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Anomalous decays of η′ and η into four pions Feng-Kun Guo,1, Bastian Kubis,1, and Andreas Wirzba2, ∗ † ‡ 1Helmholtz-Institut fu¨r Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universita¨t Bonn, D-53115 Bonn, Germany 2Institute for Advanced Simulation and Ju¨lich Center for Hadron Physics, Institut fu¨r Kernphysik, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany (Dated: January 16, 2012) We calculate the branching ratios of the yet unmeasured η′ decays into four pions, based on a combinationofchiralperturbationtheoryandvector-meson dominance. Thedecaysη′ →2(π+π−) and η′ → π+π−2π0 are P-wave dominated and can largely be thought to proceed via two ρ reso- nances;wepredictbranchingfractionsof(1.0±0.3)×10−4 and(2.4±0.7)×10−4,respectively,not muchlowerthanthecurrentexperimentalupperlimits. Thedecaysη′ →4π0 andη→4π0,incon- 2 trast, are D-wavedriven as long as conservation of CP symmetry is assumed, and are significantly 1 further suppressed; any experimental evidence for the decay η → 4π0 could almost certainly be 0 interpretedasasignalofCP violation. WealsocalculatetheCP-violatingamplitudesforη′ →4π0 2 and η→4π0 induced bytheQCD θ-term. n a PACSnumbers: 12.39.Fe,12.40.Vv,13.25.Jx J Keywords: ChiralLagrangians,Vector-mesondominance,Decays ofother mesons 3 1 I. INTRODUCTION forbidden, nor required to proceed via electromagnetic ] interactions. The reaction η 4π, in contrast, is es- h → sentially suppressed by tiny phase space: only the decay p Processesinlow-energyQCDthatinvolveanoddnum- ep- btoenrs)o,fw(hpiscehudaor-e),Gtohledrsetfoonree,boofsoodnds (inatnrdinspicospsiabrliyty,pharoe- iMntηo−42π(0Misπk±in+eMmaπt0i)ca=lly−a1l.l2oMweedV()M.Fηu−rt4hMerπm0o=re,7.t9hMefeaVct, thatanomalousamplitudesalwaysinvolvethetotallyan- h thought to be governed by the Wess–Zumino–Witten tisymmetrictensorǫ canbeusedtoshowthatnotwo [ (WZW) term [1] via chiral anomalies. While the so- µναβ pseudoscalars are allowed to be in a relative S wave: as- 2 called triangle anomaly is well tested in processes such suming they were, this would reduce the five-point func- v as π0, η γγ, and the box anomaly contributes e.g. to → tion PPPPP effectively to a four-point function SPPP 9 γπ ππ and η ππγ, the pentagon anomaly remains 4 mor→e elusive; the→simplest possible process that is usu- (whereS standsforascalarandP forapseudoscalar),in 9 ally cited is K+K π+π π0, which however has not which there are no four independent vectors left to con- .5 beenexperimentall−yt→estedy−et,andislikelytobesubject tηract tπh+eπǫ t2eπn0socranwitthhe.reTfohreedbeecaeyxspeηc′te→d 2to(πb+eπP−-)waanvde 1 to large corrections to the chiral-limit amplitude that is ′ → − dominated. As furthermore Bose symmetry forbids two 1 dictated by the WZW term. 1 neutral pions to be in an odd partial wave, η′ 4π0 1 A different set of processes involving five light pseu- and η 4π0 even require all π0 to be at least→in rel- : doscalarsisthefour-piondecaysofη andη′. Experimen- ative D→waves [4]. This, combined with the tiny phase v tal information about these is scarce: only upper limits spaceavailable,leadstothenotionofη 4π0 beingCP i X on branching ratios exist [2]; however, this may change forbidden [2, 5, 6], although strictly sp→eaking it is only r in the near future for at least some of the possible final S-wave CP forbidden. a states with the advent of high-statistics η experiments ′ suchasBES-III,WASA-at-COSY,ELSA,CB-at-MAMI- Theoutlineofthearticleisasfollows. Webeginbydis- C, CLAS at Jefferson Lab, etc. We are only aware of cussingthetwodecaychannelswithchargedpionsinthe one previous theoreticalcalculationof these decays,per- finalstate, η′ 2(π+π−) and η′ π+π−2π0, in Sec. II. → → formed in the framework of a quark model [3], whose There, we calculate the corresponding decay amplitudes partial width predictions, however, have in the mean- at leading nonvanishing order in the chiral expansion, time beenruledoutbythe experimentalupper limits, at saturate the appearing low-energy constants by vector- least for the channel η 2(π+π ). meson contributions, and calculate the corresponding ′ − → branching ratios. In Sec. III, we then construct a CP- In principle, the decays η 4π, in contradistinction ′ → conserving (D-wave) decay mechanism for η, η 4π0 tomanyotherη′decaychannels,seemnotterriblyforbid- ′ → and determine the resulting branching fractions, before denbyapproximatesymmetries: theyareneitherisospin discussing the CP-violating (S-wave) η, η 4π0 decay ′ → as induced by the QCD θ-term in Sec. IV. Finally, we summarize and conclude. The Appendices contain tech- nical details on four-particle phase space integration as ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] well as on a (suppressed) tensor-meson mechanism for ‡Electronicaddress: [email protected] η, η′ 4π0. → 2 II. η′ →2(π+π−) AND η′ →π+π−2π0 π+ π+ π+ η′ K η′ π− A. Chiral perturbation theory + ¯ π K Wewishtocalculatetheleading(nontrivial)chiralcon- − − − tribution to the anomalous decays π π π η