A direct test of $\tau$ symmetry in the neutral K meson system at KLOE-2

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4th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE2014) IOP Publishing Journal of Physics: Conference Series 631 (2015) 012018 doi:10.1088/1742-6596/631/1/012018

A direct test of T sym m etry in th e neutral K m eson system at KLOE-2

A le k s a n d e r G a jo s o n b e h a lf o f t h e K L O E -2 C o lla b o r a tio n

M arian Smoluchowski In stitu te of Physics, Jagiellonian U niversity Łojasiew icza 11, 30-348 Cracow, Poland

E-m ail: a le k s a n d e r .g a jo s @ u j.e d u .p l

A b s t r a c t . T his work presents prospects for conducting a novel d irect te st of tim e-reversal sym m etry a t th e K LOE-2 experim ent. Q u an tu m entanglem ent of n e u tra l K m eson pairs uniquely available a t KLOE-2 allows to probe th e T sym m etry directly and independently of C P violation. T his is achieved by a com parison of probabilities for a tra n sitio n and its inverse obtained th ro u g h an exchange of in itial and final states. Such tran sitio n s betw een flavor and C P-definite sta te s of th e n e u tra l kaons are only connected by th e T conjugation which ensures th e C P -independence of th e test. W hile a sim ilar m easurem ent was perform ed by th e B aB ar experim ent w ith n e u tra l B m esons, th e K LO E-2 d etecto r can te st T -violation in th e n eu tral kaons system . Such a te st requires i.a. reconstruction of th e K L ^ 3 n 0 decay accom panied by K s ^ n ± £ T v w ith good tim ing inform ation. T herefore a new reconstruction m eth o d for th e K l ^ 3 n 0 decay is also presented w hich is capable of reconstructing th is process w ith decay tim e resolution of O (1 rS ).

1. I n tr o d u c tio n

T he fact th a t C P sym m etry is not conserved in th e n e u tra l m eson system has been well known for alm ost fifty years. Conversely, th e tim e-reversal sym m etry in th is system , altho u gh its violation should follow from th e well confirm ed C P T invariance, lacked a direct experim ental evidence for decades after C P violation discovery w ith n e u tra l kaons in 1964 [1]. T he reason for th is is th a t a d irect m easurem ent of T non-invariance would require observation of a probability asym m etry betw een a process and th e same process inverted in tim e and such an experim ent is difficult to realize for any u n stab le system .

Since for n e u tra l m esons, as zero-spin particles, th e tra n sitio n s to com pare are conjugated by an exchange of initial and final states, th e oscillation phenom enon is one of th e few processes th a t can be used to o b tain b o th tran sitio n s. It was first used to te st th e tim e-reversal sym m etry by th e C P L E A R experim ent in 1998, yielding a non-zero probab ility asy m m etry in K0 ^ K0 and K0 ^ K0 oscillations [2]. However, th e fact th a t th e initial and final sta te s in this case are conjugated b o th by th e T and C P o perations has lead to a controversy as to w heth er this result can be a ttrib u te d solely to T violation independently of C P violation. W hile some auth ors pointed ou t th e role of decay as initial s ta te in teraction in th is process [3, 4] and oth ers argued th a t is not relevant in th is case [5, 6], it rem ained clear th a t a n o th e r way to directly test th e tim e reversal sym m etry violation in dependently of C P would be highly desirable [4].

A suitable m easurem ent com paring selected tra n sitio n s betw een C P -definite and flavour- definite sta te s w ith th e ir tim e-reversal conjugates was proposed for th e n e u tra l B m eson system

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by B ern abeu et al. in 2012 [7] a n d soon followed by a sim ilar proposition realizable w ith n e u tra l kaons [8]. For th e tra n s itio n s ’ reversibility, q u a n tu m entanglem ent in th e system m ust b e used which is available a t B-factories an d a t th e D A T N E 0-factory. A m ong th e form er, th e B aB ar experim ent has already m easured a T -v io latin g asy m m etry a t 14a level [9] co n stitu tin g th e first direct observation of T -v io latio n in tran sitio n s th a t are only connected by th e tim e-reversal tran sfo rm atio n . In case of K mesons, since en tangled K s K l pairs are uniquely available a t th e D A T N E collider, KLOE-2 is th e only experim ent presently able to provide th e first tim e-reversal v iolation evidence in th e n e u tra l kaon system . P rospects for such a m easurem ent a t KLOE-2 and first analysis steps are presented in th e rem aind er of th is work.

