Electromagnetic transitions of the singly charmed baryons with spin 3/2
IINHA-NTG-01/2021
Electromagnetic transitions of the singly charmed baryons with spin 3/2
June-Young Kim,
1, 2, ∗ Hyun-Chul Kim,
2, 3, † Ghil-Seok Yang, ‡ and Makoto Oka § Institut f¨ur Theoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany Department of Physics, Inha University, Incheon 22212, Republic of Korea School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of Korea Department of Physics, Soongsil University, Seoul 06978, Republic of Korea Advanced Science Research Center, Japan Atomic Energy Agency, Shirakata, Tokai, Ibaraki, 319-1195, Japan (Dated: January 27, 2021)We investigate the electromagnetic transitions of the singly charmed baryons with spin 3/2, basedon a pion mean-field approach, also known as the chiral quark-soliton model, taking into accountthe rotational 1 /N c corrections and the effects of flavor SU(3) symmetry breaking. We examine thevalence- and sea-quark contributions to the electromagnetic transition form factors and find thatthe quadrupole form factors of the sea-quark contributions dominate over those of the valence-quarkones in the smaller Q region, whereas the sea quarks only provide marginal contributions to themagnetic dipole transition form factors of the baryon sextet with spin 3/2. The effects of the flavorSU(3) symmetry breaking are in general very small except for the forbidden transition Ξ c γ → Ξ ∗ c by U -spin symmetry. We also discuss the widths of the radiative decays for the baryon sextet withspin 3/2, comparing the present results with those from other works. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] a r X i v : . [ h e p - ph ] J a n I. INTRODUCTION
It is of great importance to understand the electromagnetic (EM) structure of a baryon, since it reveals how thebaryon is shaped by its constituents. A baryon with spin 3/2 has a finite value of the electric quadrupole (E2) moment,which indicates that its charge distribution is shown to be deformed to be either a cushion-like form (oblate spheroid)or a rugby-ball-like one (prolate spheroid), depending on the signature of its charge. This implies that a singly heavybaryon with spin 3/2 may reveal a similar structure. It is also known that the effects of the vacuum polarization orthose of the pion clouds are known to contribute significantly to the E2 moment of the baryon decuplet [1]. Thisleads to an interpretation that the E2 moment of a low-lying baryon with spin 3/2 is governed by long-distance pionclouds [2]. While experimental information on the EM transitions of the singly heavy baryons is still inconclusive [3–6],there has been a great deal of theoretical works within many different approaches such as chiral perturbation theory [7–11], the quark models [12–14], QCD sum rules [15–17], and so on (see also a recent review [18]). In lattice QCD, theEM transition form factors for Ω c γ → Ω ∗ c were calculated [19, 20]. Thus, anticipating that the experimental data onthe EM transitions of the singly heavy baryons will be available in near future, it is of great interest to investigatethe structure of the EM transition form factors in a different theoretical framework.In the present work, we investigate the EM transition form factors of the low-lying singly heavy baryons withspin 3/2 within the framework of the chiral quark-soliton model ( χ QSM). The model is based on a pion mean-fieldapproach. As was proposed first by Witten [21], in the large N c limit, the light baryon can be viewed as a state of N c (the number of colors) valence quarks bound by the pion mean fields that have been produced self-consistently bythe N c valence quarks [22, 23]. The model was extended to the description of the singly heavy baryons [24–26], beingmotivated by Ref. [27]. In the limit of the infinitely heavy quark mass ( m Q → ∞ ), the spin of the heavy quark can notbe flipped, which makes the heavy-quark spin conserved. This causes also the total spin of the light quarks inside asingly heavy baryon conserved. In this limit of m Q → ∞ , the flavor of the heavy quark does not come into play. Thisis known as the heavy-quark spin-flavor symmetry [28–30]. Thus, the singly heavy baryons can be expressed withinthe SU(3) representations. That is, two light valence quarks ( ⊗ ) will allow one to have the baryon antitriplet(¯ ) and sextet ( ). The spins of the two light valence quarks can be aligned either in the spin-singlet state ( ) or inthe spin-triplet one ( ). Hence, by combining them with the spin of the heavy quark, one can have two degeneratebaryon sextets. This degeneracy can be removed by the color hyperfine interaction in order 1 /m Q [24]. Note that theinfinitely heavy quark can be regarded as the mere static color source. This indicates that the light quarks governthe dynamics inside a singly heavy baryon. Based on this heavy-quark spin-flavor symmetry, the pion mean-fieldapproach was developed also for the singly heavy baryons that can be regarded as the bound state of N c − χ QSM described various properties of the singly heavy baryons quantitativelywell, compared with the experimental data, without any free parameters [24–26, 31–34] (See also a recent review [35]).The EM form factors of the low-lying singly heavy baryons have been studied in the χ QSM [36, 37]. Since the heavy-quark mass is taken to be infinitely heavy, the heavy quark gives a constant contribution to the electric monopoleform factor constrained by the gauge invariance, whereas contributions to the magnetic dipole from factor from theheavy quark is negligible. The numerical results were in good agreement with the lattice data [38]. In the presentwork, we want to investigate the EM transition form factors of the baryon sextet with spin 3/2. While the Ω c γ → Ω ∗ c radiative decay was computed in lattice QCD, there is no work on the EM transition form factors for all possibleradiative decays for the baryon sextet with spin 3/2. Thus, we will consider for the first time the magnetic dipole(M1), and electric quadrupole (E2) and Coulomb quadrupole (C2) transition form factors for the baryon sextet ( )with spin 3/2. We will compare the results for the Ω c γ → Ω ∗ c radiative decay with those from the lattice calculation.We will compare the present numerical results for the decay rates of the radiative decays for the singly heavy baryonswith those from other theoretical works.The present work is organized as follows: In Section II, we define the M1, E2 and C2 transition form factors ofthe singly heavy baryons. In Section III, we explain explicitly how the singly heavy baryon state can be consistentlyconstructed based on the heavy-quark spin-flavor symmetry in the limit of the infinitely heavy-quark mass. We showthat the heavy-quark field can be decoupled from the singly heavy baryon and its mass contributes to the classicalmass of the singly heavy baryon in a simple manner. In Section IV, we