Three-body Final State Interaction in η→3π
Peng Guo, Igor V. Danilkin, Diane Schott, C. Fernández-Ramírez, V. Mathieu, Adam P. Szczepaniak
JJLAB-THY-15-2041
Three-body Final State Interaction in η → π Peng Guo,
1, 2, 3, ∗ Igor V. Danilkin, Diane Schott,
3, 4
C. Fern´andez-Ram´ırez, V. Mathieu,
1, 2 and Adam P. Szczepaniak
1, 2, 3 (Joint Physics Analysis Center) Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403 Physics Department, Indiana University, Bloomington, IN 47405, USA Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA Department of Physics, The George Washington University, Washington, DC 20052, USA (Dated: August 16, 2018)We present a unitary dispersive model for the η → π decay process based upon the Khuri-Treiman equations which are solved by means of the Pasquier inversion method. The descriptionof the hadronic final-state interactions for the η → π decay is essential to reproduce the availabledata and to understand the existing discrepancies between Dalitz plot parameters from experimentand chiral perturbation theory. Our approach incorporates substraction constants that are fixed byfitting the recent high-statistics WASA-at-COSY data for η → π + π − π . Based on the parametersobtained we predict the slope parameter for the neutral channel to be α = − . ± . Q = 21 . ± . PACS numbers: 13.25.Jx, 11.55.Fv, 14.65.Bt, 12.39.Fe
I. INTRODUCTION
Production of three particles plays an important rolein hadron physics. It sheds light on the reaction dynam-ics, e.g. the OZI rule, and can amplify production ofhadron resonances, with the mysterious XYZ states seenin the spectrum of charmonia and bottomonia [1] beingthe most recent examples. The need for precision analysisof final states containing three light hadrons has becomeeven more pressing given the high quality data emergingfrom the various hadron facilities around the world, in-cluding Jefferson Lab, COMPASS and BESIII [2–5]. Re-cently, significant progress has been made in analysis ofhadron-hadron interactions at low energies based on theS-matrix principles of unitarity, analyticity and crossingsymmetry [6–9]. At low energies, unitarity is an impor-tant constraint given that there is only a limited numberof contributing channels. Unitarity also determines theanalytical properties of partial waves and constraints res-onant scattering. Implementation of crossing-symmetryis much more difficult since it is related to the underlyingdynamics. However, at low energies it can be systemati-cally investigated by identifying the most important, i.e. closest to the physical region, singularities of the cross-channel amplitudes, and for example in reactions involv-ing Goldstone bosons these can be constrained by chiralsymmetry of QCD [10, 11].In this paper we focus on decays of the η meson to threepions. From the experimental side, the high-quality datafrom WASA-at-COSY [12, 13], Crystal Barrel [14, 15],and KLOE [16, 17], along with the data from CLAS [3],which is currently being analyzed, present an opportu- ∗ Electronic address: [email protected] nity for precision analysis of the Dalitz distribution. Inthe charged decay channel, η → π + π − π , we only haveaccess to the binned data from the WASA-at-COSY [12]experiment and therefore it is the only data set we usein our data-driven analysis. From the theoretical pointof view η → π decays are of interest because of isospinviolation. These decays are dominated by the intrinsicisospin breaking effects in QCD as electromagnetic ef-fects are expected to be small [18, 19]. Consequently, thedecay width for η → π is expected to be proportional tothe light quark mass difference and the decay amplitudeis often expressed in terms of the quantity, 1 /Q definedby 1 Q = m d − m u m s − ˆ m . (1)Here ˆ m = ( m u + m d ) / u and d quark masses. One determines Q by comparing a the-oretical prediction with the experimental decay widthΓ( η → π + π − π ) = 281 ±
28 eV [1]. However, it is im-portant to emphasize that this procedure requires thatthe amplitude implements chiral constraints or at leastit agrees with the leading-order chiral perturbation the-ory ( χ PT), which is where Q originates. Once Q is ex-tracted, it can be combined with the knowledge of the ˆ m and m s , e.g. from lattice simulations, to determine thelight-quark mass difference.It is necessary to consider the η → π decay ampli-tudes beyond χ PT. This is apparent when consideringcontributions to Γ( η → π + π − π ) from the first few termsin the low energy expansion. Specifically, the leading-order χ PT result, Γ LO η → π + π − π = 66 eV [20, 21], is ap-proximately four times smaller than expected. Inclu-sion of next-to-leading (one loop) corrections increasesthe theoretical prediction to Γ NLO η → π + π − π = 167 ±
50 eV a r X i v : . [ h e p - ph ] S e p [22], which is still significantly below the data. The next-to-next-to-leading calculation (two loops) has been per-formed recently [23]. It pushes the decay width furthertowards the data; however, it contains a large numberof low energy constants. In addition to the apparentpoor convergence, low orders of χ PT give an incorrectresult for the shape of the Dalitz distribution in theneutral 3 π decay. To the leading order, this distribu-tion is represented by a single parameter, α , which χ PTpredicts to be positive while the experimental result is α = − . ± . χ PT is incorporated only order byorder. To fulfill unitarity various dispersive frameworkswere developed [24, 25] with recent updates of [26, 27]and [28]. These analyses are based on the Khuri-Treiman(KT) representation [29]. In the KT approach, partialwaves are given in the elastic approximation with the left-hand cut contributions computed from cross-channel am-plitudes that are approximated by the same elastic par-tial waves as in the direct channel and are bootstrapped.Other calculations employed, for example, nonrelativis-tic effective field theory (NREFT) [30] and alternativedispersive approaches were studied in [31].The final state interactions in η → π at low en-ergies can be approximated by elastic ππ scattering.These amplitudes are available with high precision upto √ s = 1 . χ PT. We show the fitted Dalitzplot parameters for the charged decay and predict theslope parameter for the neutral decay channel. In thesecond part we match our amplitudes to χ PT in orderto extract the Q value. Conclusions are summarized inSection IV. II. FORMALISMA. Kinematics and partial wave expansion
