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Physics Letters B 779 (2018) 464–472

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

New analysis of ηπ tensor resonances measured at the COMPASS

experiment

JPAC Collaboration

A. Jackura a,b,∗,1, C. Fernández-Ramírez c,2, M. Mikhasenko d,3, A. Pilloni e,1,

V. Mathieu a,b,4, J. Nys f,4,5, V. Pauk e,1, A.P. Szczepaniak a,b,e,1,4, G. Fox g,4

aPhysics Dept., Indiana University, Bloomington, IN 47405, USA

bCenter for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47403, USA

cInstituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico

dUniversität Bonn, Helmholtz-Institut für Strahlen- und Kernph ysik , 53115 Bonn, Germany 6

eTheory Center, Thomas Jefferson National Accelerator Facility, Newport News , VA 23606, USA

fDept. of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium

gSchool of Informatics and Computing, Indiana University, Bloomington, IN 47405, USA

COMPASS Collaboration

M. Aghasyan ae, R. Akhunzyanov n, M.G. Alexeev af, G.D. Alexeev n, A. Amoroso af,ag,

V. Andrieux ai,aa, N.V. Anﬁmov n, V. Anosov n, A. Antoshkin n, K. Augsten n,y,

W. Augustyniak aj, A. Austregesilo v, C.D.R. Azevedo h, B. Badełek ak, F. Balestra af,ag,

M. Ball j, J. Barth k, R. Beck j, Y. Be dfer aa, J. Bernhard s,p, K. Bicker v,p, E.R. Bielert p,

R. Birsa ae, M. Bodlak x, P. Bo r dalo r,8, F. Bradamante ad,ae, A. Bressan ad,ae , M. Büchele o,

V.E. Burtsev ah, W.-C. Chang ab, C. Chatterjee m, M. Chiosso af,ag, I. Choi ai, A.G. Chumakov ah,

S.-U. Chung v,9, A. Cicuttin ae,10, M.L. Crespo ae,10, S. Dalla Torre ae, S.S. Dasgupta m,

S. Dasgupta ad,ae, O.Yu. Denisov ag,∗∗, L. Dhara m, S.V. Donskov z, N. Doshita am,

Ch. Dreisbach v, W. Dünnweber 11 , R.R. Dusaev ah, M. Dziewiecki al, A. Efremov n,18,

P.D. Eversheim j, M. Faessler 11, A. Ferrero aa, M. Finger x, M. Finger jr. x, H. Fischer o,

C. Franco r, N. du Fresne von Hohenesche s,p, J.M. Friedrich v,∗∗, V. Frolov n,p, E. Fuchey aa,12,

F. Gautheron i, O.P. Gavrichtchouk n, S. Gerassimov u,v, J. Giarra s, F. Giordano ai,

I. Gnesi af,ag, M. Gorzellik o,24, A. Grasso af,ag, M. Grosse Perdekamp ai, B. Grube v,

T. Grussenmeyer o, A. Guskov n, D. Hahne k, G. Hamar ae, D. von Harrach s, F.H. Heinsius o,

R. Heitz ai, F. Herrmann o, N. Horikawa w,13, N. d’Hose aa , C.-Y. Hsieh ab,14, S. Huber v,

S. Ishimoto am,15, A. Ivanov af,ag, Yu. Ivanshin n,18, T. Iwata am, V. Jary y, R. Joosten j, P. Jö r g o,

E. Kabuß s, A. Kerbizi ad,ae, B. Ketzer j, G.V. Khaustov z, Yu.A. Khokhlov z,16, Yu. Kisselev n,

F. Klein k, J.H. Koivuniemi i,ai, V.N. Kolosov z, K. Kondo am, K. Königsmann o, I. Konorov u,v,

*Corresponding author.

** Corresponding authors.

E-mail address: gerhard.mallot@cern.ch (G.K. Mallot).

1Supported by U.S. Dept. of Energy, Oﬃce of Science, Oﬃce of Nuclear Physics under contracts DE-AC05-06OR23177, DE-FG0287ER40365.

2Supported by PAPIIT- DGAPA (UNAM, Mexico) Grant No. IA101717, by CONACYT (Mexico) Grant No. 251817 and by Red Tem át ic a CONACY T de Física en Altas Energías

(Red FAE, Mexico).

