πKK̄)|exp = (9.0 ± 0.4)% B.R.(f1 → KK̄π)|th = (7.2 − 8.3)%

with gf1 = 7555 MeV and only the tree level diagrams: B.R.(f1 → KK̄π) = 9.1%

the contribution of the FSIs is very small: cancellation due to their opposite sign


f1(1285) → πKK̄ decay: results

invariant mass distribution for f1(1285) → π0K+K− as a function of MK+K− :.

......dΓ

dMK+K−=

MK+K−64π3M2

f1

∫dωK+

∑|M|2θ(1− cos2θKK̄) θ(Mf1 −ωK+ −ωK− −mπ)θ(ωK− −mK)

with ωK− = 12Mf1

(M2K+K− + M2

f1− m2

π) − ωK+

1000 1020 1040 1060 1080 1100 1120 11400.000

0.001

0.002

0.003

0.004

0.005

0.006

d/d

Min

v

MK+K-(MeV)

Total Tree diagram Phase space • shape of the tree level very different

• we can see a peak at low MK+K−

• the peak is shifted closer to the mass of the a0(980)by the FSI



invariant mass distribution for f1(1285) → π0K+K− as a function of Mπ0K+ :.

......dΓ

dMπ0K+

=M

π0K+

64π3M2f1

∫dωK+

∑|M|2θ(1− cos2θKK̄) θ(Mf1 −ωK+ −ωK− −mπ)θ(ωK− −mK)

with ωK− = 12Mf1

(M2f1

+ m2K − M2

π0K+)

640 660 680 700 720 740 760 780 8000.000

0.001

0.002

0.003

0.004

0.005

d/d

Min

v

M 0K+ (MeV)

Total Tree diagram Phase space

• now the peak is at high Mπ0K+

• the peak is shifted closer to the mass of the κ(800) bythe FSI

..

..

presence of the peak=⇒ tied to the K∗ propagator

=⇒ tied to the K̄∗K − c.c. natureof the f1(1285)


Conclusions

The assumption that the f1(1285), a0(980) and f0(980) are dynamically generatedresonances is supported by our findings:

• a branching ratio for f1(1285) → πa0(980) in qualitative agreement with experiment

• a branching ratio for f1(1285) → πKK̄ in good agreement with experiment

• the shape we obtained of the f1(1285) → π0π+π− mass distribution

−→ in agreement with what found for the η(1405) → π0π+π−

−→ in agreement with recent BES observations


Conclusions

We also observed:

• an amount of isospin breaking in f1(1285) → π0f0 in agreement with what recentlymeasured but different from other reaction

=⇒ it appears to be strongly dependent on the process

• a shape of the invariant mass distributions of f1(1285) → πKK̄ tied to the nature ofthe f1

=⇒ a measurement of these quantities would help understanding the nature ofthe f1(1285)


The molecular K∗K̄ nature of the f1(1285) and itsdecays into πa0(980) (πf0(980)) and πKK̄

Francesca Aceti

Instituto de Física Corpuscular, Universidad de Valencia - CSIC

September 17, 2015

Collaborators:J. M. Dias, Ju-Jun Xie and E. Oset


) and K · f1(1285)! a0(980)ˇ decay: formalism Vertices: f1 K (K 1) K (K).. it1 = igf1C1ϵ ϵ′...

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Transcript of ) and K · f1(1285)! a0(980)ˇ decay: formalism Vertices: f1 K (K 1) K (K).. it1 = igf1C1ϵ ϵ′...