JHEP03(2018)066 Springer March 8, 2018 , March 12, 2018 : October 17, 2017 : -violation effect February 14, 2018 : : CP Accepted Published Revised Received 1 violations in charm-baryon decays [email protected] , violation in charmed asymmetry can be used to search for CP asymmetry in neutral kaon modes can Published for SISSA by 0 https://doi.org/10.1007/JHEP03(2018)066 L K CP − CP 0 S K asymmetries and 0 [email protected] L , K − 0 S K . 3 1709.09873 asymmetries and The Authors. 0 L c Heavy Quark Physics, CP violation as a promising observable. Besides, it is studied for a new

We study the , K [email protected] 0 S,L pK − 0 → S Corresponding author. 1 School of Nuclear Science andLanzhou 730000, Technology, Lanzhou China University, E-mail: [email protected] + c Open Access Article funded by SCOAP Keywords: ArXiv ePrint: Cabibbo-suppressed amplitudes with the neutraleffect kaon is mixing. determined Once by the experiments,then new the CP-violation be direct extracted andSU(3) symmetry used will to be search tested for by the new experiments physics. in the The future. numerical results based on Abstract: with neutral kaons in thetwo-body doubly final Cabibbo-suppressed state. amplitudes The ofΛ charm-baryon decays, with thein one in these processes, induced by the interference between the Cabibbo-favored and doubly baryon decays into neutral kaons Di Wang, Peng-Fei Guo, Wen-Hui Long and Fu-Sheng Yu K JHEP03(2018)066 ] ]. 19 30 CP – (1.1) ], the 12 1 . form an anti- 3 − 0 c 10 × and Ξ 21) + c . 0 ,Ξ  + c 27 . 0  35 . ) = (2 + π ]. They also provide the essential inputs for the – 1 – − 27 ]. Charmed baryon physics is becoming intriguing ], – pK , respectively. Due to the small branching fractions, 4 11 10 20 – − → 2 + c (Λ and 10 3 2 /BR − ) 8 − , 10 π 2 + 4 − ]. In this work, we will study charmed baryon decays into neutral kaons. pK 1 29 , → asymmetry 12 28 + c [ 0 L | -flavored hadrons decaying into charmed baryons to determine the CKM matrix (Λ K cb b V asymmetry | − BR 0 S Under the flavor SU(3) symmetry, the ground states of Λ decay by the Belle collaboration [ CP K + c But none of thedecays two-body play DCS processes an has essentialsince been role the observed in to multi-body understanding date.might decays the contribute The are to dynamics two-body the difficult relatively of DCS small to charmed DCS amplitude, be hadron leading to decays, studied a in much larger theory. direct Besides, new physics are of the order ofthe 10 DCS decays areThe more first and difficult only to evidence ofΛ be DCS observed transitions of compared charmed to baryons is the found CF in a and three-body SCS decays. triplet, all ofweak which decays decay are weakly. classifiedCabibbo-suppressed into (SCS) In three decays types: the and the StandardIt the doubly can Cabibbo-favored Model Cabibbo be (CF) suppressed (SM), generally decays, (DCS) expected the the decays that [ charmed singly the branching baryon fractions of the CF, SCS and DCS modes the Belle and BESIIIwith experiments more [ data available anddecays collected provide by an Belle ideal (II), laboratoryand to BESIII test study and the the LHCb. flavor heavy-to-light SU(3) baryonic Charmeddecays symmetry transitions of baryon [ [ element Charm physics plays an importantQCD role and in studying searching the forUnlike perturbative new and the non-perturbative physics charmed with mesonstudy special decays of structure with weak in fruitful decaysand the of in results up-type experiment charmed during until quark baryons the a sector. has few past years been decades ago made when [ some little new progress measurements were both performed in by theory 1 Introduction 3 4 Numerical analysis 5 Summary Contents 1 Introduction 2 JHEP03(2018)066 . 0 0 0 0 L L L L 3 K K K K (2.1) − − − − 0 0 S 0 S S 0 S ], a new violation K K K K 49 asymmetry CP asymmetry is asymmetry in asymmetry in ] and provides CP 0 L 0 0 L L 42 K K K − . − − 0 violation in charmed S ) ) ]. In ref. [ 0 0 S S 0 0 ) and neutral kaons is L L K 48 B K K K K – CP is the promising observable 44 asymmetries have been well ]. , → B → B c c 0 S,L is the summary. 