Eu . Phys. J. C (2023) 83:954
h ps://doi.o g/10.1140/epjc/s10052-023-12131-4
Regula A icle - Theo e ical Physics
Theo e ical in e p e a ion o he (1620)and (1690) esonances
seen in +
c→−π+π+decay
Hai-Peng Li1, Gong-Jie Zhang1, Wei-Hong Liang1,2,a,E.Ose
1,3
1Depa men o Physics, Guangxi No mal Uni e si y, Guilin 541004, China
2Guangxi Key Labo a o y o Nuclea Physics and Technology, Guangxi No mal Uni e si y, Guilin 541004, China
3Depa amen o de Física Teó ica and IFIC, Cen o Mix o Uni e sidad de Valencia-CSIC Ins i u os de In es igación de Pa e na, Ap do.22085,
46071 Valencia, Spain
Recei ed: 25 Augus 2023 / Accep ed: 9 Oc obe 2023 / Published online: 24 Oc obe 2023
© The Au ho (s) 2023
Abs ac We s udy he Belle eac ion +
c→−π+π+
looking a he mass dis ibu ion o π+, whe e clea signals
o he (1620)and (1690) esonances a e seen. These
wo esonances a e gene a ed dynamically om he in e ac-
ion in coupled channels o π, ¯
K, ¯
Kand η wi hin
he chi al uni a y app oach. Ye , he weak decay p ocess a
he qua k le el, oge he wi h he had oniza ion o p oduce
pai s o mesons, does no p oduce he ππ inal s a e. In
o de o p oduce his s a e one mus make ansi ions om
he ¯
K, ¯
Kand η componen s o π, and his in e ac-
ion is wha p oduces he esonances. So, he eac ion o e s
a good es o he molecula pic u e o hese esonances.
Adding he con ibu ion o he ∗(1530)and some back-
g ound we a e able o ge a good ep oduc ion o he mass
dis ibu ion showing he signa u es o he wo esonances as
ound in he expe imen .
1 In oduc ion
The s a es a e s ill ela i ely poo ly known and only a ew
o hem ha e been obse ed [1]. One o hem, he (1690),
has a h ee s a s a us, and ano he one, he (1620), is a ed
wi h only one s a , and bo h o hem appea wi h unknown
spin-pa i y. The (1620)is obse ed in (1620)→π
decay wi h la ge s a is ical unce ain ies [2–4] and is no
ound in Re . [5]. F om he heo e ical poin o iew he
qua k models sys ema ically gi e he i s exci ed s a e, a e
he (1
2+),(3
2+)g ound s a es, a abou 1750 MeV [6–10]
and so does he algeb aic model o Re . [11]. One excep ion
is he Sky me model o Re . [12], assuming a soli on-hsys-
em, whe e wo s a es o 1
2−wi h 1616 MeV and 1658 MeV
ae-mail: [email p o ec ed] (co esponding au ho )
a e ob ained. The si ua ion wi h he (1620)s a e is simila
o ha o he (1405), whe e qua k models o e es ima e he
ene gy. In bo h cases, he ad en o chi al dynamics, imple-
men ed wi h coupled channels and uni a i y, came o he es-
cue and p oduced a global pic u e whe e hese s a es appea
na u ally [13]. In pa icula , in Re s. [14,15] he(1620)
and (1690)appea ed bo h wi h JP=1
2−, coming om
he in e ac ion o he coupled channels π,¯
K,¯
K,η.
The ad en o new acili ies as Belle, LHCb, BES has
added a new dimension o had on spec oscopy and new
s a es o old ones wi h excellen s a is ics a e being obse ed
[16–18]. In pa icula weak decays o hea y had ons ha e
added a new dimension o he in o ma ion a ailable on
dynamically gene a ed esonances, s emming om he in e -
ac ion o had on s a es [19]. In ha line, i was sugges ed
in Re . [20] o s udy he c→ππ eac ion and look o
he (1620)and (1690) esonances in he π spec um.
Such eac ion has been ecen ly measu ed a Belle [21] and
excellen signals o hese esonances ha e been obse ed.
The pu pose o he p esen wo k is o use his expe imen al
in o ma ion o u he dig in o he na u e o hese esonances.