3Also a member of the COMPASS Collaboration.

4Supported by National Science Foundation Grant PHY-1415459.

5Supported as an ‘FWO-aspirant’ by the Research Foundation Flanders (FWO-Flanders).

6Supported by BMBF – Bundesministerium für Bildung und Forschung (Germany).

https://doi.org/10.1016/j.physletb.2018.01.017

0370-2693/©2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by

SCOAP3.

JPAC Collaboration, COMPASS Collaboration / Physics Letters B 779 (2018) 464–472 465

V.F. Konstantinov z, A.M. Kotzinian ag,20, O.M. Kouznetsov n, Z. Kral y, M. Krämer v,

P. Kre m ser o, F. Krinner v, Z.V. Kroumchtein n,7, Y. Kulinich ai, F. Kunne aa, K. Kurek aj,

R.P. Kurjata al, I.I. Kuznetsov ah, A. Kveton y, A.A. Lednev z,7, E.A. Levchenko ah,

M. Levillain aa, S. Levorato ae, Y.-S. Lian ab,21, J. Lichtenstadt ac, R. Longo af,ag,

V.E. Lyubovitskij ah, A. Maggiora ag, A. Magnon ai, N. Makins ai, N. Makke ae,10 , G.K. Mallot p,

S.A. Mamon ah, B. Marianski aj , A. Martin ad,ae , J. Marzec al , J. Matoušek ad,ae,x,

H. Matsuda am, T. Matsuda t, G.V. Meshcheryakov n, M. Meyer ai,aa, W. Meyer i,

Yu.V. Mikhailov z, M. Mikhasenko j, E. Mitrofanov n, N. Mitrofanov n, Y. Miyachi am,

A. Nagaytsev n, F. Nerling s, D. Neyret aa, J. Nový y,p, W.-D. Nowak s, G. Nukazuka am,

A.S. Nunes r, A.G. Olshevsky n, I. Orlov n, M. Ostrick s, D. Panzieri ag,22, B. Parsamyan af,ag,

S. Paul v, J.-C. Peng ai , F. Pereira h, M. Pešek x, M. Pešková x, D.V. Peshekhonov n,

N. Pierre s,aa, S. Platchkov aa, J. Pochodzalla s, V.A. Polyakov z, J. Pretz k,17, M. Quaresma r,

C. Quintans r, S. Ramos r,8, C. Regali o, G. Reicherz i, C. Riedl ai, N.S. Rogacheva n,

D.I. Ryabchikov z,v, A. Rybnikov n, A. Rychter al, R. Salac y, V.D. Samoylenko z, A. Sandacz aj,

C. Santos ae, S. Sarkar m, I.A. Savin n,18, T. Sawada ab , G. Sbrizzai ad,ae, P. S c h iavon ad,ae,

T. Schlüter 19, K. Schmidt o,24, H. Schmieden k, K. Schönning p,23, E. Seder aa , A. Selyunin n,

L. Silva r, L. Sinha m, S. Sirtl o, M. Slunecka n, J. Smolik n, A. Srnka l, D. Steffen p,v,

M. Stolarski r, O. Subrt p,y, M. Sulc q, H. Suzuki am,13, A. Szabelski ad,ae,aj , T. Szameitat o,24,

P. Sznajder aj, M. Tasevsky n, S. Tessaro ae, F. Tessarotto ae, A. Thiel j, J. Tomsa x, F. Tosello ag ,

V. Tskhay u, S. Uhl v, B.I. Vasilishin ah, A. Vauth p, J. Veloso h, A. Vidon aa, M. Virius y,

S. Wallner v, T. Weisrock s, M. Wilfert s, J. ter Wolbeek o,24, K. Zaremba al, P. Zavada n,

M. Zavertyaev u, E. Zemlyanichkina n,18, N. Zhuravlev n, M. Ziembicki al

hUniversity of Aveiro, Dept. of Physics, 3810-193 Aveiro, Portugal

iUniversität Bochum, Institut für Experimentalphysik, 44780 Bochum, Germany 25,26

jUniversität Bonn, Helmholtz-Institut für Strahlen- und Kernph ysik , 53115 Bonn, Germany 25

