Measuring K0SK± interactions using Pb–Pb collisions at √sNN=2.76 TeV

Physics Le e s B 774 (2017) 64–77 Con en s lis s a ailable a ScienceDi ec Physics Le e s B www.else ie .com/loca e/physle b Measu ing K0 SK±in e ac ions using Pb–Pb collisions a √sNN =2.76 TeV .ALICE Collabo a ion a i c l e i n o a b s a c A icle his o y: Recei ed 22 May 2017 Recei ed in e ised o m 24 Augus 2017 Accep ed 4 Sep embe 2017 A ailable online 8 Sep embe 2017 Edi o : L. Rolandi We p esen he fi s e e measu emen s o em oscopic co ela ions be ween he K0 Sand K±pa icles. The analysis was pe o med on he da a om Pb–Pb collisions a √sNN =2.76 TeV measu ed by he ALICE expe imen . The obse ed em oscopic co ela ions a e consis en wi h final-s a e in e ac ions p oceeding ia he a0(980) esonance. The ex ac ed kaon sou ce adius and co ela ion s eng h pa ame e s o K0 SK−a e ound o be equal wi hin he expe imen al unce ain ies o hose o K0 SK+. Compa ing he esul s o he p esen s udy wi h hose om published iden ical-kaon em oscopic s udies by ALICE, mass and coupling pa ame e s o he a0 esonance a e es ed. Ou esul s a e also compa ible wi h he in e p e a ion o he a0ha ing a e aqua k s uc u e ins ead o ha o a diqua k. ©2017 The Au ho . Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3. 1. In oduc ion Iden ical boson em oscopy, especially o iden ical cha ged pi- ons, has been used ex ensi ely o e he yea s o s udy expe i- men ally he space– ime geome y o he collision egion in high- ene gy pa icle and hea y-ion collisions [1]. Iden ical-kaon em- oscopy s udies ha e also been ca ied ou , ecen examples o which a e he ones wi h Au–Au collisions a √sNN =200 GeV by he STAR Collabo a ion [2] (K0 SK0 S) and wi h pp a √s=7TeV and Pb–Pb collisions a √sNN =2.76 TeV by he ALICE Collabo a- ion [3–5] (K0 SK0 Sand K±K±). The pai -wise in e ac ions be ween he iden ical kaons ha o m he basis o em oscopy a e o K±K±quan um s a is ics and he Coulomb in e ac ion, and o K0 SK0 Squan um s a is ics and he final-s a e in e ac ion h ough he 0(980)/a0(980) h eshold esonances. One can also conside he case o non-iden ical kaon pai s, e.g. K0 SK±pai s. Besides he non- esonan channels which may be p esen , e.g. non- esonan elas ic sca e ing o ee-s eaming o he kaons om hei eeze-ou posi ions o he de ec o , he o he only pai -wise in e ac ion allowed o a K0 SK±pai a eeze ou om he collision sys em is a final-s a e in e ac ion (FSI) h ough he a0(980) esonance. The o he pai -wise in e ac ions p esen o iden ical-kaon pai s a e no p esen o K0 SK±pai s because: a) he e is no quan um s a is ics enhancemen since he kaons a e no iden ical, b) he e is no Coulomb e ec since one o he kaons is uncha ged, and c) he e is no s ong FSI h ough he 0 eso- E-mail add ess: [email p o ec ed]. nance since he kaon pai is in an I=1 isospin s a e, as is he a0, whe eas he 0is an I=0s a e. Ano he ea u e o he K0 SK±FSI h ough he a0 esonance is, due o he a0ha ing s angeness S=0 and he K0 Sbeing a linea combina ion o he K0and K0, K0 S=1 √2K0+K0,(1) only he K0K+pai om K0 SK+and he K0K−pai om K0 SK−ha e S=0 and hus can o m he a0 esonance. This allows he pos- sibili y o s udy he K0and K0sou ces sepa a ely since hey a e indi idually selec ed by s udying K0 SK−and K0 SK+pai s, espec- i ely. An addi ional consequence o his ea u e is ha only 50% o ei he he K0 SK−o K0 SK+de ec ed pai s will pass h ough he a0 esonance. This is aken in o accoun in he exp ession o he model used o fi he co ela ion unc ions. On he o he hand, he na u al equi emen ha he sou ce sizes ex ac ed om he K0 SK± em oscopy ag ee wi h hose ob- ained o he K0 SK0 Sand K±K±sys ems allows one o s udy he p ope ies o he a0 esonance i sel . This is in e es ing in i s own igh since many s udies discuss he possibili y ha he a0, lis ed by he Pa icle Da a G oup as a diqua k ligh unfla o ed meson s a e [6], could be a ou -qua k s a e, i.e. a e aqua k, o a “K–K molecule” [7–12]. Fo example, he p oduc ion c oss sec ion o he a0 esonance in a eac ion channel such as K0K−→a− 0should de- pend on whe he he a− 0is composed o duo dssuqua ks, he o me equi ing he annihila ion o he ss pai and he la e be- ing a di ec ans e o he qua ks in he kaons o he a− 0. The h p://dx.doi.o g/10.1016/j.physle b.2017.09.009 0370-2693/©2017 The Au ho . Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3. ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 65 esul s om K0 SK− em oscopy migh be sensi i e o hese wo di - e en scena ios. In his Le e , esul s om he fi s s udy o K0 SK± em oscopy a e p esen ed. This has been done o Pb–Pb collisions a √sNN = 2.76 TeV measu ed by he ALICE expe imen a he LHC [13]. The physics goals o he p esen K0 SK± em oscopy s udy a e he ol- lowing: 1) show o wha ex en he FSI h ough he a0 esonance desc ibes he co ela ion unc ions, 2) s udy he K0and K0sou ces o see i he e a e di e ences in he sou ce pa ame e s, and 3) es published a0mass and coupling pa ame e s by compa isons wi h published iden ical kaon esul s [5]. 