η π γγ Chiral Perturbation Theory for η 3π and η π γγJohan Bijnens Older reviews Model...

ChPT forη → 3π andη → π0γγ

Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

1/53

Chiral Perturbation Theory forη → 3π and η → π

0γγ

Johan Bijnens

Lund University

[email protected]

http://www.thep.lu.se/∼bijnens

http://www.thep.lu.se/∼bijnens/chpt.html

Hadronic Probes of Fundamental Symmetries, Amherst, 6-8 March 2014

http://www.thep.lu.se/~bijnens/

http://www.thep.lu.se/~bijnens/chpt.html


Johan Bijnens

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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

2/53

ChPT aspects of η → 3π and η → π0γγ

1 Older reviews

2 η → 3π: Some model independent comments/resultsDefinitionsExperimentWhy?

3 ChPT

4 η → 3π in ChPTLOLO and NLONNLO

5 η → π0γγp4

ExperimentLoops at p6, p8

All elseDistributions

6 Conclusions


Johan Bijnens

Older reviews

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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

3/53

Eta Physics Handbook: ETA01


Johan Bijnens

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η → 3π inChPT

η → π0γγ

Conclusions

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ETA05: Acta Phys.Slov. 56(2006) No 3

Acta Physica Slovaca 56(2006)

C. Hanhart Hadronic production of eta-mesons: Recent results and open questions , 193 (2006)C. Wilkin, U. Tengblad, G. Faldt The p d -> p d eta reaction near threshold , 205 (2006)J. Smyrski, H.-H. Adam, A. Budzanowski, E. Czerwinski, R. Czyzykiewicz, D. Gil, D. Grzonka, A. Heczko, M. Janusz, L. Jarczyk, B. Kamys, A. Khoukaz, K. Kilian, P. Klaja, J. Majewski, P. Moskal, W. Oelert, C. Piskor-Ignatowicz, J. Przerwa, J. Ritman, T. Rozek, R. Santo, T. Sefzick, M. Siemaszko, A. Taschner, P. Winter, M. Wolke, P. Wustner, Z. Zhang, W. Zipper Study of the 3He-eta system in d-p collisions at COSY-11 , 213 (2006)M. Doring, E. Oset, D. Strottman Chiral dynamics in gamma p -> pi0 eta p and related reactions , 221 (2006)H. Machner, M. Abdel-Bary, A. Budzanowski, A. Chatterjee, J. Ernst, P. Hawranek, R. Jahn, V. Jha, K. Kilian, S. Kliczewski, Da. Kirillov, Di. Kirillov, D. Kolev, M. Kravcikova, T. Kutsarova, M. Lesiak, J. Lieb, H. Machner, A. Magiera, R. Maier, G. Martinska, S. Nedev, N. Pisku\-nov, D. Prasuhn, D. Protic, P. von Rossen, B. J. Roy, I. Sitnik, R. Siudak, R. Tsenov, M. Ulicny, J. Urban, G. Vankova, C. Wilkin The eta meson physics program at GEM , 227 (2006)K. Nakayama, H. Haberzettl Photo- and Hadro-production ofÂ eta' meson , 237 (2006)S.D. Bass Gluonic effects in eta and eta' nucleon and nucleus interactions , 245 (2006)P. Klaja, P. Moskal, H.-H. Adam, A. Budzanowski, E. Czerwinski, R. Czyzykiewicz, D. Gil, D. Grzonka, M. Janusz, L. Jarczyk, B. Kamys, A. Khoukaz, K. Kilian, J. Majewski, W. Migdal, W. Oelert, C. Piskor-Ignatowicz, J. Przerwa, J. Ritman, T. Rozek, R. Santo, T. Sefzick, M. Siemaszko, J. Smyrski, A. Taschner, P. Winter, M. Wolke, P. Wustner, Z. Zhang, W. Zipper Correlation femtoscopy for studying eta meson production mechanism , 251 (2006)M.T. Pena, H. Garcilazo Study of the np -> eta d reaction within a three-body model , 261 (2006)A. Gillitzer Search for nuclear eta states at COSY and GSI , 269 (2006)A. Wronska, V. Hejny, C. Wilkin Near threshold eta meson production in the dd -> 4He eta reaction , 279 (2006)M. Bashkanov, T. Skorodko, C. Bargholtz, D. Bogoslawsky, H. Calen, F. Cappellaro, H. Clem\-ent, L. Demiroers, E. Doroshkevich, C. Ekstrom, K. Fransson, L. Geren,Â J.Greiff, L. Gustafsson, B. Hoistad, G. Ivanov,Â M. Jacewicz, E. Jiganov, T. Johansson,M.M. Kaskulov,Â S. Keleta, O. Khakimova, I. Koch, F. Kren, S. Kullander,Â A. Kupsc,A. Kuznetsov, K. Lindberg, P. Marciniewski,Â R. Meier, B. Morosov, W. Oelert, C.Pauly, Y. Petukhov,Â A. Povtorejko, R.J.M.Y. Ruber, W. Scobel, R. Shafigullin, B.Shwartz, V. Sopov, J. Stepaniak, P.-E. Tegner,Â V. Tchernyshev, P.Thorngren-Engblom, V. Tikhomirov, A. Turowiecki, G.J. Wagner, M. Wolke, A.Yamamoto,Â J. Zabierowski, I. Zartova,Â J. Zlomanczuk On the pi pi production in free and in-medium NN-collisions: sigma-channel low-mass