2 Gene a ion o he (1620)and (1690)s a es
Using he chi al uni a y app oach, we ollow closely he
wo k o Re . [14] and conside he π sca e ing wi h he
π, ¯
K, ¯
K,η coupled channels. The s-wa e sca e ing
ampli udes in ma ix o m a e gi en by means o he Be he–
Salpe e (BS) equa ion
T=[1−VG]−1V,(1)
whe e Vij, he ansi ion po en ial be ween channels, is
ob ained om he chi al Lag angians and gi en explici ly
123
954 Page 2 o 7 Eu . Phys. J. C (2023) 83 :954
Table 1 Poles o he Tma ix
wi h di e en alues o qmax (in
MeV)
qmax 630 700 750 770
Poles 1569.4+125.7i1563.7+106.1i1558.0+94.0i1555.6+89.7i
1687.9+0.7i1681.8+1.8i1674.8+2.3i1671.5+2.4i
Table 2 Couplings o he wo
gene a ed s a es o di e en
channels (wi h
qmax =630 MeV)
1569.4+i125.7π ¯
K¯
Kη
gi−2.0−1.6i1.9+0.9i0.7+0.5i−0.1−0.4i
|gi|26.64.50.60.1
1687.9+i0.7π ¯
K¯
Kη
gi0.1−0.1i0.2+0.1i−1.2+0.2i−0.8+0.1i
|gi|20.01 0.04 1.50.6
Bold alues indica e he pole posi ion o he s a es
in Re . [14]. The G unc ion is he diagonal loop unc ion
o he meson–ba yon p opaga o s. He e we di e a bi om
Re . [14] and use he cu o me hod o egula ize he loop
unc ion, gi en by
Gl=|q|<qmax
d3q
(2π)3
1
2wl(q)
Ml
El(q)
1
√s−wl(q)−El(q)+i,
(2)
whe e wl(q)=m2
l+q2,El(q)=M2
l+q2, and
ml,Mla e he meson, ba yon masses o he channel l.Gl
is egula ized cu ing wi h a h ee momen um qmax.Thisis
he only pa ame e in he heo y, and inpu om expe imen
is needed o de e mine i . The esul s om he Belle expe -
imen [21] p o ide his in o ma ion. Then he BS equa ion
o Eq. (1) is sol ed, and poles o he Tma ix in he second
Riemann shee a e sea ched o .
The e exis wo poles ound in he Tampli ude, which a e
shown in Table 1wi h di e en alues o he cu o qmax.
The wo poles co espond o he ∗ esonances gene a ed
dynamically om he coupled-channel π in e ac ion, ha -
ing he quan um numbe s o isospin I=1
2and spin-pa i y
JP=1
2−. The i s pole loca es below he ¯
K h eshold
(∼1611 MeV)and has a a he wide wid h. While he sec-
ond pole appea s nea he ¯
K h eshold (∼1689 MeV),
wi h a na ow wid h. We iden i y he i s pole and he sec-
ond pole as he (1620)and (1690)s a es, espec i ely.
Acco ding o Table 1, as he cu o qmax inc eases om 630
o 770 MeV, he posi ions o he wo poles dec ease by less
han 17 MeV. This means ha he masses o he (1620)
and (1690)s a es a e no e y much dependen on qmax.
Bu hei wid hs a e mo e sensi i e o he cu o .
To u he unde s and he molecula componen s o he
(1620)and (1690)s a es, we calcula ed he couplings o
he wo esonances o di e en channels, which a e showed
in Table 2when aking qmax =630 MeV.