2. Desc ip ion o expe imen and da a selec ion The ALICE expe imen and i s pe o mance in he LHC Run 1 (2009–2013) a e desc ibed in Re . [13] and Re . [14,15], espec- i ely. Abou 22 ×106Pb–Pb collision e en s wi h 0–10% cen ali y class aken in 2011 we e used in his analysis ( he a e age cen- ali y in his ange is 4.9% due o a sligh igge inefficiency in he 8–10% ange). E en s we e classified acco ding o hei cen- ali y using he measu ed ampli udes in he V0 de ec o s, which consis o wo a ays o scin illa o s loca ed along he beamline and co e ing he ull azimu h [16]. Cha ged pa icles we e econ- s uc ed and iden ified wi h he cen al ba el de ec o s loca ed wi hin a solenoid magne wi h a field s eng h o B =0.5T. Cha ged pa icle acking was pe o med using he Time P ojec ion Chambe (TPC) [17] and he Inne T acking Sys em (ITS) [13]. The ITS allowed o high spa ial esolu ion in de e mining he p ima y (collision) e ex. T acks we e econs uc ed and hei momen a we e ob ained wi h he TPC. A momen um esolu ion o less han 10 MeV/cwas ypically ob ained o he cha ged acks o in e - es in his analysis. The p ima y e ex was ob ained om he ITS, he posi ion o he p ima y e ex being cons ained along he beam di ec ion ( he “z-posi ion”) o be wi hin ±10 cm o he cen- e o he ALICE de ec o . In addi ion o he s anda d ack quali y selec ions, he ack selec ions based on he quali y o ack e- cons uc ion fi and he numbe o de ec ed acking poin s in he TPC we e used o ensu e ha only well- econs uc ed acks we e aken in he analysis [14,15]. Pa icle iden ifica ion (PID) o econs uc ed acks was ca - ied ou using bo h he TPC and he Time-o -Fligh (TOF) de ec- o in he pseudo apidi y ange |η| <0.8[14,15]. Fo each PID me hod, a alue was assigned o each ack deno ing he numbe o s anda d de ia ions be ween he measu ed ack in o ma ion and calcula ed alues (Nσ) [5,14,15]. Fo TPC PID, a pa ame ized Be he–Bloch o mula was used o calcula e he specific ene gy loss dE/dxin he de ec o expec ed o a pa icle wi h a gi en mass and momen um. Fo PID wi h TOF, he pa icle mass was used o calcula e he expec ed ime-o -fligh as a unc ion o ack leng h and momen um. This p ocedu e was epea ed o ou “pa icle species hypo heses”—elec on, pion, kaon and p o on—, and, o each hypo hesis, a di e en Nσ alue was ob ained pe de ec o . 2.1. Kaon selec ion The me hods used o selec and iden i y indi idual K0 Sand K± pa icles a e he same as hose used o he ALICE Pb–Pb K0 SK0 Sand K±K±analyses [5]. These a e now desc ibed below. 2.1.1. K0 Sselec ion The K0 Spa icles we e econs uc ed om he decay K0 S→ π+π−, wi h he daugh e π+and π− acks de ec ed in he TPC and TOF de ec o s. Pions wi h pT>0.15 GeV/cwe e accep ed (since o lowe pT ack finding efficiency d ops apidly) and he dis ance o closes app oach o he p ima y e ex (DCA) o he econs uc ed K0 Swas equi ed o be less han 0.3 cm in all di- ec ions. The equi ed Nσ alues o he pions we e NσTPC <3 and NσTOF <3 o p >0.8 GeV/c. An in a ian mass dis ibu- ion o he π+π−pai s was p oduced and he K0 Swas defined o be esul ing om a pai ha ell in o he in a ian mass ange 0.480 <mπ+π−<0.515 GeV/c2. 2.1.2. K±selec ion Cha ged kaon acks we e also de ec ed using he TPC and TOF de ec o s, and we e accep ed i hey we e wi hin he ange 0.14 <pT<1.5GeV/c. In o de o educe he numbe o secon- da ies ( o ins ance he cha ged pa icles p oduced in he de ec o ma e ial, pa icles om weak decays, e c.) he p ima y cha ged kaon acks we e selec ed based on he DCA, such ha he DCA ans e se o he beam di ec ion was less han 2.4 cm and he DCA along he beam di ec ion was less han 3.2 cm. I he TOF signal we e no a ailable, he equi ed Nσ alues o he cha ged kaons we e NσTPC <2 o pT<0.5 GeV/c, and he ack was e- jec ed o pT>0.5 GeV/c. I he TOF signal we e also a ailable and pT>0.5GeV/c: NσTPC <3 and NσTOF <2(0.5 <pT<0.8 GeV/c), NσTOF <1.5(0.8 <pT<1.0GeV/c), NσTOF <1(1.0 <pT< 1.5GeV/c). K0 SK±expe imen al pai pu i y was es ima ed om a Mon e Ca lo (MC) s udy based on HIJING [18] simula ions using GEANT3 [19] o model pa icle anspo h ough he ALICE de ec o s. The pu i y was de e mined om he ac ion o he econs uc ed MC simula ed pai s ha we e iden ified as ac ual K0 SK±pai s inpu om HIJING. The pai pu i y was es ima ed o be 88% o all kine- ma ic egions s udied in his analysis. 