The isospin violating η → π decay involves a∆ I = 1 interaction. The transition matrix elements, A αβγη ( s, t, u ), depends on four isospin indices, with theindex η referring to the isospin component of the in-teraction and α, β, γ to three pions. In terms of theparticle momenta the three Mandelstam variables are s = ( p + p ) = ( p − p ) , t = ( p + p ) = ( p − p ) ,and u = ( p + p ) = ( p − p ) . The Mandelstam vari-ables satisfy s + t + u = m η + m + m + m , with m η being the mass of the η , also referred to as particle i = 4and m i , i = 1 .. s -channel scattering 4 + ¯3 → t -channel scattering 4 + ¯1 → u -channel scatter-ing, 4 + ¯2 → → s -channelpartial wave expansion. In the s -channel, the amplitude A αβγη ( s, t, u ) has the following partial wave (p.w.) de-composition, A αβγη ( s, t, u ) = ∞ (cid:88) L =0 (cid:88) I (2 L + 1) P L ( z s ) P ( I ) αβηγ A IL ( s ) , (2)where P L ( z s ) is the Legendre polynomial and z s is a co-sine of the center-of-mass scattering angle θ s , z s ≡ cos θ s = s ( t − u ) + ( m − m ) ( m η − m ) λ / ( s, m η , m ) λ / ( s, m , m ) . (3)The usual K¨all´en triangle function is given by λ ( a, b, c ) = a + b + c − a b + b c + a c ) and ( I, L ) la-bel isospin and orbital angular momentum quantumnumbers in the s -channel with I + L = even due to Bosesymmetry of pions. The isospin projection operators P ( I ) αβγη are given in Appendix A. We note that at thisstage the partial waves are arbitrarily normalized. Theunitary relation, which we discuss in the following is ho-mogeneous in A and at the end we will normalize theamplitude by comparing with the experimental data.The p.w. amplitudes A IL ( s ) have both the right-handcut discontinuities demanded by the direct channel uni-tarity and left-hand cut discontinuities from exchangesin the t and u channels. We emphasize that Eq. (2) isexact in the s -channel physical region, when the infinitesum over L converges. The amplitudes in the other chan-nels are obtained by analytical continuation. Low-energyapproaches based on partial wave expansion involve trun-cation of the partial waves series at some L = L max < ∞ ,which violates analytical properties of cross-channel am-plitudes. To partially recover those, we represent theamplitude as a sum of truncated partial wave series in ¯3 ¯3 ¯34 12∆ A ( s ) 4 a ( s ) 12( a ) 12 4 a ( t ) 12 f ∗ ( s ) 12( b )= + FIG. 1: A diagrammatic representation of discontinuity relations in Eq. (10). each of the three channels [29, 34, 37–41], A αβγη ( s, t, u ) = L max (cid:88) L =0 (cid:88) I (2 L + 1) (cid:16) P L ( z s ) P ( I ) αβηγ a IL ( s )+ P L ( z t ) P ( I ) βγαη a IL ( t ) + P L ( z u ) P ( I ) γαβη a IL ( u ) (cid:17) , (4)where the amplitudes are a IL defined as having onlyright-hand discontinuities demanded by unitarity in therespective channels. The center of mass scattering anglesin the t - and the u -channel are given by z t = t ( s − u ) + ( m − m ) ( m η − m ) λ / ( t, m η , m ) λ / ( t, m , m ) ,z u = u ( t − s ) + ( m − m ) ( m η − m ) λ / ( u, m η , m ) λ / ( u, m , m ) . (5)We remark that the decomposition in Eq. (4) satisfiescrossing symmetry explicitly; however, violation of an-alyticity remains since the amplitude contains a finitenumber of high-spin partial waves in any given channel.This would be a problem at high energies but hopefullydoes not influence our low-energy analysis. What therepresentation in Eq. (4) does is to allow for unitarityto be implemented in all three channels. We also notethat decomposition in Eq.(4) is exact up to NNLO in χ PT [42, 43] and is often referred to as “reconstructiontheorem”.It is convenient to express the p.w. amplitude A IL ( s )( c.f. Eq. (2)) in terms of the amplitudes a IL ( s ) that aredefined by Eq. (4), A IL ( s ) = a RightIL ( s ) + a LeftIL ( s ) . (6)Here the amplitude a RightIL ( s ) has only the right-hand dis-continuity, a RightIL ( s ) = a IL ( s ) , (7)and the left-hand discontinuities of a LeftIL ( s ) originatefrom the exchange terms, a LeftIL ( s ) = L max (cid:88) L (cid:48) =0 (cid:88) I (cid:48) (2 L (cid:48) + 1)2 (cid:90) − dz s P L ( z s ) × (cid:0) P L (cid:48) ( z t ) C II (cid:48) st a I (cid:48) L (cid:48) ( t ) + P L (cid:48) ( z u ) C II (cid:48) su a I (cid:48) L (cid:48) ( u ) (cid:1) . (8)Here C st and C su are the standard crossing matrices andare given in Appendix A. B. Unitarity and the three-body effects in thedecay channel
In the following we consider both decay modes of the η meson, the charged decay η → π + π − π , and the neu-tral decay η → π . When comparing with experimen-tal data it is important to have an accurate descrip-tion of the phase space boundary, thus in the compu-tation of the kinematical factors we use the physical pionmasses. Elsewhere we assume the isospin limit and use m i = (2 m π + + m π ) / ≡ m π , i.e. the isospin averagedmass.The model is defined by Eq. (8) together with theelastic unitarity constraint for the right-hand disconti-nuity [44],∆ a RightIL ( s ) ≡ i (cid:16) a RightIL ( s + i(cid:15) ) − a RightIL ( s − i(cid:15) ) (cid:17) = f ∗ IL ( s ) ρ ( s ) ( a RightIL ( s ) + a LeftIL ( s )) , (9)where ρ ( s ) = (cid:112) − m π /s . The elastic ππ partialwave amplitudes are denoted by f IL and normalized byIm(1 /f IL ( s )) = − ρ ( s ). Therefore, the amplitudes a IL ( s )satisfy the relation,∆ a IL ( s ) = f ∗ IL ( s ) ρ ( s ) (cid:18) a IL ( s ) + L max (cid:88) L (cid:48) =0 (cid:88) I (cid:48) L (cid:48) + 1) K ( s ) /s × (cid:90) t + ( s ) t − ( s ) dt P L ( z s ) P L (cid:48) ( z t ) C II (cid:48) st a I (cid:48) L (cid:48) ( t ) (cid:19) . (10)The first term on the right-hand side of Eq. (10) rep-resents the contribution from the direct s -channel,4 + ¯3 → s -channel partial-wave projectionof the unitarity relation and it is illustrated in the di-agram in Fig. 1(a). The second term, illustrated inFig. 1(b), gives the contribution from the exchange con-tributions in the t -channel 4 + ¯1 → u -channel4 + ¯2 → z s to integration over t , (cid:90) − dz s · · · ) = (cid:90) t + ( s ) t − ( s ) dtK ( s ) s ( · · · ) , (11)with the integration limits t ± ( s ) corresponding to z s = ± t ± ( s ) = m η + 3 m π − s ± K ( s )2 s . (12) t − (s)t + (s) s L (m − m ) t plane egb fda chi FIG. 2: Integration contour in the complex t plane. Thearrows indicate the direction of increasing s in the intervalfrom 4 m π to ∞ . The points labeled a through i correspond tospecific values of s , with (a) t − ( ∞ ) = 0, (b) t − (( m η + m π ) ) = m π ( m π − m η ), (c) t − (( m η − m π ) ) = m π ( m η + m π ), (d) t − ( m η − m π ) = 4 m π , (e) t ± (4 m π ) = m η − m π , (f) t + ( m π ( m η + m π )) = ( m η − m π ) , (g) t + (( m η − m π ) ) = m π ( m η + m π ),(h) t + (( m η + m π ) ) = m π ( m π − m η ), and (i) t + ( ∞ ) = −∞ ,respectively. The Kacser function K ( s ) is given by the product of thetriangle functions and has the following determination[24, 45] K ( s ) = + κ ( s ) , m π ≤ s ≤ ( m η − m π ) ,i κ ( s ) , ( m η − m π ) ≤ s ≤ ( m η + m π ) , − κ ( s ) , ( m η + m π ) ≤ s < + ∞ ,κ ( s ) = | λ ( s, m η , m π ) λ ( s, m π , m π ) | / . (13)In the scattering region s ≥ ( m η + m π ) the inte-gral in Eq. (11) is well