One can see ha he (1620)s a e couples mainly o
he π and ¯
Kchannels, while he (1690)s a e couples
s ongly o he ¯
Kchannel. In Re . [14] whe e dimensional
egula iza ion is used o egula ize he loop unc ion, only
one exci ed s a e wi h mass a ound 1600 MeV, which is
iden i ied as he (1620), is gene a ed om he π in e ac-
ion in coupled channels. Howe e , he e was a s ong cusp
a ound he ¯
K h eshold ha could be iden i ied wi h he
(1690) esonance, see Fig. 1 o Re . [14]. This si ua ion
is common in heo e ical s udies when poles appea close
o a h eshold. The e is a con inui y, wi h a smoo h ansi-
ion om a e y weakly bound s a e o a sligh ly unbound
o i ual s a e, p oduced by small changes in he egula -
iza ion o he loops o he s eng h o he in e ac ion. Fo
us he e is no much di e ence be ween he sligh ly bound
and he sligh ly unbound s a es, hey co espond o he same
dynamics and he expe imen al signals a e also simila . In
he sligh ly unbound s a e, he signal is seen as a sha p peak,
ypical o a cusp. Le us men ion in passing ha some s a es
admi ed as esonances, like he a0(980)co espond o ha
case. One can see his in ecen high esolu ion expe imen s
[22], as well as in heo e ical s udies [23,24]. Thus, he e is
no p oblem o b and he (1690)also as a esonance e en
i i is a bo de line s a e be ween sligh ly bound and sligh ly
unbound.
3The+
c→π+π+− eac ion
He e we ollow closely he o malism o Re . [20]. A he
qua k le el he eac ion o p oduce he (1620)and (1690)
p oceeds as depic ed in Fig. 1, ollowed by he had oniza ion
o he inal qua ks, whe e a ¯qq s a e wi h he quan um num-
be s o he acuum is c ea ed o p oduce a meson–ba yon
pai .
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Eu . Phys. J. C (2023) 83 :954 Page 3 o 7 954
Fig. 1 Feynman diag am a qua k le el o he i s s ep o he +
c→
−π+π+decay
Fig. 2 The esca e ing mechanism o +
c→π+π+−decay
The o iginal s,uqua ks a e spec a o s in he weak p o-
cess and a e in a s a e 1
√2(su −us)[25]. The weak p o-
cess is doubly Cabibbo a o ed and he had oniza ion mus
necessa ily in ol e he c ea ed squa k a e he weak e -
ex because in o de o ha e he (1620)s a e wi h nega-
i e pa i y in +
c→π+0(1620)(1
2−), he c ea ed squa k
mus ha e o bi al angula momen um L=1. Ye , when
he pseudoscala -ba yon coupled channels o Sec . 2a e c e-
a ed in he had oniza ion, all qua ks a e in hei g ound s a e,
which o ces his squa k o pa icipa e in he had oniza ion
p ocess o ge deexci ed. Bookkeeping he meson and ba yon
s a es coming om he had oniza ion, i is ound in Re . [20]
ha he had onic s a e coming a e his p ocess is gi en by
H=K−+−1
√2¯
K00+1
√6¯
K0−1
√3η
0,(3)
whe e he ηcomponen , which plays no ole in he (1620)
and (1690) o ma ion, has been neglec ed.
I is su p ising ha he πcomponen does no appea in
Eq. (3), bu i is p ecisely his ea u e wha s esses he p o-
duc ion o a dynamically gene a ed esonance, because he
π inal s a e comes a e esca e ing o he componen s o
Eq. (3) ogi eπ in he inal s a e, and i is p ecisely his
in e ac ion he one ha p oduces he esonances. Diag am-
ma ically, his is depic ed in Fig. 2, and analy ically he decay
ampli ude co esponding o he mechanism o Fig. 2is gi en
by
=VP
4

i=1
hiGi(Min ) i,π+−,(4)
Fig. 3 Mechanism o ∗(1530)( 3
2+)p oduc ion
whe e VPis a global ac o , i,π+−a e he ansi ion ampli-
udes om channel i(each s a e o Eq. (3)) o π+−, and
hia e he weigh s o each o hese channels in Eq. (3),
hK−+=1;h¯
K00=− 1
√2;
h¯
K0=1
√6;hη0=− 1
√3.(5)
Since he I=3
2sec o does no con ibu e o gene a ing
he (1620)and (1690), we neglec i . Conside ing he
phase con en ion o he isospin mul iple s (−+,0,−),
(¯
K0,−K−),(−π+,π0,π−), and (0,−−), he i,π+−
ampli udes a e ob ained om he ampli udes gene a ed in
Eq. (1) in isospin basis by means o
K−+,π+−=−2
3 I=1/2
¯
K,π,
¯
K00,π+−=√2
3 I=1/2
¯
K,π,
¯
K0, π+−=2
3 I=1/2
¯
K, π,
η0,π+−=2
3 I=1/2
η, π.(6)
In he expe imen o Re . [21], apa om he signals o
he (1620)and (1690)one can see a clean peak o he
∗(1530)(3
2+). I is in e es ing o see how his is p oduced. I
canno be p oduced wi h he mechanism o ex e nal emission
o Fig. 1because he inal wa e unc ion is 1
√2s(su −us),
which is mixed an isymme ic and o hogonal o he ully
symme ic (ssu +sus +uss)o he ∗(1530)( 3
2+). Then
one needs in e nal emission wi h qua k ea angemen as
depic ed in Fig. 3.
