Charmed Ω c weak decays into Ω in the light-front quark model
aa r X i v : . [ h e p - ph ] N ov Charmed Ω c weak decays into Ω in the light-front quark model Yu-Kuo Hsiao, ∗ Ling Yang, † Chong-Chung Lih, ‡ and Shang-Yuu Tsai § School of Physics and Information Engineering,Shanxi Normal University, Linfen 041004, China Department of Optometry, Central Taiwan Universityof Science and Technology, Taichung 40601 (Dated: November 24, 2020)
Abstract
More than ten Ω c weak decay modes have been measured with the branching fractions relative tothat of Ω c → Ω − π + . In order to extract the absolute branching fractions, the study of Ω c → Ω − π + is needed. In this work, we predict B π ≡ B (Ω c → Ω − π + ) = (5 . ± . × − with the Ω c → Ω − transition form factors calculated in the light-front quark model. We also predict B ρ ≡ B (Ω c → Ω − ρ + ) = (14 . ± . × − and B e ≡ B (Ω c → Ω − e + ν e ) = (5 . ± . × − . The previousvalues for B ρ / B π have been found to deviate from the most recent observation. Nonetheless, our B ρ / B π = 2 . ± . B e / B π = 1 . ± .
2, whichis consistent with the current data. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION The lowest-lying singly charmed baryons include the anti-triplet and sextet states B c =(Λ + c , Ξ c , Ξ + c ) and B ′ c = (Σ (0 , + , ++) c , Ξ ′ (0 , +) c , Ω c ), respectively. The B c and Ω c baryons pre-dominantly decay weakly [1–5], whereas the Σ c (Ξ ′ c ) decays are strong (electromagnetic)processes. There have been more accurate observations for the B c weak decays in the recentyears, which have helped to improve the theoretical understanding of the decay processes [6–14]. With the lower production cross section of σ ( e + e − → Ω c X ) [4], it is an uneasy task tomeasure Ω c decays. Consequently, most of the Ω c decays have not been reanalysized since1990s [15–23], except for those in [24–29].One still manages to measure more than ten Ω c decays, such as Ω c → Ω − ρ + , Ξ ¯ K ( ∗ )0 and Ω − ℓ + ν ℓ , but with the branching fractions relative to B (Ω c → Ω − π + ) [5]. To extractthe absolute branching fractions, the study of Ω c → Ω − π + is crucial. Fortunately, theΩ c → Ω − π + decay involves a simple topology, which benefits its theoretical exploration.In Fig. 1a, Ω c → Ω − π + is depicted to proceed through the Ω c → Ω − transition, while π + is produced from the external W -boson emission. Since it is a Cabibbo-allowed processwith V ∗ cs V ud ≃
1, a larger branching fraction is promising for measurements. Furthermore,it can be seen that Ω c → Ω − π + has a similar configuration to those of Ω c → Ω − ρ + andΩ c → Ω − ℓ + ν ℓ , as drawn in Fig. 1, indicating that the three Ω c decays are all associatedwith the Ω c → Ω − transition. While Ω is a decuplet baryon that consists of the totallysymmetric identical quarks sss , behaving as a spin-3/2 particle, the form factors of theΩ c → Ω − transition can be more complicated, which hinders the calculation for the decays.As a result, a careful investigation that relates Ω c → Ω − π + , Ω − ρ + and Ω c → Ω − ℓ + ν ℓ hasnot been given yet, despite the fact that the topology associates them together.Based on the quark models, it is possible to study the Ω c decays