defined; however, when4 m π ≤ s < ( m η + m π ) , analytical continuation to thedecay region is needed. For this a positive infinitesimalimaginary part is added to the eta mass [37, 45, 46],which leads to the integration contour in the t -planeshown in Fig. 2. It is worth noting that the contouravoids the unitary cut. Finally, the amplitudes a IL ( s )are obtained by bootstrapping the dispersion reaction a IL ( s ) = 1 π (cid:90) ∞ m π ds (cid:48) ∆ a IL ( s (cid:48) ) s (cid:48) − s , (14)with a IL appearing on the right-hand side ( cf. Eq. (10))together with the input two-body scattering amplitudes, f IL ( s ).As in the standard N/D approach, the inhomogeneouspart in Eq. (10) can be accounted for writing a IL ( s ) asa product of f IL ( s ) times another function of s , whosediscontinuity is given by the s -channel projection of thecross-channel amplitudes. It is also convenient to removeany zeros of f IL ( s ), e.g. the Adler zero, since these areprocess dependent. Finally, the partial waves have kine-matical singularities, which do not contribute to the dis-continuity relation given by Eq. (10). Thus, we write a IL ( s ) = Z L ( s ) F IL ( s ) f IL ( s ) g IL ( s ) , (15) where the first factor removes the kinematical singulari-ties Z L ( s ) = (cid:20) K ( s ) s/ − m π (cid:21) L (16)and the second factor removes zeros from the ππ ampli-tude, F IL ( s ) = ( s − s ( I ) χ ) / ( s − s ( I ) A ) , L = 0 , , L > . (17)That is, we assume f IL has zeros in the S -wave only.Note that at leading order in χ PT, Adler zeros are lo-cated at s (0) A = m π / s (2) A = 2 m π in the ππ S -waveisoscalar and isotensor amplitudes, respectively, and at s (0) χ = 4 / m π for η → π . In the actual calculation weuse as input the ππ amplitudes from the phenomenolog-ical analysis of [7] which have zeros at the same positionas the leading order in χ PT; when matching η → π with χ PT we use NLO calculation which places the ze-ros in η → π at s (0) χ = 1 . m π and s (2) χ = 2 . m π in theisoscalar and isotensor channels, respectively.Finally, it follows from Eq. (10) and Eq. (15) that thefunction g IL has the discontinuity given by∆ g IL ( s ) = − θ ( − s ) ∆ f IL ( s ) f ∗ IL ( s ) g IL ( s ) (18)+ θ ( s − m π ) L max (cid:88) L (cid:48) =0 (cid:88) I (cid:48) L (cid:48) + 1) K ( s ) /s ρ ( s ) P L ( z s ) F IL ( s ) Z L ( s ) × (cid:90) t + ( s ) t − ( s ) dt P L (cid:48) ( z t ) C II (cid:48) st Z L (cid:48) ( t ) F I (cid:48) L (cid:48) ( t ) f I (cid:48) L (cid:48) ( t ) g I (cid:48) L (cid:48) ( t ) . The first term on the left-hand side takes into account theleft-hand cut of f IL ( s ); i.e. in addition to the unitary cut, g IL has a left-hand cut determined by f IL to guaranteethat there is no dynamical left-hand cut in the amplitudes a IL . The integrand in Eq. (18) is free from kinematicalsingularities in t and the function g IL ( s ) satisfies g IL ( s ) = 1 π (cid:90) ∞−∞ ds (cid:48) ∆ g IL ( s (cid:48) ) s (cid:48) − s . (19)Inserting Eq. (18) into Eq. (19) we obtain a double in-tegral equations for g IL ( s ), which can be reduced to asingle integral equation by changing the order of disper-sive integral (over s ) and the angular projection (internalover t ). The procedure, which we referred to earlier as thePasquier inversion, was developed in [34, 35] and recentlyrevisited in [36]. It leads to the following representation g IL ( s ) = − π (cid:90) −∞ ds (cid:48) s (cid:48) − s ∆ f IL ( s ) f ∗ IL ( s ) g IL ( s (cid:48) )+ 1 π (cid:90) ( M − m π ) −∞ dt L max (cid:88) L (cid:48) =0 (cid:88) I (cid:48) K IL,I (cid:48) L (cid:48) ( s, t ) × C II (cid:48) st f I (cid:48) L (cid:48) ( t ) g I (cid:48) L (cid:48) ( t ) , (20)where the kernel function K IL,I (cid:48) L (cid:48) ( s, t ) is given explicitlyin Appendix B. The left-hand cut contribution to g IL ( s )is largely unknown. Since we are primarily interested inthe physical decay region we therefore parametrize con-tributions to g IL from integration over s <
0. In thesimplest approximation these are reduced to a constant.A more elaborated representation could, for example, in-volve a conformal map of the s -plane cut along the neg-ative real axis onto a unit circle [47]. However, in theanalysis of the data we find the simple approximation tobe sufficient: g IL ( s ) = g IL ( s ) + 1 π (cid:90) ( M − m π ) dt L max (cid:88) L (cid:48) =0 (cid:88) I (cid:48) C II (cid:48) st × (cid:0) K IL,I (cid:48) L (cid:48) ( s, t ) − K IL,I (cid:48) L (cid:48) ( s , t ) (cid:1) f I (cid:48) L (cid:48) ( t ) g I (cid:48) L (cid:48) ( t ) . (21)This equation can now be solved using standard matrixinversion methods with the subtraction constants g IL ( s )as fitting parameters. The subtraction point is arbitraryand we choose it to coincide with the Adler zero of theLO χ PT s = 4 / m π . After solving the integral equationfor g IL ( s ), we compute a IL ( s ) from Eq. (15). Finally,to compare with the experimental data we convert theisospin amplitudes to the charge amplitude, A C ( s, t, u )for the η → π + π − π and A N ( s, t, u ) for the neutral case.These are given by Eq. (4), A C ( s, t, u ) = L max (cid:88) L =0 (2 L + 1)2 (cid:20) P L ( z s ) ( a L ( s ) − a L ( s ))+ P L ( z t ) ( a L ( t ) + a L ( t )) − P L ( z u ) ( a L ( u ) − a L ( u )) (cid:21) ,A N ( s, t, u ) = L max (cid:88) L =0 (2 L + 1)3 (cid:20) P L ( z s ) ( a L ( s ) + 2 a L ( s ))+ ( s → t ) + ( s → u ) (cid:21) . (22) III. NUMERICAL RESULTS
In this section we present our results for the decays η → π + π − π and η → π . We study the systematic un-certainties of the model by using different sets of partialwaves, i.e. varying L max and maximal isospin. We havefound that partial waves with ( L ≥
2) are negligible inthe physical decay region, 4 m π ≤ s ≤ ( m η − m π ) . Asinput we use two-pion scattering amplitudes from theanalysis of [7]. The parameters of the fit are the sub-traction constants, g IL ( s ), for each contributing partialwave. Our aim is to fix these by fitting η → π + π − π decay using the high statistic WASA-at-COSY data [12]and by matching to NLO χ PT [22]. The results for the η → π decay mode will then constitute a prediction,which we compare with the Dalitz plot distribution from[48]. We investigate the role of cross-channel exchanges, a.k.a. final-state interactions in the decay region, by per-forming two analyses. In the first, we do not includecross-channel effects and approximate g IL ( s ) in Eq. (21)by a constant, setting g IL ( s ) = g IL ( s ). It correspondsto a traditional isobar model, but with a fully incorpo-rated two-pion interaction. In the second, we includecross-channel rescattering effects and solve Eq. (21). Inthe following we refer to the two cases as “two-body” and“three-body”, respectively. A. Fitting WASA-at-COSY data η → π + π − π In this subsection we summarize the results