Convergence properties of η → 3π in low energy QCD
NNuclear and Particle Physics Proceedings 00 (2019) 1–5
Nuclear andParticle PhysicsProceedings
Convergence properties of η → π in low energy QCD Mari´an Koles´ar a, ∗ , Jiˇr´ı Novotn´y a a Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University, Prague, Czech republic
Abstract
Even today, the convergence of the decay widths and some of the Dalitz plot parameters of the η → π decaysseems problematic in low energy QCD. We provide an overview of the current experimental and theoretical situationwith historical background and summarize our recent results, which explore the question of compatibility of experi-mental data with a reasonable convergence of a carefully defined chiral series in the framework of resummed chiralperturbation theory.
1. Overview
From the very beginning it was understood that the η → π decays are isospin breaking processes. Initially,the decays were thought to be of electromagnetic origin[1, 2], generated by the isospin breaking virtual photonexchange H QED ( x ) = − e (cid:90) dyD µν ( x − y ) T ( j µ ( x ) j ν ( y )) . (1)Simultaneously, however, it was discovered that the de-cays are almost forbidden in the framework of QED (theSutherland theorem [3, 4]), which was met with somedisbelief [2].The early calculations [1, 2], applying current alge-bra and PCAC, related the η - π matrix elements to thedi ff erence of squared kaon masses or kaon and pionmasses, respectively. In fact, the latter resembles thelater Dashen’s theorem, which cannot be justified byelectrodynamics [4]. In spite of that, the obtained valuefor the neutral channel decay rate were of approximatelycorrect order of magnitude, Γ =
160 eV.Hence it became apparent that there has to be a sourceof isospin breaking beyond the term (1) [5]. As isknown today, strong interactions break isospin via thedi ff erence between the masses of the u and d quarks H IBQCD ( x ) = m d − m u d ( x ) d ( x ) − ¯ u ( x ) u ( x )) . (2) ∗ Speaker
The work [5] collected all the relevant current algebraterms contributing to the decays and can be consideredto be the first to provide the correct leading order cal-culation. The obtained value for the neutral decay ratedid not significantly change though and turned out to bemuch lower than the experimental value then available( Γ =
164 eV vs 750 ±
200 eV). There remained a signifi-cant discrepancy, as was concluded in [5].When a systematic approach to low energy hadronphysics was born in the form of chiral perturbation the-ory ( χ PT) [6, 7, 8], it was quickly applied to the η → π decays [9]. The one loop corrections were very sizable,the result for the decay width of the charged channel was Γ + = ±
50 eV, compared to the current algebra predic-tion of 66 eV. However, already at that time there werehints that the experimental value is still much larger(340 ±
100 eV), thus “resurrecting the puzzle” the the-ory aimed to solve. The current PDG value [10] is Γ + exp = ±
12 eV . (3)In the case of the neutral channel, the average is [10] Γ = ±
17 eV . (4)After the e ff ective theory was extended to includevirtual photon exchange generated by (1) [11], it wasshown that the next-to-leading electromagnetic correc-tions to the Sutherland’s theorem are very small as well[12, 13]. The theory thus seems to converge really a r X i v : . [ h e p - ph ] S e p . Koles´ar, J. Novotn´y / Nuclear and Particle Physics Proceedings 00 (2019) 1–5 η → π + π − π a b d Cr.Barrel ’98 [23] − . ± .
07 0 . ± .
11 0 .
06 (input)KLOE ’08 [24] − . ± .
020 0 . ± .
012 0 . ± . − . ± .
018 0 . ± .
042 0 . ± . − . ± .
017 0 . ± .
017 0 . ± . − . ± .
004 0 . ± .
006 0 . ± . − . ± .
014 0 . ± .
023 0 . ± . χ PT ’07 [14] − . ± .
075 0 . ± .