into Ω − with theΩ c → Ω − transition form factors. However, the validity of theoretical approach needs to betested, which depends on if the observations, given by B (Ω c → Ω − ρ + ) B (Ω c → Ω − π + ) = 1 . ± . > . , B (Ω c → Ω − e + ν e ) B (Ω c → Ω − π + ) = 2 . ± . , (1)can be interpreted. Since the light-front quark model has been successfully applied to theheavy hadron decays [27, 30–46], in this report we will use it to study the Ω c → Ω − transition2 a )Ω c Ω − u ¯ dcss sssW π + , ρ + ( b )Ω c Ω − ν e , ν µ e + , µ + css sssW FIG. 1. Feynman diagrams for (a) Ω c → Ω − π + ( ρ + ) and (b) Ω c → Ω − ℓ + ν ℓ with ℓ + = e + or µ + . form factors. Accordingly, we will be enabled to calculate the absolute branching fractionsof Ω c → Ω − π + ( ρ + ) and Ω c → Ω − ℓ + ν ℓ , and check if the two ratios in Eq. (1) can be wellexplained. II. THEORETICAL FRAMEWORKA. General Formalism
To start with, we present the effective weak Hamiltonians H H,L for the hadronic andsemileptonic charmed baryon decays, respectively [47]: H H = G F √ V ∗ cs V ud [ c (¯ ud )(¯ sc ) + c (¯ sd )(¯ uc )] , H L = G F √ V ∗ cs (¯ sc )(¯ u ν v ℓ ) , (2)where G F is the Fermi constant, V ij the Cabibbo-Kobayashi-Maskawa (CKM) matrix ele-ments, c , the effective Wilson coefficients, (¯ q q ) ≡ ¯ q γ µ (1 − γ ) q and (¯ u ν v ℓ ) ≡ ¯ u ν γ µ (1 − γ ) v ℓ . In terms of H H,L , we derive the amplitudes of Ω c → Ω − π + ( ρ + ) and Ω c → Ω − ℓ + ν ℓ as [48, 49] M h ≡ M (Ω c → Ω − h + ) = G F √ V ∗ cs V ud a h Ω − | (¯ sc ) | Ω c ih h + | (¯ ud ) | i , M ℓ ≡ M (Ω c → Ω − ℓ + ν ℓ ) = G F √ V ∗ cs h Ω − | (¯ sc ) | Ω c i (¯ u ν ℓ v ℓ ) , (3)where h = ( π, ρ ), ℓ = ( e, µ ), and a = c + c /N c results from the factorization [50], with N c the color number. 3ith B ′ c ( B ′ ) denoting the charmed sextet (decuplet) baryon, the matrix elements of the B ′ c → B ′ transition can be parameterized as [28, 43] h T µ i ≡ h B ′ ( P ′ , S ′ , S ′ z ) | ¯ qγ µ (1 − γ ) c | B ′ c ( P, S, S z ) i = ¯ u α ( P ′ , S ′ z ) " P α M γ µ F V + P µ M F V + P ′ µ M ′ F V ! + g αµ F V γ u ( P, S z ) − ¯ u α ( P ′ , S ′ z ) " P α M γ µ F A + P µ M F A + P ′ µ M ′ F A ! + g αµ F A u ( P, S z ) , (4)where ( M, M ′ ) and ( S, S ′ ) = (1 / , /
2) represent the masses and spins of ( B ′ c , B ′ ), respec-tively, and F V,Ai ( i = 1 , , ..,
4) the form factors to be extracted in the light-front quarkmodel. The matrix elements of the meson productions are defined as [5] h π ( p ) | (¯ ud ) | i = if π q µ , h ρ ( λ ) | (¯ ud ) | i = m ρ f ρ ǫ µ ∗ λ , (5)where f π ( ρ ) is the decay constant, and ǫ µλ is the polarization four-vector with λ denoting thehelicity state. B. The light-front quark model
The baryon bound state B ′ ( c ) contains three quarks q , q and q , with the subscript c for q = c . Moreover, q and q are combined as a diquark state q [2 , , behaving as a scalar oraxial-vector. Subsequently, the baryon bound state | B ′ ( c ) ( P, S, S z ) i in the light-front quarkmodel can be written as [31] | B ′ ( c ) ( P, S, S z ) i = Z { d p }{ d p } π ) δ ( ˜ P − ˜ p − ˜ p ) × X λ ,λ Ψ SS z (˜ p , ˜ p , λ , λ ) | q ( p , λ ) q [2 , ( p , λ ) i , (6)where Ψ SS z is the momentum-space wave function, and ( p i , λ i ) stand for