of the fitto the recent WASA-at-COSY data on η → π + π − π [12],where binned Dalitz plot is given. Up to a normalizationfactor, the Dalitz plot distribution is given by the ampli-tude squared, d Γ ds dt ∝ | A ( s, t ) | . (23)It is convenient to express the amplitude in terms of twoindependent, dimensionless variables ( x, y ) which are lin-early related to the Mandelstam variables by x = √ m η Q c ( t − u ) ,y = 32 m η Q c (cid:0) ( m η − m π ) − s (cid:1) − , (24)where Q c = m η − m + π − m π (for the neutral decay weuse Q n = m η − m π ). A general property of these vari-ables is that the physical region of the Dalitz plot liesinside the unit circle x + y ≤ x = y = 0.We fit our model to the data [12] by minimizing the χ defined by χ = N (cid:88) bins (cid:32) | A | data − | A C ( { g IL ( s ) } ) | ∆ | A | data (cid:33) , (25)over the set of subtraction constants, g IL ( s ). InEq. (25), | A | data is the acceptance-corrected number ofevents in each of the N = 59, ∆ x = ∆ y = 0 . x = y = 0and ∆ | A | data is the statistical uncertainty. Note, thatsince Eq. (21) is linear in g IL , the parameter g ( s ) canbe factored out and fixed by the overall normalization.Since normalization of the data is arbitrary the absolutevalue of g ( s ) is irrelevant. Therefore, in Table I, whichsummarized fit results, when presenting results of two-body fits we quote ( g bIL ( s ) ± ∆ g bIL ( s )) /g b ( s ). Whenpresenting results of three-body fit we quote ( g bIL ( s ) ± ∆ g bIL ( s )) /g b ( s ), where g b ( s ) is the central value ob-tained in the two-body fit with the same number of par-tial waves. We do the latter to illustrate the relative TABLE I: Results of two-body and three-body fits for different wave sets. g ( s ) /g (2 b )00 g ( s ) /g (2 b )00 g ( s ) /g (2 b )00 χ /d.o.f. ( I, L ) = (0 , . ± .
002 – – 2 . . ± .
002 – – 15(
I, L ) = (0 , , (2 , . ± .
003 0 . ± .
01 – 1 . . ± .
003 0 . ± .
01 – 1 . I, L ) = (0 , , (1 , . ± .
002 – 0 . ± .
009 1 . . ± .
005 – 0 . ± .
009 0 .
95 (Set 1)(
I, L ) = (0 , , (2 , , (1 , . ± . − . ± .
05 0 . ± .
07 0 . . ± .
01 0 . ± .
03 0 . ± .
04 0 .
90 (Set 2)TABLE II: Dalitz plot parameters for η → π + π − π . Set 1 and Set 2 correspond to ( I, L ) = (0 , , (1 ,
1) and(
I, L ) = (0 , , (2 , , (1 ,
1) cases respectively (see Table I). a b d f g
WASA-at-COSY [12] − . ± .
018 0 . ± . ± .
037 0 . ± . ± .
018 0 . ± .
037 –KLOE [16] − . ± . +0 . − . . ± . ± .
010 0 . ± . +0 . − . . ± . ± .
02 –CBarrel [14] − . ± .
07 0 . ± .
11 0 . ± .
04 (fixed) – –Layter et al. [49] − . ± .
014 0 . ± .
03 0 . ± .
03 – –Gormley et al. [50] − . ± .
02 0 . ± .
03 0 . ± .
04 – –TheorySet 1 − . ± .
030 0 . ± .
010 0 . ± .
005 0 . ± . − . ± . − . ± .
035 0 . ± .
014 0 . ± .
003 0 . ± . − . ± . − .
371 0 .
452 0 .
053 0 .
027 –NNLO [23] − . ± .
075 0 . ± .
102 0 . ± .
057 0 . ± .
160 –Kambor et al. [24] − .
16 0 . ... .
26 0 . ... .
10 – –NREFT [30] − . ± .
014 0 . ± .
023 0 . ± .
003 0 . ± . − . ± . change in normalization between two- and three-bodyfits.In the first fit we use a single, scalar-isoscalar, a par-tial wave. In this case, the model gives a parameter freeprediction for the event distribution. We observe that the( I, L ) = (0 ,
0) amplitude provides the dominant contri- bution that covers approximately 90% of the Dalitz plot.The calculated χ /d.o.f. for the two-body and three-body cases are 2 . x and y axes. The error bars associated withthe model originate from the uncertainties in the pion-pion amplitude f IL [7] and from the statistical error infitting the overall normalization.In the next step, we add the isospin-2 S -wave. In thiscase we fit two parameters, one gives the overall normal-ization and the other contributes to a modification of theshape of the Dalitz plot. The resulting parameters and χ /d.o.f are given in Table I. In both, the two- and three-body fits we find that the model slightly underestimatesthe data. The inclusion of the second ( I, L ) = (2 ,
0) wavesignificantly improves χ and also drastically reduces thedifference in the fit quality between the two- and three-body cases pertinent in the fit with the single ( I, L ) = 0wave.In the spirit of keeping the number of free parame-ters as low as possible, we considered another set of twowaves, (
I, L ) = (0 , , (1 , S and P waves. In this case there is alsoone parameter that affects the shape of the Dalitz distri-bution and we find χ /d.o.f = 1 .
45 and χ /d.o.f = 0 . P -wave contri-bution over the isospin-2 S -wave. The results of the fitare shown in Fig. 3.We now turn to the case when a complete set of S and P waves is incorporated, i.e. ( I, L ) = (0 , , (2 , , (1 , χ /d.o.f around 0 . I, L ) = (0 ,
0) am-plitude, the three-body fit converges poorly indicatingimportance of higher partial waves that are brought inby the cross-channel exchanges. Thus apparent conver-gence of the two-body fit in this case is deceptive. Withany combination of higher partial waves all calculatedthree-body χ /d.o.f are quite similar to the two-bodyfits, except for the case when only ( I, L ) = (0 , , (1 , x = y = 0 is used to parametrize the η decay distri-bution. For the charged decay it leads to | A C ( x, y ) | | A C (0 , | = 1 + a y + b y + c x + d x + e xy + f y + g x y + · · · . (26)The charge conjugation symmetry, x → − x requiresterms odd in x to vanish, i.e. c = e = 0. In TableII we give the Dalitz plot parameters from our three-body fits based on the ( I, L ) = (0 , , (1 ,
1) (set 1) and(
I, L ) = (0 , , (2 , , (1 ,
1) (set 2) wave sets. For com-parison we quote the results of next-to-leading-order(NLO) and next-to-next-to leading order (NNLO) of χ PT [22, 23], the dispersive analysis from [24], NREFT[30] and alternative dispersive approach [31]. We also in-clude Dalitz parameters extracted from direct fits to theexperimental data [12, 14, 16, 49, 50]. The most recentanalyses where performed by the WASA-at-COSY [12]and KLOE [16] collaborations. As expected, our Dalitz plot parameters are consistent with the WASA-at-COSYparameters within the error bars. We also observe thatcentral values of the fit tend toward the KLOE results. η → π The results obtained in the charged mode can beused to predict the Dalitz plot parameters for the neu-tral channel. The Dalitz parameters are defined ascoefficients in the expansion around the center of theDalitz plot using the polar coordinates x = √ z cos φ and y = √ z sin φ in Eq. (24) | A N ( z, φ ) | | A N (0 , | = 1 + 2 α z + 2 β z / sin 3 φ + · · · . (27)The slope parameter α has been extracted from severalexperiments, while to the best of our knowledge, thereis no determination of β or higher moments. In Ta-ble III we compare our findings with the experimentalmeasurements and other theoretical predictions. The av-erage of experimental results compiled by the PDG is α = − . ± . α (Set 1) = − .