enhancement and pi0 pi0 / pi+ pi- asymmetry , 285 (2006)K. Schonning for the CELSIUS/WASA collaboration Production of omega in pd -> 3He omega at kinematic threshold , 299 (2006)J. Bijnens Decays of eta and eta' and what can we learn from them? , 305 (2006)B. Borasoy, R. Nissler Decays of eta andÂ eta' within a chiral unitary approach , 319 (2006)E. Oset, J. R. Pelaez, L. Roca Discussion of the eta -> pi0 gamma gamma decay within a chiral unitary approach , 327(2006)C. Bloise on behalf of the KLOE collaboration Perspectives on Hadron Physics at KLOE with 2.5 fb^-1 , 335 (2006)T.Capussela for the KLOE collaboration Dalitz plot analysis of eta into 3pi final state , 341 (2006)A. Starostin The eta and eta' physics with crystal ball , 345 (2006)S. Schadmand for the WASA at COSY collaboration WASA at COSY , 351 (2006)M. Lang for the A2- and GDH-collaborations Double-polarization observables, eta-meson and two-pion photoproduction , 357 (2006)M. Jacewicz, A. Kupsc for CELSIUS/WASA collaboration Analysis of eta decay into pi+ pi- e+ e- in the pd -> 3He eta reaction , 367 (2006)F. Kleefeld Coulomb scattering and the eta-eta' mixing angle , 373 (2006)C. Pauly, L. Demirors, W. Scobel for the CELSIUS-WASA collaboration Production of 3pi0 in pp reactions above the eta threshold and the slope parameter alpha ,381 (2006)R. Czyzykiewicz, P. Moskal, H.-H. Adam, A. Budzanowski, E. Czerwinski, D. Gil, D. Grzonka, M. Janusz, L. Jarczyk, B. Kamys, A. Khoukaz, K. Kilian, P. Klaja, B. Lorentz, J. Majewski, W. Oelert, C. Piskor-Ignatowicz, J. Przerwa, J. Ritman, H. Rohdjess, T. Rozek, R. Santo, T. Sefzick, M. Siemaszko, J. Smyrski, A. Taschner, K. Ulbrich, P. Winter, M. Wolke, P. Wustner, Z. Zhang, Z. Zipper The analysing power for the pp -> pp eta reaction at Q=10 MeV , 387 (2006)A. Nikolaev for the A2 and Crystal Ball at MAMI collaborations Status of the eta mass measurement with the Crystal Ball at MAMI , 397 (2006)B. Di Micco for the CLOE collaboration The eta -> pi0 gamma gamma, eta/eta' mixing angle and status of eta mass measurement at KLOE , 403 (2006)


Johan Bijnens

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Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

5/53

Definitions: η → 3π

Reviews: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/0202242]

JB, Acta Phys. Slov. 56(2005)305 [hep-ph/0511076]

ηpη

π+ pπ+

π− pπ−

π0 pπ0

s = (pπ+ + pπ−)2 = (pη − pπ0)2

t = (pπ− + pπ0)2 = (pη − pπ+)2

u = (pπ+ + pπ0)2 = (pη − pπ−)2

s + t + u = m2η + 2m2

π+ +m2π0 ≡ 3s0 .

〈π0π+π−out|η〉 = i (2π)4 δ4 (pη − pπ+ − pπ− − pπ0) A(s, t, u) .