3. Analysis me hods 3.1. Expe imen al co ela ion unc ions This analysis s udies he momen um co ela ions o K0 SK±pai s using he wo-pa icle co ela ion unc ion, defined as C(k∗)=A(k∗)/B(k∗)(2) whe e A(k∗)is he measu ed dis ibu ion o pai s om he same e en , B(k∗)is he e e ence dis ibu ion o pai s om mixed e en s, and k∗is he magni ude o he momen um o each o he pa icles in he pai es ame (PRF), k∗=(s−m2 K0−m2 K±)2−4m2 K0m2 K± 4s(3) whe e, s=m2 K0+m2 K±+2EK0EK±−2 pK0· pK±(4) and mK0(EK0) and mK±(EK±) a e he es masses ( o al ene gies) o he K0 Sand K±, espec i ely. The denomina o B(k∗)was o med by mixing K0 Sand K±pa - icles om each e en wi h pa icles om en o he e en s. The e exes o he mixed e en s we e cons ained o be wi hin 2 cm o each o he in he z-di ec ion. A cen ali y cons ain on he mixed e en s was ound no o be necessa y o he na ow cen- ali y ange, i.e. 0–10%, used in his analysis. Co ela ion unc ions we e ob ained sepa a ely o wo di e en magne ic field o ien a- ions in he expe imen and hen ei he a e aged o fi sepa a ely, depending on he fi ing me hod used (see below). Co ela ion unc ions we e measu ed o h ee o e lapping/non- exclusi e pai ans e se momen um (kT=|pT,1+pT,2|/2) bins: all kT, kT<0.675 and kT>0.675 GeV/c. The mean kT alues o hese h ee bins we e 0.675, 0.425 and 0.970 GeV/c, espec i ely. 66 ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 Fig. 1. Examples o aw K0 SK+co ela ion unc ions o he h ee kTbins wi h linea fi s o he baseline a la ge k∗. S a is ical unce ain ies a e shown. Fig. 1 shows sample aw K0 SK+co ela ion unc ions o hese h ee bins o one o he magne ic field o ien a ions. One can see he main ea u e o he em oscopic co ela ion unc ion: he supp es- sion due o he s ong final-s a e in e ac ions o small k∗. In he highe k∗ egion, he e ec s o he a0appea o no be p esen and hus could be used as a e e ence, i.e. “baseline”, o he a0-based model fi ed o C(k∗)in o de o ex ac he sou ce pa ame e s. Also shown in he figu e a e linea fi s o he baseline o la ge k∗. The e ec s on C(k∗)by he a0 esonance a e mos ly seen in he k∗<0.2 GeV/c egion, whe e he wid h o he a0 egion eflec s he size o he kaon sou ce (see equa ions below). Co ela ion unc ions we e co ec ed o momen um esolu ion e ec s using HIJING calcula ions. HIJING was used o c ea e wo co ela ion unc ions: one in e ms o he gene a o -le el k∗and one in e ms o he simula ed de ec o -le el k∗. Because HIJING does no inco po a e final-s a e in e ac ions, weigh s we e calcu- la ed using a 9 h-o de polynomial fi in k∗ o an expe imen al co ela ion unc ion and we e used when filling he same-e en dis ibu ions. These weigh s we e calcula ed using k∗. Then, he a io o he “ideal” co ela ion unc ion o he “measu ed” one ( o each k∗bin) was mul iplied o he da a co ela ion unc ions be- o e he fi p ocedu e. This co ec ion mos ly a ec ed he lowes k∗bins, inc easing he ex ac ed sou ce pa ame e s by se e al pe - cen . 3.2. Final-s a e in e ac ion model The K0 SK±co ela ion unc ions we e fi wi h unc ions ha include a pa ame e iza ion which inco po a es s ong FSI. I was assumed ha he FSI a ises in he K0 SK±channels due o he nea - h eshold esonance, a0(980). This pa ame e iza ion was in- oduced by R. Lednicky and is based on he model by R. Lednicky and V.L. Lyuboshi z [20,21] (see also Re . [2] o mo e de ails on his pa ame e iza ion). Using an equal emission ime app oxima ion in he PRF [20], he elas ic K0 SK± ansi ion is w i en as a s a iona y solu ion − k∗( ∗)o he sca e ing p oblem in he PRF. The quan i y  ∗ ep- esen s he emission sepa a ion o he pai in he PRF, and he − k∗ subsc ip e e s o a e e sal o ime om he emission p ocess. A la ge dis ances his has he asymp o ic o m o a supe posi ion o a plane wa e and an ou going sphe ical wa e, − k∗( ∗)=e−i k∗· ∗+ (k∗)eik∗ ∗ ∗,(5) whe e (k∗)is he s-wa e K0K−o K0K+sca e ing ampli ude whose con ibu ion is he s-wa e iso ec o a0 esonance (see Eq. (11) in Re . [2]), Table 1 The a0masses and coupling pa ame e s, all in GeV ( aken om Re . [2]). Re e ence ma0γa0K¯ Kγa0πη Ma in [7] 0.974 0.333 0.222 An onelli [8] 0.985 0.4038 0.3711 Achaso 1 [9] 0.992 0.5555 0.4401 Achaso 2 [9] 1.003 0.8365 0.4580 (k∗)=γa0→KK m2 a0−s−i(γa0→KKk∗+γa0→πηkπη).(6) In Eq. (6), ma0is he mass o he a0 esonance, and γa0→KK and γa0→πη a e he couplings o he a0 esonance o he K0K−(o K0K+) and πη channels, espec i ely. Also, s =4(m2 K0+k∗2)and kπη deno es he momen um in he second decay channel (πη) (see Table 1). The co ela ion unc ion due o he FSI is hen calcula ed by in eg a ing − k∗( ∗)in he Koonin–P a equa ion [22,23] C( k∗)=d3 ∗S( ∗)− k∗( ∗) 2 ,(7) whe e S( ∗)is a one-dimensional Gaussian sou ce unc ion o he PRF ela i e dis ance  ∗wi h a Gaussian wid h Ro he o m S( ∗)∼e− ∗ 2/(4R2).(8) Equa ion (7) can be in eg a ed analy ically o K0 SK±co ela ions wi h FSI o he one-dimensional case, wi h he esul C(k∗)=1+λα1 2 (k∗) R 2 +2R (k∗) √πRF1(2k∗R) −I (k∗) RF2(2k∗R),(9) whe e F1(z)≡√πe−z2e fi(z) 2z;F2(z)≡1−e−z2 z.