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We present a new amplitude analysis of the ηπ D-wave in the reaction π⁻p→ηπ⁻p measured 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 a2′(1700), and perform a comprehensive analysis of their pole positions. For the mass and width of the a2 we find M=(1307±1±6) MeV and Γ=(112±1±8) MeV, and for the excited state a2′ we obtain M=(1720±10±60) MeV and Γ=(280±10±70) MeV, respectively.
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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. nnweber 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 πp ηπpmeasured 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 πp η()πpreaction, 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 =−aJM[23]. That is,
the M=0amplitude is forbidden and the two M1amplitudes
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 πp ηπpvia 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 tteff. 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, t0, 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
dsI(s)=
tmax
tmin
dt p |a(s,t)|2Np|a(s)|2.(1)
Here, I(s)is the intensity distribution of the Dwave, p =
λ1/2(s, m2
η, m2
π)/(2s)the ηπ breakup momentum, and q =
λ1/2(s, m2
π, teff)/(2s), 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(ss).(4)
Here, the function D0(s)is real for s >sth and can be parameter-
ized as
D0(s)=c0c1sc2
c3s.(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
c3s
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
K1(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 πp 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 CDDand (b) using CDDand 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
[GeV2]
c01.5129 (fixed)GeV2a00.471
c10.532 ±0.006 GeV2a10.134
c20.253 ±0.007 GeV2a21.484
c32.38 ±0.02 GeV2a30