〈π0π0π0out|η〉 = i (2π)4 δ4 (pη − p1 − p2 − p3) A(s1, s2, s3)

A(s1, s2, s3) = A(s1, s2, s3) + A(s2, s3, s1) + A(s3, s1, s2)

Obervables: Γ(η → π+π−π0) and r = Γ(η→π0π0π0)Γ(η→π+π−π0)


Johan Bijnens

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Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

6/53

Definitions: Dalitz plot

x =√3T+ − T−

Qη=

√3

2mηQη(u − t)

y =3T0

Qη− 1 =

3 ((mη −mπo )2 − s)

2mηQη− 1

iso=

3

2mηQη(s0 − s)

Qη = mη − 2mπ+ −mπ0

T i is the kinetic energy of pion πi

z =2

3

∑

i=1,3

(

3Ei −mη

mη − 3mπ0

)2

Ei is the energy of pion πi

|M|2 = A20

(

1 + ay + by2 + dx2 + fy3 + gx2y + · · ·)

|M|2 = A20 (1 + 2αz + · · · )

Note: neutral, next order: x and y appear separately


Johan Bijnens

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Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

7/53

Relations

Expand amplitudes and use isospin: JB, Ghorbani, arXiv:0709.0230

M(s, t, u) = A(

1 + a(s − s0) + b(s − s0)2 + d(u − t)2 + · · ·

)

M(s, t, u) = A(

3 + (b + 3d)(

(s − s0)2 + (t − s0)

2 + (u − s0)2))

Gives relations (Rη = (2mηQη)/3)

a = −2Rη Re(a) , b = R2η

(

|a|2 + 2Re(b))

, d = 6R2η Re(d) .

α =1

2R2η Re

(

b + 3d)

=1

4

(

d + b − R2η |a|2

)

≤ 1

4

(

d + b − 1

4a2)

equality if Im(a) = 0


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Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

8/53

Relations

Consequences:

Relations between the charged and neutral decay

Relations between r and Dalitz plot(see also Gasser, Leutwyler, Nucl. Phys. B 250 (1985) 539)

If you can calculate Im(a) then relation:nonrelativistic pion EFTSchneider, Kubis and Ditsche, JHEP 1102 (2011) 028 [1010.3946].


Johan Bijnens

Older reviews

Modelindependent

Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

9/53

Definitions: Dalitz plot

0.08

0.1

0.12

0.14

0.16

0.08 0.1 0.12 0.14 0.16

t [G

eV

2]

s [GeV2]

mpiavmpidiff

u=ts=u

t-thresholdu thesholds threshold

x=y=0

x variation:vertical

y variation:parallel tot = u


Johan Bijnens

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Modelindependent

Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

10/53

Experiment: Decay rates

Width: determined from Γ(η → γγ) and Branching ratiosUsing the PDG12 partial update 2013 numbers

Γ(η → π+π−π0) = 300± 12 eV (in JB,Ghorbani 295 ± 17 eV)

r : 1.426 ± 0.026 (our fit)1.48 ± 0.05 (our average)


Johan Bijnens

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Modelindependent

Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

11/53

Experiment: charged

Exp. a b d f

WASA (prel) −1.104(3) 0.144(3) 0.073(3) 0.153(6)KLOE (prel) −1.074(23)(3) 0.179(27)(8) 0.059(25)(10) 0.089(58)

KLOE −1.090(5)(+8−19) 0.124(6)(10) 0.057(6)(+7

−16) 0.14(1)(2)Crystal Barrel −1.22(7) 0.22(11) 0.06(4) (input)Layter et al. −1.08(14) 0.034(27) 0.046(31)Gormley et al. −1.17(2)(21) 0.21(3) 0.06(4)

Crystal Barrel: d input, but a and b insensitive to d

Large correlations: KLOE:

a b d f

a 1 −0.226 −0.405 −0.795b 1 0.358 0.261d 1 0.113f 1


Johan Bijnens

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Modelindependent

Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

12/53

Experiment: charged

But very good agreement:

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.5

0 0.5

1 0

0.5

1

1.5

2

2.5

3

KLOE 08KLOE prelWASA prel

yx


Johan Bijnens

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Modelindependent

Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

13/53

Experiment: neutral

See talks by Kupsc and Unverzagt

Exp. α

GAMS2000 −0.022 ± 0.023SND −0.010 ± 0.021 ± 0.010

Crystal Barrel −0.052 ± 0.017 ± 0.010Crystal Ball (BNL) −0.031 ± 0.004WASA/CELSIUS −0.026 ± 0.010 ± 0.010

KLOE −0.0301 ± 0.0035+0.0022−0.0035

WASA@COSY −0.027 ± 0.008 ± 0.005Crystal Ball (MAMI-B) −0.032 ± 0.002 ± 0.002Crystal Ball (MAMI-C) −0.032 ± 0.003

All experiments in good agreement


Johan Bijnens

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Modelindependent

Definitions

Experiment

Why?