This obse a ion is ele an because he ex e nal emis-
sion p ocess has bigge s eng h han in e nal emission, bu
he had oniza ion also leads o a educ ion wi h espec o a
p ocess ha equi es no had oniza ion, as he one in Fig. 3.As
a consequence o his, one can quali a i ely unde s and ha
he in eg a ed s eng hs o he exci a ion o he ∗(1530)and
(1620)a e simila , as one can obse e in he expe imen .
We a e only conce ned he e abou he (1620)and (1690),
bu o compa e ou esul s wi h expe imen we will accoun
o he con ibu ion o he ∗(1530)empi ically.
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954 Page 4 o 7 Eu . Phys. J. C (2023) 83 :954
4 Resul s
The di e en ial wid h dis ibu ion in e ms o he π+−
in a ian mass is gi en by
d
dMin =1
(2π)3
1
4M2
c
pπ+˜pπ+| |2,(7)
whe e is gi en by Eq. (4) o (1620)and (1690).The
ampli udes o Eq. (6) con ain he con ibu ion o bo h eso-
nances. In Eq. (7)pπ+,˜pπ+a e he momen um o he π+
p oduced in he weak p ocess, Fig. 1, and he momen um
o he π+p oduced in he inal π+−s a e, in he cand
π+− es ames, espec i ely,
pπ+=λ1/2(M2
c,m2
π,M2
in )
2Mc
,
˜pπ+=λ1/2(M2
in ,m2
π,M2
−)
2Min
.
The exis ence o wo π+in he inal s a e o he decay,
π+π+−, poses in p inciple p oblems in he iden i ica ion
since hey a e iden ical pa icles. Howe e , in he analysis
o he expe imen , a mos use ul s a egy is ollowed, dis-
inguishing he πLand πHwhe e πLco esponds o he
pion wi h lowe momen um and πH o he pion wi h highe
momen um. I is an in e es ing kinema ical exe cise o see
ha bo h o (1620)and (1690),aswellas(1530)p o-
duc ion, πHco esponds o he pion p oduced in he weak
e ex (Figs. 1and 3), while πLco esponds o he one o
π+−decay o he (1620),(1690),(1530)s a es in
he ange o in a ian masses ha we s udy. The o mula
o Eq. (7), wi h π+co esponding o πHand ˜pπ+ o πL,
has Min o med om he π+
Land −, consis en ly wi h
he o me discussion. Hence, a measu emen o d
dMin wi h
M2
in =(pπ+
L+p−)2p o ides he magni ude o be com-
pa ed o ou esul s. This magni ude is ac ually p o ided in
he analysis o Re . [21], allowing a di ec compa ison o ou
esul s wi h he da a. No e ha his dis inc ion was no done
in Re . [20], whe e he 0π0decay mode was sugges ed o
a oid his p oblem.
Ye , he da a con ain also he signal o (1530)p oduc-
ion and some backg ound. We pa ame ize hem in he ol-
lowing way. The ∗(1530)(3
2+)p oduc ion can be depic ed
as in Fig. 2subs i u ing he loop unc ion by he ∗(1530)
p opaga o . Bu he c(1
2+)π+(3
2+) e ex equi es p-
wa e by angula momen um conse a ion and he (3
2+)
also decays in p-wa e in o π. Thus we mus in oduce an
ex a | |2o he ype
| |2=B2|pπ+|2|˜pπ+|2
1
Min −M(1530)+i(1530)/2
2.