kUniversität Bonn, Physikalisches Institut, 53115 Bonn, Germany 25

lInstitute of Scientiﬁc Instruments, AS CR, 61264 Brno, Czech Republic 27

mMatrivani Institute of Experimental Research & Education, Calcutta-700 030, India28

nJoint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia 18

oUniversität Freiburg, Physikalisches Institut, 79104 Freiburg, Germany 25,26

pCERN, 1211 Geneva 23, Switzerland

qTech nical University in Liberec, 46117 Liberec, Czech Republic 27

rLIP, 1000-149 Lisbon, Portugal 29

sUniversität Mainz, Institut für Kernphysik , 55099 Mainz, Germany25

tUniversity of Miyazaki, Miyazaki 889-2192, Japan 30

uLebedev Physical Institute, 119991 Moscow, Russia

vTech nisch e Universität München, Physik Dept., 85748 Garching, Germany 25,11

wNagoya University, 464 Nagoya, Japan 30

xCharles University in Prague, Faculty of Mathematics and Physics, 18000 Prague, Czech Republic 27

yCzech Te chn ic al University in Prague, 16636 Prague, Czech Republic 27

zState Scientiﬁc Center Institute for High Energy Physics of National Research Center ‘Kurchatov Institute’, 142281 Protvino, Russia

aa IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France 26

ab Academia Sinica, Institute of Physics, Ta ip ei 11529, Tai wan 31

ac Tel Aviv University, School of Physics and Astronomy, 69978 Tel Aviv, Israel 32

ad University of Trieste, Dept. of Physics, 34127 Trieste, Italy

ae Trieste Section of INFN, 34127 Trieste , Italy

af University of Turin, Dept. of Physics, 10125 Turin, Italy

ag Torino Section of INFN, 10125 Turin, Italy

ah Tomsk Polytechnic University, 634050 Tomsk, Russia 33

ai University of Illinois at Urbana-Champaign, Dept. of Physics, Urbana, IL 61801-3080, USA34

aj National Centre for Nuclear Research, 00-681 Warsaw, Poland 35

ak University of Warsaw, Faculty of Physics, 02-093 Warsaw, Poland 35

al Warsaw University of Tec hn olo gy, Institute of Radioelectronics, 00-665 Warsaw, Poland 35

am Yamagata University, Yamagata 992-8510, Japan 30

a r t i c l e i n f o a b s t r a c t

Article history:

Received 10 July 2017

Received in revised form 20 November 2017

Accepted 8 January 2018

Avail abl e online xxxx

Editor: M. Doser

We present a new amplitude analysis of the ηπ D-wave in the reaction π−p →ηπ−pmeasured by

COMPASS. Employing an analytical model based on the principles of the relativistic S-matrix, we ﬁnd

two resonances that can be identiﬁed with the a2(1320)and the excited a

2(1700), and perform a

comprehensive analysis of their pole positions. For the mass and width of the a2we ﬁnd M=(1307 ±

466 JPAC Collaboration, COMPASS Collaboration / Physics Letters B 779 (2018) 464–472

1 ±6)MeV and =(112 ±1 ±8)MeV, and for the excited state a

2we obtain M=(1720 ±10 ±60)MeV

and =(280 ±10 ±70)MeV, respectively.

©2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

The spectrum of hadrons contains a number of poorly de-

termined or missing resonances, the better knowledge of which

is of key importance for improving our understanding of Quan-

tum Chromodynamics (QCD), the fundamental theory of the strong

interaction. Active research programs in this direction are being

pursued at various experimental facilities, including the COMPASS

and LHCb experiments at CERN [1–4], CLAS/CLAS12 and GlueX at

JLab [5–7], BESIII at BEPCII [8], BaBar, and Belle [9]. In order to

connect the experimental observables like angular and momentum

distributions of ﬁnal-state particles with the corresponding degrees

of freedom of the strong interaction an amplitude analysis of the

experimental data is required. Traditionally, the mass-dependence

of partial-waves is described by a coherent sum of Breit–Wigner

amplitudes and, if needed, a phenomenological background. While

generally providing a good ﬁt to the data, such a procedure, how-

ever, violates fundamental principles of S-matrix theory. In order

to better constrain the form of the amplitude, more reliable reac-

tion models which fulﬁll the principles of unitarity and analytic-

ity (which originate from probability conservation and causality,

7Deceased.