43 [ B B 34 5 , CP mixing and direct meson decays into neutral kaons, pK σ Γ( 31 ]. In order to estimate the 0 3 . − D → asymmetries are studied in section 27 K ) + Γ( ) – + c , we discuss the 0 0 S S − 12 2 0 K K CP ]. Therefore, it is important to search for with 3 ]. In the baryon sector, the only signal of K 34 – 30 + → B → B . And section π c c 31 – 2 – 4 + asymmetry could be obtained and used to search B B π Γ( Γ( − CP ≡ pπ ) decaying into light baryon ( ) c violation in B → 0 S,L 0 b ]. In this work, we investigate the CP K 41 meson systems [ – → B B c 35 , B ( asymmetry, we analyze the decays of charm-baryon anti-triplet into 33 R CP asymmetry -violation effect is found existing in 0 L CP K − violation can occur in charmed baryon decays into neutral kaons. It is an essential 0 S meson decays [ asymmetry is found in Λ This paper is organized as follows. In section Numerically, the dynamics of charmed baryon decays is always difficult to describe, CP K -violation effect also exist in charmed baryon decays. Once the new effect was well D favored (CF) and the doublyasymmetry, Cabibbo-suppressed an (DCS) amplitudes observable leads to toasymmetry search the in for charmed the baryondefined DCS ( as transitions. Specifically, the 2 In the charmed hadron decays into neutral kaons, the interference between the Cabibbo- consistent with the measured data and can be testedcharmed by baryon Belle decays II, into neutral BESIII kaonsresponding and and time-dependent LHCb. its and search time-integrated for theThe DCS numerical transitions. analysis is The cor- given in section due to the sizable non-factorizableasymmetry contributions [ and light baryon octet andthose pseudoscalar universal octet parameters based on are the extracted flavor from SU(3) the symmetry, in available which data. Our results are well in charm decays. Theamplitudes new with effect the is mixing induced ofCP by final-state the mesons. interference In between thisdetermined the work, in CF we experiment, and will the DCS direct showfor that new the physics. new CP baryon decays has notin been charm observed decays up intomeasurable to neutral now. kaons have Inexcept be some for studied the literatures, known [ the indirect charmed baryon decays, and findto that search the for one two-body in DCS Λ charm-baryon decay amplitudes. element to interpret thea matter-antimatter window asymmetry in to the searchestablished Universe in for [ kaon new and physics beyond the SM. the two-body DCS amplitudes of charm-baryonon decays. the DCS Except for processes, the theasymmetry direct two-body measurement DCS in amplitudes charmed can baryon alsoinduced be by decays revealed the into by interference the neutral betweenin the kaons. CF and The DCS amplitudes, which has been studied violation compared to the SM prediction [ JHEP03(2018)066 f 0 L δ K (2.5) (2.6) (2.7) (2.2) (2.3) (2.4) − ) and and 0 S f 6= 0, can DCS K r δ f , ( r ) eigenstates , decays read as CF DCS , . f , δ ) ) δ 0 δ asymmetry shall  + violating effect in i CF CF K CP 0 L 0 δ δ cos + + K DCS K f φ | CP ( r ), namely CF CF → B − i 2 φ φ e + , c 0 ( ( S i i 0 S,L i B e e asymmetry can be further K 0 K ' − 0 DCS L . The parameters K CF CF 03)% | T 2 . 2 K T T | is expected to be proportional |  CF 0 f 2 2 f , δ → B 2 f − 1 1 ) iδ iδ  1 ) = r √ √ f c . e 0 S e − √ 0 δ 0 B = 0, the f f + K ( 08 − + r K r . φ = f ) ) ( R 0 pK r i i DCS e δ ] 0 L  f DCS DCS 9 → B → 1 + δ δ 1 + r K | = . The | c + + 52 + c ), being measurable in experiments. 4 . ≡ f B − | 2 + − ( is mode independent in the SM. It is found δ 2 ) ) − 2 DCS DCS | ), and the weak phase is negligibly small, i.e., | 0 0 A φ φ 10 φ f 2 states can be referred as the f – 3 – ( ( , = (1 i i δ iδ − K K (10 × e e e and 0 L  e = 0. In the SM, i O f . f exp K 0 4) (10 r i f , r . ) and the DCS amplitude of CF 0 ) r → B → B 0 0 DCS DCS S K φ 0 − c c T T − K CF has a branching fraction of the order of one percent, and  asymmetry is not sensitive to the and δ 2 2 B B ∼ O 1 − K 1 is measured as [ pK 1 1 0 ( ( 2 L | + −| 0 | L . 