2. A T - s y m m e t r y t e s t in d e p e n d e n t o f C P

T h e principle of th e tim e-reversal sy m m etry test a t KLO E-2 is b ased on defining tran sitio n s betw een sta te s of n e u tra l kaons being eith er sta te s w ith definite strangeness {K0 (S = + 1 ), K0 (S = + 1 )} or eigenstates of th e C P o p e ra to r which can be expressed using th e form er as:

|K + } = T [|K°> + |K°}] (C P = + 1 ), (1)

|K_} = T [|K°} - |K°}] (C P = - 1 ) . (2)

S ta te of a n e u tra l kaon can be identified in one of these two bases a t th e m om ent of its decay by observation of th e decay final sta te . W ith an assum ption of th e A Q = A S rule (well teste d in sem ileptonic kaon decays [10]), each charge of secondary lepto n can only be c reated from a strangeness-definite s ta te of th e decaying kaon, as shown in th e diagram s in Figure 1. In th e following, final sta te s w ith a positively a n d negatively charged leptons will be ind icated as I + and l _ , respectively.

F ig u r e 1. D iagram s of sem ileptonic decays of n e u tra l kaons. T he A S = A Q rule guarantees th a t th e s ta te w ith a positively-charged lepton comes from K° decay w hereas K° always decays into a final s ta te w ith a lep to n of negative charge.

O n th e o th er hand, hadronic final sta te s which are C P-even (e.g. two pions, hereafter nam ed th e n n final sta te ) can only b e pro du ced from th e K+ kaon s ta te a n d C P -o d d sta te s like 3n°

come from decays of K _ . It can be shown th a t C P violation m ay be safely neglected in these considerations [8].

A lthough th e above facts allow for identification of th e decaying kaon, observing a tra n sitio n betw een two kaon sta te s also requires tagging of th e kaon s ta te also a c ertain tim e before its decay. T his can be achieved using n e u tra l kaon pairs available a t KLOE-2 w hich exhibit q u a n tu m entanglem ent of th e ir states. Once th e s ta te of th e first decaying kaon is identified by observation of its decay, its entan gled an d still-living p a rtn e r is known to be in an orthogonal s ta te a t th e sam e tim e. Consequently, observation of its decay after a tim e interval A t can lead to th e observation of a kaon tra n sitio n w here b o th initial and final s ta te are tagged in th e strangeness or C P-basis.

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3. T im e -r e v e r s a l s y m m e tr y v io la t io n o b s e r v a b le s a t K L O E -2

T here are four possible tra n sitio n s betw een flavour and C P -definite sta te s of ne u tra l K m esons, as listed in Table 1. As these tra n sitio n s are not irreversible, th ere can be a T -co n ju g ated process defined for each of them . C om parison of rates betw een each tra n sitio n and its tim e-reversal co njugate would c o n stitu te a d irect T sym m etry te s t and its independence of C P violation follows from a com parison of th e T -a n d C P -conjugated processes shown in Table 1, which are not identical.

T a b le 1. Four possible tra n sitio n s betw een flavour and C P -definite sta te s of n e u tra l kaons and th e ir tim e-reversal conjugates o b tained by exchange of initial and final states. T he T -co n ju g ated tra n sitio n s are not identical w ith th e C P conjugates given in th e last colum n. E ach of th e tra n sitio n s is experim entally identified by a tim e-ordered pair of decays given in parentheses.

T ransition T -co n ju g ate C P -con jug ate 1 K0 ^ K + ( C , n n ) K + ^ K0 ( 3 n V + ) K0 ^ K+

2 K0 ^ K - (£- , 3n 0) K - ^ K0 ( n n ,f + ) K0 ^ K - 3 K0 ^ K + ( f + ,n n ) K + ^ K0 ( 3 n V - ) K0 ^ K+

4 K0 ^ K - (£+, 3n0) K - ^ K0 ( n n , f - ) K0 ^ K -

T he tim e-reversal asy m m etry can be defined using ratio s of probabilities of tra n sitio n s at tim e A t to probabilities of th e ir T-inverses a t th e sam e tim e. Four th eo retical ratios m ay be defined this way:

E ach of th e tra n sitio n s used in th e above ratios is experim entally identified by a tim e-ordered p air of kaon decays given in brackets in Table 1. For tra n sitio n s 1 and 3, th e conjugated process would involve th e first kaon to decay into 3n0, for which little sta tistic s is expected as th e search for C P -v iolatin g decay K S ^ 3 n0 a t K L O E yielded no can d idates [11]. For th e rem aining tra n sitio n s 2 and 4, however, large sta tistic s is available a t K LO E-2 and th e ir probabilities can be m easured th ro u g h num bers of double kaon decays to sta te s f , f2 separated by tim e A t, denoted as I ( f i, f 2; A t). E x p erim en tal equivalents of ratios R2 and R4 are th u s defined as follows:

T he th eo retical ratios (R ^(A t)) are easily ex tra cte d from R e^ p and R e4xp as shown in Eq. 4, 5 by using coefficients dep en den t on n e u tra l kaon branching fractions and w idths [8]:

C ( l- , 3n 0) ^ C (l+, 3n0) ^ B R (Kl ^ 3 n 0)

C ( n n ,l+ ) “ C (n n , l - ) “ B R (Ks ^ n n )

rS

^, ^{( )}

all of which have been m easured by K L O E [12] and whose precision should be fu rth e r im proved by K LO E-2 [13].

R _ P [K0(0) ^ K + (A t)] R (A t) _ P [K0(0) ^ K - ( A t)]

R i( t) P [K + (0) ^ K0( A t) ] , R2( t) P [ K - (0) ^ K0( A t) ] ,

r _ P[K 0(0) ^ K + ( A t) ] , _ P[K 0(0) ^ K - ( A t) ] . ( ) P [K + (0) ^ K0(At)] P [ K - (0) ^ K0(At)]

Rexp(At) _ I(l , 3n ; A t ) _ R (A t) '' C (l , 3n ) (4) R ( A t ) _ I ( n n , l + ; A t ) _ R 2(A t) X C ( n n ,l+ ) , (4) RexP (A t) _ I(l+, 3n0; A t ) _ r (A t) w C ( l+, 3n0) (5) R ( A t ) _ I ( n n , l - ; A t ) _ R ( A t ) X C ( n n , l - ) ' (5)

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F ig u r e 2. E xpected behaviour of th e R i ratios as a function of decay tim e difference. T he figure was a d a p te d from [8].

Finally, d eterm in atio n of an asy m p to tic behaviour of th eo retical ratios R2(A t) and R4(A t) would m easure violation of tim e-reversal sym m etry in th e n e u tra l kaon system . Figure 2 shows th e expected dependence of these ratio s w ith assum ed tim e-reversal violation. If T was conserved, th e ratios should ten d to u nity for large tim e intervals A t. Indeed, it can be shown [8] th a t asy m ptotic discrepancy of R2(A t) and R4(A t) from 1 is related to th e T -v io latin g p a ra m ete r e as:

R2(A t » t s) ^ 1 - 4Re, (7)

R 4(A t » t s ) ~ 1 + 4-Re. (8)

Therefore, a m easurem ent of R e^ v and R <4xp as functions of A t for large tim e differences at KLO E-2 will provide m easurem ent of degree of T sym m etry violation in th e n e u tra l K m eson system .

4. K L O E -2 a n d D A T N E 4-1. The D A & N E 0-fa cto ry

D A T N E is an electron-positron collider located at th e accelerator com plex of IN FN N ational L a b o rato ry of F rascati (LN F). It is com posed of two sep arate storage rings, each storing a beam of 0.51GeV. T he rings intersect a t two regions, one of which provides events to th e K L O E d e te c to r (Figure 3). T he center-of-m ass energy of th e colliding beam s (y% ^1020M eV ) is th e m ass of th e phi-m eson resonance, which is produced w ith a cross-section of ab o u t 3^b.

T he 0-m esons are produced alm ost a t rest ( ^ « 0.015), w ith only a sm all m om entum com ponent in th e direction p erp en d icular to th e beam axis. T heir alm ost im m ediate decays (t<£ = 1.55 ± 0.01 x 10-22s) provide pairs of K m esons, eith er charged (w ith a branching fraction of 48.9%) or n e u tra l (34.2%). T he K + K - and K LK S pairs are widely used to stu d y kaon properties as th e possibility of tagging kaons by th e ir p a rtn e r’s decay allows absolute branching ratios to be m easured. Furth erm o re, th e n e u tra l kaon pairs provided by D A T N E have an ad ditio n al unique feature. D ue to q u a n tu m num bers conservation in a strong 0 m eson decay, th e K SKl pairs are produced in an anti-sym m etric zero-strangeness state:

|i) = - = (|K ° (+ p )> \ K ° ( - p )> - \K°(+p)> | K ° ( - p ) » , (9) or as expressed in th e basis of C P eigenstates:

|i) = - (|K + (+ p)> |K _ ( - p ) ) - |K - ( + j S)> |K + ( - j S) ) ) . (10)