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

14/53

Why is η → 3π interesting?

Pions are in I = 1 state =⇒ A ∼ (mu −md ) or αem

αem effect is small

but is there via (mπ+ −mπ0) in kinematicsLowest order vanishes (current algebra)αm and αms smallBaur, Kambor, Wyler, Nucl. Phys. B 460 (1996) 127

η → π+π−π0γ needs to be included directlyDitsche, Kubis, Meissner, Eur. Phys. J. C 60 (2009) 83 [0812.0344]

Estimates the corrections of α(mu −md ) as wellConclusion: at the precision I will discuss not relevantException: Cusps and Coulomb at π+π− thresholds

So η → 3π gives a handle on mu −md


Johan Bijnens

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η → 3π inChPT

η → π0γγ

Conclusions

15/53

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral

Symmetry Spontaneous Breakdown (without η′)

Power counting: Dimensional counting in momenta/massesExpected breakdown scale: Resonances, so Mρ or higher

depending on the channel

Power counting in momenta: Meson loops

p2

1/p2

∫

d4p p4

(p2)2 (1/p2)2 p4 = p4

(p2) (1/p2) p4 = p4


Johan Bijnens

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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

16/53

Lagrangians

U(φ) = exp(i√2Φ/F0) parametrizes Goldstone Bosons

Φ(x) =

π0

√2

+η8√6

π+ K+

π− − π0

√2

+η8√6

K 0

K− K 0 −2 η8√6

.

LO Lagrangian: L2 =F 204 {〈DµU

†DµU〉+ 〈χ†U + χU†〉} ,

DµU = ∂µU − irµU + iUlµ ,left and right external currents: r(l)µ = vµ + (−)aµ

Scalar and pseudoscalar external densities: χ = 2B0(s + ip)quark masses via scalar density: s = M+ · · ·〈A〉 = TrF (A)


Johan Bijnens

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η → 3π inChPT

η → π0γγ

Conclusions

17/53

Lagrangians

L4 = L1〈DµU†DµU〉2 + L2〈DµU

†DνU〉〈DµU†DνU〉+L3〈DµU†DµUD

νU†DνU〉+ L4〈DµU†DµU〉〈χ†U + χU†〉+L5〈DµU†DµU(χ†U + U†χ)〉+ L6〈χ†U + χU†〉2

+L7〈χ†U − χU†〉2 + L8〈χ†Uχ†U + χU†χU†〉−iL9〈FR

µνDµUDνU† + F L

µνDµU†DνU〉

+L10〈U†FRµνUF

Lµν〉+ H1〈FRµνF

Rµν + F LµνF

Lµν〉+ H2〈χ†χ〉

Li : Low-energy-constants (LECs)Hi : Values depend on definition of currents/densities

These absorb the divergences of loop diagrams: Li → LriRenormalization: order by order in the powercounting


Johan Bijnens

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Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

18/53

Lagrangians

Lagrangian Structure:

2 flavour 3 flavour 3+3 PQChPT

p2 F ,B 2 F0,B0 2 F0,B0 2

p4 l ri , hri 7+3 Lri ,H

ri 10+2 Lri , H

ri 11+2

p6 c ri 52+4 C ri 90+4 K r

i 112+3

p2: Weinberg 1966

p4: Gasser, Leutwyler 84,85

p6: JB, Colangelo, Ecker 99,00


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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

19/53

Chiral Logarithms

The main predictions of ChPT:

Relates processes with different numbers of pseudoscalars

Chiral logarithms (and perturbative FSI,. . . )

m2π = 2Bm+

(

2Bm

F

)2 [ 1

32π2log

(2Bm)

µ2+ 2l r3(µ)

]

+ · · ·

M2 = 2Bm

B 6= B0, F 6= F0 (two versus three-flavour)


Johan Bijnens

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η → 3π inChPT

η → π0γγ

Conclusions

20/53

LECs and µ

l r3(µ)

li =32π2

γil ri (µ)− log

M2π

µ2.

Independent of the scale µ.

For 3 and more flavours, some of the γi = 0: Lri (µ)

µ :

mπ, mK : chiral logs vanish

pick larger scale

1 GeV then Lr5(µ) ≈ 0 large Nc arguments????

compromise: µ = mρ = 0.77 GeV


Johan Bijnens

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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

21/53

Expand in what quantities?