(10) In he abo e equa ions αis he ac ion o K0 SK±pai s ha come om he K0K−o K0K+sys em, se o 0.5 assuming symme y in K0and K0p oduc ion [2], Ris he adius pa ame e om he sphe ical Gaussian sou ce dis ibu ion gi en in Eq. (8), and λis he co ela ion s eng h. The co ela ion s eng h is uni y in he ideal case o pu e a0- esonan FSI, pe ec PID, a pe ec Gaussian kaon sou ce and he absence o long-li ed esonances which decay in o kaons. No e ha he o m o he FSI e m in Eq. (9) di e s om ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 67 he o m o he FSI e m o K0 SK0 Sco ela ions (Eq. (9) o Re . [2]) by a ac o o 1/2due o he non-iden ical pa icles in K0 SK±co - ela ions and hus he absence o he equi emen o symme ize he wa e unc ion gi en in Eq. (5). As seen in Eq. (6), he K0K−o K0K+s-wa e sca e ing am- pli ude depends on he a0mass and decay couplings. In he p esen wo k, we ha e aken he alues used in Re . [2] which ha e been ex ac ed om he analysis o he a0→πη spec a o se e al expe imen s [7–10], shown in Table 1. The ex ac ed a0mass and decay couplings ha e a ange o alues o he a - ious e e ences. Excep o he Ma in e e ence [7], which ex- ac s he a0 alues om he eac ion 4.2 GeV/cinciden mo- men um K−+p →+(1385)π−ηusing a wo-channel B ei – Wigne o mula, he o he e e ences ex ac he a0 alues om he adia i e φ-decay da a, i.e. φ→π0ηγ , om he KLOE col- labo a ion [24]. These la e h ee e e ences apply a model ha assumes, a e aking in o accoun he φ→π0ρ0→π0ηγ back- g ound p ocess, ha he φdecays o he π0ηγ final s a e h ough he in e media e p ocesses φ→K+K−γ→a0γo φ→K+K−→ a0γ, i.e. he “cha ged kaon loop model” [9]. The main di e ence be ween hese analyses is ha he An onelli e e ence [8] as- sumes a fixed a0mass in he fi o his model o he π0ηda a, whe eas he Achaso 1 and Achaso 2 analyses [9] allow he a0 mass o be a ee pa ame e in he wo di e en fi s made o he da a. I is assumed in he p esen analysis ha hese decay couplings will also be alid o K0K−and K0K+sca e ing due o isospin in a iance. Co ela ion unc ions we e fi ed wi h all ou o hese cases o see he e ec on he ex ac ed sou ce pa ame- e s. 3.3. Fi ing me hods In o de o es ima e he sys ema ic e o s in he fi ing me hod used o ex ac Rand λusing Eq. (9), wo di e en me hods, judged o be equally alid, ha e been used o handle he e ec s o he baseline: 1) a sepa a e linea fi o he “baseline egion,” ol- lowed by fi ing Eq. (9) o he co ela ion unc ion di ided by he linea fi o ex ac he sou ce pa ame e s, and 2) a combined fi o Eq. (9) and a quad a ic unc ion desc ibing he baseline whe e he sou ce pa ame e s and he pa ame e s o he quad a ic unc ion a e fi ed simul aneously. The sou ce pa ame e s a e ex ac ed o each case om bo h me hods and a e aged, he symme ic sys em- a ic e o o each case due o he fi ing me hod being one-hal o he di e ence be ween he wo me hods. Bo h fi ing me hods will now be desc ibed in mo e de ail. 3.3.1. Linea baseline me hod In he “linea baseline me hod,” o he all kT, kT<0.675 and kT>0.675 GeV/cbins he a0 egions we e aken o be k∗<0.3, k∗<0.2 and k∗<0.4GeV/c, espec i ely. In he highe k∗ egion i was assumed ha e ec s o he a0we e no p esen and hus can be used as a e e ence, i.e. “baseline”, o he a0-based model fi ed o C(k∗), which was a e aged o e he wo magne ic field o ien a ions used in he expe imen , o ex ac he sou ce pa am- e e s. Fo he h ee kTbins, linea fi s we e made in he k∗ anges 0.3–0.45, 0.2–0.45 and 0.4–0.6GeV/c, espec i ely, and he co - ela ion unc ions we e di ided by hese fi s o emo e baseline e ec s ex ending in o he low-k∗ egion. These anges we e aken o define he baselines since he measu ed co ela ion unc ions we e ound o be linea he e. Fo la ge alues o k∗ he co ela- ion unc ions became non-linea . The baseline was s udied using HIJING MC calcula ions which ake in o accoun he de ec o cha - ac e is ics as desc ibed ea lie . The C(k∗)dis ibu ions ob ained om HIJING do no show supp essions a low k∗as seen in Fig. 1 bu a he show linea dis ibu ions o e he en i e anges in k∗ shown in he figu e. HIJING also shows he baseline becoming non- linea o la ge alues o k∗, as seen in he measu emen s. The MC gene a o code AMPT [25] was also used o s udy he baseline. AMPT is simila o HIJING bu also includes final-s a e esca e - ing e ec s. AMPT calcula ions also showed linea baselines in he k∗ anges used in he p esen analysis, becoming non-linea o la ge k∗. Bo h HIJING and AMPT quali a i ely show he same di- ec ion o changes in he slopes o he baseline s. kTas seen in he da a, bu AMPT mo e accu a ely desc ibed he slope alues hem- sel es, sugges ing ha