π+(p )π (p )π+(p )π (p ) , FIG. 1: Feynman diagrams contributing to η′ → 2(π+π−) ′ → 1 − 2 3 − 4 (and similarly to η′ →π+π−2π0) at O(p6). The thick dot in η′ π+(p1)π0(p2)π−(p3)π0(p4) . (1) theright diagram denotesa vertex from L(6) . → odd The amplitudes can be written in terms of the invariant variables s = (p +p )2, i, j = 1,...,4, which are that only contribute for the singlet field η : ij i j 0 subject to the constraint (6) =iCWǫ χ uµuνuαuβ Lodd 1 µναβh − i s12+s13+s14+s23+s24+s34 =Mη2′ +8Mπ2 (2) iC˜1Wǫ χ uµuνuαuβ µναβ − 3 h −ih i (in the isospin limit of equal pion masses). The five- +CWǫ hγµ[u ,uνuαuβ] meson vertices of the WZW term can be deduced from 12 µναβh γ i the Lagrangian C˜W 12ǫ hγµ[u ,uνuα] uβ +... , (6) µναβ γ − 3 h ih i N ǫ LWP5ZW = 24c0πµ2νFαβπ5 (cid:10)ϕ∂µϕ∂νϕ∂αϕ∂βϕ(cid:11)+..., (3) w(niethgletchteinguseuxatlercnhairlacluvrrieelnbtesi)n, uuµ==expi((uiϕ†∂/µ2uF−),uh∂µu†=) π µν u + u with X = ∂ X + [Γ ,X] and Γ = whereNcisthenumberofcolorsandwillbetakentobe3 ∇1(µu ν∂ u∇+νu∂µ u ) (n∇egµlecting aµgain extµernal currenµts). inthis paper, F =92.2MeVis the piondecayconstant, 2 † µ µ † π Furthermore, we use χ = u χu uχ u, where χ = and ... denotesthetraceinflavorspace. Forsimplicity, † † † h i 2Bdiag(m ,m ,m )+.−.. contains−the quark mass ma- we refrain from spelling out the WZW term in its full, u d s trix and B is related to the quark condensate according chirally invariant form. Furthermore, to B = q¯q /F2. The decay amplitudes at (p6) take −h i π O η0 + η8 + π0 π+ K+ the compact forms √3 √6 √2 √ϕ2 = Kπ−− √η03 +K√¯η860− √π02 √η03K−02√η68 . A(η0/8 =→Nπ3+√cǫπ3µ−νFαππ5β+pπµ1−p)ν2p=α3p−β4A(cid:2)F(η00//88(s→12)π++πF00π/8−(πs034)) (4) (s ) (s ) , We assume a simple, one-angle ηη′ mixing scheme, −F0/8 14 −F0/8 23 (cid:3) (s)= 16√2 CWr(µ) C˜Wr(µ) s , η =cosθ η sinθ η , F0 − (cid:16) 12 − 12 (cid:17) P 8 P 0 | i | i− | i 1 η =sinθ η +cosθ η , (5) (s)= s 4M2 J¯ (s) | ′i P| 8i P| 0i F8 8π2F2(cid:26) − K KK π (cid:0) (cid:1) and use the standard mixing angle θP =arcsin( 1/3) s 2logMK + 1 16CWr(µ)s , 19.5◦. As we are going to present what in som−e sens≈e − 16π2(cid:16) µ 3(cid:17)(cid:27)− 12 − corresponds to a leading-order calculation of the decay 1 4M2 amplitudes,weregardthemoreelaboratetwo-anglemix- J¯ (s)= (1 σ arccotσ ), σ = K 1, KK 8π2 − K K K r s − ing schemes [7] as beyond the scope of this study; we (7) expect the errormade thereby to be coveredby our gen- erous final uncertainty estimate. with the scale-dependent renormalized low-energy con- The flavor structure of Eq. (3) is such that there are stantsCWr(µ)andC˜Wr(µ). Therearenoloopcontribu- 12 12 nodirectcontributionstoη, η 4π,andthe decayam- tions to the η amplitudes at this order, since at (p4) ′ 0 → O plitudes vanish at leading order (in the anomalous sec- the anomalous five-pseudoscalar term (3) (the left ver- tor) (p4). Nonvanishing contributions only occur at tex of the loop diagram in Fig. 1) contributes only to O (p6),wheretheamplitudesaregivenbysumsof(kaon) the octet case. Equation (7) is scale-independent with lOoops and countertermcontributions from the (p6) La- the β function for CWr(µ) obtained in Ref. [8], if we O 12 grangian of odd intrinsic parity [8], see Fig. 1. Only demand C˜W to havethe same infinite partand scale de- 12 two different structures ( CW, CW) remain when ex- pendence as CW. A numerical estimate for the finite ∝ 1 12 12 ternal currents are switched off. Ref. [8] only considers part CWr(M ) will be obtained by resonance saturation 12 ρ the Goldstone boson octet; we add terms C˜W, C˜W through vector-mesoncontributions. ∝ 1 12 3 B. Resonance saturation from hidden local we find symmetry N (c c ) (4) = c 1− 2 ǫ ϕ∂µϕ∂νϕ∂αϕ∂βϕ , (12) Resonance saturation for the (p6) chiral Lagrangian LP5,V 128π2Fπ5 µναβ(cid:10) (cid:11) O of odd intrinsic parity has been studied in great gener- which exactly cancels the second term inside the square ality recently in Ref. [9]. Here however we opt for the brackets in Eq. (8). simpler, but on the other hand more predictive hidden- If we extend the equation of motion Eq. (9) to next- local-symmetryscheme[10–13],whichhastheadditional to-leading order in the derivative expansion, advantage of having been tested phenomenologically in great detail [14]. 1 ∂2 In the framework of hidden local symmetry (HLS), Vµ = 8igF2 (cid:18)1− M2(cid:19)[∂µϕ,ϕ] , (13) there are four additional terms involving vector-meson π V fields,withcoefficientsci(i=1,...,4),inadditiontothe where MV is the vector-meson mass, we can derive the WZWactionforanomalousprocesses[10,12];asalready vector-meson contribution to the five-meson vertices at noted in Ref. [15], only three independent combinations (p6). Inserting Eq. (13) into Eq. (10), we find of these contribute to low-energy amplitudes at (p6). O O HLS amplitudes for any given anomalous process con- N (c c +c ) (6) = c 1− 2 3 ǫ tain two kinds of contributions: contact terms and reso- LP5,V 128π2F5M2 µναβ π V nanceexchangeterms. The contacttermshavethe same ∂λ∂µϕ ∂ ϕ,∂νϕ∂αϕ∂βϕ form as those derived from the WZW action, but with a × λ modifiedcoefficient(seebelow); thegauge-invariantcon- (cid:10) 2∂2ϕ∂(cid:2)µϕ∂νϕ∂αϕ∂βϕ .