Expansion is in momenta and masses

But is not unique: relations between masses(Gell-Mann–Okubo) exists

Express orders in terms of physical masses and quantities(Fπ, FK )?

Express orders in terms of lowest order masses?

E.g. s + t + u = 2m2π + 2m2

K in πK scattering

Relative sizes of order p2, p2, p4,. . . can vary considerably

I prefer physical masses

Thresholds correct

Chiral logs are from physical particles propagating


Johan Bijnens

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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

22/53

LECs

Some combinations of order p6 LECs are known as well:curvature of the scalar and vector formfactor, two morecombinations from ππ scattering (implicit in b5 and b6)

General observation:

Obtainable from kinematical dependences: known

Only via quark-mass dependence: poorely known


Johan Bijnens

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Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

23/53

C ri

Most analysis use (i.e. almost all of mine):C ri from (single) resonance approximation

π

πρ,S

→ q2

π

π|q2| << m2

ρ,m2

S

=⇒C ri

Motivated by large Nc : large effort goes in this

Ananthanarayan, JB, Cirigliano, Donoghue, Ecker, Gamiz, Golterman, Kaiser,

Knecht, Peris, Pich, Prades, Portoles, de Rafael,. . .


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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

24/53

C ri

LV = −1

4〈VµνV

µν〉+ 1

2m2

V 〈VµVµ〉 − fV

2√2〈Vµν f

µν

+ 〉

− igV

2√2〈Vµν [u

µ, uν ]〉+ fχ〈Vµ[uµ, χ−]〉

LA = −1

4〈AµνA

µν〉+ 1

2m2

A〈AµAµ〉 − fA

2√2〈Aµν f

µν

− 〉

LS =1

2〈∇µS∇µS −M2

SS2〉+ cd〈Suµuµ〉+ cm〈Sχ+〉

Lη′ =1

2∂µP1∂

µP1 −1

2M2

η′P21 + i dmP1〈χ−〉 .

fV = 0.20, fχ = −0.025, gV = 0.09, cm = 42 MeV, cd = 32 MeV,

dm = 20 MeV, mV = mρ = 0.77 GeV,mA = ma1 = 1.23 GeV,

mS = 0.98 GeV, mP1 = 0.958 GeV

fV , gV , fχ, fA: experiment

cm and cd from resonance saturation at O(p4)


Johan Bijnens

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ChPT

η → 3π inChPT

η → π0γγ

Conclusions

25/53

C ri

Problems:

Weakest point in the numerics

However not all results presented depend on this

Unknown so far: C ri in the masses/decay constants and

how these effects correlate into the rest

No µ dependence: obviously only estimate

What we do/did about it:

Vary resonance estimate by factor of two

Vary the scale µ at which it applies: 600-900 MeV

Check the estimates for the measured ones

Again: kinematic can be had, quark-mass dependencedifficult


Johan Bijnens

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Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

26/53

Lri and C ri

Full NNLO fits of the Lri

Amoros,JB,Talavera, 2000, 2001 (fit 10)simple C r

i

JB, Jemos, 2011

simple C ri

JB,Ecker,2014, to be published

Continuum fit with more input for C ri

Numerics presented for η → 3π is with fit 10JB,Ghorbani, 2007

Would expect no major changes from that


Johan Bijnens

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Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

27/53

Lowest order

ChPT:Cronin 67: A(s, t, u) =B0(mu −md )

3√3F 2

π

{

1 +3(s − s0)

m2η −m2

π

}

with Q2 ≡ m2s−m2

m2d−m2

uor R ≡ ms−m

md−mum = 1

2 (mu +md )

A(s, t, u) =1

Q2

m2K

m2π

(m2π −m2

K )M(s, t, u)

3√3F 2

π

,

A(s, t, u) =

√3

4RM(s, t, u)

LO: M(s, t, u) =3s − 4m2

π

m2η −m2

π

M(s, t, u) =1

F 2π

(

4

3m2

π − s

)


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

28/53

η → 3π: p2 and p4

Γ (η → 3π) ∝ |A|2 ∝ Q−4 allows a PRECISE measurement

Q2 form lowest order mass relation: Q ≈ 24

=⇒ Γ(η → π+π−π0)LO ≈ 66 eV

m2K+ −m2

K 0 ∼ Q−2 at NNLO: Q = 20.0± 1.5

=⇒ Γ(η → π+π−π0)LO ≈ 140 eV

At order p4 Gasser-Leutwyler 1985:

∫

dLIPS |A2 + A4|2∫

dLIPS |A2|2= 2.4 ,

(LIPS=Lorentz invariant phase-space)

Major source: large S-wave final state rescattering

Experiment: 300± 12 eV (PDG 2012/13)


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

29/53

η → 3π: LO, NLO, NNLO, NNNLO,. . .