show briefly how to compute them withinthe framework of the χ QSM. In Section V, we first compare the numerical results for Ω c + γ → Ω ∗ c with those of thecorresponding lattice data. We also examine the dependence of the EM transition form factors of Ω c + γ → Ω ∗ c onthe pion mass. We then scrutinize the valence- and sea-quark contributions separately and show that the sea quarksor the Dirac continuum play a crucial role in describing the E2 and C2 transition form factors of the baryon sextetwith spin 3/2, which can be interpreted as the pion clouds. We also study the effects of the explicit breaking of flavorSU(3) symmetry breaking on the EM transition form factors of the baryon sextet with spin 3/2. We compare thepresent results for the decay rates of the radiative decays for the singly heavy baryons with spin 3/2 with those fromother works. Finally, we summarize the results from the present work and draw conclusions. II. EM TRANSITION FORM FACTORS OF THE BARYON SEXTET WITH SPIN 3/2
To describe the EM transition from a singly heavy baryon with spin 1/2 to that with spin 3/2, Bγ ∗ → B ∗ , weassume that the baryon with spin 3/2 is at rest. In this rest frame, we define the four-momenta for the baryon withspin 3/2, the baryon with spin 1/2 and the photon respectively as p B ∗ , p B , and q , which are explicitly written as p B ∗ = ( M B ∗ , ) , p = ( E B , − q ) , q = ( ω q , q ) , (1)where q and ω q denote the three-momentum and energy of the virtual photon. The energy-momentum relation isgiven by E B = M B + | q | and E B ∗ = M B ∗ . Using this relation, we can express the momentum and the energy of thevirtual photon as follows | q | = (cid:18) M B ∗ + M B + Q M B ∗ (cid:19) − M B , ω q = (cid:18) M B ∗ − M B − Q M B ∗ (cid:19) , (2)where Q = − q > V µ ( x ) = ¯ ψ ( x ) γ µ ˆ Q ψ ( x ) + ¯Ψ h ( x ) γ µ Q h Ψ h ( x ) , (3)where ψ ( x ) and Ψ h ( x ) denote respectively the light and heavy quarks. The first term in Eq. (3) is the EM currentfor the light quarks with the charge operator defined by the charges of the light quarks ˆ Q = diag(2 / , − / , − / Q h . If one considers thecharm quark, then Q h = 2 /
3. In the case of the bottom baryon, we have Q h = − /
3. In the present work, we willconsider only the charmed baryons. The transition EM matrix element between B ∗ and B is then parametrized interms of the three real EM transition form factors (cid:104) B ∗ ( p (cid:48) , λ (cid:48) ) | V µ (0) | B ( p, λ ) (cid:105) = i (cid:114) u β ( p (cid:48) , λ (cid:48) )Γ βµ u ( p, λ ) . (4) λ and λ (cid:48) denote the helicities of the baryons with spin 1/2 and 3/2, respectively. u β ( p, λ (cid:48) ) and u ( p, λ ) stand for theRarita-Schwinger and Dirac spinors, respectively. Γ βµ in Eq. (4) denote the three real EM transition form factors:Γ βµ = G ∗ M ( Q ) K βµM + G ∗ E ( Q ) K βµE + G ∗ C ( Q ) K βµC , (5)where G ∗ M , G ∗ E , and G ∗ C are known respectively as the magnetic dipole transition form factor, the electric quadrupoleone, and the Coulomb quadrupole one. The corresponding Lorentz tensors K βµM are written as K βµM = − M B ∗ + M B )2 M B [( M B ∗ + M B ) + Q ] ε βµστ P σ q τ , K βµE = −K βµM − M B ∗ | q | M B ∗ + M B M B ε βσνγ P ν q γ ε µσαδ p B ∗ α q δ iγ , K βµC = − M B ∗ | q | M B ∗ + M B M B q β [ q P µ − q · P q µ ] iγ . (6)The Lorentz tensors are required to satisfy the gauge-invariant identities q µ K βµM ,E ,C = 0, which arises from theconservation of the EM current.The EM transition form factors