final-s a e esca e ing plays a ole in he kTdependence o he baseline slope. The sys ema ic unce ain ies on he ex ac ed sou ce pa ame e s due o he assump ion o lin- ea i y in hese k∗ egions we e es ima ed om HIJING o be less han 1%. Fig. 2 shows examples o K0 SK+and K0 SK−co ela ion unc- ions di ided by linea fi s o he baseline wi h Eq. (9) using he Achaso 2 pa ame e s. One can see he main ea u e o he em- oscopic co ela ion unc ion: he supp ession due o he s ong final-s a e in e ac ions o small k∗. As seen, he a0FSI pa ame e - iza ion gi es an excellen ep esen a ion o he “signal egion” o he da a, i.e. he supp ession o he co ela ion unc ions in he k∗ ange 0 o abou 0.15 GeV/c. 3.3.2. Quad a ic baseline me hod In he “quad a ic baseline me hod,” Rand λa e ex ac ed as- suming a quad a ic baseline unc ion by fi ing he p oduc o a quad a ic unc ion and he Lednicky equa ion, Eq. (9), o he aw co ela ion unc ions o each o he wo magne ic field o ien a- ions used in he expe imen , such as shown in Fig. 1, i.e., C i aw(k∗)=a(1−bk∗+ck∗2)C(k∗)(11) whe e C(k∗)is gi en by Eq. (9), and a, band ca e fi pa ame- e s. Eq. (11) is fi o he same k∗ anges as shown in Fig. 1, i.e. 0–0.45 GeV/c o all kTand kT<0.675 GeV/c, and 0–0.6GeV/c o kT>0.675 GeV/c. The fi s o he expe imen al co ela ion unc- ions a e ound o be o simila good quali y as seen o he linea baseline me hod fi s shown in Fig. 2. 3.4. Sys ema ic unce ain ies Sys ema ic unce ain ies on he ex ac ed sou ce pa ame e s we e es ima ed by a ying he anges o kinema ic and PID cu alues on he da a by ±10% and ±20%, as well as om MC simu- la ions. The main sys ema ic unce ain ies on he ex ac ed alues o Rand λdue o a ious sou ces, no including he baseline fi - ing me hod, a e: a) k∗fi ing ange: 2%, b) single-pa icle and pai cu s (e.g. DCA cu s, PID cu s, pai sepa a ion cu s): 2%–4% o R and 3%–8% o λ, and c) pai pu i y: 1% on λ. Combining he indi- idual sys ema ic unce ain ies in quad a u e, he o al sys ema ic unce ain ies on he ex ac ed sou ce pa ame e s, no including he baseline fi ing me hod con ibu ion, a e in he anges 3%–5% o Rand 4%–8% o λ. As men ioned ea lie , o he wo fi ing me hods, he sou ce pa ame e s a e ex ac ed o each case om bo h me hods and a - e aged, he symme ic sys ema ic e o o each case due o he fi ing me hod being one-hal o he di e ence be ween he wo me hods. The baseline fi ing me hod sys ema ic e o hus ob- ained is added in quad a u e wi h he sys ema ic e o s gi en abo e. I is ound ha he size o he baseline fi ing me hod sys- ema ic e o s a e abou 50% la ge o Rand o simila magni ude o λas hose quo ed abo e o he non-fi ing-me hod sys ema ic e o s. 68 ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 Fig. 2. Examples o K0 SK+and K0 SK−co ela ion unc ions di ided by linea fi s o he baseline wi h he Lednicky pa ame e iza ion using he Achaso 2 [9] pa ame e s. S a is ical (lines) and he linea sum o s a is ical and sys ema ic unce ain ies (boxes) a e shown. 4. Resul s and discussion Fig. 3 shows sample esul s o he Rand λpa ame e s ex- ac ed in he p esen analysis om K0 SK± em oscopy using he Achaso 1 pa ame e s. The le column compa es K0 SK+and K0 SK− esul s om he quad a ic baseline fi me hod, and he igh col- umn compa es esul s a e aged o e K0 SK+and K0 SK− o he quad a ic baseline fi s and he linea baseline fi s. As i is usually he case in em oscopic analyses, he fi ed Rand λpa ame e s a e co ela ed. The fi ing (s a is ical) unce ain ies a e aken o be he ex eme alues o he 1σfi con ou s in R s. λ. S a is- ical unce ain ies a e plo ed o all esul s. I is seen in he figu e ha he Rand λ alues o K0 SK−ha e a sligh endency o be la ge han hose o K0 SK+. Such a di e ence could esul om he K−–nucleon sca e ing c oss sec ion being la ge han ha o K+–nucleon (see Fig. 51.9 o Re . [6]), possibly esul ing in mo e final-s a e esca e ing o he K−. Since he di e ence is no sig- nifican once sys ema ic unce ain ies a e aken in o accoun , K0 SK+ and K0 SK−a e a e aged o e in he final esul s. The di e ence in he ex ac ed pa ame e s be ween he wo baseline fi ing me h- ods is also seen o be small, and is accoun ed o as a sys ema ic e o , as desc ibed ea lie . The esul s o he Rand λpa ame e s ex ac ed in he p esen analysis om K0 SK± em oscopy, a e aged o e he wo baseline fi me hods and a e aged o e K0 SK+and K0 SK−, a e p esen ed in Table 2 and in Figs. 4 and 5. Fi esul s a e shown o all ou pa- ame e se s gi en in Table 1. Figs. 4 and 5 also show compa isons wi h iden ical kaon esul s o he same collision sys em and en- e gy om ALICE om Re . [5]. S a is ical and o al unce ain ies a e shown o all esul s. As shown in Fig. 4, bo h Achaso pa ame e se s, wi h he la ge a0masses and