(cid:3) (14) − structionofthe HLSLagrangiandensity guaranteesthat (cid:11) the additional, c -dependent contributions will be can- The first term is exactly of the form of the Lagrangian i celed by vector-meson exchange in the low-energy limit. term CW in Eq. (6). For the second term, we use ∝ 12 In the following, we again for simplicity reasons refrain the equation of motion for the Goldstone bosons, which, from properly defining all the HLS Lagrangian terms in neglectinghigherordersinthefields,reads(comparee.g. their chirally invariant forms, but only quote the terms Ref. [8]) relevant for vertices of five pseudoscalars; the full La- grangians can be retrieved e.g. from Refs. [12, 13]. ∂2ϕ=−21{ϕ,χ}+ 13hϕχi , (15) Thecontacttermsforfive-pseudoscalarverticescanbe so we also identify a vector-meson contribution to CWr read off from the Lagrangian 1 and C˜Wr. Our results read altogether 1 N ǫ 15 HLS = c µναβ 1 (c c ) ϕ∂µϕ∂νϕ∂αϕ∂βϕ . LP5 240π2F5 (cid:20) − 8 1− 2 (cid:21) CWr(M )=C˜Wr(M )= 2CWr(M ) π (cid:10) (cid:11) 1 V 1 V − 12 V (8) N (c c +c ) c 1 2 3 The low-energy limit of the vector-meson-exchange con- = − , (16) 128π2M2 tribution can be obtained by integrating out the heavy V fields: substituting the leading-order equation of motion where we have indicated the conventional assumption of of the vector-mesonfields the resonance-saturation hypothesis to be valid roughly 1 at the resonance scale, µ = M (which in the following V V = [∂ ϕ,ϕ], (9) µ 8igF2 µ wewillidentifywiththemassoftheρ,Mρ =775.5MeV). π The numerical values of the HLS coupling constants are where g is the universalvector-mesoncoupling constant, oftentakentobegivenbyc c c 1[10],fairlycon- 1 2 3 into the HLS Lagrangians [13, 14] sistent with more elaborate−phe≈nome≈nological fits that yield c c =1.21, c =0.93 [14]. = Ncc3g2ǫ ∂µVν∂αVβϕ , Inpr1in−cip2le,thiscom3pletesthetasktoprovidethenec- LVVP −8π2F µναβ π (cid:10) (cid:11) essaryinputforanevaluationofthechiralrepresentation = iNc(c1−c2−c3)gǫ Vµ∂νϕ∂αϕ∂βϕ , of the decay amplitude, Eq. (7). We observe, however, LVPPP − 32π2F3 µναβ the following. First, evaluating the slope of (the largely π (cid:10) (cid:11) (10) linear function) (s) in Eq. (7) with this input (using 8 J¯ (0)=1/(96πF2M2)), we find K′ K K where the vector-meson nonet (with ideal mixing) is defined as 8π2(4πFπ)2×F8′(0) Vµ = √12 √ρ0µ2Kρ+−√ωµ2 −√ρ0µK2¯ρ+µ+0√ωµ2 KKφµ∗µ∗+0 , (11) Num=er3i(cca1ll−y,ct2h+eficr3s)t(t42eπrMFmπρ2)is2a−bo(cid:16)u1t+6.27l×og(cMM1−Kρc(cid:17)2+. c3()1/72),  µ∗− µ∗ µ  and the second is 0.1. Hence, at the scale µ = Mρ, the 4 slope is entirely dominated by the vector-meson contri- terms, see Eq. (10). As the phenomenological values of bution, and the kaon loops are negligible. the HLS coupling constants suggest c c c 2c , 1 2 3 3 − − ≪ Second, the maximal value for the kinematical invari- the box anomaly yields the lesser part of the two, and ants in η′ 4π allowed by phase space is √sij the decays are dominated by the triangle-anomalyterm. → ≤ Mη′ 2Mπ 680MeV,thereforereplacingtheρpropaga- − ≈ torbyitsleadinglinearapproximationisnotphenomeno- logically reliable. Even deviations induced by the finite C. Branching ratios width oftheρ,Γ =149.1MeV,willbeclearlyvisible. In ρ thefollowing,wewillthereforeusethefullvector-meson- We calculate the partial widths of the decays η ′ exchangeamplitudesasderivedfromtheHLSformalism, π+π π+π and η π+π0π π0 using → − − ′ − → withtheρ-mesonpropagatorsincludingthewidth,which 1 in additionis expectedto be a verygoodestimate ofthe Γ(η 4π)= (η 4π)2dΦ , (21) ′ ′ 4 higher-order pairwise P-wave interaction of the pions in → 2SMη′ Z |A → | the final state (of course neglecting any crossed-channel where the evaluation of the four-particle phase space effects). They are given by Φ is discussed in detail in Appendix A. S is a sym- 4 metry factor—S = 4 for the 2(π+π ) final state, and 1 − AV(η8 →π+π−π+π−)= √2AV(η0 →π+π−π+π−) S(=η 2 for4πth)e=π+√π2−2(πη0 one4.πN)oatnedthtahtewstiathndtahredrmelaixtiinong 0 8 1 A → A → =−AV(η8→π+π0π−π0)=−√2AV(η0 →π+π0π−π0) TacocoorbdtinaigntobEraqn.c(h5i)n,gwerahtaiovse,Aw(ηe′n→or4mπa)li=zeAt(hηe8 →par4tπia).l = 1N6√cǫ3µπν2αFβπ5pµ1pν2pα3pβ4(cid:26)(c1−c2−c3)(cid:20)DρM(sρ212) wpthaiadrtttihcbsleybduyastitanhggertothouetpa,mlΓwoηs′itd=tph(r0eoc.1fis9te9h±esin0ηg.′0le0a9sm)qMeuaeosVuterde[2m]b.eynNttohtoeef + Mρ2 Mρ2 Mρ2 this width alone,Γη′ =(0.226±0.017±0.014)MeV[16], D (s ) − D (s ) − D (s )(cid:21) our predictions for the branching fractions would be re- ρ 34 ρ 14 ρ 23 M4 M4 duced by more than 10%. Given the observation of ρ ρ +2c (18) Eq. (17), we neglect the kaon-loop contributions alto- 3 (cid:20)D (s )D (s ) − D (s )D (s )(cid:21)(cid:27) ρ 12 ρ 34 ρ 14 ρ 23 gether and evaluate the matrix elements using Eq. (18). Ncǫµναβ pµpνpαpβ (c c ) s12 Inordertoaccountfortrivialisospin-breakingeffectsdue ≃ 16√3π2Fπ5 1 2 3 4(cid:26) 1− 2 (cid:20)Dρ(s12) to phase space corrections, we calculate the branching ratio for η π+π0π π0 using an average pion mass s34 s14 s23 ′ → − + M =(M +M )/2,whileweemploythechargedpion D (s ) − D (s ) − D (s )(cid:21) π π+ π0 ρ 34 ρ 14 ρ 23 mass for the decay into four charged pions. All results Mρ2(s12+s34) Mρ2(s14+s23) are first quoted as a function of the coupling constants +c , (19) 3(cid:20)Dρ(s12)Dρ(s34) − Dρ(s14)Dρ(s23)(cid:21)(cid:27) c1 −c2 and c3, before inserting two sets of values: (i) c c = c = 1, and (ii) c c = 1.21, c = 0.93 [14]. 