8Also at Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal.

9Also at Dept. of Physics, Pusan National University, Busan 609-735, Republic of

Korea and at Physics Dept., Brookhaven National Laboratory, Upton, NY 11973, USA.

10 Also at Abdus Salam ICTP, 34151 Trieste, Italy.

11 Supported by the DFG cluster of excellence ‘Origin and Structure of the Uni-

verse’ (www.universe-cluster.de) (Germany).

12 Supported by the Laboratoire d’excellence P2IO (France).

13 Also at Chubu University, Kasugai, Aichi 487-8501, Japan.

14 Also at Dept. of Physics, National Central University, 300 Jhongda Road, Jhongli

32001, Taiwan.

15 Also at KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan.

16 Also at Moscow Institute of Physics and Technology, Moscow Region, 141700,

Russia.

17 Present address: RWTH Aachen Univer sity, III. Physikalisches Institut, 52056

Aachen, Germany.

18 Supported by CERN-RFBR Grant 12-02-91500.

19 Present address: LP-Research Inc., Tokyo , Japan.

20 Also at Yerev an Physics Institute, Alikhanian Br. Street, Yer ev an , Armenia, 0036.

21 Also at Dept. of Physics, National Kaohsiung Normal University, Kaohsiung

County 824, Taiwan.

22 Also at University of Eastern Piedmont, 15100 Alessandria, Italy.

23 Present address: Uppsala Univers ity, Box 516, 75120 Uppsala, Sweden.

24 Supported by the DFG Research Training Group Programmes 1102 and 2044

(Germany).

25 Supported by BMBF – Bundesministerium für Bildung und Forschung (Ger-

many).

26 Supported by FP7, HadronPhysics3, Grant 283286 (European Union).

27 Supported by MEYS, Grant LG13031 (Czech Republic).

28 Supported by B. Sen fund (India).

29 Supported by FCT – Fundação para a Ciência e Tecnologia, COMPETE and QREN,

Grants CERN/FP 116376/2010, 123600/2011 and CERN/FIS-NUC/0017/2015 (Portu-

gal).

30 Supported by MEXT and JSPS, Grants 18002006, 20540299, 18540281 and

26247032, the Daiko and Yam ad a Foundations (Japan).

31 Supported by the Ministry of Science and Technology (Taiwan).

32 Supported by the Israel Academy of Sciences and Humanities (Israel).

33 Supported by the Russian Federation program “Nauka” (Contract No.

0.1764.GZB.2017) (Russia).

34 Supported by the National Science Foundation, Grant no. PHY-1506416 (USA).

35 Supported by NCN, Grant 2015/18/M/ST2/00550 (Poland).

respectively) should be applied. When resonances dominate the

spectrum, which is the case studied here, unitarity is especially

important since it constrains resonance widths and allows us to

determine the location of resonance poles in the complex energy

plane of the multivalued partial wave amplitudes.

In 2014, COMPASS published high-statistics partial-wave analy-

ses of the π−p →η()π−preaction, at pbeam =191 GeV [2]. The

waves with odd angular momentum between the two pseudoscalar

particles in the ﬁnal state have manifestly spin-exotic quantum

numbers and were found to exhibit structures that may be com-

patible with a hybrid meson [10,11]. The even angular-momentum

waves show strong signals of non-exotic resonances. In particular,

the D-wave of ηπ , with IG(JPC ) =1−(2++), is dominated by the

peak of the a2(1320)and its Breit–Wigner parameters were ex-

tracted and presented in Ref. [2]. The D-wave also exhibits a hint

of the ﬁrst radial excitation, the a

2(1700)[12].

In this letter we present a new analysis of the ηπ D-wave

based on an analytical model constrained by unitarity, which

extends beyond the simple Breit–Wigner parameterization. Our

model builds on a more general framework for a systematic anal-

ysis of peripheral meson production, which is currently under

development [13–15]. Using the 2014 COMPASS measurement as

input, the model is ﬁtted to the results of the mass-independent

analysis that was performed in 40 MeV wide bins of the ηπ mass.