decay couplings, appea o gi e R alues ha ag ee bes wi h hose ob ained om iden ical-kaon em oscopy. The An- onelli pa ame e se appea s o gi e sligh ly lowe alues. Com- pa ing he measu ed R alues be ween K0 SK0 Sand K±K±in Fig. 4 hey a e seen o ag ee wi h each o he wi hin he unce ain ies. In ac , he only eason o he em oscopic K0 SK± adii o be di e en om he K0 SK0 Sand K±K±ones would be i he K0 Sand K±sou ces we e displaced wi h espec o each o he . This is no expec ed be- cause he collision dynamics is go e ned by s ong in e ac ions o which he isospin symme y applies. The esul s o he co ela ion s eng h pa ame e s λa e shown in Fig. 5. The λpa ame e s om K0 SK±and K±K±a e co ec ed o expe imen al pu i y [5]. The K0 SK0 Spai s ha e a high pu i y o >90%, so he co esponding co ec ion was neglec ed [5] (see he ea lie discussion on pu i y). S a is ical and o al unce ain ies a e shown o all esul s. The K0 SK±λ alues, wi h he excep ion o he Ma in pa ame- e s, appea o be in ag eemen wi h he λ alues o he iden ical kaons. All o he λ alues a e seen o be measu ed o be abou 0.6, i.e. less han he ideal alue o uni y, which can be due o he con ibu ion o kaons om K∗decay (∼50 MeV, whe e  is he decay wid h) and om o he long-li ed esonances (such as he D-meson) dis o ing he spa ial kaon sou ce dis ibu ion away om he ideal Gaussian which is assumed in he fi unc ion [26]. One would expec ha he K0 SK±λ alues ag ee wi h hose om he iden ical kaons i he FSI o he K0 SK±wen solely h ough he a0 esonan channel since his analysis should see he same sou ce dis ibu ion. In o de o ob ain a mo e quan i a i e compa ison o he p esen esul s o Rand λwi h he iden ical kaon esul s, he χ2/nd is calcula ed o Rand λ o each pa ame e se , χ2 ω/nd =1 nd 3  i=1 [ωi(K0 SK±)−ωi(KK)]2 σ2 i (12) ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 69 Fig. 3. Sample esul s o he Rand λpa ame e s ex ac ed in he p esen analysis om K0 SK± em oscopy using he Achaso 1 pa ame e s. The le column compa es K0 SK+ and K0 SK− esul s om he quad a ic baseline fi me hod, and he igh column compa es esul s a e aged o e K0 SK+and K0 SK− o he quad a ic baseline fi s and he linea baseline fi s. S a is ical unce ain ies a e plo ed o all esul s. Table 2 Fi esul s o Rand λex ac ed in he p esen analysis om K0 SK± em oscopy a e aged o e K0 SK+and K0 SK−. S a is ical and sys ema ic e o s a e also shown. Pa ame e s R( m) o λAll kTkT<0.675 GeV/ck T>0.675 GeV/c Achaso 2 R5.17 ±0.16 ±0.41 6.71 ±0.40 ±0.42 4.75 ±0.18 ±0.36 λ0.587 ±0.034 ±0.051 0.651 ±0.073 ±0.076 0.600 ±0.040 ±0.034 Achaso 1 R4.92 ±0.15 ±0.39 6.30 ±0.40 ±0.43 4.49 ±0.18 ±0.30 λ0.650 ±0.038 ±0.056 0.723 ±0.087 ±0.091 0.649 ±0.048 ±0.038 An onelli R4.66 ±0.17 ±0.46 5.74 ±0.36 ±0.26 4.07 ±0.18 ±0.29 λ0.624 ±0.044 ±0.058 0.703 ±0.085 ±0.077 0.613 ±0.052 ±0.037 Ma in R3.29 ±0.12 ±0.35 4.46 ±0.25 ±0.20 2.90 ±0.11 ±0.41 λ0.305 ±0.020 ±0.033 0.376 ±0.041 ±0.037 0.296 ±0.021 ±0.030 whe e ωis ei he Ro λ, i uns o e he h ee kT alues, he num- be o deg ees o eedom aken is nd =3 and σiis he sum o he s a is ical and sys ema ic unce ain ies on he i h K0 SK±ex ac ed pa ame e (No e ha he all kTbin indeed con ains he kaon pai s ha make up he kT<0.675 GeV/cand kT>0.675 GeV/cbins, bu in addi ion i con ains an equal numbe o new pai combina- ions be ween he kaons in he kT<0.675 GeV/cand kT>0.675 GeV/cbins. So o he pu poses o his simple compa ison, we ap- p oxima e he all kTbin as being independen .) The linea sum o he s a is ical and sys ema ic unce ain ies is used o σi o be consis en wi h he linea sum o he s a is ical and sys ema ic un- ce ain ies plo ed on he poin s in Figs. 4 and 5. The quan i y ωi(KK)is de e mined by fi ing a quad a ic o he iden ical kaon esul s and e alua ing he fi a he a e age kT alues o he K0 SK± measu emen s. Table 3 summa izes he esul s o each pa ame e se and he ex ac ed p- alues. As seen, he Achaso 2, Achaso 1 and An onelli pa ame e se s a e consis en wi h he iden ical kaon esul s o bo h Rand λ. The Ma in pa ame e se is seen o ha e anishingly small p- alues o bo h Rand λand is hus in clea Table 3 Compa isons o Rand λ om K0 SK±wi h iden ical kaon esul s. Pa ame e s χ2 R/nd Rp- alue χ2 λ/nd λp- alue λ(K0 SK±) λ(KK) Achaso 2 0.456 0.713 0.248 0.863 1.04 ±0.17 Achaso 1 0.583 0.626 0.712 0.545 1.14 ±0.20 An onelli 1.297 0.273 0.302 0.824 1.09 ±0.20 Ma in 14.0 0.000 22.2 0.000 0.55 ±0.10 disag eemen wi h he iden ical kaon esul s, as can easily be seen by examining Figs. 4 and 5. In o de o quan i a i ely es ima e he size o he non- esonan channel p esen , he a io λ(K0 SK±) λ(KK)has been calcula ed o each pa ame e s se , whe e he a e age is o e he h ee kT alues and he unce ain y is calcula ed om he a e age o he s a