102 0 . ± . η → π + π − π . slowly for the decays. At last, the two loop χ PT calcu-lation [14] has succeeded to provide a reasonable pre-diction for the decay widths.Meanwhile, experimental data are being gatheredwith increasing precision in order to make more detailedanalysis of the Dalitz plot distribution possible. Com-parison of the recent experimental information with theNNLO χ PT results can be found in tables 1 and 2, withthe conventionally defined Dalitz plot parameters de-fined as η → π π + π − : | A | = A (1 + ay + by + dx + . . . ) (5) η → π : | A | = A (1 + α z + . . . ) , (6)where x ∼ u − t , y ∼ s − s , z ∼ x + y and s is the Dalitzplot center s = / M η + M π ). For the sake of brevity,we added the systematic and statistical uncertainties insquares. As can be seen, a tension between χ PT and ex-periments appears to be in the charged decay parameter b and the neutral decay parameter α .Alternative approaches were developed in order tomodel the amplitudes more precisely, namely disper-sive approaches [15, 16, 17, 18, 19] and non-relativistice ff ective field theory [20, 21, 22]. These more or lessabandon strict equivalence to χ PT and succeed in re-producing a negative sign for α (see table 2)Thence comes our motivation to ask whether it is pos-sible to carefully define an amplitude with reasonableconvergence properties which would reproduce the ex-perimental data for the decay widths and the Dalitz plotparameters. In other words, we aim to investigate thequestion of compatibility of the experimental data witha reasonable convergence of the chiral series.
2. Calculation
There is a long standing suspicion that chiral per-turbation theory might posses slow or irregular con- η → π π π α Crystal Barrel ’98 [28] − . ± . − . ± . − . ± . / WASA ’07 [31] − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . χ PT ’07 [14] + . ± . η → π . vergence in the case of the three light quark flavours[36, 37]. An alternative method, now dubbed resummed χ PT [38, 39], was developed in order to incorporatesuch a possibility. The starting point is the realizationthat the standard approach to χ PT, as a usual treatmentof perturbation series, implicitly assumes good conver-gence properties and hides the uncertainties associatedwith a possible violation of this assumption. The re-summed procedure uses the same standard χ PT La-grangian and power counting, but only expansions de-rived linearly from the generating functional are con-sidered safe. All subsequent manipulations are carriedout in a non-perturbative algebraic way. The expansionis done explicitly to next-to-leading order and higherorders are collected in remainders. These are not ne-glected, but retained as sources of error, which have tobe estimated.The working hypothesis of the resummed approachis that only a limited set of safe observables, as definedabove, has the property of global convergence, i.e. thatthe NNLO remainders are of a natural order of magni-tude. Observables derived from the safe ones by meansof nonlinear relations do not in general satisfy the cri-teria for global convergence due to the possible irregu-larities of the chiral series. Therefore, it is necessary toexpress such dangerous observables in terms of the safeones in a non-perturbative way.Our calculation is described in depth in [40]. Whatwe present here is only a brief excerpt, meant as a sum- . Koles´ar, J. Novotn´y / Nuclear and Particle Physics Proceedings 00 (2019) 1–5 mary of the basic steps and obtained results.Within the formalism, we start by expressing thecharged decay amplitude in terms of the 4-point Greenfunctions G i jkl . We compute at first order in isospinbraking. In this case the amplitude takes the form F π F η A ( s , t , u ) = G + − − ε π G + − + ε η G + − + ∆ (6) G D , (7)where ∆ (6) G D is the direct higher order remainder to thecomplete 4-point Green function. The physical mixingangles to all chiral orders and first in isospin brakingcan be expressed in terms of quadratic mixing terms ofthe generating functional to NLO and related indirectremainders ε π,η = − F F π ,η ( M (4)38 + ∆ (6) M ) − M η,π ( Z (4)38 + ∆ (6) Z ) M η − M π . (8)In this approximation the neutral decay channel ampli-tude can be related to the charged one as A ( s , t , u ) = A ( s , t , u ) + A ( t , u , s ) + A ( u , s , t ) . (9)As dictated by the method, O ( p ) parameters appearinside loops, while physical quantities in outer legs.Due to the leading order masses in loops, such a strictlyderived amplitude has an incorrect analytical structure,cuts and poles are placed in unphysical positions. Toaccount for this, we carefully modify the amplitude us-ing a NLO dispersive representation. The procedure isdescribed in detail in [40].The next step is the treatment of the low energy con-stants (LECs). The leading order ones, as well as thequark masses, are expressed in terms of convenient pa-rameters Z = F F π , X = F B ˆ mF π M π , r = m s ˆ m , R = ( m s − ˆ m )( m d − m u ) , (10)where ˆ m = ( m u + m d ) /