1 2 3 1 2 3 − − where We refrain from employing the errors given in the fits in Ref. [14]: the uncertainties in the HLS coupling con- D (s)=M2 s iM Γ (s) , ρ ρ − − ρ ρ stants are well below what we estimate to be the overall M s 4M2 3/2 uncertainty of our prediction. The results are Γ (s)= ρ − π Γ (20) ρ √s(cid:18)Mρ2−4Mπ2(cid:19) ρ B η′ →2(π+π−) (cid:0) (cid:1) is the inverse ρ propagator, and we have neglected = 0.15(c c )2+0.47(c c )c +0.37c2 10 4 the width term in the transformation from Eq. (18) to h 1− 2 1− 2 3 3i× − Eq. (19) in order to demonstrate the correct chiral di- = 1.0, 1.1 10−4 , (22) × mension (p6) of the vector-meson contribution explic- η(cid:8)′ π+π(cid:9)−2π0 O B → itly. Expanding the resonance propagators in Eq. (19) (cid:0) (cid:1) = 0.35(c c )2+1.09(c c )c +0.87c2 10 4 and comparing to Eq. (7) easily leads back to the cou- h 1− 2 1− 2 3 3i× − plingconstantestimateforC1W2r foundontheLagrangian = 2.3, 2.5 10−4 . (23) level in Eq. (16). × (cid:8) (cid:9) At this point, we can try to answer the introductory We therefore find that the uncertainties due to the HLS question on which parts of the WZW anomaly action— coupling constants are small. We wish to point out that triangle, box, or pentagon—the decays η 2(π+π ) although ππ P-wave dynamics are usually well approx- ′ − and η π+π 2π0 yield information. As th→e pentagon imated by the ρ resonance, and crossed-channel effects ′ − → anomaly only enters via the kaon-loop contributions, we areexpectedtooccurratheratthe10%level(asinferred have found above that its significance for the decays un- fromstudies ofdecayssuchasω 3π, φ 3π [17]),the → → derinvestigationhereisnegligible;thevector-mesoncon- present study in some sense still amounts to a leading- tributionsarederivedfromthetriangleandbox-anomaly order calculation: SU(3)-breaking effects of the order of 5 Fη/Fπ 1.3 [18] may occur, and in the treatment of A. Pion-loop contribution ≈ the η (η ), we have implicitly evoked the 1/N expan- ′ 0 c sion. We therefore deem a generic uncertainty of 30% realistic, and quote our predictions accordingly as Our decay mechanism for η, η 4π0 is built on the ′ → observation that there is a specific diagrammatic contri- η 2(π+π ) =(1.0 0.3) 10 4 , B ′ → − ± × − bution that we can easily calculate, and that, in partic- (cid:0)η π+π 2π0(cid:1)=(2.4 0.7) 10 4 . (24) ular, comprises the complete leading contribution to the ′ − − B → ± × (cid:0) (cid:1) imaginary part of the decay amplitude. This is given by These are to be compared to the current experimental π+π intermediate states, and hence harks back to the − upper limits [2, 19] results of the previous section. As argued above, it ap- pearsatchiral (p10): η π+π 2π0 ascalculatedin Bexp η′ →2(π+π−) <2.4×10−4 , Eq.(7)to (p6O),followed0/b8y→rescatt−eringπ+π− π0π0, (cid:0)η π+π 2π0(cid:1)<2.6 10 3 , (25) where theOS wave does not contribute, and D an→d higher exp ′ − − B (cid:0) → (cid:1) × partial waves start to appear at (p4) [20]; see Fig. 2 O hence signalsof these decaysoughtto be within reachof for illustration. We calculate this first in the follow- modern high-statistics experiments soon. ing approximation: given the numerical dominance of 0 π 0 0 III. η, η′ →4π0 π π ′ π+ η, η′ f2 π0 η, η AswehavementionedintheIntroduction,theP-wave mechanism described in the previous section, proceed- π0 − f2 ing essentiallyviatwoρ intermediateresonances,cannot π contribute to the 4π0 final states. In fact, we can show 0 0 0 π π π that the D-wave characteristic of η, η 4π0 suppresses ′ these decays to (p10) in chiral power→counting, that is FIG. 2: Left: Pion-loop contribution to η, η′ → 4π0. The O to the level of three loops in the anomalous sector. This black circle denotes an effective local η, η′ → π+π−2π0 cou- is,inparticular,duetotheflavorandisospinstructureof plingatO(p6),theblacksquareaneffectivelocal D-waveππ the anomaly, which does not contain five-meson vertices scattering vertex at O(p4). Right: η, η′ → 4π0 through two including 2π0 at leading order ( (p4)), and to the chiral intermediate f2 mesons. O structure of meson–meson scattering amplitudes, which only allows for S and P waves at tree level ( (p2)). As O a complete three-loop calculation would be a formidable taskandiscertainlybeyondthescopeofourexploratory study, we instead consider the decay mechanisms shown the counterterm contribution in Eq. (7), the amplitudes inFig.2. AsshowninAppendixB,thecontributionfrom F0/8(s) are taken to be linear, F0/8(s) ≈ F0′/8(s)s, ne- two f mesons is negligible in comparison to the pion glecting