is i- cal+sys ema ic unce ain ies on he K0 SK±pa ame e s. These alues a e shown in he las column o Table 3. Dis ega ding he Ma in alue, he smalles alue his a io can ake wi hin he unce ain- 70 ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 Fig. 4. Sou ce adius pa ame e , R, ex ac ed in he p esen analysis om K0 SK± em oscopy a e aged o e K0 SK+and K0 SK−and he wo baseline fi me hods ( ed symbols), along wi h compa isons wi h iden ical kaon esul s om ALICE [5] (blue symbols). S a is ical (lines) and he linea sum o s a is ical and sys ema ic unce ain ies (boxes) a e shown. (Fo in e p e a ion o he colo s in his figu e, he eade is e e ed o he web e sion o his a icle.) Fig. 5. Co ela ion s eng h pa ame e , λ, ex ac ed in he p esen analysis om K0 SK± em oscopy a e aged o e K0 SK+and K0 SK−and he wo baseline fi me hods ( ed symbols), along wi h compa isons wi h iden ical kaon esul s om ALICE [5] (blue symbols). S a is ical (lines) and he linea sum o s a is ical and sys ema ic unce ain ies (boxes) a e shown. (Fo in e p e a ion o he colo s in his figu e, he eade is e e ed o he web e sion o his a icle.) ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 71 ies is 0.87 ( om he Achaso 2 pa ame e s) which would hus allow a mos a 13% non- esonan con ibu ion. The esul s o his s udy p esen ed abo e clea ly show ha he measu ed K0 SK±ha e dominan ly unde gone a FSI h ough he a0 esonance. This is ema kable conside ing ha we measu e in Pb– Pb collisions he a e age sepa a ion be ween he wo kaons a eeze ou o be ∼5 m, and due o he sho - anged na u e o he s ong in e ac ion o ∼1 m his would seem o no encou - age a FSI bu a he encou age ee-s eaming o he kaons o he de ec o esul ing in a “fla ” co ela ion unc ion. A dominan FSI is wha migh be expec ed i he a0would be a ou -qua k, i.e. e aqua k, s a e o a “K–K molecule.” The e appea s o be no cal- cula ions in he li e a u e o he e aqua k s. diqua k p oduc ion c oss sec ions o he in e ac ion KK →a0, bu quali a i e a gu- men s compa ible wi h he a0being a ou –qua k s a e can be made based on he p esen measu emen s. The main a gumen in a o o his is ha he eac ion channel K0K−→a− 0(K0K+→a+ 0) is s ongly a o ed i he a− 0(a+ 0) is composed o dssu(dssu) qua ks such ha a di ec ans e o he qua ks in he kaons o he a− 0(a+ 0) has aken place, since his is an “OZI supe allowed” eac- ion [12]. The “OZI ule” can be s a ed as “an inhibi ion associa ed wi h he c ea ion o annihila ion o qua k lines” [12]. Thus, adi- qua k a0final s a e is less a o ed acco ding o he OZI ule since i would equi e he annihila ion o he s ange qua ks in he kaon in e ac ion. This would allow o he possibili y o a significan non- esonan o ee-s eaming channel o he kaon in e ac ion ha would esul in a λ alue below he iden ical-kaon alue by dilu ing he a0signal. As men ioned abo e, he collision geome y i sel also supp esses he annihila ion o he s ange qua ks due o he la ge sepa a ion be ween he kaons a eeze ou . No e ha his assumes ha he C(k∗)dis ibu ion o a non- esonan channel would be mos ly “fla ” o “mono onic” in shape and no showing a s ong esonan -like signal as seen o he a0in Fig. 1 and Fig. 2. This assump ion is clea ly ue in he ee-s eaming case, which is assumed in Eq. (9) in se ing α=0.5due o he non- esonan kaon combina ions. A simila a gumen , namely ha he success o he “cha ged kaon loop model” in desc ibing he adia i e φ-decay da a a o s he a0as a e aqua k s a e, is gi en in Re . [9]. 5. Summa y In summa y, em oscopic co ela ions wi h K0 SK±pai s ha e been s udied o he fi s ime. This new em oscopic me hod was applied o da a om cen al Pb–Pb collisions a √sNN =2.76 TeV by he LHC ALICE expe imen . Co ela ions in he K0 SK±pai s a e p oduced by final-s a e in e ac ions which p oceed h ough he a0(980) esonance. The a0 esonan FSI is seen o gi e an excel- len ep esen a ion o he shape o he signal egion in he p esen s udy. The di e ences be ween K0K+and K0K− o he ex ac ed R and λ alues a e ound o be insignifican wi hin he unce ain ies o he p esen s udy. The h ee la ge a0mass and decay pa ame e se s a e a o ed by he compa ison wi h he iden ical kaon esul s. The p esen esul s a e also compa ible wi h he in e p e a ion o he a0 esonance as a e aqua k s a e. This wo k should p o ide a cons ain on models ha a e used o p edic kaon–kaon in e - ac ions [27,28]. I will be in e es ing o apply K0 SK± em oscopy o o he collision ene gies, e.g. he highe LHC ene gies now a ail- able, and bomba ding species, e.g. p o on–p o on collisions, since he di e en sou ce sizes encoun e ed in hese cases will p obe he in e ac ion o he K0 Swi h he K±in di e en sensi i i y anges (i.e. see he Rdependence in Eq. (9)). Acknowledgemen s The