0 2 i − | √ √ S 0 S A A 0 6 λ CF → K = K are then pK φ DCS − → B K ( i | + c ) = pK − i 0 φ L ) = ) = 0 c e  | ∼ → 0 0 0 S L S K B (Λ 0 K = S,L → 2 ud r ] = ( + c K K K 1 CF ], the V K φ √ + c B T ud ∗ , 40 cs CP| → B V = → B ∗ → B c asymmetry could be used to search for the DCS decays in charmed cs CF /V i → B ) = c c B T 0 0 S ) is the magnitude of the CF (DCS) amplitude, and 0 ], the ratio between the DCS and CF amplitudes are defined as c L us /V B B / ( ( V B K K K 40 | us ( ∗ A A cd ] and DCS V mixing [ R DCS − V T ∗ | 0 cd T S 40 0 ( → B 0 S V ) are the relative strong and weak phases, respectively. The decay amplitudes K c K = K − B [ CF f − ( DCS T r 0 A φ → B ( Arg The The CF amplitude of c K B ≡ CF it can be expected that Λ thus might be measuredasymmetry with can large reveal data the sample. DCS amplitude Therefore, of the Λ corresponding baryon decays. If the DCSvanish. transition Thereby, is a absent, i.e., non-zerobe experimental taken result as of thebranching evidence fraction for of Λ the DCS transitions of charmed baryons. In fact, since the reduced as [ which is expected to be of the order of where depend on the individualthat processes, if while the DCS amplitudeto vanishes, the ratio φ Similar as ref. [ of where φ the under the convention Due to the fact that the JHEP03(2018)066 . ) 0 0 S ), 0 0 t π K ( π 0 pK (3.1) (3.2) + K K ], there → violation )), where 49 → + c − ( ) or t π + ( decays is of a CP ∗ + 0 π K detected with a 0 S,L nK asymmetry in the → 0 L )( pK → K t ( ]. In experiments, the . In the latter mode, + c CP → ) are actually included + K 30 − [ + c B asymmetry in the decays with ◦ nK pπ 0 L → is a small complex parameter mixing should be taken into 05 0 . K S,L c  → → 0 0 mixing system, with the values ], while a partial-wave analysis B ( − , 0 + c 0 pK K  10 ) ) , 0 S Γ( [ t t and ∆ i ◦ into the baryon octet and a vector K − ( ( 0 , K mixing. As pointed out in [ + → ) ≡ − 52 0 − + c with an additional strange quark are 0 . 2 ππ ππ K π | K ) | 0 + c K t Γ Γ and Λ 0 c  + q K | ( K 0 asymmetry in Λ − − → ππ = 43 0 pK ) ) + 0 L i ∓ t t Γ  + c 0 pK ( ( K can be produced due to the energy limit. At K φ and Ξ 2(1 + → . The time-dependent K ππ ππ | + c − − + c → – 4 – + c p p Γ Γ π 0 S / + c ) + = K )) and ≡  violation in π decays into the light baryon octet and a pseudoscalar i ) and Λ − ) − t ) reads as − π asymmetry in charm decays, and the effect from the + c ( 0 . Among them, the decays of Λ S,L − π + asymmetry in Λ CP + π + π K CP -violation effects in the charmed baryon decays. 0 = (1 | L + CP K K A asymmetry in the and its phase q π 0 K → 3 CP ), )( violation effects in charmed baryon decays. Such that the → − → t 2 − , ∆ ( | ( ) is recognized as a time-evolved neutral kaon CP 0 0 0  S )( t | 10 ∗ t ) suffer from the low efficiencies of detections of neutrons or ( K ( CP K K 0 × K + K pK pπ → B 2(1 + , ∆ ), the indirect → → c 011) → B + . ( p B ∗ is a promising observable to search for the two-body DCS amplitude of 0 + c c + / 2.2 . For the two-body Λ ) -violation effects in charmed meson decays, i.e., the indirect B Γ(  ∆  + c nK 0 0 S,L mixing, the direct ≡ , CP K 0 0 ) 228 ∗ pK . t asymmetry ( K states are → = (1+ pK → ππ is the time difference between the charmed baryon decays and the neutral kaon 0 − p L c + = (2 0 t K c + Unlike eq. ( In the following discussions, we will see that the CP violation can occur in the charm decays into neutral kaons, induced by the interference | state is actually reconstructed by →  K | 0 S + c with Γ the intermediate state and characterizing the indirect of K decay chain of account for the study of and where exist three in interference between two treealso (CF worthwhile and to DCS) study amplitudes the with neutral kaon mixing. It is 3 CP between the CF and DCS amplitudes with the The modes of Λ in the measured three-bodyrequires DCS much decay more Λ data. Inpriority short, to the investigate the two-bodyespecially DCS at charm-baryon-decay Belle amplitudes (II). in experiments, neutron is always difficultcan to be be revealed detected byhigh in the efficiency experiments. at Belle Onmeson, (II). In and the the the contrary, decay Λ baryonΛ modes decuplet of and Λ aand pseudoscalar Λ meson, the DCS transitions include of Λ charmed baryon decays. At BESIII,Belle only (II) Λ and LHCb,smaller the than productions Λ of Ξ meson, the only two DCS modes are Λ JHEP03(2018)066 , 0   ) ) ] in ] as ) = pK − − (3.6) (3.7) (3.8) (3.3) (3.4) (3.5) , i.e., − 48 π π 49 f π – r + + → . + π π )) can be