(8)
Ob iously his e m does no in e e e wi h he s-wa e p o-
duc ion o he (1620)and (1690). We can conside he
e ec o 3
2+backg ound by adding a e m C2 o he p op-
aga o squa e in Eq. (8). In addi ion we can also add a 1
2+
backg ound, which will ha e s-wa e in he weak e ex and
p-wa ein heπ+− inal s a e. This e m also does no in e -
e e wi h he o he wo. Al oge he we can calcula e d
dMin
using Eq. (7) and subs i u ing
| |2→| |2+B2|pπ+|2|˜pπ+|2

1
Min −M(1530)+i(1530)/2
2+C2+D2|˜pπ+|2.
(9)
As o possible backg ound om 1
2−we do no need o ake
i in o accoun in he egion o he (1620)and (1690)
esonances. We ha e seen ha 1
2−comes om esca e ing
and he chi al ampli udes a e good in a easonable ange o
ene gies, con aining much mo e in o ma ion han jus he
poles o he esonances. The e a e i e pa ame e s in o al,
qmax,V2
p,B2,C2,D2,
which a e used o i he expe imen al da a om Belle [21].
The pa ame e VP egula es he s eng h o he (1620)
and he pa ame e qmax de e mines he posi ion and ela-
i e s eng h o he (1620) o he (1690). The o he h ee
pa ame e s i he (1530)signal and he backg ound.
Wi h he aim o seeing mo e clea ly he e ec o he
indi idual pa ame e s on he inal esul , we p esen he
esul s wi h di e en se s o pa ame e s. In Figs. 4,5,6,7
and 8, we show he −π+
Lin a ian mass dis ibu ion o
+
c→π+
Hπ+
L−decay wi h di e en se s o pa ame e s,
and he compa ison wi h he expe imen al da a om Belle
[21], whe e he black poin s wi h e o ba a e Belle da a
and he blue lines a e ou heo e ical esul s. The ag eemen
wi h he da a is qui e good, bu one inds ha he esul s a e
inconsis en wi h expe imen al da a below 1520 MeV.
I can be seen ha he pa ame e Bmainly de e mines
he s eng h o he (1530), om Figs. 4and 5. Likewise,
one can know ha he pa ame e s C2and D2a ec he global
s eng h and he pa ame e D2mainly a ec s he high ene gy
egion, when compa ing Figs. 4,6and 7. The posi ion and
ela i e s eng h o he (1620) o he (1690)change when
qmax and V2
pchange as shown in Figs. 4and 8. We calcula e
he χ2in he in a ian mass o [1530,1750]MeV, so as o
pick ou be e se s o pa ame e s. The χ2/nisshownin he
igu e.
In he wo k we conside ha (1620)and (1690)a e
molecula s a es, wi h I(JP)=1
2(1
2−). The heo e ical
esul s a e in good ag eemen wi h he expe imen al da a in
he ange o [1530,1750]MeV, and suppo he molecula
123
Eu . Phys. J. C (2023) 83 :954 Page 5 o 7 954
Fig. 4 −π+
Lin a ian mass dis ibu ion o +
c→π+
Hπ+
L−decay.
[pa ame e s: qmax =750 MeV, V2
P=8.1×107,B2=9×10−2,
C2=2.2×10−5,D2=22]
Fig. 5 −π+
Lin a ian mass dis ibu ion o +
c→π+
Hπ+
L−decay
[pa ame e s: qmax =750 MeV, V2
P=8.1×107,B2=8.3×10−2,
C2=2.4×10−5,D2=22]
s a e pic u e o he exci ed hype on esonances (1620)and
(1690).