The a2and a

2resonance parameters are extracted in the single-

channel approximation and the coupled-channel effects are esti-

mated by including the ρπ ﬁnal state. We determine the statistical

uncertainties by means of the bootstrap method [16–20], and as-

sess the systematic uncertainties in the pole positions by varying

model-dependent parameters in the reaction amplitude.

2. Reaction model

We consider the peripheral diffractive production process

πp →ηπ p(Fig. 1(a)), which is dominated by Pomeron (P) ex-

change at high energies of the incoming beam particle. This allows

us to assume factorization of the “top” vertex, so that the πP →

ηπ amplitude resembles an ordinary helicity amplitude [21]. It is

a function of sand t1, the ηπ invariant mass squared and the

invariant momentum transfer squared between the incoming pion

and the η, respectively. It also depends on t, the momentum trans-

fer between the nucleon target and recoil. In the Gottfried–Jackson

(GJ) frame [22], the Pomeron helicity in πP →ηπ equals the ηπ

total angular momentum projection M, and the helicity ampli-

tudes aM(s, t, t1)can be expanded in partial waves aJM(s, t)with

total angular momentum J=L. The allowed quantum numbers

of the ηπ partial waves are JP=1−, 2+, 3−, .... The exchanged

Pomeron has natural parity. Parity conservation relates the am-

plitudes with opposite spin projections aJM =−aJ−M[23]. That is,

the M=0amplitude is forbidden and the two M=±1amplitudes

are given, up to a sign, by a single scalar function.

The assumption about the Pomeron dominance can be quanti-

ﬁed by the magnitude of unnatural partial waves. In the analysis

of Ref. [2], the magnitude of the L =M=0 wave, which also ab-

sorbs other possible reducible backgrounds, was estimated to be

<1%. We are unable to further address the nature of the exchange

from the data of Ref. [2], since the analysis was performed at a sin-

JPAC Collaboration, COMPASS Collaboration / Physics Letters B 779 (2018) 464–472 467

Fig. 1. (a) Rea ction diagram of π−p →ηπ−pvia Pomeron exchange. (b) Unitarity

diagram: the πP →ηπ amplitude is expanded in partial waves in the s-channel of

the ηπ system, aJM(s), with J=Land t→teff. Unitarity relates the imaginary part

of the amplitude to ﬁnal state interactions that include all kinematically allowed

intermediate states n.

gle beam energy and integrated over the momentum transfer t.36

Analyzes such as Ref. [24] suggest that fexchange could also con-

tribute. Since in our analysis we do not discriminate between dif-

ferent natural-parity exchanges, we consider an effective Pomeron

which may be a mixture of pure Pomeron and f. The patterns of

azimuthal dependence in the central production of mesons [25–29]

indicate that at low momentum transfer, t∼0, the Pomeron be-

haves as a vector [30,31], which is in agreement with the strong

dominance of the |M| =1 component in the COMPASS data.37

The COMPASS mass-independent analysis [2] is restricted to

partial waves with L =1to 6 and |M| =1(except for L =2 where

also the |M| =2 wave is taken into account). The lowest-mass ex-

changes in the crossed channels of πP →ηπ correspond to the a

(in the t1channel) and the f(in the u1channel) trajectories,

thus higher partial waves are not expected to be signiﬁcant in the

ηπ mass region of interest, √s<2GeV. Systematic uncertainties

due to truncation of higher waves were found to be negligible in

Ref. [34].

In order to compare with the partial-wave intensities measured

in Ref. [2], which are integrated over tfrom tmin =−1.0GeV

2

to tmax =−0.1GeV

2, we use an effective value for the momen-

tum transfer teff =−0.1GeV

2and aJM(s) ≡aJM(s, teff). The effect

of a possible teff dependence is taken into account in the esti-

mate of the systematic uncertainties. The natural-parity exchange

partial-wave amplitudes aJM(s)can be identiﬁed with the ampli-

tudes Aε=1

LM (s)as deﬁned in Eq. (1) of Ref. [2], where ε=+1is the

reﬂectivity eigenvalue that selects the natural-parity exchange.