momentum andhelicity of the constituent (di)quark, with i = 1 , q and q [2 , , respectively. The tildenotations represent that the quantities are in the light-front frame, and one defines P =( P − , P + , P ⊥ ) and ˜ P = ( P + , P ⊥ ), with P ± = P ± P and P ⊥ = ( P , P ). Besides, ˜ p i aregiven by ˜ p i = ( p + i , p i ⊥ ) , p i ⊥ = ( p i , p i ) , p − i = m i + p i ⊥ p + i , (7)4ith m = m q , m = m q + m q ,p +1 = (1 − x ) P + , p +2 = xP + ,p ⊥ = (1 − x ) P ⊥ − k ⊥ , p ⊥ = xP ⊥ + k ⊥ , (8)where x and k ⊥ are the light-front relative momentum variables with k ⊥ from ~k = ( k ⊥ , k z ),ensuring that P + = p +1 + p +2 and P ⊥ = p ⊥ + p ⊥ . According to e i ≡ q m i + ~k and M ≡ e + e in the Melosh transformation [30], we obtain x = e − k z e + e , − x = e + k z e + e , k z = xM − m + k ⊥ xM ,M = m + k ⊥ − x + m + k ⊥ x . (9)Consequently, Ψ SS z can be given in the following representation [41–44]:Ψ SS z (˜ p , ˜ p , λ , λ ) = A ( ′ ) q p · ¯ P + m M ) ¯ u ( p , λ )Γ ( α ) S,A u ( ¯ P , S z ) φ ( x, k ⊥ ) , (10)with A = vuut m M + p · ¯ P )3 m M + p · ¯ P + 2( p · p )( p · ¯ P ) /m , Γ S = 1 , Γ A = − √ γ ǫ/ ∗ ( p , λ ) , and A ′ = vuut m M m M + ( p · ¯ P ) , Γ αA = ǫ ∗ α ( p , λ ) , (11)where the vertex function Γ S ( A ) is for the scalar (axial-vector) diquark in B ′ c , and Γ αA for theaxial-vector diquark in B ′ . We have used the variable ¯ P ≡ p + p to describe the internalmotions of the constituent quarks in the baryon [32], which leads to ( ¯ P µ γ µ − M ) u ( ¯ P , S z ) = 0,different from ( P µ γ µ − M ) u ( P, S z ) = 0. For the momentum distribution, φ ( x, k ⊥ ) is presentedas the Gaussian-type wave function, given by φ ( x, k ⊥ ) = 4 πβ ! / s e e x (1 − x ) M exp − ~k β , (12)where β shapes the distribution. 5sing | B ′ c ( P, S, S z ) i and | B ′ ( P, ′ S ′ , S ′ z ) i from Eq. (6) and their components in Eqs. (10),(11) and (12), we derive the matrix elements of the B ′ c → B ′ transition in Eq. (4) as h ¯ T µ i ≡ h B ′ ( P ′ , S ′ , S ′ z ) | ¯ qγ µ (1 − γ ) c | B ′ c ( P, S, S z ) i = Z { d p } φ ′ ( x ′ , k ′⊥ ) φ ( x, k ⊥ )2 q p +1 p ′ +1 ( p · ¯ P + m M )( p ′ · ¯ P ′ + m ′ M ′ ) × X λ ¯ u α ( ¯ P ′ , S ′ z ) h ¯Γ ′ αA ( p/ ′ + m ′ ) γ µ (1 − γ )( p/ + m )Γ A i u ( ¯ P , S z ) , (13)with m = m c , m ′ = m q and ¯Γ = γ Γ † γ . We define J µ j = ¯ u (Γ µβ ) j u β and ¯ J µ j = ¯ u (¯Γ µβ ) j u β with j = 1 , , ...,
4, where (Γ µβ ) j = { γ µ P β , P ′ µ P β , P µ P β , g µβ } γ , (¯Γ µβ ) j = { γ µ ¯ P β , ¯ P ′ µ ¯ P β , ¯ P µ ¯ P β , g µβ } γ . (14)Then, we multiply J j ( ¯ J j ) by h T i ( h ¯ T i ) as F j ≡ J j · h T i and ¯ F j ≡ ¯ J j · h ¯ T i with h T i and h ¯ T i in Eqs. (4) and (13), respectively, resulting in [43] F j = T r (cid:26) u β ¯ u α " P α M γ µ F V + P µ M F V + P ′ µ M ′ F V ! + g αµ F V γ ¯ u (Γ β µ ) j (cid:27) , ¯ F j = Z { d p } φ ′ ( x ′ , k ′⊥ ) φ ( x, k ⊥ )2 q p +1 p ′ +1 ( p · ¯ P + m M )( p ′ · ¯ P ′ + m ′ M ′ ) × X λ T r (cid:26) u β ¯ u α h ¯Γ ′ αA ( p/ ′ + m ′ ) γ µ ( p/ + m )Γ A i u (¯Γ β µ ) j (cid:27) . (15)In the connection of F j = ¯ F j , we construct four equations. By solving the four equations,the four form factors F V , F V , F V and F V can be extracted. The form factors F Ai can beobtained in the same way. C. Branching fractions in the helicity basis