IN Gasser,Leutwyler, 1985 (√2.4 = 1.55):

about half: ππ-rescatteringother half: everything else

ππ-rescattering important Roiesnel, Truong, 1981

Dispersive approach (next talk): resum all ππ

assume rescattering + rest separable:

LO

NLO

NNLO

· · ·

NLO

NNLO

· · ·· · ·

NNLO

· · ·· · ·· · ·

· · ·· · ·· · ·· · ·

→ ππ-rescattering

dispersive does this all the way

↑ Other effects


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

30/53

Why look at it this way?

LO

NLO

NNLO

· · ·

NLO

NNLO

· · ·· · ·

NNLO

· · ·· · ·· · ·

· · ·· · ·· · ·· · ·

→ ππ-rescattering

dispersive does this all the way

↑ Other effects

δπ = 0.3, δO = 0.3

LO = 1

NLO = δπ + δO = 0.6

NNLO = δ2π + δπδO + δ2O = 0.27

Squared: 1 → 2.6 → 3.5

Underlying other is: 1 + 0.3 + 0.09

Goal: remove dispersive from ChPT, then add again viadispersion relations (but now all boxes)

Problem: Separation is not trivial


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

31/53

Two Loop Calculation: why?

In Kℓ4 dispersive gave about half of p6 in amplitude

Same order in ChPT as masses for consistency check onmu/md

Check size of 3 pion dispersive part

At order p4 unitarity about half of correction

Technology exists:

Two-loops: Amoros,JB,Dhonte,Talavera,. . .

Dealing with the mixing π0-η: Amoros,JB,Talavera 01

Done: JB, Ghorbani, arXiv:0709.0230

Dealing with the mixing π0-η: extended to η → 3π


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

32/53

Diagrams

(a) (b) (c) (d)

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Include mixing, renormalize, pull out factor√3

4R , . . .

Two independent calculations (comparison lots of work)

You have to carefully define which LO (M or M)

You have to carefully define which NLO

Integrals only in numerical form: (g) is the hardest one


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

33/53

η → 3π: M(s, t = u)

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=t)

s [GeV2]

Li fit 10 and Ci

Re p2

Re p2+p

4

Re p2+p

4+p

6

Im p4

Im p4+p

6

Along t = u

-2

0

2

4

6

8

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=t)

s [GeV2]

Li fit 10 and Ci

Re p6 pure loops

Re p6 Li

r

Re p6 Ci

r

sum p6

Along t = u parts


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

34/53

η → 3π: M(s, t = u)

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

M(s

,t,u

=t)

s [GeV2]

Li = Ci = 0

Re p2

Re p2+p

4

Re p2+p

4+p

6

Im p4

Im p6

Along t = u

Lri = C ri = 0

-10

0

10

20

30

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s

,t,u

=t)

s [GeV2]

Li fit 10 and Ci

Re p2+p

4

µ= 0.6 GeV

µ= 0.9 GeV

Re p2+p

4+p

6

µ= 0.6 GeV

µ= 0.9 GeV

Along t = u: µ dependenceI.e. where C r

i (µ) estimated


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

35/53

Neutral decay

A20 α

LO 1090 0.000NLO 2810 0.013

NLO (Lri = 0) 2100 0.016NNLO 4790 0.013NNLOq 4790 0.014

NNLO (C ri = 0) 4140 0.011

NNLO (Lri = C ri = 0) 2220 0.016

dispersive (KWW) — −(0.007—0.014)tree dispersive — −0.0065

absolute dispersive — −0.007Borasoy — −0.031

error 160 0.032

experiment: α = −0.032 with small error

NNLO ChPT gets a00 in ππ correct


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

36/53

Theory: charged

A20 a b d f

LO 120 −1.039 0.270 0.000 0.000NLO 314 −1.371 0.452 0.053 0.027

NLO (Lri = 0) 235 −1.263 0.407 0.050 0.015NNLO 538 −1.271 0.394 0.055 0.025

NNLOp (y from T 0) 574 −1.229 0.366 0.052 0.023NNLOq (incl (x , y)4) 535 −1.257 0.397 0.076 0.004NNLO (µ = 0.6 GeV) 543 −1.300 0.415 0.055 0.024NNLO (µ = 0.9 GeV) 548 −1.241 0.374 0.054 0.025