In the following we consider the single, J=2, |M| =1 natural-

parity partial wave, which we denote by a(s), and ﬁt its mod-

ulus squared to the measured (acceptance-corrected) number of

events [2]:

36 For example, Ref. [24] suggested a dominance of f2exchanges for a2(1320)pro-

duction. To probe this, one should analyze the tand total energy dependences. We

note here that COMPASS has published data in the 3πchannel, which are binned

both in 3πinvariant mass and momentum transfer t[3], which may give further

insight into the production process.

37 At low t, the Pomeron trajectory passes through J=1, while at larger, posi-

tive t, the trajectory is expected to pass though J=2where it would relate to the

tensor glueball [32,33].

dσ

d√s∝I(s)=

tmax

tmin

dt p |a(s,t)|2≡Np|a(s)|2.(1)

Here, I(s)is the intensity distribution of the Dwave, p =

λ1/2(s, m2

η, m2

π)/(2√s)the ηπ breakup momentum, and q =

λ1/2(s, m2

π, teff)/(2√s), which will be used later, is the πbeam

momentum in the ηπ rest frame with λ(x, y, z) =x2+y2+z2−

2xy −2xz −2yz being the Källén triangle function. Since the physi-

cal normalization of the cross section is not determined in Ref. [2],

the constant Non the right hand side of Eq. (1) is a free parame-

ter.

In principle, one should consider the coupled-channel problem

involving all the kinematically allowed intermediate states (see

Fig. 1(b)). For the 2++ states, the PDG reports the 3π(ρπ, f2π)

and ηπ ﬁnal states as dominant decay channels [12]. Far from

thresholds, a narrow peak in the data is generated by a pole in

the closest unphysical sheet, regardless of the number of open

channels. The residues (related to the branching ratios) depend

on the individual couplings of each channel to the resonance, and

therefore their extraction requires the inclusion of all the relevant

channels. However, the pole position is expected to be essentially

insensitive to the inclusion of multiple channels. This is easily un-

derstood in the Breit–Wigner approximation, where the total width

extracted for a given state is independent of the branchings to in-

dividual channels. Thus, when investigating the pole position, we

restrict the analysis to the elastic approximation, where only ηπ

can appear in the intermediate state. We will elaborate on the ef-

fects of introducing the ρπ channel, which is known to be the

dominant one of the decay of a2(1320)[12], as part of the sys-

tematic checks.

In the resonance region, unitarity gives constraints for both the

ηπ interaction and production. Denoting the ηπ →ηπ scattering

D-wave by f(s), unitarity and analyticity determine the imaginary

part of both amplitudes above the ηπ threshold sth =(mη+mπ)2:

Im ˆ

a(s)=ρ(s)ˆ

f∗(s)ˆ

a(s), (2)

Im ˆ

f(s)=ρ(s)|ˆ

f(s)|2,(3)

with ρ(s) =2p5/

√sbeing the two-body phase space factor that

absorbs the barrier factors of the D-wave. From the analysis of

kinematical singularities [35–37] it follows that the amplitude a(s)

appearing in Eq. (1) has kinematical singularities proportional to

K(s) =p2q, and f(s)has singularities proportional to p4. The re-

duced partial waves in Eqs. (2) and (3) are free from kinematical

singularities, and deﬁned by e.g. ˆ

a(s) =a(s)/K(s), ˆ

f(s) =f(s)/p4.

Note that Eq. (2) is the elastic approximation of Fig. 1(b).

We write ˆ

fin the standard N-over-Dform, ˆ

f(s) =N(s)/D(s),

with N(s)absorbing singularities from exchange interactions, i.e.