tiny curvature effects from the kaon loops; and 2 loop. We therefore focus on the pion loop as shown in we approximate ππ rescattering by a phenomenological the left panel of Fig. 2. It represents a decay mechanism D wave, thus improving on the leading chiral represen- that, we believe, ought to capture at least the correct tation, but neglecting G and higher partial waves. We order of magnitude of the corresponding partial width. find 1 N (c c +c ) ǫ (η 4π0)= (η 4π0)= c 1− 2 3 µναβpµpνpαpβ (s ,s ,s ,s ;s ) A 8 → √2A 0 → − 8π √3Fπ5 1 2 3 4nG 12 23 14 34 13 + (s ,s ,s ,s ;s ) (s ,s ,s ,s ;s ) (s ,s ,s ,s ;s ) 12 14 23 34 24 13 23 14 24 12 13 14 23 24 34 G −G −G (s ,s ,s ,s ;s ) (s ,s ,s ,s ;s ) , 12 24 13 34 14 12 13 24 34 23 −G −G o v w x+y 16(t0(s) t2(s)) (v,w,x,y;s)= − − 2 − 2 (s 4M2)2J¯ (s) G M2 3(s 4M2)2 (cid:26) − π ππ ρ − π 1 M 1 s2 8 2 s2 10sM2+30M4 L+ log π + sM2+15M4 , − (cid:0) − π π(cid:1)(cid:16) 16π2 µ (cid:17) 16π2(cid:18)15 − 3 π π(cid:19)(cid:27) 6 1 σ 1+σ 4M2 µd 4 1 1 J¯ (s)= 1 log iπ , σ = 1 π , L= − + (γ 1 log4π) . ππ 8π2(cid:26) − 2(cid:16) 1 σ − (cid:17)(cid:27) r − s 16π2(cid:26)d 4 2 E − − (cid:27) − − (26) tI(s) is the partial wave of angular momentum ℓ=2 for DespiteD-wavescatteringnearthresholdnotbeingdom- 2 the appropriate isospin quantum number I; the expres- inated by the f (1270),1 we still use Eq. (28) to invoke 2 sion 16tI(s)(s 4M2) 2 = aI + (s 4M2), with the the f . This is because at somewhathigher energies,the D-waves2catter−inglenπg−thaI,is2theOrefor−efinitπe atthresh- I = 02ππ D wave dominates over the I = 2 component 2 old. NotefurthermorethattI = (p4)inchiralcounting, andwillbewellapproximatedbythef (1270)resonance. such that the chiral order o2f EqO. (26) is indeed (p10). One may wonder whether, in particu2lar, for η 4π0, O → L contains the infinite part of the divergent loop dia- whichstaysclosetoππ thresholdthroughouttheallowed gramin the usual way, using dimensional regularization. phasespace,thisapproximationmaynotleadtosizeable Of course, this individual loop contribution is both di- errors. We havecheckedfor the numericalresultsfor the vergent and scale-dependent: only the imaginary part is branching fraction discussed below that, employing the complete (to this order) and in that sense well-defined fullOmnèsfunctionaccordingtoEq.(27)withthephase and finite. We display the full expressionhere as we will parameterizationprovidedinRef.[23],the branchingra- use the scale dependence as a rough independent consis- tiochangesbyabout10%,wellbelowtheaccuracywecan tency check below. aim for here. On the other hand, within the η 4π0 ′ → decay, we stay sufficiently far below the resonance en- Without the knowledge of counterterms of an order as high as (p10), one cannot make a quantitative pre- ergy that the phase of the D wave can still be neglected. O WiththecorrespondencebetweenEqs.(28)and(29),we diction using the loop amplitude derived in the above. concludethatthef (1270)contributiontotheamplitude Hence, we have to resort to a certain phenomenological 2 can be estimated as representation. The imaginarypart of Eq.(26), which is complete at (p10) as mentioned, is used to establish a 1 O (η 4π0)= (η 4π0) connection to a one-f2 exchange in the s channel. Note Af2 8 → √2Af2 0 → that the f (1270) exchange dominates the available ππ 2 N (c c +c ) ǫ scattering phase shifts in the I = 0, ℓ = 2 channel, see = c 1− 2 3 µναβpµpνpαpβ − 24π2 √3F5 1 2 3 4 e.g. Ref. [21]. π terWinegwciollnptrriobcueteidontoaesstfiomllaotwest.heNfuellglDec-wtinavgeaπgπairnescaanty- ×nGf2(s12,s23,s14,s34;s13,s24) (s ,s ,s ,s ;s ,s ) crossed-channel effects, rescattering of two pions can be −Gf2 13 23 14 24 12 34 summed by the Omnès factor, (s ,s ,s ,s ;s ,s ) , −Gf2 12 24 13 34 14 23 o v w x+y ΩI(s)=exp s ∞ δℓI(z)dz , (27) Gf2(v,w,x,y;s,t)= − M−ρ2 ℓ (cid:26)π Z4Mπ2 z(z−s−iǫ)(cid:27) Mf22 + Mf22 , (30) ×(cid:20)M2 s M2 t(cid:21) f2 − f2 − where δI is the ππ scattering phase shifts in the channel ℓ withisospinI andangularmomentumℓ. Nearthreshold, neglecting for simplicity the width of the f2, which is its imaginary part can be approximated as justified in the kinematic regime accessible in η′ 4π0. → Note that, due to the special symmetry of the amplitude, Eq. (30) can be rewritten identically by employing ImΩIℓ(s)≈δℓI(s) 1+O(σ2) ≈σtIℓ(s) 1+O(σ2) a “twice-subtracted” version of the resonance term, i.e. (cid:8) (cid:9) (cid:8) ((cid:9)28) replacing , (neglecting the shift fromunity in Ω(4M2), whichis jus- Gf2 →Gf′′2 π tified in the D wave for our intended accuracy), while in v w x+y the approximationof a phase being dominated by a nar- Gf′′2(v,w,x,y;s,t)= −M2−M2 ρ f2 rowresonanceofmassM andwidthΓ,theOmnèsfactor s2 t2 is given by + , (31) ×(cid:20)M2 s M2 t(cid:21) f2 − f2 − M2exp iδI(s) ΩI(s) ℓ , ℓ ≈ (M2 s)2(cid:0)+M2Γ(cid:1)2(s) − pM s 4M2 ℓ+1/2 1 It is dominated by the low-energy constant ℓ¯2 from the O(p4) Γ(s)= − π Γ . (29) Lagrangian [20], or by t-channel vector-meson exchange in the √s M2 4M2 (cid:16) − π(cid:17) spiritofresonancesaturation[22]. 