023 and α (Set 2) = − . α = Q n Q c (cid:18) d + b − a − Im(¯ a ) (cid:19) ≤ Q n Q c (cid:18) d + b − a (cid:19) , (28)where the factors Q c , Q n were defined below Eq. (24).Note that we only take Q c (cid:54) = Q n in the overall normal-ization while we use Q c = Q n when solving dispersionrelations for the partial wave amplitudes. Here, the com-plex parameters ¯ a is the coefficient of the linear term inthe expansion of the charged amplitude A C ( x, y ), A C ( x, y ) ∝ a y + ... (29)Using the Dalitz plot parameters from WASA-at-COSYand KLOE collaborations one finds α WASA ≤ − . , α KLOE ≤ − . . (30)The large difference in the upper limits is due to thedifference in the b parameter which differs by a factorof two between the two data sets. As pointed out in[30] the value for Im(¯ a ) can be sizable due to ππ fi-nal state interactions. Our results confirm this find-ing and we obtain Im(¯ a ) = − . ± .
03. Nevertheless,since ( α WASA ) max = − .
006 is quite large the Im(¯ a ) term -1 -0.5 0 0.5 1 x -1 -0.5 0 0.5 1 y -1 -0.5 0 0.5 1 x -1 -0.5 0 0.5 1 y -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 y -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 y FIG. 3: Upper and middle panels are the x - and y -projection plots. Black circles are the data. Red squares and blue squaresrepresent results of the two-body and three-body fits, respectively. The fits are performed on the Dalitz distribution [12] sownin the bottom left panel using a single, ( I, L ) = (0 ,
0) wave (upper panels) and two waves, (
I, L ) = (0 , , (1 ,
1) (central panels).For better visualization fit results are shifted horizontally (three-body to right and two-body to left) from the experimentalpoints. The bottom right panel is the Dalitz distribution from the three-body fit with (
I, L ) = (0 , , (1 ,
1) waves.
TABLE III: Dalitz plot parameters for η → π . Set1 and Set 2 correspond to ( I, L ) = (0 , , (1 ,
1) and(
I, L ) = (0 , , (2 , , (1 ,
1) cases respectively (see Table I). α β
GAMS-2000 [51] − . ± .
023 –Crystal Barrel, LEAR [52] − . ± .
020 –Crystal Ball, BNL [15] − . ± .
004 –SND [53] − . ± .
023 –CELSIUS-WASA [13] − . ± .
014 –WASA-at-COSY [54] − . ± .
009 –MAMI-B [55] − . ± .
004 –MAMI-C [48] − . ± .
003 –KLOE [17] − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . .
013 –NNLO [23] +0 . ± .
032 –Kambor et al. [24] − . ... − .
014 –NREFT [30] − . ± . − . ± . et al. [31] − . ± .
004 – alone can not be responsible for lowering α to the PDGvalue. Once the KLOE data become available [56] itwould be very interesting to perform a combined fit ofthe WASA-at-COSY and KLOE measurements.The neutral channel does not depend on the P -waveamplitude contributing to the charged decay mode andit contains only even partial waves. Unfortunately, usingthe charge mode we could not find sensitivity to the D -wave which was omitted from the Table I. Finally inFig 4, we compare our results with the recent MAMI-Cmeasurement [48]. The R ( z ) function is determined as R ( z ) = (cid:82) π dφ θ ( ϕ ( s, t, u )) | A N ( z,φ ) | | A N (0 , | (cid:82) π dφ θ ( ϕ ( s, t, u )) , (31)where ϕ ( s, t, u ) = s t u − m π ( m η − m π ) = 0 (32)defines the boundary of the Dalitz plot distribution and θ ( x ) is the step function. We observe that a cusp around R (cid:72) Z (cid:76) FIG. 4: Comparison of R ( z ) plot from [48] (black points) withour predictions from Table III that correspond to Set 1 (blueband) and Set 2 (red band). z (cid:39) .
765 appears in R ( z ) for nonzero β . This is a kine-matical effect which reflects the fact that for larger z thephase space distribution in the Dalitz plot is no longercircular. We find our results for Sets 1 and 2 provide asatisfactory agreement with the data. B. Matching to χ PT and the Q -value We remind that the data in [12] were normalized tothe center of the Dalitz plot and therefore our modelonly predicts the Dalitz plot distributions for the chargedand neutral decays. The overall normalization can befixed by comparing the experimental decay widths withthe phase space integral over the corresponding squaredamplitudes, Γ = N (cid:90) dx dy | A ( x, y ) | | A (0 , | , (33)with the boundaries of the integral determined by thephase space. We emphasize that the quantity Q definedin Eq. (1) enters into the normalization constant N . Inorder to determine Q one has to match the model, dis-persive amplitude, with χ PT where Q is defined.As discussed in Sec. I, the χ PT [23] series seems toconverge rather slowly and the question arises to whichorder of the χ PT should one match the model. It wouldbe desirable to find a matching point where on the χ PTside contributions, from powers of Mandelstam invari-ants, are small. Therefore, matching the amplitudes inthe physical region may not be the best option. Up toNNLO the chiral amplitude satisfies the decomposition ofEq. (4), and up to this order matching is simplified sinceit is sufficient to match the single variable, partial waveamplitudes a IL ( s ). The χ PT amplitude for the chargeddecay, up to NNLO can be written in the form A CχP T ( s, t, u ) = − Q m K ( m K − m π )3 √ m π F π M ( s, t, u ) , (34)0 -1 -0.5 0 0.5 1 x -1 -0.5 0 0.5 1 y M (cid:72) s (cid:76) (cid:45) M (cid:72) s (cid:76) (cid:45) (cid:45) (cid:45) M (cid:72) s (cid:76) FIG. 5: Upper panels: x - and y -projections of the Dalitz plots. Black circles represent the data. The red squares and blue squaresare model results using amplitudes with only two-body and including three-body correlations, respectively. The amplitudeswere computed using three partial wave components with (I,L)=(0,0), (2,0), (1,1). For better visualization fit results are shiftedhorizontally (three-body to right and two-body to left) from the experimental points. Bottom panels: The comparison of theNLO χ PT amplitudes M I ’s (black curves), with the two-body (red curves) and three-body (blue curves) dispersive amplitudes.Real parts are shown with solid lines and imaginary with dashed lines. In all figures the unknown couplings were fixed bymatching to NLO χ PT (see Eq.(38)).TABLE IV: Values of Q from different calculations.Theory Q Set 1 21 . ± . . ± . N f = 2 + 1) a [57] 22 . ± . . . et al. [24] 22 . ± . et al. [31] 23 . ± . a Here and in the following we combined in quadrature the errorsquoted in [57]. where F π = 92 . M ( s, t, u ) = M ( s ) − M ( s ) + M ( t ) + M ( u )+ ( s − u ) M ( t ) + ( s − t ) M ( u ) . (35)Explicit expressions for the functions M I at various or-ders in the chiral expansion can be found in [23]. Com-paring Eq. (22) and Eqs. (34), (35) one finds a ( s ) = 3 N χP T M ( s ) ,a ( s ) = 2 N χP T M ( s ) , (36) a ( s ) = 23 N χP T K ( s ) s M ( s ) , where N χP T = − Q m K ( m K − m π )3 √ m π F π . (37)The NNLO χ PT calculation was performed in [23].The order O ( p ) LECs were estimated using a reso-nance saturation model and error analysis was not pro-vided. Given that uncertainties in the low energy con-stants entering M I ’s at the NNLOs are not quantita-tively settled in the following we choose to match our1dispersive calculation with the NLO χ PT result. Inthis case one can use the NLO relations between decayconstants and meson masses which reduces the numberof low energy constants in the chiral amplitude to one, L = ( − . ± . · − [58]. We choose the matchingpoint to coincide with the subtraction point in Eq. (21),which in turn was chosen to coincide with the Adler zeroin the LO χ PT amplitude. In that case the determinedparameters from matching are the same for the two-bodyand three-body scenarios.In the following we consider two methods for matchingthe dispersive analysis with χ PT. In the first case we useEq. (37) together with the χ PT NLO amplitudes M I ’sto compute the overall normalization and the parameters g IL ( s ), which in turn completely determine dispersiveamplitudes of our model. We find, g ( s ) = 16 . N χP T ,g ( s ) /g ( s ) /g ( s ) = 1 / . / (0 . ± . . (38)This confirms that the amplitude ( I, L ) = (0 ,