NNLO (C ri = 0) 465 −1.297 0.404 0.058 0.032

NNLO (Lri = C ri = 0) 251 −1.241 0.424 0.050 0.007

dispersive (KWW) — −1.33 0.26 0.10 —tree dispersive — −1.10 0.33 0.001 —

absolute dispersive — −1.21 0.33 0.04 —error 18 0.075 0.102 0.057 0.160

KLOE 08 −1.090 0.124 0.057 0.14

NLO to NNLO changes, but no large ones

Error: ∆|M(s, t, u)|2 = |M(6) M(s, t, u)|


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

37/53

Experiment vs Theory: charged

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.5

0 0.5

1 0

0.5

1

1.5

2

2.5

3

KLOE 08LO

NLONNLO

yx


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

38/53

Experiment : relative

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.5

0 0.5

1-0.02

0

0.02

0.04

0.06

0.08

0.1

KLOE 08KLOE prel/KLOE 08WASA prel/KLOE 08

yx


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

39/53

Experiment vs Theory: relative

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

KLOE 08 LO/KLOE 08

NLO/KLOE 08NNLO/KLOE 08

yx


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

40/53

r and decay rates

sin ǫ =√3

4R +O(ǫ2)

Γ(η → π+π−π0) = sin2 ǫ · 0.572 MeV LO ,

sin2 ǫ · 1.59 MeV NLO ,

sin2 ǫ · 2.68 MeV NNLO ,

sin2 ǫ · 2.33 MeV NNLOC ri = 0 ,

Γ(η → π0π0π0) = sin2 ǫ · 0.884 MeV LO ,

sin2 ǫ · 2.31 MeV NLO ,

sin2 ǫ · 3.94 MeV NNLO ,

sin2 ǫ · 3.40 MeV NNLOC ri = 0 .


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

41/53

r and decay rates

r ≡ Γ(η → π0π0π0)

Γ(η → π+π−π0)

rLO = 1.54

rNLO = 1.46

rNNLO = 1.47

rNNLOC ri=0 = 1.46

PDG 2013

r = 1.48 ± 0.05 our average .

r = 1.426 ± 0.026 our fit ,

Reasonable agreement


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

LO

LO and NLO

NNLO

η → π0γγ

Conclusions

42/53

R and Q from eta → 3π

LO NLO NNLO NNLO (C ri = 0)

R (η) 18.9 31.5 40.9 38.2R (Dashen) 44 44 37 —

R (Dashen-violation) 36 37 32 —Q (η) 16.5 21.3 24.3 23.4

Q (Dashen) 24 24 22 —Q (Dashen-violation) 22 22 20 —

LO from R =m2

K 0 +m2K+ − 2m2

π0

2(

m2K 0 −m2

K+

) (QCD part only)

NLO and NNLO from masses: Amoros, JB, Talavera 2001

Q2 =ms + m

2mR = 14.4R

(ms/m = 27.8 used for η → 3π)


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

43/53

η → π0γγ

Not a full review, pointing out certain things

Long ago: γγ → π0π0 only loop at order p4

JB, Cornet, 1988, Donoghue, Holstein, Lin, 1988

Same argument goes for η → π0γγ

Neutral particles and gauge invariance: A ∝ FµνFµν

But chiral invariance requires U and U† and neutral partscommute with Q

Ametller, JB, Bramon, Cornet, Phys. Lett. B 276 (1992) 185

π-loop suppressed by mu −md

K -loop very small: the loop integral sγγ small is 124m2

K

total at p4:π-loops: 0.00084 eVK -loops: 0.00245 eVsum: 0.00389 eV

Note that this decay does not break isospin


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

44/53

Experiment

Large background from η → 3π0 with missed photons

104BR Γ(η → π0γγ) [eV]

GAMS 2000 7.1 ± 1.4 0.92 ± 0.18CBALL (BNL) 2.2 ± 0.5 0.29 ± 0.07

CBALL (MAMI-B) 2.21 ± 0.24 ± 0.47 0.29 ± 0.07KLOE 0.84 ± 0.27 ± 0.14 0.11 ± 0.04

Results on the spectrum exists: talk by Unverzagt


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

45/53

General structure

The amplitude has two structures: A and B

η(P) → π0(p)γ(q1)γ(q2):