π 46 + c − , )) asymmetry . The weak → → π , 0 → − 0 +
mt 34 π ) , K 0 π K CP ( + ( K 31 π mt A A → ( -violation effect is ) ) in the numerator ) t t t )( A → ( ( ( t ]sin(∆ without kaon mixing (  0 ) − CP − [ 0 t , g g K CP e ( K ) ] sin(∆ q p K 0 t q p A  ( p , [ , i.e., t with neutral kaon mixing ) is from the interference K ) is independent of −R ) ) meson. t )+ − )+ m t → B ) ( S 0 0 /D → L ( − π I − Γ → c ) is the direct i 0 i π 2 + K K t π + c ) mt int CP B 0 ( ( K CP t p + π ( φ. − + meson decays [ ( A ) + S π K 0 A S π , while the strong phase is from A dir CP  m → → sin D ( K int CP mt → i A → f 0 A 0 − 0 + c ]cos(∆ δ e  ], K [ K K , a schematic description of the chain 1 2 ( ( ) + sin 47 . The first term m t 1 2 A A  f ( ] cos(∆ I | ) ) r  ( ) of the t  t t [ | ) t t ( L ( t e dir CP Γ L S L + . Equivalently, the new + asymmetry is approximated as Γ Γ − A R (Γ g Γ i g 2 – 5 – − e −  −  t − π e S ) e − ) Γ . It is clear that 0 L t + ) + CP 0 with kaon mixing − S t S m ) as an example. π K e ( ( Γ )+ 0 K i − m 2 0 ]. The amplitude of asymmetry in − φ ) = 2 − π → t e K CP e − − ] ( pK + → B )  49 1 2 A t → B [ , t cos L π c h S c ( CP dir CP → f Γ B 0 m m 47 B δ ( ' A − ( I + c ) = → K e t ) A ≡ ] A t ( )( f  cos ( t [ δ  + → ( m e f g r , and the doubly Cabibbo-suppressed amplitude of Λ 0 )) = CP R ) and the width Γ sin 2 − − pK A L K , and Λ ) originates from the interference between the two tree (CF and DCS) describe the flavor preserving and flavor changing time evolutions, φ π 2, ∆ π t − − ( / m + + − ) ( π → π (1 ) = 2 π g cos L t + t int CP ), the time-dependent S violation in neural kaon mixing [ f ( S + c π ) is from the (daughter) kaon mixing, − r A → Γ 0 t m → ( π 4 + Γ − )( K CP and → t e ) + − CP S ( A t ) int CP π ( -violation effect. It arises from the mother decay and the daughter mixing. + t 0 ( (Γ K g A ) = 0 ) = → t t K ≡ CP ( ( K 0 ) has been missed in the literature of studying → B t D ( K → Neglecting the tiny direct c int CP → ( B A 0 0 ( CP int A K without kaon mixing also from the interference effect betweenK the amplitudes of Λ phase of A which the DCS amplitudesa are new assumed toIts be physical zero, meaning anddecay, can was taking first be pointed Λ depicted outeffect in in between figure [ the Cabibbo-favored amplitude of Λ The third term amplitudes with the neutral kaon mixing, with Γ independent on the DCS amplitude.induced The by second the term interference between the CF and DCS amplitudes, with denotes the with the mass −A in which respectively, deduced as decays in the kaon rest frame [ JHEP03(2018)066 and (3.9) f (3.10) ) isn’t r t ). Such ( 4 ) from the − int CP t asymmetry ). ( A − (10 = 0 and 0, only 0. In order to π int CP t + CP ) in the relevant A t π → → ( ∼ O . φ φ i → φ ) int CP mode, in which both t )( t ( A ) and ( − t ( ). Besides, int CP K φ. t 0 pK ( ) vanish at + A ) and K CP t 0 2 ( sin → K A − D f ) + + c δ int CP t → ) ( (10 t A ) ( is well determined in experiment. t sin ( dir CP D  f 0 ) A r ∼ O t D ( ]. Unlike the SCS processes, in which f ) and t r ) + ( 49 t 0 ( dt F 0 K CP ), since t ∞ K CP A 0 ( since = 0) = 2 R A t 5 – 6 – h ( int CP ) which is tiny in the SM due to the smallness of − ) t t A ( ( = 0). Thereby, an observation with nonvanishing dir CP t A dir CP ( asymmetry cannot discriminate new physics due to the A dt F ). Thus it is worthwhile to study CP ∞ 0 A 3.5 R CP . Its mechanism is more complicated than for the ordinary = 0) = asymmetry is covered by the bare asymmetry with a time- f δ t ( ) = ), it is found that violation CP ∞ CP ) indicates new physics effect because of its tiny value in the SM. ), introduced to take into account relevant experimental effects, , 3.8 can be obtained by the data of branching fractions, which will be t A − ( ( (0 CP π f . ), neglecting the weak phase in charm, i.e., setting F δ asymmetry in, for instance, the ∼ + 4 = 0) is of order of 10 CP . Schematic description of the chain decay Λ π ) 3.8 t A ( CP and → 3.6 asymmetry in charm decays, since it doesn’t vanish as f )( CP t r ( A CP K Figure 1 violation, seen in eq. ( = 0) would be the signal of new physics [ → B t CP ( The time-integrated From eqs. ( c are required