Al hough ou conce n is abou he (1620)and (1690)
esonances, we make he e a sho discussion conce ning he
disag eemen in he lowe pa o he spec um. We ha e ied
o o e come his p oblem by adding some new ing edien s:
(1) Ex a 1
2−backg ound;
(2) Conside a ion o he ene gy dependence o he ∗(1530)
wid h;
Fig. 6 −π+
Lin a ian mass dis ibu ion o +
c→π+
Hπ+
L−decay
[pa ame e s: qmax =750 MeV, V2
P=8.1×107,B2=9×10−2,
C2=1.3×10−4,D2=22]
Fig. 7 −π+
Lin a ian mass dis ibu ion o +
c→π+
Hπ+
L−decay
[pa ame e s: qmax =750 MeV, V2
P=8.1×107,B2=9.0×10−2,
C2=2.2×10−5,D2=26]
(3) E ec s o symme iza ion o he ampli ude. Al hough,
as we men ioned, in ou esonance ampli udes he π+
H
and π+
Lwe e pe ec ly iden i ied in he ange o ene gies
s udied, his is no he case o he backg ound. Then we
pe o med he calcula ion
d
dM2
in (π+
L−)=d2
dM2
12 dM2
23
θ(p1−p3)dM2
12
+d2
dM2
12 dM2
23
θ(p3−p1)dM2
23,(10)
123

954 Page 6 o 7 Eu . Phys. J. C (2023) 83 :954
Fig. 8 −π+
Lin a ian mass dis ibu ion o +
c→π+
Hπ+
L−decay
[pa ame e s: qmax =770 MeV, V2
P=7.6×107,B2=9×10−2,
C2=2.2×10−5,D2=23]
using he PDG o mula wi h he o de o he pa icles
π+(1), π−(2), π+(3), whe e
p1=λ1/2(M2
c,m2
π,M2
23)
2Mc
,
p3=λ1/2(M2
c,m2
π,M2
12)
2Mc
.(11)
Using he di e en ing edien s, we we e unable o sol e he
p oblem.
Al hough we a e no conce ned abou his issue he e, he
discussion made can se e o u u e s udies ying o ind
an explana ion o his p oblem. I would be mos in e es ing
o see i an app oach like he one o Re s. [26,27] going
beyond he Weinbe g–Tomozawa in e ac ion used he e, has
some hing o say abou his issue.
I is in e es ing o ema k ha he peak o igina ed by he
calcula ion in d/dMin in he igu es does no exac ly co -
espond o he pole posi ion o he esonance in he second
Riemann shee shown in Table 1. This is a qui e common ea-
u e in he PDG [1] o esonances which a e no na ow. I is
also wo h men ioning ha he (1690)signal has appea ed
wi h a clea in e e ence wi h he (1620)and does no ha e
a B ei –Wigne (BW) shape. I a he looks like Im(iBW),
wi h a as change o sign in he egion o he pole. I is in e -
es ing o no e ha he da a wi h high p ecision also show a
as jump om one bin o he nex in ha egion. The molec-
ula pic u e o he (1620)and (1690) ha we ha e used
in his wo k is suppo ed by a la ge numbe o pape s in he
li e a u e in addi ion o Re s. [14,15] men ioned in he In o-
duc ion. The (1620)can be in e p e ed as ¯
Kmolecula
s a e wi h I(JP)=1
2(1
2−)in Re . [28], unde he ame-
wo k o he one-boson-exchange (OBE) model. This wo k
gi es he binding ene gy o he ¯
Ksys em as −2.9MeV.
The (1620)is also assumed o be a ¯
K-¯
Kmolecula
s a e in Re . [29], and he decay wid hs o his s a e decaying
in o π and ππ a e calcula ed h ough iangle diag ams
in an e ec i e Lag angian app oach, showing ha he ¯
K
channel p o ides abou (50 −68)% o he o al decay wid h,
while he ¯
Kchannel p o ides he es . Simila ly, (1620)
is conside ed a s-wa e ¯
Ko ¯
Kbound s a e, based on he
Be he–Salpe e equa ion, in Re . [30], and he decay wid hs
o (1620)→πa e 36.94 MeV and 9.35 MeV o he ¯
K
and ¯
Kbound s a es, espec i ely. Howe e , he (1690)
is no discussed in Re s. [28–30]. In ou wo k, he (1620)
is conside ed as a molecula s a e wi h π and ¯
Kas i s
main componen s, and he (1690)is conside ed as mos ly
a molecula s a e o ¯
K, e y close o h eshold, which could
be equally conside ed as a i ual s a e, wi hou changing he
ea u es o he mass dis ibu ion. An ad an age o ou wo k
is ha we can discuss he (1620)and (1690)s a es a he
same ime.
Thewo ko Re .[14] is epea ed in Re . [31], by using he
chi al uni a y app oach. When selec ing an app op ia e se o
sub ac ion cons an s, a pole o 1682 −0.8iMeV is ound,
co esponding o (1690). The coupling cons an s appea
o be domina ed by ¯
K, which is in line wi h ou esul s
ob ained by he cu o me hod o egula ize he loop unc ion.