0) is domi-nant. In the lower panel of Fig. 5, we compare the χ PTamplitudes with the dispersive ones, the latter obtainedusing the subtraction constants from Eq. (38). Compar-ing with the WASA-at-COSY data shown in the upperpanel in Fig. 5, we find that the dispersive amplitudefixed by Eq. (38) gives χ /d.o.f. of approximately 13 . . χ /d.o.f. prevents us fromextracting the Q -value using this method.To extract the Q -value we therefore use the χ PT ampli-tudes to determine the overall normalization only, whilefor the subtraction constants g IL ( s ) we use the re-sults from the fit of the WASA-at-COSY data describedin the previous section. We find Q (Set 1) = 21 . ± . Q (Set 2) = 21 . ± . Q -values are somewhat smaller comparedto [23, 24, 31], and within 1 σ from the recent ( N f = 2+1)lattice computations [57]. We note that lattice calcu-lations of electromagnetic correction for N f = 2 + 1are not yet available, while for N f = 2 these were re-ported in [ ? ]. The lattice result given in Table IV de-pends on the input value for the light quark mass ratio, m u /m d = 0 . ± .
03 which is the LO χ PT result re-duced by a factor of 8(4)% chosen as an estimate of thecorrection from higher-orders chiral effects [57]. Alterna-tively, using the extracted Q -value and the N f = 2 + 1lattice result for m s / ˆ m = 27 . ± .
44 [57] we can esti-mate m u /m d . We find m u m d = 0 . ± .
02 (39)as an average between Sets 1 and 2. Another useful quan-tity that can be calculated from our Q and m s / ˆ m is the so-called R -value given by R = m s − ˆ mm d − m u = 2 Q (cid:16) m s ˆ m (cid:17) − = 32 . ± . . (40) IV. CONCLUSIONS
In this paper, a new data driven dispersive analysisof η → π was performed. The hadronic final stateinteractions were incorporated using the Khuri-Treimanequation, which was solved using Pasquier inversion tech-nique. To the best of our knowledge it is the first timesuch an approach has been used in analysis of the η de-cays. In an earlier study [36], we illustrated the prosand cons of the Pasquier technique using a toy modelwith known exact solutions. The main limitation of thismethod is related to the treatment of the left-hand cuts,which in general are not known. We approximated themby a constant which is absorbed in the subtraction con-stants. As it was shown in [36], this approximation worksvery well, when the physical region does not dependstrongly on the accurate form of the left-hand cut. Onthe other hand the advantage of the Pasquier inversion isthat it eliminates the need for specifying the high-energybehavior of the absorptive parts in the physical region.In the analysis of the η → π decays presented here, wehave shown that with a single real parameter ( g ) andthe physical ππ partial-wave amplitudes [7] it is possibleto reproduce the Dalitz distribution of the charged η de-cay mode [12]. We have also verified that including morepartial waves leads fits with comparable χ /d.o.f. Theresulting Dalitz parameters, averaged over the variouscombinations of partial waves considered in this paperare, a = 1 . ± . , b = 0 . ± . ,d = 0 . ± . , f = 0 . ± . , (41) g = 0 . ± . . These are consistent, within 1 σ with the analysis ofWASA-at-COSY having central values shifted towardsvalues obtained from analysis by the KLOE Collabora-tion, which were not include in our fits. Based on theanalysis of the charged decay we made a prediction forthe slope parameter of the Dalitz distribution in the neu-tral decay channel, α = − . ± . . (42)This value is above the PDG value of α exp = − . ± . b parameter beingsignificantly larger than in the earlier KLOE analysis [16].We expect that in the future this issue will be resolvedonce the new KLOE data [56] become available allowinga simultaneous fit of both data sets.2Another useful test of the amplitudes is provided bythe ratio of neutral and charged decay rates. In theisospin limit this ratio does not depend on the normaliza-tion, and if the small electromagnetic isospin breaking isalso ignored [59], it depends only on the integrated Dalitzplot distributions. From our amplitude we find r = Γ( η → π )Γ( η → π + π − π ) = 1 . ± . , (43)which is consistent with the experimental value of r exp =1 . ± .
02 [1]. We have also compared our amplitudeswith the NLO χ PT results and found the Q -value of Q = 21 . ± . . (44)The error is of the statistical origin. It was computedthrough standard error propagation of the uncertaintiesarising from the ππ phase shifts, the L coefficient, theexperimental decay width Γ( η → π + π − π ) and the sta-tistical error in fitting the Dalitz plot. Inelasticity andhigher partial waves are also potential sources of uncer-tainties [60].Using the extracted Q -value and recent averages fromthe N f = 2 + 1 lattice computation for ˆ m = 3 . ± . m s = 93 . ± .
24, [57] we estimate the up and downquark masses to be m u = 2 . ± .
14 MeV ,m d = 4 . ± .
08 MeV . (45)The method for amplitude construction presented inthis work can be directly applied to decays of heaviermeson, e.g. η (cid:48) and used, for example, to test reliability ofthe isobar model. It can also be extended to incorporatecouple-channels, which might be more relevant in decaysof heavier mesons.All the material, including data and code are availablein an interactive form online [61]. Acknowledgments
We would like to thank B. Kubis, V. Mokeev, E. Passe-mar and M. R. Pennington for useful discussions. In ad-dition I. V. D. acknowledges discussions with G. Colan-gelo and H. Leutwyler. This material is based upon worksupported in part by the U.S. Department of Energy, Of-fice of Science, Office of Nuclear Physics under contractDE-AC05-06OR23177. This work was also supported inpart by the U.S. Department of Energy under Grant No.DE-FG0287ER40365, National Science Foundation un-der Grants PHY-1415459 and PHY-1205019, and IU Col-laborative Research Grant.After submission of our manuscript a new η → π analysis by the BESIII Collaboration became available[5]. The values for the Dalitz plot parameters (exceptthe parameter f ) of the charged η decay are compat-ible with our results within the error bars. We note that the determined b BESIII = 0 . ± . ± .