can be extracted experimentally by using the helicity amplitudes. The transverseand Coulomb helicity amplitudes are defined respectively in terms of the spatial and temporal components of the EMcurrent A λ = − e (cid:112) ω q (cid:90) d re i q · r (cid:15) +1 · (cid:104) B ∗ (3 / , λ ) | ¯ ψ ( r ) ˆ Q γ ψ ( r ) | B (1 / , λ − (cid:105) ,S / = − e (cid:112) ω q √ (cid:90) d re i q · r (cid:104) B ∗ (3 / , / | ¯ ψ ( r ) ˆ Qγ ψ ( r ) | B (1 / , / (cid:105) , (7)where λ is the corresponding value of the helicity of the baryon B ∗ with spin 3/2, i.e. λ = 3 / /
2. Note that thetransverse photon polarization vector is defined as ˆ (cid:15) = − / √ , i, A / = − e (cid:112) ω q c ∆ ( G ∗ M − G ∗ E ) , A / = − e (cid:112) ω q √ c ∆ ( G ∗ M + G ∗ E ) , S / = e (cid:112) ω q | q | c ∆ M B ∗ G ∗ C , (8)where c ∆ = (cid:113) M B M B ∗ | q | (cid:113) Q ( M B ∗ + M B ) . Then, we are able to express the EM transition form factors inversely bythe transition amplitudes G ∗ M ( Q ) = − c ∆ (cid:90) d r j ( | q || r | ) (cid:104) B ∗ (3 / , / | [ˆ r × V ] | B (1 / , − / (cid:105) ,G ∗ E ( Q ) (cid:39) − c ∆ (cid:90) d r (cid:114) π ω q | q | (cid:18) ∂∂r rj ( | q || r | ) (cid:19) (cid:104) B ∗ (3 / , / | Y (ˆ r ) V | B (1 / , − / (cid:105) ,G ∗ C ( Q ) = 4 c ∆ M B ∗ | q | (cid:90) d r √ πj ( | q || r | ) (cid:104) B ∗ (3 / , / | Y (ˆ r ) V | B (1 / , / (cid:105) . (9)Note that we neglect a term that provides a tiny correction to the E2 transition form factor at low-energy regions,which implements the current conservation.From the form factors, the well-known quantities R EM and R SM , which are defined respectively as R EM ( Q ) = − G ∗ E ( Q ) G ∗ M ( Q ) , R SM ( Q ) = − | q | M B ∗ G ∗ C ( Q ) G ∗ M ( Q ) , (10)can be obtained. The decay width is expressed in terms of the helicity amplitudes [39]:Γ( B ∗ → Bγ ) = ω q π M B M B ∗ (cid:0) | A / | + | A / | (cid:1) = α EM
16 ( M B ∗ − M B ) M B ∗ M B (cid:0) | G ∗ M (0) | + 3 | G ∗ E (0) | (cid:1) (11)with the EM fine structure constant α EM . III. A SINGLY HEAVY BARYON IN THE CHIRAL QUARK-SOLITON MODEL
The pion mean-field approach or the χ QSM has one great virtue. The model allows one to describe both lightbaryons and singly heavy baryons on an equal footing. While various properties of singly heavy baryons were in-vestigated in the previous works based on the χ QSM, it was not discussed formally how a singly heavy baryon canbe explicitly constructed in the χ QSM. Thus, before we compute the EM transition form factors of singly heavybaryons, we first want to show how a singly heavy baryon can be formulated in the present approach. Let us firstdefine the normalization of the baryon state (cid:104) B ( p (cid:48) , J (cid:48) ) | B ( p, J ) (cid:105) = 2 p δ J (cid:48) J (2 π ) δ (3) ( p (cid:48) − p ). In the large N c limit,this normalization can be expressed as (cid:104) B ( p (cid:48) , J (cid:48) ) | B ( p, J ) (cid:105) = 2 M B δ J (cid:48) J (2 π ) δ (3) ( p (cid:48) − p ), where M B is a baryon mass.Since a singly heavy baryon consists of the N c − N c − | B, p (cid:105) = lim x →−∞ exp( ip x ) N ( p ) (cid:90) d x exp( i p · x )( − i Ψ † h ( x , x ) γ ) J † B ( x , x ) | (cid:105) , (cid:104) B, p | = lim y →∞ exp( − ip (cid:48) y ) N ∗ ( p (cid:48) ) (cid:90) d y exp( − i p (cid:48) · y ) (cid:104) | J B ( y , y )Ψ h ( y , y ) , (12)where N ( p )( N ∗ ( p (cid:48) )) stands for the normalization factor depending on the initial (final) momentum. J B ( x ) and J † B ( y )denote