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T his initial sta te of th e n e u tra l kaon system produced by D A T N E exhibits q u a n tu m entanglem ent of th e two kaon sta te s in th e genuine E P R sense [14]. W h a t follows is th a t w hereas sta te of each kaon is undefined un til th e observation of th e first decay, m easurem ent of one of th e sta te s (by observation of its decay) guarantees th e p a rtn e r particle to be in th e o rthogonal s ta te at th e sam e tim e.

T his feature allows for a broad range of kaon interferom etry based tests of fundam ental sym m etries [15] such as C P T and Lorentz S ym m etry te sts [16]. M oreover, it is th e p ro p erty which opens a possibility of a tim e-reversal sym m etry at KLOE-2.

in tersecting a t two points, at one of which th e K L O E

d etecto r is located. T he figure was a d a p te d from [12]. F ig u r e 4. L ongitudinal section ° f th e K L O E d etector. T h e figure was a d a p te d from [1 2].

4.2. The K L O E detector

T ho K L O E (K LO ng E xperim ent) d etecto r is shaped as a b arrel surrounding one of th e D A TN E in teraction points, w ith a radius of 2 m and length of alm ost 3.5 m. Its large size is d icta te d by th e m ean p a th travelled by th e long-lived n e u tra l kaons produced th ere, which is ab o u t 3.4 m.

T he d etecto r is c o n stitu te d by a cylindrical drift cham ber surrou n ded by an electrom agnetic calorim eter (see th e longitudinal d etecto r section in Figure 4) which provides good coverage a round th e in teractio n point (98% of solid angle).

T he drift cham ber [17] uses a gas m ix ture based on helium (90%) and isobutane (10%) which ensures sm all m aterial budget in order to prevent K L regeneration and p hoton conversion in th e gas. T he resolution of th e K L O E D rift C ham ber (DC) is 150 ^ m in th e transv erse plane and 2 m m in th e z direction. M om entum of charged particles can be reco n stru cted w ith th e relative resolution of 0.4% w hereas vertexing resolution is of th e order of 1 m m [17]. T he whole detecto r is im m ersed in a m agnetic field of 0.52 T provided by a superco nd uctin g coil.

T he calorim eter [18] is a sam pling electrom agnetic d etecto r w ith scintillating fibers as active m aterial and passive lead layers to enhance electrom agnetic shower production. It provides an excellent tim in g resolution and high d etection efficiency for photons in th e 20-500 M eV energy range [18], which is crucial for th e reco nstruction presented in section 5.2. S p atial and tim e resolution of th e K L O E calorim eter is given below:

54 p s 5.7% E 1.2 cm

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Ą.3. Upgrade o f K L O E to K L O E -2

K L O E s ta rte d its op eratio n a t th e D A T N E collider in 1999 and was tak in g d a ta collecting a to ta l in teg rated lum inosity of 2.5 fb-1 which corresponds to ab o u t 1010 of produced f m esons. D uring K L O E runs D A T N E reached a peak value of instan tan eo u s lum inosity of 1.4 ■ 1032 c m - 2 s - 1 .

In th e recent years, th e K L O E d etecto r has undergone a th o rou g h upgrade to s ta rt new m easurem ents as K LO E-2. T he upgrades involved th e ad d itio n of new calorim eters at small angles arou nd th e beam axis to increase acceptance for photons and in stru m en t th e final focusing region [19, 20]. M oreover, new sets of d etectors have been installed close to th e beam line for tagging 7 7 events th ro u g h d etectio n of scattered high and low energy e + e - [21, 22]. Finally, a novel tracking device was co n stru cted and installed in th e region betw een th e in teractio n point and drift cham ber inner wall. This K LO E-2 Inner Tracker is a pioneer co n stru ctio n of a cylindrical Gas E lectro n M ultiplier (G EM ) track er [23] com posed of 4 layers of a triple-G E M d e te c to r barrel-shaped aro u nd th e K LO E-2 in teraction point. Its ad d itio n to K LO E-2 improves v ertexing capabilities and increases acceptance for track s w ith low transverse m om entum .