One can present the amplitude of Ω c → Ω − h + (Ω − ℓ + ν ℓ ) in the helicity basis of H λ Ω λ h ( ℓ ) [28,43], where λ Ω = ± / , ± / − baryon, and λ h,ℓ thoseof h + and ℓ + ν ℓ . Substituting the matrix elements in Eqs. (3) with those in Eqs. (4) and (5),the amplitudes in the helicity basis now read √ M h = ( i ) P λ Ω ,λ h G F V ∗ cs V ud a m h f h H λ Ω λ h and √ M ℓ = P λ Ω ,λ ℓ G F V ∗ cs H λ Ω λ ℓ , where H λ Ω λ f = H Vλ Ω λ f − H Aλ Ω λ f with f = ( h, ℓ ). Explicitly, H V ( A ) λ Ω λ f is written as [28] H V ( A ) λ Ω λ f ≡ h Ω − | ¯ sγ µ ( γ ) c | Ω c i ε µf , (16)6ith ε µh = ( q µ / √ q , ǫ µ ∗ λ ) for h = ( π, ρ ). For the semi-leptonic decay, since the ℓ + ν ℓ systembehaves as a scalar or vector, ε µℓ = q µ / √ q or ǫ µ ∗ λ . The π meson only has a zero helicitystate, denoted by λ π = ¯0. On the other hand, the three helicity states of ρ are denoted by λ ρ = (1 , , − λ ℓ = λ π or λ ρ . Subsequently, we expand H V ( A ) λ Ω λ f as H V ( A ) ¯0 = vuut Q ± q Q ∓ M M ′ ! ( F V ( A )1 M ± ∓ F V ( A )2 ¯ M + ∓ F V ( A )3 ¯ M ′− ∓ F V ( A )4 M ) , (17)for ε µf = q µ / √ q , where M ± = M ± M ′ , Q ± = M ± − q , and ¯ M ( ′ ) ± = ( M + M − ± q ) / (2 M ( ′ ) ).We also obtain H V ( A ) = ∓ q Q ∓ F V ( A )4 ,H V ( A ) = − s Q ∓ " F V ( A )1 Q ± M M ′ ! − F V ( A )4 ,H V ( A ) = vuut Q ∓ q F V ( A )1 Q ± M ∓ M M ′ ! ∓ (cid:18) F V ( A )2 + F V ( A )3 MM ′ (cid:19) | ~P ′ | M ′ ∓ F V ( A )4 ¯ M ′− , (18)for ε µf = ǫ µ ∗ λ , with | ~P ′ | = q Q Q − / (2 M ). Note that the expansions in Eqs. (17) and (18)have satisfied λ Ω c = λ Ω − λ f for the helicity conservation, with λ Ω c = ± /
2. The branchingfractions then read B h ≡ B (Ω c → Ω − h + ) = τ Ω c G F | ~P ′ | πm c | V cs V ∗ ud | a m h f h H h , B ℓ ≡ B (Ω c → Ω − ℓ + ν ℓ ) = τ Ω c G F | V cs | π m c Z ( m Ω c − m Ω ) m ℓ dq | ~P ′ | ( q − m ℓ ) q H ℓ , (19)where H π = (cid:12)(cid:12)(cid:12) H ¯0 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H − ¯0 (cid:12)(cid:12)(cid:12) ,H ρ = (cid:12)(cid:12)(cid:12) H (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H − − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) H − − (cid:12)(cid:12)(cid:12) ,H ℓ = m ℓ q ! H ρ + 3 m ℓ q H π , (20)with τ Ω c the Ω c lifetime. III. NUMERICAL ANALYSIS
In the Wolfenstein parameterization, the CKM matrix elements are adopted as V cs = V ud = 1 − λ / λ = 0 . ± . ABLE I. The Ω c → Ω − transition form factors with F (0) at q = 0, where δ ≡ δm c /m c = ± . F (0) a bF V .
54 + 0 . δ − .
27 1 . F V . − . δ − .
00 96 . F V .
33 + 0 . δ .
96 9 . F V .
97 + 0 . δ − .
53 1 . F (0) a bF A .
05 + 1 . δ − .
66 1 . F A − .
06 + 0 . δ − .
15 71 . F A − . − . δ − .
01 5 . F A − .
44 + 0 . δ − . − . Ω c baryon and the decay constants ( f π , f ρ ) = (132 , c , c ) = (1 . , − .
51) at the m c scale [47], we determine a . In the generalized factorization, N c is taken as an effective color number with N c = (2 , , ∞ ) [28, 29, 46, 50], in order toestimate the non-factorizable effects. For the Ω + c ( css ) → Ω − ( sss ) transition form factors,the theoretical inputs of the quark masses and parameter β in Eq. (15) are given by [34, 40] m = m c = (1 . ± .
05) GeV , m ′ = m s = 0 .