the Ioffe-type current of the N c − J B ( x ) = 1( N c − (cid:15) α ··· α Nc − Γ f ··· f Nc − ( T T Y )( JJ Y R ) ψ f α ( x ) · · · ψ f Nc − α Nc − ( x ) ,J † B ( y ) = 1( N c − (cid:15) α ··· α Nc − Γ f ··· f Nc − ( T T Y )( JJ (cid:48) Y R ) ( − iψ † ( y ) γ ) f α · · · ( − iψ † ( y ) γ ) f Nc − α Nc − , (13)where f · · · f N c − and α · · · α N c − denote respectively the spin-isospin and color indices. Γ ( T T Y )( JJ Y R ) are matriceswith the quantum numbers ( T T Y )( JJ Y R ) for the corresponding baryon. For example, a singly heavy baryon Σ + c can be identified as the state with J = 1 / T = 1, T = 0, and Y = 2 /
3. The right hypercharge Y R for singlyheavy baryons is constrained by the number of the valence quarks. Note that Y R = N c / Y R = ( N c − / Y R = 1 with N c = 3 allows one to get the lowest-lying representations for the SU(3) baryons, i.e., the baryon octet ( ) and decuplet ( ) for the light baryons. On theother hand, we find the baryon antitriplet (¯ ), sextet ( ) and so on [24, 35]. ψ f k α k ( x ) denotes the light-quark fieldand Ψ h ( x ) stands for the heavy-quark field. In the limit of m Q → ∞ , a singly heavy baryon satisfies the heavy-quarkflavor symmetry. Then the heavy-quark field can be written asΨ h ( x ) = exp( − im Q v · x ) ˜Ψ h ( x ) , (14)where ˜Ψ h ( x ) is a rescaled heavy-quark field almost on mass-shell. It carries no information on the heavy-quark mass inthe leading order approximation in the heavy-quark expansion. v denotes the velocity of the heavy quark [28–30, 40].We can show explicitly that the normalization factor N ∗ ( p (cid:48) ) N ( p ) correctly turns out to be 2 M B . The normalizationof the baryon state can be computed as follows: (cid:104) B ( p (cid:48) , J (cid:48) ) | B ( p, J ) (cid:105) = 1 Z eff N ∗ ( p (cid:48) ) N ( p ) lim x →−∞ lim y →∞ exp ( − iy p (cid:48) + ix p ) × (cid:90) d xd y exp( − i p (cid:48) · y + i p · x ) (cid:90) D U D ψ D ψ † D ˜Ψ h D ˜Ψ † h J B ( y )Ψ h ( y )( − i Ψ † h ( x ) γ ) J † B ( x ) × exp (cid:20)(cid:90) d z (cid:110) ( ψ † ( z )) fα ( i / ∂ + iM U γ + i ˆ m ) fg ψ gα ( z ) + Ψ † h ( z ) v · ∂ Ψ h ( z ) (cid:111)(cid:21) = 1 Z eff N ∗ ( p (cid:48) ) N ( p ) lim x →−∞ lim y →∞ exp ( − iy p (cid:48) + ix p ) × (cid:90) d xd y exp( − i p (cid:48) · y + i p · x ) (cid:104) J B ( y )Ψ h ( y )( − i Ψ † h ( x ) γ ) J † B ( x ) (cid:105) , (15)where Z eff represents the low-energy effective QCD partition function defined as Z eff = (cid:90) D U exp( − S eff ) . (16) S eff is called the effective chiral action (E χ A) defined by S eff = − N c Tr ln [ i / ∂ + iM U γ + i ˆ m ] . (17) (cid:104) ... (cid:105) in Eq. (15) denotes the vacuum expectation value of the baryon correlation function. M represents the dynamicalquark mass that arises from the spontaneous breakdown of chiral symmetry. The U γ denotes the chiral field that isdefined by U γ ( z ) = 1 − γ U ( z ) + U † ( z ) 1 + γ U ( z ) = exp[ iπ a ( z ) λ a ] , (19)where π a ( z ) represents the pseudo-Nambu-Goldstone (pNG) fields and λ a are the flavor Gall-Mann matrices. ˆ m designates the mass matrix of current quarks ˆ m = diag( m u , m d , m s ). Note that we deal with the strange currentquark mass m s perturbatively. Thus, we will consider it when we make a zero-mode quantization for a collectivebaryon state. The propagators of a light quark in the χ QSM [22] is given by G ( y, x ) = (cid:28) y (cid:12)(cid:12)(cid:12)(cid:12) i / ∂ + iM U γ + im ( iγ ) (cid:12)(cid:12)(cid:12)(cid:12) x (cid:29) = Θ( y − x ) (cid:88) E n > e − E n ( y − x ) ψ n ( y ) ψ † n ( x ) − Θ( x − y ) (cid:88) E n < e − E n ( y − x ) ψ n ( y ) ψ † n ( x ) , (20)where Θ( y − x ) stands for the Heaviside step function. Here, m represents the average mass of the up and downcurrent quarks: m = ( m u + m d ) / E n is the energy eigenvalues of the single-quark state given by Hψ n ( x ) = E n ψ n ( x ) , (21)where H denotes the one-body Dirac Hamiltonian in the presence of the pNG boson fields, which is defined by H = γ γ i ∂ i + γ M U γ + γ ¯ m . (22)The heavy-quark propagator in the limit of m Q → ∞ is expressed as G h ( y, x ) = (cid:28) y (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:12)(cid:12)(cid:12)(cid:12) x (cid:29) = Θ( y − x ) δ (3) ( y − x ) . (23)Using these quark propagators and taking the limit of y − x = T → ∞ , we can derive the baryon correlation function (cid:104) J B ( y )Ψ h ( y )( − i Ψ † h ( x ) γ ) J † B ( x ) (cid:105) as follows [22, 41]: (cid:104) J B ( y )Ψ h ( y )( − i Ψ † h ( x ) γ ) J † B ( x ) (cid:105) ∼ exp [ −{ ( N c − E val + E sea + m Q } T ] = exp[ − M B T ] . (24)Since the result for the correlation function given in Eq. (24) is canceled with the term exp ( − iy p (cid:48) + ix p ) =exp[ M B T ] in the large N c limit, i.e., − ip (cid:48) = − ip = M B = O ( N c ). Thus, the normalization factor becomes N ∗ ( p (cid:48) ) N ( p ) = 2 M B . Using this normalization and Eq. (24), we are able to produce the classical mass of the singlyheavy baryon correctly to be M B = ( N c − E val + E sea + m Q , (25)which was already defined in a previous work [25]. IV. EM TRANSITION FORM FACTORS FROM THE CHIRAL QUARK-SOLITON MODEL
In the present Section, we will present here only the final expressions of the EM transition form factors, sincedetailed formalisms of how to derive the form factors of the SU(3) baryons can be found in previous works. For adetailed calculation, we refer to Refs. [42–44] (see also a review [41]). The EM current for the heavy quark given inthe second term of Eq. (3) can be expressed in terms of the effective heavy quark field [45] − i Ψ † h ( x ) γ µ Q h Ψ h ( x ) = − i exp( − im Q v · x ) ˜Ψ † h ( x ) (cid:20) v µ + i m Q ( ←− ∂ µ − −→ ∂ µ ) + 12 m Q σ µν ( ←− ∂ µ + −→ ∂ µ ) (cid:21) Q h ˜Ψ h ( x ) ≈ − i exp( − im Q v · x ) ˜Ψ † h ( x ) v µ Q h ˜Ψ h ( x ) . (26)This indicates that the heavy quark does not contribute to the EM transition form factors of the singly heavy baryons.It only gives a constant contribution to their electric form factors, which yields the correct charges corresponding tothe singly heavy baryon. Thus, we can simply consider the light-quark current to compute the matrix element of theEM current [43] (cid:104) B ∗ , p (cid:48) | V µ (0) | B, p (cid:105) = 1 Z eff lim T →∞ exp (cid:20) − i ( p (cid:48) + p ) T (cid:21) (cid:90) d xd y exp( − i p (cid:48) · y + i p · x ) (cid:90) D U D ψ D ψ † × J B ∗ ( y , T / − iψ † (0)) γ µ ˆ Q ψ (0) J † B ( x , − T /