T he K LO E-2 d e te c to r re sta rte d op eratio n in 2014 in view of collecting a sam ple of th e order of 10 fb-1 in th e next years. T he physics program m e of K LO E-2 is rich [13] and includes th e possibility to conduct th e first direct T sym m etry te s t w ith n e u tra l kaons described in th is work.

5. P r o s p e c t s for e x p e r im e n ta l r e a liz a tio n o f t h e T t e s t at K L O E -2 5.1. Required reconstruction o f events

In o rder to m easure th e double decay rates used in R e2xp and R e4xp (Eq. 4-5) reco n structio n m ust be perform ed for th e following two classes of events, each com posed of a p air of kaon decays sep arated by a tim e interval of A t:

f ^ K s K l ^ 2n £±n, (1 2)

f ^ K SK L ^ n ± £ T v 3n0. (13)

As th e tim e-reversal sym m etry te st observables (Eq. 4-5) are tim e-d epen dent, it is im p o rta n t to recon struct th e tim e interval betw een decays w ith good accuracy. In tu rn , th e resolution of kaon decay tim es in th e asy m pto tic p late au region of A t » t s should be O (1 rS ) which to tra n sla tes to decay point sp atial resolution O(1 cm ).

F igure 5 schem atically shows th e recon struction of these two classes of events. In case of a tw o-pion final sta te , charged pions ( n + n - ) m ay be chosen ra th e r th a n n0n0 to profit from pion track s reco nstru cted by th e K L O E d rift cham ber. B o th kaon decay vertices can th e n be easily recon stru cted using charged particle tracks (Figure 5, left). T he o th er class of processes, shown in Figure 5, right, is significantly m ore challenging to recon struct as one of th e decays, K l ^ 3 n 0, only involves n e u tra l particles. Moreover, as th e kinem atics of th e accom panying K S decay is not closed due to a m issing neutrino, reconstru ction of K L ^ 3n0 m ust rely solely on inform ation on clusters created in th e EM C by photons originating in n0 decays. Therefore a special vertex reconstructio n m eth o d was devised for th is decay.

5.2. New K l ^ 3n0 reconstruction m ethod fo r K SK L ^ n ± e T ve 3 n0

T he new recon structio n m eth od presented in th is work aim s a t providing sp atial co ordinates of th e K l decay point and K L decay tim e by using only th e photons from K L ^ 3n0 ^ 67 recorded by th e electrom agnetic calorim eter (Figure 6).

E ach EM C clu ster contains inform ation on its location as well as recording tim e ( X i , Y i , Z i , T i ). Thus, one m ight consider a set of possible points a t which th e corresponding p h oto n originated. Such a set c o n stitu tes a sphere centered a t clu ster position w ith a radius equal to p a th length travelled by th e 7. T he la tte r, however, is unknow n due to its dependence on th e unknow n decay tim e of th e kaon t, which is also th e 7 creatio n tim e if th e n0 lifetime

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F ig u r e 5. Schem atic view of th e two classes of processes whose reco n structio n is required to m easure R e^ p and R e4xp for th e T te st. C ontinuous lines d enote DC track s of charged particles while dashed and d o tte d lines correspond to n e u tra l p articles. Process shown in th e right poses a recon stru ction challenge due to th e K L ^ 3n° decay involving n e u tra l particles only.

F ig u r e 6. T he K l ^ 3n° ^ 67 process in th e cross-section view of th e detector. T he n e u tra l kaon nor th e pions are recorded by th e tracking detecto rs and th e only experim ental inform ation on this decay are up to six clusters in th e K L O E electrom agnetic calorim eter (grey b a n d ) created by photons to which th e n eu tral pions decay alm ost im m ediately.

is neglected (Figure 7). T he equ ation describing th e sphere of origin points for i-th calorim eter cluster is:

(Ti — t )2c2 = (X i — x)2 + (Yi — y)2 + (Zi — z)2. (14) Such a sphere can be defined for each recorded EM C cluster. T he K L decay v ertex is th e com m on origin of all th e photons and therefore can be reco n structed as th e intersection of th e p h oton spheres as shown in F igu re 6. If at least 4 photons are recorded, th e ir corresponding system of equations (Eq. 14) can be solved analytically. T his yields two solutions for th e K L decay v ertex coordinates and decay tim e, am ong which th e physical solution is identified by a set of c riteria and th e o th er one is rejected as a m ath em atical a rtifa c t. Possible presence of up to six recorded photons is used to num erically solve an overdeterm ined system of equations to o b tain im proved accuracy.