to predict the values of B and thus sensitive to new physics, it is necessary to extract CP f dependent function, large, which results inA a larger the precise measurement of the ambiguities in estimating thein penguin amplitudes, the measurable direct In the SM, an order is farLHCb upgrade. beyond the However, in precision some limit new of physics the models, forthcoming the experiments, weak phase Belle difference II can and be processes. In eq. ( δ Fortunately, discussed in section the oscillation and decaythe take direct place ininvestigate the the mother direct particle φ total the (mother) charm decays, mixing-induced JHEP03(2018)066 . ) f 0 S δ ) + K = 0. + (3.16) (3.14) (3.15) (3.13) (3.11) (3.12) asym- 2 mt cos t π S φ f Γ represent δ → , − CP . ) f asymmetry e r cos(∆   f +  sin ] cos 0 L δ   2 x = D [ ) φ [ x ( K t 2 e x 2 t sin Γ t 2 R − ( R Γ φ − φ s 1+ 0 S f e ] cos − r  f f e K − [ δ δ . cos cos ) f m f f f 1 f ) = δ δ t r δ I t sin sin δ . ) ] ], since ( ( ], we get f  φ φ S s s [ r ). Such an order is much cos  49 cos sin 4 47 3 /τ cos 018 f f m 2 φ f φ . δ δ φ t cos cos violation in the neutral kaon = 1 and . φ r I − 0 1 [ ∼− − 1 f f 2 2 1 4 sin sin e r r φ t cos cos  ] ] cos − − − φ φ S cos   ] f f CP )], and [ [ − 1 f Γ /D. ≈ −  f δ r sin e r [  t < t , cos cos − r S ] 2 (10 m 016 y e f 2 e mt  2 . f f R [ I /τ δ r r O 0 cos ] ] int CP R 1 , corresponds to the direct 2   t or − = m 2 [ [ f , A ' − e 2( −  ), the terms proportional to − r and sin 1 I ]+2 1 1 m ] e 2 t t are the  R 2 f 4 , is a sub-leading contribution which is at I  + f sin(∆ [ + ] Γ [ δ 2 f )( r f < t < t e − δ 2 − φ − 022 − m x 3.12 ]+2 r 47 ] . e x/y  1 ] x  dir CP [ I R 1 φ [ t t > t  cos , − e [ m A sin cos f ] + 2 L − e ) – 7 – I R  f r τ φ sin 2 ' − [ + δ R ) = 0 2 f e x ] + mt 2 1 0 0 δ  2  Γ (1 + [  R 0 S,L K CP term in the denominator cannot be neglected, since cos e ) S ] cos 2 + 2 1+ , the interference between charm decays and neutral ' − 2 τ  t sin f A K f [ t R ( − φ  r φ ) f − ( δ + / 2 ] r c m 1 [cos(∆  π  φ ' = 2 t  0 I [ S,L − cos ) = − Γ cos ) S f 1 t and × m K τ → ) r f − L ( sin φ 1 δ e τ I f ' t decay, the term in the denominator is one order of magnitude f F + r ( ]  ) δ violation in charm decays. c 0 S D cos ,  L → B cos  1 -violation effect and can be neglected [ violation then reads as ( 41  ), and those without τ ) = t f 2 K c f t 2 sin is tiny in the SM, r t R r ( − + B f CP  , t c 2 ( φ CP r π 1 1 CP 2  t 2 R t   − term can be determined by the measurement of the ( Γ, is expected to be of the order of S → 1 f τ ×  δ int CP + m/ ), the 2 S ' int CP A  τ D ∆ ) cos A 1 2 t -violation effect, 4  3.14 φ ≡ ( , t 1 1 t x ) is reduced as t ( CP CP )]. The first term, being independent of cos ( A f CP r mt CP 3.10 In eq. ( In the limitation of A A In the case of smaller than the new is measured to be small [ the new the same order asThe the 2 term insince the the denominator weak by phase expansion to be 4 Considering the sizes of kaon mixing larger than the direct Under the approximation of The time-integrated in which sin(∆ metry in charm decays. Inthe the rest new part effect of eq. ( mixing. Eq. ( In this work, we adopt the approximation in [ JHEP03(2018)066 , 0 0 S 15 D O pK (3.17) (3.18) + 6 → O + c = 87 in the asymmetries , we propose , ) for the CF ) =  . ) )] ud int CP ud 0 S can be measured CP + A )(¯ )(¯ K π 0 S ) ) /d.o.f + sc sc + 0 0 2 violation in neutral S S are non-vanishing in π π χ )] pK K K − + + + = 0 in the flavor SU(3) → int CP CP π → K π π ). It can be measured at f ], the fitted results show A + − decays, like the difference δ + + c → → → D 39 ( D pK + + + and D , p, π D D D → f ). The 0 ( S ( ( − δ -violation effect violation is negligible. Therefore, asymmetries and + c ) K 0 ( CP int CP raw S meson decays using the topological 0 L 3.14 (Λ with those in CF channels, canceling A A A CP K CP CP D pK − − − raw − A violation in Λ raw ) ) ) 0 S A → 0 0 0 S S S A K − + c K CP ) pK pK asymmetry. + 0 (Λ S π 0 – 8 – L D → →  pK K ]. In charmed baryon decays, it will be found in is unobservable, → · + c + c f − 0 + → 56 δ (Λ (Λ , 0 S K CP D . + c ( A K CP int CP . However, the SU(3) symmetry seems to work well in sin 55 (Λ int CP , − A A − raw ∝ π A 49 A ≡ ' raw , the contribution from the denominator can only be deter- + [ 0 S ) π A int CP − + and A pK f and ) = [ δ → , p, π + − 0 S + c ) for the DCS modes. These operators can be decomposed into irre- K K 65 for a global fit with only CF processes, while ( . . As discussed above, the direct + , p, π us 0 S 0 S K CP )(¯ ] and thus K -meson decays [ K = 0 ) into light baryon octet and pseudoscalar octet. A ¯ ( + dc 54 → D + c -violation effect can be revealed in ∆ – ∆ π 0 CP in the last line is the denominator of eq. ( ,Λ A 50 D → CP , + c /d.o.f ∆ D Ξ + 2 40 χ − D The nonleptonic charmed decays are induced by the operators (¯ Note that sizable flavor SU(3) breaking effect is always expected in charmed hadron The time-dependent and time-integrated , 0 c the following that theinducing CF non-vanishing and DCS amplitudes are different in themodes, SU(3) and decomposition, ( ducible representations of flavor SU(3) symmetry group. For example, (¯ case with only SCSkaons processes. in Since the we CF focusthe and on global DCS the fit. processes charmed baryon Thusdecays in decays into large this into neutral SU(3) work, neutral kaons breaking the inlimit effects SCS which [ can processes the be strong are avoided.the phase not difference charged involved Besides, in unlike the decays. It is foundbetween to be verythe large CF in processes. the SCSdiagrammatic For processes example, approach of in under thethat the analysis SU(3) of symmetry in ref. [ in charmed baryon decays into neutraland kaons DCS based processes on are the model-independently flavorexperimental SU(3) analyzed data. symmetry. with In The the this CF parameters work,(Ξ extracted we from focus on the decays of the charmed-baryon anti-triplet 4 Numerical analysis In the this section, we numerically estimate the and the new LHCb by combination of thethe raw production asymmetries and detection asymmetries, as where kaon mixing is mode-independent and thus cancelled between the ones in Λ mined by the measurement of the by Belle II andan LHCb. observable, In order to extract the new But in the case of Λ JHEP03(2018)066 C ' 23 θ / 2 C 18 (4.5) (4.6) (4.1) (4.2) (4.3) (4.4) ) and θ )] are the 2 1 W B , m ( m s 06) . , 0 /α  ) cd (6) = 2 tan and b T 97 c 33 . d b m B . So then five free (0 ( H i . s + M m e c a always occur together α π , is also well determined    B (shown in table − g 8 h    Ξ η 06) + c . (6) 3 )]. The perturbative QCD 0 , 0 / + 3Λ c ab and ] → 2 sd /  K T K 2 H p n 0 c f . One can find the magnitudes )(¯ p , respectively, where the DCS g . Since only the relative phases 27 π , , 43 abc p + − . uc 2 1  + − to 8 K 21 d b = , η 0 0 |A| π and Ξ B 6 ) + (¯ = (0 c c Σ 1 B d by a factor of [ 20 − ab √ 0 B S is the Cabibbo angle and tan 2 g 1 ud m in the fit to avoid large correlation, and M √ | 0 K 15 + 0 + + c C + )(¯ g ac πm ¯ and Ξ Λ θ 0 p π K Ξ O − Σ 2 | T f sc π + 0 S Λ 2 [(¯ ,T → as real. At the current stage, the available = 1 f ) 6 (6) 1 2 √ 1 0 c ) = + c e pK √ over ab – 9 – = − = Λ 0 M 6 H , ranges from h are free complex parameters to be extracted from 8 → , h Σ f 15 η + c O f 2 6 Ξ 1 O 03 1 + c . Then we find + . √ √ → B − f − − 0 d b − ]. In an approximation of neglecting the contributions c , − Ξ Σ + 0 c − B π  e, f, g M K 0 Λ + c d 58 h , π Γ( 6 )] and 26 B 1 2 . √ 1 = 57 = (Ξ sd ac √ (6) = 2 for Cabibbo-allowed modes, and c g T )(¯    = 0 5 [ T 22    . | uc 2 = (6) H mode is highly suppressed. The parameter f (¯ = | a b ab 0 S ∼ a − b make sense, we take B H ) , e 25 M g pK 17, and the phase of / ud . = 03 . 18 )(¯ 0 . The coefficients )] eff | are mostly fixed by Λ and = 0 b sc  H | ud [(¯ m f f 1 2 ( V | 67 s , ∗ . cs is the center-of-mass momentum of the final state particles, , the effective Hamiltonian is expressed as [ e ) in the CF amplitudes, we use = c /α g /d.o.f 15 /V 6 parameters, with the representations given in table ) p 2 = 0 and c + O O us g χ e | The partial decay width of charmed baryon decays is m V , f e ( | ∗ f s cd can be obtained through , V α with of contribution in the relevant data include five branchingthe fractions ratio of of CF branchingparameters decays fractions are of fixed between Λ by Ξ sixas data ( via a global fit.g Notice that where masses of charmed baryon ande light baryons. Thebetween decay amplitudes are expressed by the for doubly Cabibbo-suppressed| modes, where data of branching fractions. The non-zero elements and the pseudoscalar octet with the charmed-baryon anti-triplet the light baryon octet, corrections enhance the coefficient[ of from with JHEP03(2018)066 22 30 84 81 67 80 . (%) . . 