7 0 π the finite counterterm by 0 π η, η′ ρ+ π+ a02−a22 M4 0.22 , (33) 16π × f2 ≈ − − ρ π π0 where we have used a0 = 1.75 10 3M 4, a2 = 0.17 2 × − π− 2 × 10 3M 4[24]. Inotherwords,thescaledependencesug- 0 − π− π geststheorderofmagnitudeofourcountertermestimate using f saturation to be reasonable. FIG. 3: Pion-loop contribution to η, η′ → 4π0 via ρ± in- 2 termediatestates;seethevector-mesondominatedamplitude discussed in Sec. II. The black square denotes an effective local D-waveππ scattering vertex at O(p4). B. Pion-loop contribution improved: including vector propagators which makes the correct chiral dimension of the resonance contribution manifest. As a rough final consistency check, we compare the We have seen in Sec. II on the P-wave dominated, orderofmagnitudeofachiralcounterterminducedbythe (partially) charged four-pion final states that the lead- f exchange, see Eq. (31) in the low-energy limit s, t 2 ing approximationin an expansion of the ρ meson prop- ≪ Mf22 , with the scale running of such a counterterm as agators is not a sufficient description of these decays, necessitated by the logµ dependence in Eq. (26). If we given the available phase space in η decays. With the ′ only retain the scattering lengths in the D-wave partial η π+π 2π0 transitions entering the decay mecha- 0/8 − waves,therelevantparttobecomparedtoEq.(31)(that nism→forη 4π0 asdescribedintheprevioussection, 0/8 does not cancel in the full amplitude) is thisdeficitwo→uldbefullyinheritedinourestimateofthe all-neutralfinalstates. Infact,the imaginarypartofthe d µ (v,w,x,y;s)+ (v,w,x,y;t) corresponding diagram including the full ρ propagators, dµ G G (cid:2) (cid:3) seeFig.3,canevenbecalculatedexactly,usingCutkosky v w x+y s2+t2 rules; however, the resulting expressions are extremely = − − a0 a2 . (32) 3M2 2− 2 8π2 involvedandnotveryilluminating. Itturnsout,though, ρ (cid:0) (cid:1) that the main effects of the not-so-large vector-meson Comparing the numerical prefactors, we find that the mass can be approximated by the following expression scale dependence is suppressed versus the estimate for for the imaginary part: 1 N ǫ Im (η 4π0)= Im (η 4π0)= c µναβpµpνpαpβ (c c c ) Im ρ(s ,s ,s ,s ;s ) A 8 → √2 A 0 → −8π √3Fπ5 1 2 3 4n 1− 2− 3 h G1 12 23 14 34 13 +Im ρ(s ,s ,s ,s ;s ) Im ρ(s ,s ,s ,s ;s ) Im ρ(s ,s ,s ,s ;s ) G1 12 14 23 34 24 − G1 13 23 14 24 12 − G1 13 14 23 24 34 Im ρ(s ,s ,s ,s ;s ) Im ρ(s ,s ,s ,s ;s ) − G1 12 24 13 34 14 − G1 12 13 24 34 23 i +2c Im ρ(s ,s ,s ,s ;s )+Im ρ(s ,s ,s ,s ;s ) Im ρ(s ,s ,s ,s ;s ) 3h G2 12 23 14 34 13 G2 12 14 23 34 24 − G2 13 23 14 24 12 Im ρ(s ,s ,s ,s ;s ) Im ρ(s ,s ,s ,s ;s ) Im ρ(s ,s ,s ,s ;s ) , − G2 13 14 23 24 34 − G2 12 24 13 34 14 − G2 12 13 24 34 23 io M2(v w) M2(x y) σ Im ρ(v,w,x,y;s)= ρ − ρ − t0(s) t2(s) + σ7 , G1 (cid:20) M2 1(v+w) 2 − M2 1(x+y) 2(cid:21) 2 − 2 3π O ρ − 2 ρ − 2 (cid:0) (cid:1) (cid:0) (cid:1) M(cid:0) 4 M2(v w (cid:1)x+y(cid:0)) vy+wx (cid:1) σ Im ρ(v,w,x,y;s)= ρ ρ − − − t0(s) t2(s) + σ7 . (34) G2 M2(cid:0) 1(v+w) 2 M2 1(x+y)(cid:1)2 2 − 2 3π O ρ − 2 ρ − 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) We find, furthermore, that the neglected terms indicated as (σ7) are also suppressed in inverse powers of M , ρ starting at (M 6) comparedto the leading terms of (M 2)Oin the above. Numerically,the indicated higher-order O ρ− O ρ− corrections in σ2 are found to be small, less than about 10% all over phase space. However, the corrections by the remnants of the ρ propagators are large compared to the limit M , given the available phase space and the ρ → ∞ highpowerof these propagatorsinthe denominator. Using the same trick as inthe previoussectionto transformthe 8 imaginary part into an estimate for the whole (resonance-dominated)partial wave via the Omnès function, we arrive at 1 N ǫ (η 4π0)= (η 4π0)= c µναβpµpνpαpβ (c c c ) ρ (s ,s ,s ,s ;s ) A 8 → √2A 0 → −24π2√3Fπ5 1 2 3 4n 1− 2− 3 hGf2,1 12 23 14 34 13 + ρ (s ,s ,s ,s ;s ) ρ (s ,s ,s ,s ;s ) ρ (s ,s ,s ,s ;s ) Gf2,1 12 14 23 34 24 −Gf2,1 13 23 14 24 12 −Gf2,1 13 14 23 24 34 ρ (s ,s ,s ,s ;s ) ρ (s ,s ,s ,s ;s ) −Gf2,1 12 24 13 34 14 −Gf2,1 12 13 24 34 23 i +2c ρ (s ,s ,s ,s ;s )+ ρ (s ,s ,s ,s ;s ) ρ (s ,s ,s ,s ;s ) 3hGf2,2 12 23 14 34 13 Gf2,2 12 14 23 34 24 −Gf2,2 13 23 14 24 12 ρ (s ,s ,s ,s ;s ) ρ (s ,s ,s ,s ;s ) ρ (s ,s ,s ,s ;s ) , −Gf2,2 13 14 23 24 34 −Gf2,2 12 24 13 34 14 −Gf2,2 12 13 24 34 23 io M2(v w) M2(x y) M2 ρ (v,w,x,y;s)= ρ − ρ − f2 , Gf2,1 (cid:20) Mρ2− 12(v+w) 2 − Mρ2− 21(x+y) 2(cid:21)Mf22 −s M(cid:0) 4 M2(v w (cid:1)x+y(cid:0)) vy+wx (cid:1)M2 ρ (v,w,x,y;s)= ρ ρ − − − f2 . (35) Gf2,2 Mρ2(cid:0)− 12(v+w) 2 Mρ2− 21(x+y)(cid:1)2 Mf22 −s (cid:0) (cid:1) (cid:0) (cid:1) Note that this result is far from the one-f dominance Γ =(1.30 0.07)keV [2], we find 2 η ± estimate, with a f coupling constant M 2 as the previous section su2ggested. Expanding ∝Eq. (ρ3−5) simul- η 4π0 B → tleaandeionugsltyeramro(ucnodrretshpeonlidminitgstMoρch→iral∞d,imMenf2sio→n ∞(p,1t0h)e) =(cid:0) 0.4(c1−(cid:1) c2)2+1.1(c1−c2)c3+1.0c23 ×10−30 isnotdominatedbytermsofO(Mρ−2Mf−24),butaOlsocon- =(cid:2)2.4, 2.6 ×10−30 , (cid:3) (37) tainsothertermsofO(Mρ−4Mf−22)andO(Mρ−6). Inother in oth(cid:8)er words(cid:9), the D-wave characteristic of the decay words,Eq. (30) is numerically no reasonable approxima- combined with tiny phase space leads to an enormous tiontoEq.