I t should be stressed th a t th is reconstru ction m ethod directly yields th e kaon decay tim e in ad d itio n to vertex location, which elim inates th e need to calculate it w ith kaon travelled p a th length and m om entum which would introduce ad d ition al u n certain ty to its evaluation. In conjunction w ith th e excellent tim in g properties of th e K L O E electrom agnetic calorim eter, this recon stru ction yields a good resolution of long-lived kaon decay tim e alth ou gh being based on calorim etric inform ation only.

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F ig u r e 7. For each of th e EM C clusters its position and recording tim e is reco nstru cted by th e calorim eter w ith a good resolution (see 11). T his allows to define a set of possible origin points of th e incident p hoton as a sphere centered at th e clu ster whose radius is pro p o rtion al to th e difference of cluster tim e and K L decay tim e.

F ig u r e 8. T he K L decay vertex is also a com m on origin points of th e 6 photons which hit th e calorim eter. Therefore, its location can be found analytically as th e intersection of a t least 4 of th e spheres defined for each EM C clusters.

5.3. Resolution o f K L decay tim e reconstruction

T he reco n stru ction was tested using a MC sim ulated sam ple of K L ^ 3n0 events (generated w ith th e official K L O E software [24]) including com plete d e te c to r response and ru n-by -ru n d a ta tak in g conditions. T im e resolution, crucial for th e tim e-reversal sym m etry te s t a t K LO E-2, was studied as a function of th e decay v ertex distance from th e 0-m eson decay vertex, i.e. th e p a th length travelled by th e kaon before its decay. T he result, shown in F igure 9 proves th a t this reconstru ction algorithm has an alm ost c o n stan t resolution a t th e level of 2ts independently of th e decay vertex location. T his is a prom ising result for th e fu tu re te s t of tim e-reversal sym m etry a t K LO E-2 and studies on th e required processes will continue tow ards this test.

T his will allow T to be teste d in th e asy m p to tic region of A t » ts as sta te d in Section 5.1. This new recon struction m eth od for th e challenging Kl ^ 3 n0 decay opens th e way for K LO E-2 to provide th e first d irect evidence for tim e-reversal sym m etry violation in th e n e u tra l kaon system .

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F ig u r e 9. T he K L decay tim e resolution as a function of th e decay distan ce from 0 decay point.

A c k n o w le d g m e n ts

We w arm ly th a n k our form er K L O E colleagues for th e access to th e d a ta collected d u rin g th e K L O E d a ta tak in g cam paign. We th a n k th e D A T N E tea m for th eir efforts in m ain tainin g low background running conditions and th e ir collaboration du ring all d a ta taking. We w ant to th a n k our technical staff: G .F. F ortugno and F. Sborzacchi for th e ir d edication in ensuring efficient op eratio n of th e K L O E com puting facilities; M. Anelli for his continuous a tte n tio n to th e gas system and d e te c to r safety; A. B alla, M. G a tta , G. C orradi and G. P ap alin o for electronics m aintenance; M. Santoni, G. Paoluzzi and R. Rosellini for general d e te c to r sup p ort; C. Piscitelli for his help during m ajo r m aintenance periods.

T his work was su p p o rted in p a rt by th e E U In te g rate d In fra stru ctu re In itiativ e H adro n Physics P ro je c t u n d er c o n tra ct num ber R II3-C T - 2004-506078; by th e E u ro p ean Com m ission u nd er th e 7th Fram ew ork P ro gram m e th ro u g h th e Research Infra

structures action of th e Capacities Program m e, Call: FP7-IN FR A ST R U C T U R E S-2008- 1, G ran t A greem ent No. 227431; by th e Polish N atio nal Science C entre th ro u g h th e G ran ts No. 2 0 1 1 /0 3 /N /S T 2 /0 2 6 4 1 , 2 0 1 1 /0 1 /D /S T 2 / 00748, 2 0 1 1 /0 3 /N /S T 2 /0 2 6 5 2 , 2 0 1 3 /0 8 /M /S T 2 /0 0 3 2 3 , D E C -2 0 1 4 /1 2 /S /S T 2 /0 0 4 5 9 , 2 0 1 3 /1 1 /B /S T 2 /0 4 2 4 5 and by th e Foun

d a tio n for Polish Science th ro u g h th e M PD program m e and th e project H O M IN G PLU S B IS /2011-4/3.

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