004 isconsiderably lower than WASA-at-COSY result. It con-firms the expected correlation to the slope parameterin the neutral decay channel, which turned out to be α BESIII = − . ± . ± . Appendix A: Isospin algebra
In Eq. (4) the isospin factors are given by P (0) αβγη = 13 δ αβ δ γη , P (1) αβγη = 12 ( δ αγ δ βη − δ αη δ βγ ) , (A1) P (2) αβγη = 12 ( δ αγ δ βη + δ αη δ βγ ) − δ αβ δ γη , which satisfy, (cid:88) ηγ P ( I ) αβηγ P ( I (cid:48) ) ηγα (cid:48) β (cid:48) = P ( I (cid:48) ) αβα (cid:48) β (cid:48) δ II (cid:48) , (cid:88) ηγ P ( I ) βγαη P ( I (cid:48) ) ηγα (cid:48) β (cid:48) = P ( I (cid:48) ) αβα (cid:48) β (cid:48) [ C st ] II (cid:48) , (A2) (cid:88) ηγ P ( I ) γαβη P ( I (cid:48) ) ηγα (cid:48) β (cid:48) = P ( I (cid:48) ) αβα (cid:48) β (cid:48) [ C su ] II (cid:48) . Here α, β, γ, η are the Cartesian isovector indices andisospin crossing matrices C st and C su , are given by C st = / / / / − / / − / / , C su = / − / − / / / / / / . Appendix B: Kernel functions
The kernel functions in Eq. (20) are determined as K IL,I (cid:48) L (cid:48) ( s, t ) = 2 (2 L (cid:48) + 1) × ( θ ( t ) ∆ IL,I (cid:48) L (cid:48) ( s, t ) − θ ( − t ) Σ IL,I (cid:48) L (cid:48) ( s, t )) , (B1)with∆ IL,I (cid:48) L (cid:48) ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − s ρ ( s (cid:48) ) ( s (cid:48) / − m π ) L F IL ( s (cid:48) ) K L +1 ( s (cid:48) ) /s (cid:48) × F I (cid:48) L (cid:48) ( t ) K L (cid:48) ( t )( t/ − m π ) L (cid:48) P L ( z s (cid:48) ) P L (cid:48) ( z t ) , (B2)andΣ IL,I (cid:48) L (cid:48) ( s, t ) = (cid:90) ∞ s + ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − s ρ ( s (cid:48) ) ( s (cid:48) / − m π ) L F IL ( s (cid:48) ) K L +1 ( s (cid:48) ) /s (cid:48) × F I (cid:48) L (cid:48) ( t ) K L (cid:48) ( t )( t/ − m π ) L (cid:48) P L ( z s (cid:48) ) P L (cid:48) ( z t ) , (B3)3 s − (t) (m − m ) C’ (m + m ) s + (t) d ba f gis planec e h FIG. 6: Integration contour C (cid:48) in the complex s plane afterPasquier inversion. The black wiggle lines represent cutsattached to two branch points: ( m η ± m π ) in s -plane. Thepoints labeled by a − i correspond to (a) s − (0) = −∞ ,(b) s − (4 m π ) = m η − m π , (c) s − ( m η − m π ) = 4 m π ,(d) s ± (( m η − m π ) ) = m π ( m π + m η ), (e) s + ( m π ( m η + m π )) = ( m η − m π ) , (f) s + (4 m π ) = m η − m π ,(g) s + (0) = ∞ , (h) s + ( m π ( m π − m η )) = ( m η + m π ) , and(i) s + ( −∞ ) = ∞ , respectively. where the contour C (cid:48) is shown in Fig. 6 (see [36] for moredetails). These kernel functions can be computed analyt-ically, what significantly speeds up numerical computa-tions. In the calculations presented in this paper, onlythe functions ∆ IL,I (cid:48) L (cid:48) ( s, t ) are needed and their analyti-cal representations are below in terms of∆ L,L (cid:48) ( s, t ) ≡ K L (cid:48) ( t ) (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − s U ( s (cid:48) ) × ( s (cid:48) − m π ) L K L ( s (cid:48) ) P L ( z s (cid:48) ) P L (cid:48) ( z t ) , (B4)so that for L = 0,∆ I ,I (cid:48) L (cid:48) ( s, t ) = 4 L (cid:48) − L F I (cid:48) L (cid:48) ( t )( t − m π ) L (cid:48) (B5) × (cid:34) ∆ ,L (cid:48) ( s, t ) F I ( s ) − s ( I ) χ − s ( I ) A s − s ( I ) χ ∆ ,L (cid:48) ( s ( I ) χ , t ) (cid:35) , and otherwise ( L (cid:54) = 0),∆ IL,I (cid:48) L (cid:48) ( s, t ) = 4 L (cid:48) − L F I (cid:48) L (cid:48) ( t )( t − m π ) L (cid:48) ∆ L,L (cid:48) ( s, t ) . (B6)The square root function U ( z ) is given by U ( z ) = (cid:113) ( z − ( m η − m π ) )( z − ( m η + m π ) ) (B7)in the complex z plane. Here and in what fol-lows, the phase convention for U ( z ) is chosen by U ( s ± i
0) = ( ∓ , i, ± ) | U ( s ) | for s ∈ (( −∞ , ( m η − m π ) ],[( m η − m π ) , ( m η + m π ) ] , [( m η + m π ) , ∞ )) respec-tively. The kinematic factor K ( s ) / ( s ρ ( s )) is given by thevalue of U ( s ) right below the two cuts attached to branchpoints s = ( m η ± m π ) , i.e. K ( s ) / ( s ρ ( s )) = U ( s − i s and t the physical values of ∆ L,L (cid:48) ( s, t )correspond to the limit s + i t + i • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = 1 U ( s ) (cid:20) ln (cid:12)(cid:12)(cid:12)(cid:12) R ( s, t ) + U ( s ) U ( t ) R ( s, t ) − U ( s ) U ( t ) (cid:12)(cid:12)(cid:12)(cid:12) − i π θ ( ϕ ( s, t )) (cid:21) ,R ( s, t ) = − m η + ( s − m π )( t − m π ) + m η ( s + t ) ,ϕ ( s, t ) = s t ( m η + 3 m π − s − t ) − m π ( m η − m π ) . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = 2 t ∆ a ( t ) + t (2 s + t − m η − m π ) ∆ , ( s, t ) , ∆ a ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) = − ln (cid:0) m η − m π + t + U ( t ) (cid:1) m η t . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = 1 s (cid:0) ∆ , ( s, t ) − ∆ , (0 , t ) (cid:1) , ∆ , ( s, t ) = ∆ , ( s, t ) + 2 ( t + m π ( m η − m π )) ∆ (+) b ( s, t )+ ( m η − m π ) ∆ ( − ) b ( s, t )+ 2 ( t + m π ( m η − m π ))( m η − m π ) ∆ c ( s, t ) , ∆ ( ± ) b ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − s U ( s (cid:48) ) 1 s (cid:48) − ( m η ± m π ) = 1 s − ( m η ± m π ) (cid:16) ∆ , ( s, t ) − ∆ ( ± ) d ( t ) (cid:17) , ∆ c ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − s U ( s (cid:48) )= ∆ , ( s, t ) U ( s ) + 14 m π m η ∆ ( − ) d ( t ) s − ( m η − m π ) − m π m η ∆ (+) d ( t ) s − ( m η + m π ) , ∆ ( ± ) d ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − ( m η ± m π ) U ( s (cid:48) )= U ( t ) m η ( m η ± m π )( t ± m π ( m η ∓ m π )) . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = ts (cid:0) ∆ , ( s, t ) − ∆ , (0 , t ) (cid:1) , ∆ , ( s, t ) = 2 ∆ a ( t ) + 4 ( t + m π ( m η − m π ))∆ (+) d ( t )+ 2 ( m η − m π ) ∆ ( − ) d ( t )+ 4 ( t + m π ( m η − m π ))( m η − m π ) ∆ e ( t )+ (2 s + t − m η − m π ) ∆ , ( s, t ) , ∆ e ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) = ∆ (+) d ( t ) − ∆ ( − ) d ( t )4 m π m η . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = 6 t ∆ f ( s, t ) + 6 t (2 s + t − m η − m π ) ∆ a ( t )+ 3 t (2 s + t − m η − m π ) − U ( t )2 ∆ , ( s, t ) , ∆ f ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) ( s (cid:48) − s )= U ( t ) + ( m η + m π − s ) ∆ a ( t ) . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = −
12 ∆ i ( s, t ) + 32 ∆ ( − ) j ( s, t )+ 6 ( t + m π ( m η − m π )) ∆ ( − ) j ( s, t ) − ∆ ( − ) l ( t ) s − ( m η + m π ) + 3 ( t + m π ( m η − m π )) m η m π ∆ (+) j ( s, t ) − ∆ (+) l ( t ) s − ( m η − m π ) − t + m π ( m η − m π )) m η m π ∆ ( − ) j ( s, t ) − ∆ ( − ) l ( t ) s − ( m η + m π ) , ∆ i ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) ( s (cid:48) − s ) s (cid:48) − m π U ( s (cid:48) )= (cid:18) − m π s (cid:19) ∆ c ( s, t ) + 4 m π s ∆ c (0 , t ) , ∆ ( ± ) j ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − s U ( s (cid:48) ) 1( s (cid:48) − ( m η ± m π ) ) = ∆ , ( s, t ) − ∆ ( ± ) d ( t )( s − ( m η ± m π ) ) − ∆ ( ± ) k ( t ) s − ( m η ± m π ) , ∆ ( ± ) k ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) 1( s (cid:48) − ( m η ± m π ) ) = ± m π ( m η ± m π )( t ± m π ( m η ∓ m π ))(4 m η m π ) × U ( t ) ϕ (( m η ± m π ) , t ) (cid:32) m π U ( t ) + 3 m η t ( t − m π ) ϕ (( m η ± m π ) , t ) (cid:33) , ∆ ( ± ) l ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) 1 s (cid:48) − ( m η ± m π ) = ± ∆ ( ± ) k ( t ) − ∆ c ( t )4 m η m π . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = t (2 s + t − m η − m π )∆ , ( s, t ) − t ∆ e ( t ) + 4 m π t ∆ c (0 , t )+ 3 t ∆ ( − ) k ( t ) + 6 t ( t + m π ( m η − m π )) ∆ ( − ) l ( t )+ 6 t ( t + m π ( m η − m π )) ∆ m ( t ) , ∆ m ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) = ∆ (+) l ( t ) − ∆ ( − ) l ( t )4 m η m π . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = 1 s (cid:0) ∆ , ( s, t ) − ∆ , (0 , t ) (cid:1) , ∆ , ( s, t ) = (cid:20) t + m π ( m η − m π )) s − ( m η + m π ) + ( m η − m π ) s − ( m η − m π ) + 2 ( t + m π ( m η − m π ))( m η − m π ) U ( s ) (cid:21) ∆ , ( s, t ) − m π m η t + m π ( m η − m π ))( m η + m π ) s − ( m η + m π ) ∆ (+) g ( t )+ 14 m π m η t − m π ( m η + m π ))( m η − m π ) s − ( m η − m π ) ∆ ( − ) g ( t ) , ∆ ( ± ) g ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) − ( m η ± m π ) U ( s (cid:48) ) × t (2 s (cid:48) + t − m η − m π ) − U ( t )2= 6 t ∆ ( ± ) h ( t ) + 6 t ( m η ± m η m π − m π + t )∆ a ( t )+ 3 t ( m η ± m η m π − m π + t ) − U ( t )2 ∆ ( ± ) d ( t ) , ∆ ( ± ) h ( t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) U ( s (cid:48) ) ( s (cid:48) − ( m η ± m π ) )= U ( t ) ∓ m π m η ∆ a ( t ) . • ( L , L (cid:48) ) = ( , ):∆ , ( s, t ) = 3 t (2 s + t − m η − m π ) − U ( t )2 ∆ , ( s, t )+ 6 t ∆ n ( s, t ) + 6 t (2 s + t − m η − m π )∆ o ( t ) , ∆ n ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) (1 − ss (cid:48) ) 1 U ( s (cid:48) ) × s (cid:48) (2 t + s (cid:48) − m η − m π ) − U ( s (cid:48) )2= 32 (cid:104) ∆ ( − ) d ( t ) − ( s − ( m η − m π ) )∆ ( − ) k ( t ) (cid:105) + 6 ( t + m π ( m η − m π )) × (cid:34) ∆ ( − ) k ( t ) − ( s − ( m η + m π ) ) ∆ e ( t ) − ∆ ( − ) k ( t )4 m η m π (cid:35) + 6 ( t + m π ( m η − m π )) (cid:20) ∆ (+) k ( t ) − ∆ e ( t )4 m η m π + ( s − ( m η − m π ) ) 2∆ e ( t ) − ∆ (+) k ( t ) − ∆ ( − ) k ( t )(4 m η m π ) (cid:21) − (cid:104) ∆ (+) d ( t ) − ( s − ( m η − m π ) )∆ e ( t ) (cid:105) + 2 m π [∆ e ( t ) − s ∆ c (0 , t )] , o ( s, t ) = (cid:90) s + ( t ) s − ( t ) ( C (cid:48) ) ds (cid:48) s (cid:48) U ( s (cid:48) ) × s (cid:48) (2 t + s (cid:48) − m η − m π ) − U ( s (cid:48) )2= 32 ∆ ( − ) k ( t ) + 6 ( t + m π ( m η − m π )) ∆ e ( t ) − ∆ ( − ) k ( t )4 m η m π − t + m π ( m η − m π )) e ( t ) − ∆ (+) k ( t ) − ∆ ( − ) k ( t )(4 m η m π ) − (cid:0) ∆ e ( t ) − m π ∆ c (0 , t ) (cid:1) . [1] J. Beringer et al. , Phys. Rev. D , 010001 (2012).[2] M. Battaglieri, Int.J.Mod.Phys. E19 , 837 (2010).[3] P. Eugenio, JLAB-E04-005 (2003).[4] C. Adolph et al. , Phys.Lett.
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