“forces” acting between ηπ also known as left-hand cuts, and D(s)

containing the right-hand cuts that are associated with direct-

channel thresholds. Unitarity leads to a relation between Dand N,

Im D(s) =−ρ(s)N(s), with the general once-subtracted integral so-

lution

D(s)=D0(s)−s

π

∞

sth

dsρ(s)N(s)

s(s−s).(4)

Here, the function D0(s)is real for s >sth and can be parameter-

ized as

D0(s)=c0−c1s−c2

c3−s.(5)

Note that the subtraction constant has been absorbed into c0of

D0(s). The rational function in Eq. (5) is a sum over two so-called

468 JPAC Collaboration, COMPASS Collaboration / Physics Letters B 779 (2018) 464–472

Castillejo–Dalitz–Dyson (CDD) poles [38], with the ﬁrst pole lo-

cated at s =∞(CDD∞) and the second one at s =c3. The CDD

poles produce real zeros of the amplitude ˆ

fand they also lead to

poles of ˆ

fin the complex plane (second sheet). Since these poles

are introduced via parameters like c1, c2, rather than being gen-

erated through N(cf. Eq. (4)), they are commonly attributed to

genuine QCD states, i.e. states that do not originate from effective,

long-range interactions such as pion exchange [39]. In order to ﬁx

the arbitrary normalization of N(s)and D(s), we set c0to O(1),

since it is expected to be of the order of the a2mass squared ex-

pressed in units of GeV2. One also expects c1to be approximately

equal to the slope of the leading Regge trajectory [40]. The quark

model [41] and lattice QCD [42] predict two states in the energy

region of interest, so we use only two CDD poles. It follows from

Eq. (4) that the singularities of N(s)(which originate from the ﬁ-

nite range of the interaction) will also appear on the second sheet

in D(s), together with the resonance poles generated by the CDD

terms. We use a simple model for N(s), where the left-hand cut is

approximated by a higher-order pole,

ρ(s)N(s)=gλ5/2(s,m2

η,m2

π)

(s+sR)n.(6)

Here, gand sReffectively parameterize the strength and inverse

range of the exchange forces in the D-wave, respectively. The

power n =7is our model for the left-hand singularities in N(s).

This includes the effects of the ﬁnite range of interaction, i.e. the

regularization of the threshold singularities due to K(s) =p2q. The

parameterization of N(s)removes the kinematical 1/ssingularity

in ρ(s). Therefore, dynamical singularities on the second sheet are

either associated with the particles represented by the CDD poles,

or the exchange forces parameterized by the higher order pole

in N(s).

The general parameterization for ˆ

a(s), which is constrained by

unitarity in Eq. (2), is obtained following similar arguments and is

given by a ratio of two functions

ˆ

a(s)=n(s)

D(s),(7)

where D(s)is given by Eq. (4) and brings in the effects of ηπ

ﬁnal-state interactions, while n(s)describes the exchange interac-

tions in the production process πP →ηπ and contains the asso-

ciated left-hand singularities. In both the production process and

the elastic scattering no important contributions from light-meson

exchanges are expected since the lightest resonances in the t1and

u1channels are the a2and f2mesons, respectively. Therefore, the

numerator function in Eq. (7) is expected to be a smooth function

of sin the complex plane near the physical region, with one ex-

ception: the CDD pole at s =c3produces a zero in ˆ

a(s). Since a

zero in the elastic scattering amplitude does not in general imply

a zero in the production amplitude, we write n(s)as

n(s)=1

c3−s

np

j

ajTj(ω(s)), (8)

where the function to the right of the pole is expected to be

analytical in snear the physical region. We parameterize it us-

ing the Chebyshev polynomials Tj, with ω(s) =s/(s +) ap-

proximating the left-hand singularities in the production process,

πP →ηπ. The real coeﬃcients ajare determined from the ﬁt to

the data. In the analysis, we ﬁx =1GeV

2. We choose an ex-

pansion in Chebyshev polynomials as opposed to a simple power

series in ωto reduce the correlations between the ajparam-

eters. Since we examine the partial-wave intensities integrated

over the momentum transfer t, we assume that the expansion

coeﬃcients are independent of t. The only t-dependence comes

from the residual kinematical dependence on the breakup momen-

tum q.

A comment on the relation between the N-over-D method and

the K-matrix parameterization is worth making. If one assumes

that there are no left-hand singularities, i.e. let N(s)be a con-

stant, then Eq. (4) is identical to that of the standard K-matrix

formalism [43]. Hence we can relate both approaches through

K−1(s) =D0(s). It is also worth noting that the parameterization

in Eq. (5) automatically satisﬁes causality, i.e. there are no poles on

the physical energy-sheet.