2. The standard approach tacitly as-sumes values of X and Z close to one, which means thatthe leading order terms should dominate the expansion.However, recent fits [41] indicate much lower values.A possibility of a non-standard scenario of spontaneouschiral symmetry breaking is thus still open.At next-to-leading order, the LECs L - L are al-gebraically reparametrized in terms of pseudoscalarmasses, decay constants and the free parameters X , Z and r using chiral expansions of two point Green func-tions, similarly to [38]. Because expansions are for-mally not truncated, each generates an unknown higherorder remainder. We still don’t have a similar procedure for L - L .Therefore we collect a set of standard χ PT fits [42, 43,44, 41] and by taking their mean and spread, while ig-noring the much smaller reported error bars, we obtainan estimate of their influence. As is shown in [40], theresults depend on these constants only very weakly. Theerror bands given below include the estimated uncer-tainties in L - L .The O ( p ) and higher order LECs, notorious for theirabundance, are collected in a relatively smaller numberof higher order remainders. We have a direct remainderto the 4-point Green function and eight indirect ones -three related to each the pseudoscalar masses and thedecay constants, two to the mixing angles. The last stepleading to numerical results is their estimate. We use anapproach based on general arguments about the conver-gence of the chiral series [38], which leads to ∆ (4) G = (0 ± . G , ∆ (6) G = (0 ± . G , (11)where G stands for any of our 2-point or 4-point Greenfunctions, which generate the remainders. This is inprinciple an assumption. Hence we test the compatibil-ity of this assumption of a reasonably good chiral con-vergence of trusted quantities with experimental data ina statistical sense.
3. Summary of results
Our results depend, besides the remainders, on sev-eral free parameters - the chiral condensate, the chiraldecay constant, the strange quark mass and the di ff er-ence of the light quark masses. They are expressed interms of the parameters X , Z , r and R , respectively.The quark mass parameters have been fixed from latticeQCD averages [45]: r = ± R = ± . Koles´ar, J. Novotn´y / Nuclear and Particle Physics Proceedings 00 (2019) 1–5 Γ + [ M e V ] The charged width Γ + as a function of X for Z = - - -
202 X a Dalitz plot parameter a as a function of X for Z = b Dalitz plot parameter b as a function of X for Z = - - α Dalitz plot parameter α as a function of X for Z = Figure 1: Parameters Γ + , a , b and α as a function of X for Z = . Γ + [ M e V ] The charged width Γ + as a function of X for Z = - - -
202 X a Dalitz plot parameter a as a function of X for Z = b Dalitz plot parameter b as a function of X for Z = - - α Dalitz plot parameter α as a function of X for Z = Figure 2: Parameters Γ + , a , b and α as a function of X for Z = . . Koles´ar, J. Novotn´y / Nuclear and Particle Physics Proceedings 00 (2019) 1–5 found a strong dependence of the widths on X and Z andan appearance of both compatibility ( < σ C.L.) and in-compatibility ( > σ C.L.) regions. Such a behavior isnot necessarily in contradiction with the global conver-gence assumption and, moreover, it might be promisingfor constraining the parameter space and an investiga-tion of possible scenarios of the chiral symmetry break-ing [46].As for the Dalitz plot parameters, a and d can be de-scribed very well too, within 1 σ C.L. As an example,results for a are depicted in the figures.However, when b and α are concerned, we find a mildtension for the whole range of the free parameters, atless than 2 σ C.L. This marginal compatibility is not en-tirely unexpected. In the case of derivative parameters,obtained by expanding the amplitude in a specific kine-matic point, in our case the center of the Dalitz plot, anddepending on NLO quantities, the global convergenceassumption is questionable, as discussed in [40]. Also,the distribution of the theoretical uncertainties is foundto be significantly non-gaussian, so the consistency can-not be simply judged by the 1 σ error bars.The marginal compatibility in the case of the pa-rameters b and α can be interpreted in two ways -either some of the higher order corrections are indeedunexpectedly large or there is a specific configurationof the remainders, which is, however, not completelyimprobable. This warrants a further investigation ofthe higher order remainders by including additionalinformation. Work is under way in analyzing ππ rescattering e ff ects and resonance contributions, somepreliminary results can be found in [47]. Acknowledgment:
This work was supported by the Czech ScienceFoundation (grant no. GACR 15-18080S).
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