M = A[(ǫ1 · ǫ2)(q1 · q2)− (q2 · ǫ1)(q1 · ǫ2)]+ B [−(ǫ1 · ǫ2)(P · q1)(P · q2)− (P · ǫ1)(P · ǫ2)(q1 · q2)

+(P · ǫ1)(q1 · ǫ2)(P · q1) + (q2 · ǫ1)(P · ǫ2)(P · q2)]

At p4 only A nonzero

At order p6 pure pion diagrams suppressed by mu −md

There exists a partial two-loop calculation(i.e. difficult 2-loop part not done) Jetter hep-ph/9508407

Finds the expected small corrections from the loops

The decay rate is quite sensitive to interferences


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

46/53

Moreloops

First pure pion loop contribution that has no isospinbreaking is with a double WZW term (i.e. p8)

WZW2:π-loops: 0.00005 eVK -loops: 0.0022 eVsum: 0.0025 eV

Same size as p4 loops

Better look for other contributions

But also a way to check a nontrivial process withresonance saturation


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

47/53

What else should we look at?

Higher order ChPT LECs dominated by resonancesGasser,Leutwyler 84, Ecker, Gasser, Pich, de Rafael 1989,

Donoghue, Ramirez, Valencia 1989

Possible major contributions (isospin conserving):

ω, ρ

η π0

a0, a2

η π0

Determine the couplings from experiment/other theory(I will use 1992 numbers)

Keep full propagators and/or restrict to p6 part only

Models might have more contributions


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

48/53

Fix couplings

VPγ vertex from ω → πγ

The product of a0π0η and a0γγ from γγ → π0η

Same with a0 → a2 (and choose an off-shell a2 propagator)

Note that much of these data have changed since 1992(but reasonably compatible)


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

49/53

Results

Included Γ(η → π0γγ) [eV]

ρ0, ω p6 0.18ρ0, ω 0.31 VMD

ρ0, ω, a0, a2 p6 0.16-0.20 interference sign not knownρ0, ω, π,K 0.42 parts with fixed signs+a0, a2 0.35-0.50 signs again

+uncertainty 0.22-0.62 large Nc , p6 and other loops

We concluded Γ(η → π0γγ) = 0.42 ± 0.20 eV


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

50/53

Other models

Chiral quark model to p6: about 0.15 eVJB,Dawson,Valencia,1991 (same counterterm as γγ → π0π0)

ENJL and variations is very popularsome are: Bel’kov,Lanyov, Scherer, 1995, Bellucci, Bruno 1995, JB,

Fayyazzudin, Prades, 1996, Nemoto, Oka, Takizawa 96, Radzhabov, Volkov

2006

Numbers last reference; models differ in η-η′-mixing

model 1 model 2

vector mesons 0.17 0.20scalar meson 0.03 0.12vector+scalar 0.10 0.12

box 0.28 0.35box+vector 0.78 0.95

total 0.53 0.45

Interferences important


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

51/53

Other approaches/models

Use the chiral unitary approach Oset, Pelaez, Roca, 2003,2008

Generates the a0 dynamically and thus fixes its signs (+)

Also included newer V → Pγ experiments

Γ(η → π0γγ) = 0.33± 0.08 eV

I think a2 is still missing

There are also purely dispersive approaches


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

p4

Experiment

Loops at p6, p8

All else

Distributions

Conclusions

52/53

Distributions

Oset, Pelaez, Roca,,Phys. Rev. D 77 (2008) 073001 [0801.2633]

0 0.1 0.2 0.3 0.4Mγγ [GeV]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

dΓ/d

Mγγ

[eV

/GeV

]

χ-loops resummed

VMD

VMD +χ

+VMD loops

+WZW χ

Conclusions: Model testing good with spectraMπ0γ less sensitive but can help


Johan Bijnens

Older reviews

Modelindependent

ChPT

η → 3π inChPT

η → π0γγ

Conclusions

53/53

Conclusions

Overview of ChPT for η → 3π

ChPT at two loops has sizable but not unusual correctionsDoes not reproduce at present the Dalitz plotIf married with dispersive, indications that the ’other’series might work well

Overview of η → π0γγ

Chiral loops are small (if not resummed to generateresonances)VMD is main partOthers do help: note scalars are important and their signs

η π γγ Chiral Perturbation Theory for η 3π and η π γγJohan Bijnens Older reviews Model...

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Transcript of η π γγ Chiral Perturbation Theory for η 3π and η π γγJohan Bijnens Older reviews Model...