Mo e ecen ly, ou mo e wo ks add ess he na u e o hese
esonances om he molecula pe spec i e. In Re . [32], he
(1620)is ob ained using he chi al uni a y app oach and
he ¯
Ksca e ing leng h is e alua ed. In Re . [33], he same
amewo k is used and he (1690)is ob ained in addi-
ion o ano he esonance, he (2120), explaining why he
obse ed wid hs a e so small. In Re . [34], he same app oach
is used, he (1690)is also gene a ed and he ela ed eac ion
+
c→K+¯
K0is s udied. In Re . [26], he chi al uni a y
app oach is again used o s udy he meson–ba yon in e ac ion
in he S=−2 sec o including nex o leading o de and Bo n
e ms in he in e ac ion. In ha wo k bo h he (1620)and
(1690)a e gene a ed wi h simila esul s and conclusions
ega ding he na u e o he s a es, as exposed in he p esen
wo k. In ou wo k we dynamically gene a e he (1620)
and (1690)s a es, wi h only one pa ame e qmax.Wemus
s ess ha he only pa ame e o he chi al uni a y app oach
a his leading o de al eady de e mines a he same ime he
posi ion o he wo esonances, hei wid hs and he ela i e
s eng h o he p oduc ion a es o he wo esonances, a eal
challenge o any heo y.
123
Eu . Phys. J. C (2023) 83 :954 Page 7 o 7 954
5 Conclusions
In his wo k, we ha e s udied −π+
Lin a ian mass dis-
ibu ion o +
c→π+
Hπ+
L−decay, conside ing he
(1620)and (1690)as meson–ba yon molecula s a es,
unde he amewo k o he chi al uni a y app oach. We
can dynamically gene a e double esonance s a es simul a-
neously, (1620)and (1690), om he s-wa e in e ac ion
o π and o he coupled channels, when using he cu o
me hod o egula ize he G unc ions. We p o ide he cou-
pling cons an s o (1620)and (1690) o di e en chan-
nels in Table 2. F om he coupling cons an s, i can be known
ha (1620)is a molecula s a e wi h main componen s π
and ¯
K, and (1690)is a molecula s a e wi h main com-
ponen ¯
K. The mechanism o +
c→π+MB decay was
analyzed in Re . [20]. We ollow closely his wo k and ind
ha he meson and ba yon s a es om had oniza ion a e
he weak p ocess do no con ain he inal s a e π+−,soi
mus come om esca e ing. Bu wha migh be a p oblem
o desc ibe he eac ion becomes ac ually a s ong poin o
look in o he na u e o he (1620)and (1690) esonances,
which a e p oduced by his inal s a e in e ac ion in he chi al
uni a y app oach.
In o de o compa e wi h he expe imen al da a om Belle,
we ake in o accoun con ibu ions om ∗(1530)and o he
backg ounds, and ge he −π+
Lin a ian mass dis ibu ion
o +
c→π+
Hπ+
L−decay. Resul s in compa ison wi h
expe imen al da a s ongly suppo ha he (1620)and
(1690)a e molecula s a es, wi h spin-pa i y being bo h
1
2−. O he eac ions equi ing he need o inal s a e in e ac-
ion o p oduce hese esonances would be mos welcome o
u he suppo his pic u e.
Acknowledgemen s This wo k is pa ly suppo ed by he Na ional
Na u al Science Founda ion o China unde G an No. 11975083 and
No. 12365019, and by he Cen al Go e nmen Guidance Funds o
Local Scien i ic and Technological De elopmen , China (No. Guike
ZY22096024). This wo k is also pa ly suppo ed by he Spanish
Minis e io de Economia y Compe i i idad (MINECO) and Eu o-
pean FEDER unds unde Con ac s No. FIS2017-84038-C2-1-P B,
PID2020-112777GB-I00, and by Gene ali a Valenciana unde con-
ac PROMETEO/2020/023. This p ojec has ecei ed unding om
he Eu opean Union Ho izon 2020 esea ch and inno a ion p og amme
unde he p og am H2020-INFRAIA-2018-1, g an ag eemen No.
824093 o he STRONG-2020 p ojec .
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