ALICE Collabo a ion would like o hank all i s enginee s and echnicians o hei in aluable con ibu ions o he cons uc- ion o he expe imen and he CERN accele a o eams o he ou s anding pe o mance o he LHC complex. The ALICE Collab- o a ion g a e ully acknowledges he esou ces and suppo p o- ided by all G id cen e s and he Wo ldwide LHC Compu ing G id (WLCG) collabo a ion. The ALICE Collabo a ion acknowledges he ollowing unding agencies o hei suppo in building and un- ning he ALICE de ec o : A. I. Alikhanyan Na ional Science Labo a- o y (Ye e an Physics Ins i u e) Founda ion (ANSL), S a e Commi - ee o Science and Wo ld Fede a ion o Scien is s (WFS), A menia; Aus ian Academy o Sciences and Na ionals i ung ü Fo schung, Technologie und En wicklung, Aus ia; Minis y o Communica- ions and High Technologies, Na ional Nuclea Resea ch Cen e , Aze baijan; Conselho Nacional de Desen ol imen o Cien ífico e Tecnológico (CNPq), Uni e sidade Fede al do Rio G ande do Sul (UFRGS), Financiado a de Es udos e P oje os (Finep) and Fun- dação de Ampa o à Pesquisa do Es ado de São Paulo (FAPESP), B azil; Minis y o Science & Technology o China (MSTC), Na- ional Na u al Science Founda ion o China (NSFC) and Minis y o Educa ion o China (MOEC), China; Minis y o Science, Edu- ca ion and Spo s and C oa ian Science Founda ion, C oa ia; Min- is y o Educa ion, You h and Spo s o he Czech Republic, Czech Republic; The Danish Council o Independen Resea ch Na u- al Sciences, he Ca lsbe g Founda ion and Danish Na ional Re- sea ch Founda ion (DNRF), Denma k; Helsinki Ins i u e o Physics (HIP), Finland; Commissa ia à l’Ene gie A omique (CEA) and Ins i- u Na ional de Physique Nucléai e e de Physique des Pa icules (IN2P3) and Cen e Na ional de la Reche che Scien ifique (CNRS), F ance; Bundesminis e ium ü Bildung, Wissenscha , Fo schung und Technologie (BMBF) and GSI Helmhol zzen um ü Schwe i- onen o schung GmbH, Ge many; Gene al Sec e a ia o Resea ch and Technology, Minis y o Educa ion, Resea ch and Religions, G eece; Na ional Resea ch, De elopmen and Inno a ion Office, Hunga y; Depa men o A omic Ene gy Go e nmen o India (DAE) and Council o Scien ific and Indus ial Resea ch (CSIR), New Delhi, India; Indonesian Ins i u e o Science, Indonesia; Cen o Fe mi – Museo S o ico della Fisica e Cen o S udi e Rice che En ico Fe mi and Is i u o Nazionale di Fisica Nuclea e (INFN), I aly; Ins i u e o Inno a i e Science and Technology, Nagasaki Ins i u e o Applied Science (IIST), Japan Socie y o he P omo ion o Science (JSPS) KAKENHI and Japanese Minis y o Educa ion, Cul u e, Spo s, Sci- ence and Technology (MEXT), Japan; Consejo Nacional de Ciencia y Tecnología (CONACYT), h ough Fondo de Coope ación In e na- cional en Ciencia y Tecnología (FONCICYT) and Di ección Gene al de Asun os del Pe sonal Academico (DGAPA), Mexico; Nede landse O ganisa ie oo We enschappelijk Onde zoek (NWO), Ne he lands; The Resea ch Council o No way, No way; Commission on Science and Technology o Sus ainable De elopmen in he Sou h (COM- SATS), Pakis an; Pon ificia Uni e sidad Ca ólica del Pe ú, Pe u; Min- is y o Science and Highe Educa ion and Na ional Science Cen- e, Poland; Ko ea Ins i u e o Science and Technology In o ma- ion and Na ional Resea ch Founda ion o Ko ea (NRF), Republic o Ko ea; Minis y o Educa ion and Scien ific Resea ch, Ins i u e o A omic Physics and Romanian Na ional Agency o Science, Tech- nology and Inno a ion, Romania; Join Ins i u e o Nuclea Re- sea ch (JINR), Minis y o Educa ion and Science o he Russian Fede a ion and Na ional Resea ch Cen e Ku cha o Ins i u e, Rus- sia; Minis y o Educa ion, Science, Resea ch and Spo o he Slo ak Republic, Slo akia; Na ional Resea ch Founda ion o Sou h A ica, Sou h A ica; Cen o de Aplicaciones Tecnológicas y Desa - ollo Nuclea (CEADEN), Cubaene gía, Cuba, Minis e io de Ciencia e Inno acion and Cen o de In es igaciones Ene gé icas, Medioambi- en ales y Tecnológicas (CIEMAT), Spain; Swedish Resea ch Council (VR) and Knu & Alice Wallenbe g Founda ion (KAW), Sweden; Eu- opean O ganiza ion o Nuclea Resea ch, Swi ze land; Na ional Science and Technology De elopmen Agency (NSDTA), Su ana ee 72 ALICE Collabo a ion / Physics Le e s B 774 (2017) 64–77 Uni e si y o Technology (SUT) and Office o he Highe Educa- ion Commission unde NRU p ojec o Thailand, Thailand; Tu kish A omic Ene gy Agency (TAEK), Tu key; Na ional Academy o Sci- ences o Uk aine, Uk aine; Science and Technology Facili ies Coun- cil (STFC), Uni ed Kingdom; Na ional Science Founda ion o he Uni ed S a es o Ame ica (NSF) and Uni ed S a es Depa men o Ene gy, Office o Nuclea Physics (DOE NP), Uni ed S a es o Ame - ica. 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