3. Methodology

We ﬁt our model to the intensity distribution for π−p →ηπ −p

in the D-wave (56 data points) [2], as deﬁned in Eq. (1), by min-

imizing χ2. We ﬁx the overall scale, N=106(see Eq. (1)), and

ﬁt the coeﬃcients aj(see Eq. (8)), which are then expected to be

O(1), and also the parameters in the D(s)function. In the ﬁrst

step we obtain the best ﬁt for a given total number of parameters,

and in the second step we estimate the statistical uncertainties us-

ing the bootstrap technique [16–20]. That is to say, we generate

105pseudodata sets, each data point being resampled according

to a Gaussian distribution having as mean and standard deviation

the original value and error, and we repeat the ﬁt for each set.

In this way, we obtain 105different values for the ﬁt parameters,

and we take the means and standard deviations as expected values

and statistical uncertainties, respectively. The use of the bootstrap

method allows us to determine the correlations between the pole

positions and the production parameters, provided as supplemen-

tal material. As expected, the production parameters are highly

correlated among each other, but their correlations with the pole

positions are rather low. This justiﬁes the choice of Chebyshev

polynomials; similar studies with a standard polynomial expansion

showed larger correlations between production and resonance pa-

rameters.

In order to assess the systematic uncertainties we study the

dependence of the pole parameters on variations of the model.

Speciﬁcally, we change i)the number of CDD poles from 1 to 3,

ii)the total number of terms npin the expansion of the numera-

tor function n(s)in Eq. (8), iii)the value of sRin the left-hand-cut

model, iv)the value of teff of the total momentum transfered, and

v)the addition of the ρπ channel to study coupled-channel ef-

fects.

In order to determine sR, we scan the model with various val-

ues of sR, ranging from 0.1 to 10.0 GeV2, and ﬁnd that values

near sR=1.5GeV

2give a minimum in χ2. This choice is also

justiﬁed by phenomenological studies where the ﬁnite range of

strong interactions is of the order of 1 GeV. The ﬁt with CDD∞

only, shown in Fig. 2(a), for sR=1.5GeV

2and np=6(with a

total of 9 parameters), captures neither the dip at 1.5GeV nor

the bump at 1.7GeV. In contrast, the ﬁt with two CDD poles

(11 parameters), shown in Fig. 2(b), captures both features, giving

a χ2/d.o.f. =86.17/(56 −11) =1.91. The χ2/d.o.f. is somewhat

large, due to the small statistical uncertainties of the data. How-

ever, the residuals do not show any systematic deviation, which

supports the quality of the ﬁt (see residuals normalized bin-by-

bin to the corresponding uncertainty in Fig. 2). The parameters

corresponding to the best ﬁt with two CDD poles are given in

Table 1. The addition of another CDD pole does not improve the

ﬁt, as a ﬁt to the intensity only is incapable of indicating any

further resonances. Speciﬁcally the residue of the additional pole

JPAC Collaboration, COMPASS Collaboration / Physics Letters B 779 (2018) 464–472 469

Fig. 2. Intensity distribution and ﬁts to the JPC =2++ wave for different number of CDD poles, (a) using only CDD∞and (b) using CDD∞and the CDD pole at s =c3. Red

lines are ﬁt results with I(s)given by Eq. (1). Data is taken from Ref. [2]. The inset shows the a

2region. The error bands correspond to the 3σ(99.7%) conﬁdence level. The

lower plot shows the residuals normalized bin-by-bin to the corresponding uncertainty. The dashed lines indicate the 3σdeviations. (For interpretation of the colors in the

ﬁgure(s), the reader is referred to the web version of this article.)

Tabl e 1

Best ﬁt denominator and production parameters for the ﬁt with two CDD poles,

sR=1.5GeV

2, N=106, c0=(1.23)2, and the number of expansion parameters

np=6, leading to χ2/d.o.f. =1.91. Denominator uncertainties are determined from

a bootstrap analysis using 105random ﬁts. We report no uncertainties on the pro-

duction parameters as they are highly correlated.

Denominator parameters Production parameters

[GeV−2]

c01.5129 (ﬁxed)GeV2a00.471

c10.532 ±0.006 GeV−2a10.134

c20.253 ±0.007 GeV2a2−1.484

c32.38 ±0.02 GeV2a30