06 07 09 09 08 . 34 12 . . 17 17 17 ...... 4 4 1 1 1 10 0 0 0 0 0 0 0 0 0 0 decays ]. The ∼ ∼      ∼   ∼ ∼    0 ∼ , SU(3) , we only 30 + c 00 06 07 24 36 f 01 . . . . . 0.31 2.24 . BR 0 . The ratio of is much larger C θ 2 2 χ (%) 0.12 0.50 0.08 1 0.10 1.27 0.07 1.27 0.07 1.30      exp and those of Ξ 0.50 BR 0 L occurs without ]. ) 0.50 ) 0.47 g g g 30 pK + 2 + 2 → f f + c + 2 + 2 in the final states, the associated doubly , including the uncertainties from the ) 0.20 ) 0.23 e e f ) 0 ) 0 g g ) 1 ) 1.58 4 4 e e 2 2 g g 0 S,L 2 2 − − ( ( (2 (2 − − ) 1.29 ) 1.30 K − − 2 2 ) 1.24 g g ( ( 1 1 12 12 f f 1 1 g 2 √ √ 2 2 ) 0 028 is also included in the global fit. Our results 2 2 √ √ 2 1 1 . g √ √ ) significantly exceeds the experimental upper . The numerical results of branching fractions 0 + 2 − − − − 0 − ( ( − – 10 – π ) + ) f f − 1 2 1 2  , compared to the experimental data [ ) + ) + 2 e e f 2 g f pπ ) + ) g g 2 2 2 e f e 2 2 g g − 1 2 2 + 2 − 2 − − − ) + ) 210 − and − ( ( (2 (2 → . e e + 4 + 4 − e e e 2 2 ( C C C C π cannot be extracted from the available data, and thus f f + c θ θ θ θ 2 (2 (2 (2 − − 1 2 2 2 2 − √ ( ( C C 2 f (Λ 1 θ θ 2 6 . For those modes in which ) = 0 + 4 + 4 √ 1 1 0 c 2 2 √ √ B + tan tan tan tan e e π 2 2 2 2 2 2 1 1 1 1 − tan tan √ √ √ √ − − ( ( Ξ 1 2 1 2 are identical to each other due to the isospin symmetry. From C C and Ξ θ θ → 0 2 2 + c 0 c π + (Ξ tan tan ,Ξ + c 12 12 1 1 √ √ /BR ) and Σ 0 S 0 0 S L 0 − + 0 0 S L + + 0 + K + 0 0 0 0 S L S L + π K K π π π π π Λ K K K K π K + 0 K 0 0 0 0 + 0 + 0 − + pK pK Λ . Branching fractions and amplitude representations for the Cabibbo-favored charmed Λ Λ Ξ Ξ Σ Σ Ξ Σ Σ Σ Ξ Σ Σ → Σ , it is clear that our results are well consistent with the data. It is plausible that the → → → 0 c → → → → → → 1 → → → → → → Modes Representation → → + c 0 c + c 0 c + c 0 c (Ξ 0 c + c 0 c 0 c + c + c + c 0 c + c c + Ξ Ξ Ξ Λ Λ Λ Ξ Ξ Ξ Ξ + c Λ Λ Λ Ξ Ξ Ξ Besides, in ref. [27], the authorsanalysis studied similarly charmed to baryon decays our underthan work, the ours, SU(3) but symmetry and including their thebound. result SCS The of processes. predictions on Their can the be branching fractions tested of by Λ experiments in the future. give a range dueother to fitted the parameters ambiguity and of fromDCS the the amplitudes phase lifetimes of do of not charmedΛ affect baryons. the The fitting exceptions verytable in the much. Besides, theCF branching and fractions DCS of transitions of charm baryons are well expressed by flavor SU(3) symmetry. by global fitting.is The free phase in of theare full given range in between theuncertainties in last our column predictionsthe in include lifetimes table the of errors Λ from the global fit and the ones from Cabibbo-suppressed amplitudes areBR taken into account withare given the in factor the of last column, tan compared to the experimental data [ Table 1 baryon decays. For the modes with neutral kaons JHEP03(2018)066 . , ) ) 1. 3 f L δ − S τ ) ) is 4.6 0 0 S S 10  K 2 60 39 × . . pK t and the + ) 2 1 asymme- Σ 0  23 in solution → 0 1 since the . asymmetry 0 , while the L S,L K CP S decays in the 4 3 390 5 S → τ . ∼ − ∼ − K 0 K L + c − A 0 S − 0 + + c K  − K (Λ 57 91 10 Σ ≈ − ∼ . . 1 (Ξ 0 t S 4 2 − ] asymmetries. The asymmetries, seen ( ×  int CP → [ violating effect was K − − 0 S → B 2. The denominator CP e A 0 113 L 8) c . CP S + c K A . are obviously different R 0 B K CP A 6 2 0 S (Ξ CP − − ) R decays. ∼ K 0 S ) in 0 S 1 and 1, within the experimental 3 ' − L . . K . There are two solutions of τ S 0 ) K S,L 05 10 0 0 . . → B 3 K Σ 0 0 2 due to its proportion to sin K CP  0 c S,L S 2 A B   016 t K → . ) is (0 0 → B 0 0 c 0 S asymmetries 42 50 c  , is of the order of 10 Σ . . ) =  B or even 0 (Ξ 3 2 S 0 S 1 and pK τ int CP → ) from the total − − 2 S K CP CP A 0 c − 091  → . A 1 (Ξ 3.14 ]. Thus if the new t + c ( → B R . It is found that the ) 49 c (Λ 0 S 2 CP violation in charmed baryon decays into neutral – 11 – B ) ( A K 05 04 dir CP asymmetries. The results of . . Λ ). For example, the range of 0 A 0 0 S,L 004 0 CP 0 L 5 asymmetries in . CP K 0 A −   → can be revealed by the subtraction of K 0 L Λ 0 c  K are the same in solutions of − 13 58 (10 . . int CP (Ξ f → − 0 3 3 S = 0, O δ A 0 c 2 are obtained by flipping the signs of all phase parameters of 0 037 S − − asymmetry effect, . K f CP S is asymmetries 0 r K (Ξ cos A . − R φ CP dir CP CP ) asymmetries, could discriminate these solutions. In the absence of A 0 S ) cos and the one for ones due to the interference between the CF and DCS amplitudes, ), are given in table 67 09 violation are sensitive to new physics which may contribute to the f . . 0 4 S,L r 087 pK 0 CP 2 3 L . Table 2 0 S,L 2 − for simplification) are presented in table 0 K term in denominator of eq. ( K − → , one can find the branching fractions of pK CP 10 decay, the promising observable to search for two-body DCS amplitudes ∼ ∼ − ∼ − f 1 ) could be obtained from the corresponding + c CP δ × → 2 is obtained by flipping the signs of all phase parameters in -violation effect are opposite between the solutions of A → B 0 15 55 S,L S (Λ → B . . , where the sets of ). The new + c 010 asymmetry c . 3 3 3 3.14 cos 37) c CP 0 B − . int CP pK CP (Λ − − φ B 3 asymmetries for each parameter set. We label the results obtained from eq. ( − A ( . Time-integrated 3.15 A R CP → R ∼ cos 1, and + c The numerical results of the time-integrated From table 1 2 CP f S in eq. ( r 16 S S . 2 1. The direct (0 S DCS amplitudes with adetermined large in weak experiment, the phase direct kaons [ can be obtained and used to search for new physics. − signs of while the ones of D in eq. ( direct The new solution with opposite strongare phases proportional contributes to equivalently the toestablish cosine branching the fractions of above that thethe strong DCS phases. contributions, The i.e., measurements in the future, to (denoted by the by which results in measurable tries, in Λ of charmed baryons,capability. can reach the order of 10 units of 10 from the Table 3 JHEP03(2018)066 0 0 L L ] K K ] − − ] Phys. 0 S 0 S , K 0 K π + π asymmetries asymmetries asymmetries + 0 0 L L π − K K CP ]. Σ − − arXiv:1608.00407 [ 0 → S 0 S arXiv:1702.05279 + c K K [ SPIRE arXiv:1707.00089 Λ [ IN Ann. Rev. Nucl. Part. Sci. , ][ and branching fraction (2016) 232002 0 ]. (2017) 111102 φpπ (2017) 051102 117 → + c D 95 SPIRE Λ D 96 IN arXiv:1611.04382 ][ ]. [ Charm Meson Decays Evidence for the singly-Cabibbo-suppressed decay Measurement of the absolute branching fraction for Observation of the decay Measurement of Singly Cabibbo Suppressed Decays – 12 – -violation effect is found in charmed baryon decays SPIRE Phys. Rev. Phys. Rev. Lett. Search for IN CP , Phys. Rev. 0 , (2017) 42 ][ , − 0 pπ K ), which permits any use, distribution and reproduction in decay mode as a promising method to search for the two- pπ + → + π B 767 + c 0 S,L pK − Λ arXiv:1705.11109 [ K → pK + c → → Λ + c CC-BY 4.0 + c Λ arXiv:0802.2934 Phys. Lett. This article is distributed under the terms of the Creative Commons and [ , (2017) 388 µ − ν π mixing. Once it is determined in experiments, the direct and search for + + ]. ]. ]. collaboration, M. Ablikim et al., collaboration, M. Ablikim et al., collaboration, M. Ablikim et al., collaboration, M. Ablikim et al., 0 µ collaboration, B. Pal et al., Λ pπ pη K B 772 violations. violation in charmed baryon decays into neutral kaons. Induced by the interference → → → (2008) 249 − SPIRE SPIRE SPIRE 0 + c + c + c IN IN IN BESIII Λ [ BESIII Λ BESIII Λ BESIII Lett. 58 Belle measurement of [ [ CP CP K M. Artuso, B. Meadows and A.A. Petrov, [5] [6] [3] [4] [1] [2] Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. References We are grateful to Lei Li andin Xiao-Rui part Lyu for by helpful the discussions. National11375076, This Natural work 11505083 Science was and supported Foundation U1732101, of ChinaUniversities and under under the Grants Grant Fundamental No. No. Research Funds 11347027, lzujbky-2015-241 for and the lzujbky-2017-97. 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