(35)evenforthedecayη 4π0,withitstiny suppressionofthe CP-allowedη 4π0 decay. We again → phase space available. → compare these estimates to the available experimental upper limits [6, 25], η 4π0 <5 10 4 , exp ′ − C. Branching ratios B → × (cid:0)η 4π0(cid:1)<6.9 10 7 ; (38) exp − B → × (cid:0) (cid:1) We calculate the partialwidth usingEq.(21) withthe further improvementsof these experimentalupper limits symmetryfactorS =4!. Noteagainthat (η′ 4π0)= areplanned (see e.g.Ref. [26] for η 4π0). In this case, A → (η 4π0), assuming standard mixing. We employ → 8 our predictions are smaller than those by several orders A → the amplitude as given in Eq. (35) as our “best guess” of magnitude. foranestimateofthebranchingfraction. Withthesame The uncertainties of Eqs. (36) and (37) are hard to numerical input as in Sec. IIC (except using the neutral assess. The generic SU(3) and 1/N error of about 30% c pion mass everywhere), we find assumedinSec.IICisprobablytoosmall,ashere,wedo notevenhaveacompleteleading-ordercalculationatour η′ 4π0 disposal. We therefore rather assume these numbers to B → =(cid:0) 0.4(c (cid:1)c )2+1.6(c c )c +1.7c2 10 8 be the correct orders of magnitude, without quantifying 1− 2 1− 2 3 3 × − the uncertainty of the prediction any further. =(cid:2)3.7, 3.9 10 8 , (cid:3) (36) − × (cid:8) (cid:9) for the two sets of coupling constants ci. Note that the IV. CP-VIOLATING η, η′ →4π0 DECAYS useoftheamplitude(30)leadstoabranchingfractionof the order of 4 10−11, i.e. almost 3 orders of magnitude Given the smallness of the branching fractions pre- × smaller. dicted for η 4π0, η 4π0 via a D-wave dominated, ′ → → We can trivially also calculate the branching fraction CP-conservingdecaymechanismintheprevioussection, forη 4π0,theonlyη 4πdecaysthatiskinematically it is desirable to compare these numbers with possible → → allowed. We again employ the amplitude (35), and note CP-violatingcontributionsthatmay,onthe otherhand, thatmixing accordingto Eq.(5) suggests (η 4π0)= avoid the huge angular-momentum suppression. One A → √2 (η 4π0). Normalizedto the totalwidth ofthe η, such CP-violating mechanism that is expected to affect 8 A → 9 strong-interactionprocessesisinducedbytheso-calledθ- findtheleadingcontributionstothe η decayamplitudes ′ term,anadditionaltermintheQCDLagrangiannecessi- with charged pions in the final state at (p6). Utiliz- O tated for the solution of the U(1) problem. The θ-term ing the framework of hidden local symmetry for vector A violates P and CP symmetry and may induce observ- mesons,weassumethatvector-mesonexchangesaturates able symmetry-violating effects, in particular, in flavor- the (p6) low-energy constants, and find that the (P- O conserving processes. Its effective-Lagrangian treatment wave)decayamplitudeisentirelygovernedbyρinterme- includes a term that can be rewritten as (see Ref. [27] diate states. The dominant contribution is hence given and references therein) by the triangle anomaly via η ρρ (with numerically ′ → subleading boxterms), not by the pentagonanomaly. In Lθ =iθ¯0 Fπ21M2η20(cid:26)hU −U†i−log(cid:16)ddeettUU†(cid:17)(cid:27) , tηh′i→s wπa+y,πt−h2eπb0raarnechpirnegdifcrtaecdtitoonsbefor η′ → 2(π+π−) and iϕ U =u2 =exp F , (39) η′ 2(π+π−) =(1.0 0.3) 10−4 , (cid:16) π(cid:17) B → ± × which, in addition to the well-known η 2π ampli- B (cid:0)η′ →π+π−2π0(cid:1)=(2.4±0.7)×10−4 , (42) → (cid:0) (cid:1) tude [27, 28], alsoinduces a CP-violatingη 4π ampli- → respectively. The former is only a factor of 2 smaller tude, than the current experimental upper limit, so should 1 be testable in the near future with the modern high- (η 4π0)= (η 4π0) ACP 8 → √2ACP 0 → statistics facilities. = (η 4π0)= 1 (η 4π0)= Mη20θ¯0 . muPcrhedmicotrioendsifffiorcutlhte, dasecBayosseinstyomfomuertrnyeurterqaulipreiosntshaemre CP ′ CP A → √2A → −3√3Fπ3 to emerge in relative D-waves (assuming CP conserva- (40) tion), suppressing the amplitudes to (p10) in chiral O power counting. We here do not even obtain the full We will use Mη0 ≈ Mη′ for numerical evaluation. The leading-orderamplitudes,asthesewouldrequireathree- fact that this amplitude is a constant makes the phase loop calculation. We estimate the decay via a charged- space integration almost trivial, with the results for the pion-loop contribution with D-wave pion–pion charge- branching fractions exchangerescattering;analternativemechanismthrough (ηCPV 4π0)=5 10 5 θ¯2 , two f2 mesons is found to be completely negligible in B −→ × − × 0 comparison, based on an estimate of the tensor–tensor– (η CPV 4π0)=9 10 2 θ¯2 . (41) pseudoscalarcouplingconstantintheframeworkofQCD B ′ −→ × − × 0 sum rules. Because of these phenomenological approxi- We remark that we do not consider the branching ratio mations, the