38 GeV , m = 2 m s = 0 .
76 GeV ,β c = 0 .
60 GeV , β s = 0 .
46 GeV , (21)where β c ( s ) is to determine φ ( ′ ) ( x ( ′ ) , k ( ′ ) ⊥ ) for Ω c (Ω − ). We hence extract F Vi and F Ai inTable I. For the momentum dependence, we have used the double-pole parameterization: F ( q ) = F (0)1 − a ( q /m F ) + b ( q /m F ) , (22)with m F = 1 .
86 GeV. Using the theoretical inputs, we calculate the branching fractions,whose results are given in Table II.
IV. DISCUSSIONS AND CONCLUSIONS
In Table II, we present B π and B ρ with N c = (2 , , ∞ ). The errors come from the formfactors in Table I, of which the uncertainties are correlated with the charm quark mass. Bycomparison, B π and B ρ are compatible with the values in Ref. [28]; however, an order ofmagnitude smaller than those in Refs. [20, 22], whose values are obtained with the total decaywidths Γ π ( ρ ) = 2 . a (11 . a ) × s − and Γ π ( ρ ) = 1 . a (4 . a ) × s − , respectively.We also predict B e = (5 . ± . × − as well as B µ ≃ B e , which is much smaller than thevalue of 127 × − in [24]. Only the ratios R ρ/π and R e/π have been actually observed sofar. In our work, R ρ/π = 2 . ± . ABLE II. Branching fractions of (non-)leptonic Ω c decays and their ratios, where R ρ ( e ) /π ≡B ρ ( e ) / B π . The three numbers in the parenthesis correspond to N c = (2 , , ∞ ), and the errors comefrom the uncertainties of the form factors in Table I. B ( R ) our work Ref. [20] Ref. [22] Ref. [28] Ref. [24] data [4, 5]10 B π (5 . ± . , . ± . , . ± .
0) (56 . , . , .
9) (36 . , . , .
6) ( − , − , B ρ (14 . ± . , . ± . , . ± .
6) (307 . , . , .
5) (126 . , . , .
1) ( − , − , B e . ± . B µ . ± . R ρ/π . ± . . ± . > . R e/π (1 . ± . , . ± . , . ± .
1) 2 . ± . value and the most recent observation. We obtain R e/π = 1 . ± . N c = 2 to beconsistent with the data, which indicates that ( B π , B ρ ) = (5 . ± . , . ± . × − with N c = 2 are more favorable.The helicity amplitudes can be used to better understand how the form factors contributeto the branching fractions. With the identity H V ( A ) − λ Ω − λ f = ∓ H V ( A ) λ Ω λ f for the B ′ c ( J P = 1 / + ) to B ′ ( J P = 3 / + ) transition [28], H π in Eq. (20) can be rewritten as H π = 2( | H V ¯0 | + | H A ¯0 | ).From the pre-factors in Eq. (17), we estimate the ratio of | H V ¯0 | / | H A ¯0 | ≃ .
05, which showsthat H A ¯0 dominates B π , instead of H V ¯0 . More specifically, it is the F A term in H A ¯0 thatgives the main contribution to the branching fraction. By contrast, the F A , terms in H A ¯0 largely cancel each other, which is caused by F A M − ≃ F A ¯ M ′− and a minus sign between F A and F A (see Table I); besides, the F A term with a small F A (0) is ignorable.Likewise, we obtain H ρ = 2( | H Vρ | + | H Aρ | ) for B ρ , where | H V ( A ) ρ | = | H V ( A ) | + | H V ( A ) | + | H V ( A ) | . We find that | H Aρ | is ten times larger than | H Vρ | . Moreover, H A is similar to H A ¯0 , where the F A , terms largely cancel each other, F A is ignorable, and F A gives the maincontribution. While F A and F A in H A have a positive interference, giving 20% of B ρ , F A in H A singly contributes 35%. In Eq. (20), the factor of m ℓ /q with m ℓ ≃ H ℓ ≃ H ρ . Therefore, B ℓ receives the main contributions from the F A terms in H A , H A and H A , which is similar to the analysis for B ρ .9n summary, we have studied the Ω c → Ω − π + , Ω − ρ + and Ω c → Ω − ℓ + ν ℓ decays, whichproceed through the Ω c → Ω − transition and the formation of the meson π + ( ρ + ) or leptonpair from the external W -boson emission. With the form factors of the Ω c → Ω − transition,calculated in the light-front quark model, we have predicted B (Ω c → Ω − π + , Ω − ρ + ) = (5 . ± . , . ± . × − and B (Ω c → Ω − e + ν e ) = (5 . ± . × − . While the previous studieshave given the R ρ/π values deviating from the most recent observation, we have presented R ρ/π = 2 . ± . R e/π = 1 . ± . ACKNOWLEDGMENTS
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