CP-conserving branching ratios thus ob- estimate for η′ 4π0 in Eq. (41) reliable in any sense: tained, → given the available phase space and the possibility of strong S-wave ππ final-state interactions, it could easily η 4π0 4 10 8 , ′ − be enhanced by an order of magnitude. Were θ¯0 a quan- B(cid:0)η →4π0(cid:1)∼3×10 30 , (43) tity of natural size, Eq. (41) would demonstrate the en- B → ∼ × − (cid:0) (cid:1) hancement of the CP-violating S-wave mechanism com- should only be taken as order-of-magnitude estimates. pared to the CP-conserving D-wave one, see Eqs. (36) It thus turns out that the CP-conserving decay width and(37). WithcurrentlimitsontheQCDvacuumangle of η 4π0 is so small that any signal to be observed derived from neutron electric dipole moment measure- → would indicate CP-violating physics. For the latter, we ments, θ¯ . 10 11 [29], these branching fractions are 0 − calculate one specific example using the QCD θ-term. already bound beyond anything measurable; however, we note that for η 4π0, the suppression of the CP- → conserving D-wave mechanism, see Eq. (37), is so strong that it is even smaller than the CP-violating (S-wave) Acknowledgments one in Eq. (41) if the current bounds are inserted for θ¯ . 0 We would like to thank Andrzej Kup´sć for initiating this project and for discussions, and Maurice Benayoun V. SUMMARY AND CONCLUSIONS for useful communications concerning Ref. [14]. Partial financial support by the Helmholtz Association through In this article, we have calculated the branching frac- funds providedto the VirtualInstitute “Spin andstrong tions of the η and η decays into four pions. These pro- QCD” (VH-VI-231), by the DFG (SFB/TR 16, “Subnu- ′ cessesofoddintrinsicparityareanomalous,and—aslong clear Structure of Matter”), and by the project “Study asCP symmetryisassumedtobeconserved—forbidthe ofStronglyInteractingMatter” (HadronPhysics2,Grant pions to be in relative S-waves. We organize the am- No. 227431) under the Seventh Framework Program of plitudes according to chiral power-counting rules, and the EU is gratefully acknowledged. 10 Appendix A: Four-body phase space integration The n-body phase space is defined as q dΦ (P;p ,...,p ) n 1 n n n d3p ≡(2π)4δ4(cid:16)P −Xi=1pi(cid:17)iY=1(2π)32ip0i . (A1) p′3 π θ Using the recursive relation [2] p∗1 θ1∗ − 3′ θ 13 dΦn(P;p1,...,pn)=dΦj(q;p1,...,pj) ϕ′3 dq2 dΦn j+1(P;q,pj+1,...,pn) , (A2) ϕ∗1 × − 2π we have k dΦ (P;p ,...,p ) 4 1 4 dq2dk2 FIG.4: Thesolidanglesofparticle1(3)inthecmfof1and2 =dΦ (q;p ,p )dΦ (k;p ,p )dΦ (P;q,k) 2 1 2 2 3 4 2 2π 2π (3and4). θ istheanglebetweenthemomentumofparticle 13 1 M−m3−m4 M−√s12 1 in the cmf of 1 and 2 and the momentum of particle 3 in = (8π2)4M Z d√s12Z d√s34 thecmf of 3 and 4. m1+m2 m3+m4 dΩ dΩ dΩp p q , (A3) ×Z ∗1 ′3 | ∗1|| ′3|| | to k. One can define θ as the angle between the direc- 1∗ tionsofq andp ,andθ astheonebetweenk= q and where m , i = 1...4 are the masses associated with the ∗1 3′ − i p . The angle between p and p , θ , is related to the final-stateparticlesofmomentump ,M isthemassofthe ′3 ∗1 ′3 13 i solid angles Ω and Ω by decaying particle, s =q2, s =k2. dΩ =dϕ dcosθ ∗1 ′3 12 34 ∗1 ∗1 1∗ isthesolidangleofparticle1inthecenter-of-massframe cosθ = cosθ cosθ sinθ sinθ cos(ϕ +ϕ ) . (cmf) of particles 1 and 2, dΩ′3 is the solid angle of par- 13 − 1∗ 3′ − 1∗ 3′ ′3 ∗1(A6) ticle 3 in the cmf of 3 and 4, and dΩ is the solid angle of The angles are shown for illustration in Fig. 4. It is the 1,2systemintherestframeofthe decayingparticle. obvious that the integration dΩ as well as the one over The three-momenta are given by either ϕ or ϕ are trivial, such that Eq. (A3) simplifies ∗1 ′3 λ1/2(s ,m2,m2) λ1/2(s ,m2,m2) to p = 12 1 2 , p = 34 3 4 , | ∗1| 2√s | ′3| 2√s 12 34 dΦ (P;p ,...,p ) λ1/2(M2,s ,s ) 4 1 4 |q|= 2M12 34 , (A4) = 1 M−m3−m4d√s M−√s12d√s (8π2)3M Z 12Z 34 withtheusualKällénfunctionλ(x,y,z) x2+y2+z2 m1+m2 m3+m4 ≡ − 2(xy+xz+yz). dcosθ dcosθ dϕ p p q . (A7) ×Z 1∗ 3′ ′3| ∗1|| ′3|| | Denoting quantities in the cmf of1 and2 (3 and 4)by (), one can relate them with those in the rest frame of ∗ ′ the decayparticlebyLorentztransformation. Explicitly, Appendix B: Tensor-meson contributions to pµ1 ={γ12(p01∗+β12·p∗1),γ12(β12p01∗+p∗1k),p∗1⊥} , η, η′ →4π0 pµ = γ (p0 β p ),γ (β p0 p ), p , 2 { 12 2∗− 12· ∗1 12 12 2∗− ∗1k − ∗1⊥} pµ = γ (p0 +β p ),γ (β p0 +p ),p , 1. Amplitude, decay width 3 { 34 3′ 34· ′3 34 34 3′ ′3k ′3⊥} pµ4 ={γ34(p04′−β34·p′3),γ34(β34p04′−p′3k),−p′3⊥} , InthisAppendix,wediscussanalternative,resonance- (A5) driven decay mechanism for the decays η, η 4π0, ′ → namely, via two f (1270) tensor mesons, see the right where β = q/q0 (β = k/k0) is the velocity of the 2 12 34 panel of Fig. 2. The Lagrangian for the tensor–tensor– 1,2 (3,4) system in the rest frame of the decay particle, pseudoscalar interaction reads and γ = (1 β2 ) 1/2. Moreover, p are 12(34) − 12(34) − ∗1 ( ) the components of p parallel (perpendicular) tokq⊥, and g ∗1 = TTPǫ g ∂µTναTρβ∂σϕ , (B1) p′3 ( ) arethe components ofp′3 parallel(perpendicular) LTTP √2 µνρσ αβh i k ⊥