2) exp (cid:20)(cid:90) d z ( ψ † ( z )) fα ( i / ∂ + iM U γ + i ˆ m ) fg ψ gα ( z ) (cid:21) (27)Since we ignore the meson fluctuation, the integration over the pNG fields can be carried out easily. However, thereare the rotational and translational zero modes that are not at all small, so that we need to integrate exactly over thesezero modes. This is known as the collective zero-mode quantization. For details about the zero-mode quantization inthe SU(3) χ QSM, we refer to Refs. [23, 41].Having taking into account the rotational 1 /N c and linear m s corrections, we obtain the magnetic dipole formfactors G B → B ∗ ∗ M as G B → B ∗ ∗ M ( Q ) = − c ∆ (cid:90) d r √ j ( | q || r | ) G B → B ∗ M ( r ) , (28)where the corresponding magnetic dipole densities G B → B ∗ M ( r ) are defined as G B → B ∗ M ( r ) = (cid:18) Q ( r ) + 1 I Q ( r ) (cid:19) (cid:104) B ∗ | D (8) Q | B (cid:105) − √ I X ( r ) (cid:104) B ∗ | D (8) Q J | B (cid:105)− I X ( r ) (cid:104) B ∗ | d pq D (8) Qp J q | B (cid:105) − m s (cid:18) K I X ( r ) − M ( r ) (cid:19) (cid:104) B ∗ | D (8)83 D (8) Q | B (cid:105)− √ m s (cid:18) K I X ( r ) − M ( r ) (cid:19) (cid:104) B ∗ | d pq D (8)8 p D (8) Qq | B (cid:105)− m s M ( r ) (cid:104) B ∗ | D (8) Q | B (cid:105) + 23 m s M ( r ) (cid:104) B ∗ | D (8)88 D (8) Q | B (cid:105) . (29)The explicit expressions for the densities Q i , X , and M i can be found in Appendix A. The (cid:104) B ∗ | ... | B (cid:105) stands for thematrix elements of collective operators [34], of which the explicit values are found in Appendix B. I i and K i standfor the moments and anomalous moments of inertia [41]. The expression for the electric quadrupole form factors isgiven as G B → B ∗ ∗ E ( Q ) = c ∆ (cid:90) d r (cid:114) ω q | q | (cid:18) ∂∂r rj ( | q || r | ) (cid:19) G B → B ∗ E ( r ) , (30)with the electric quadrupole densities G B → B ∗ E ( r ) G B → B ∗ E ( r ) = − I I E ( r ) (cid:16) (cid:104) B ∗ | D (8) Q J | B (cid:105) − (cid:104) B ∗ | D (8) Qi J i | B (cid:105) (cid:17) − √ m s (cid:18) K I I E ( r ) − K E ( r ) (cid:19) (cid:16) (cid:104) B ∗ | D (8)83 D (8) Q | B (cid:105) − (cid:104) B ∗ | D (8)8 i D (8) Qi | B (cid:105) (cid:17) . (31)The explicit expressions for I E ( r ) and K E ( r ) can be found in Appendix A. The Coulomb quadrupole form factor G B → B ∗ ∗ C is written as G B → B ∗ ∗ C ( Q ) = c ∆ √ (cid:90) d r M B ∗ | q | j ( | q || r | ) G B → B ∗ C ( r ) , (32)where G B → B ∗ C ( r ) is simply the same as G B → B ∗ E ( r ). Note that for the E2 and C2 form factors, the leading contributionsin the large N c expansion vanish, so that the rotational 1 /N c corrections take over the role of the leading-ordercontributions.In order to scrutinize each contribution, it is more convenient to decompose the densities into three different terms G B → B ∗ ( M ,E ,C ( r ) = G B → B ∗ (0)( M ,E ,C ( r ) + G B → B ∗ (op)( M ,E ,C ( r ) + G B → B ∗ (wf)( M ,E ,C ( r ) . (33)The first term represents the SU(3) symmetric terms including both the leading and rotational 1 /N c contributions, thesecond one denotes the linear m s corrections arising from the current-quark mass term of the effective chiral action,and the last terms come from the collective baryon wave functions. When the effects of the flavor SU(3) symmetrybreaking are considered, a collective baryon wave function is not any longer in a pure state but becomes a state mixedwith higher representations. Thus, there are two different terms that provide the effects of flavor SU(3) symmetrybreaking. The explicit expressions of these three terms for the M1 form factors are then given as follows G / → / (0) M ( r ) = 14 √ Q B → B ∗ (cid:18) Q ( r ) + 1 I Q ( r ) + 12 1 I X ( r ) (cid:19) , (34) G / → / (op) M ( r ) = − m s √ (cid:18) Q Λ c → Σ ∗ c − Q Ξ c → Ξ ∗ c + 3 (cid:19) (cid:18) K I X ( r ) − M ( r ) (cid:19) − m s √ (cid:18) Q Λ c → Σ ∗ c −Q Ξ c → Ξ ∗ c + 2 (cid:19) (cid:18) K I X ( r ) − M ( r ) (cid:19) + m s √ (cid:18) − Q Λ c → Σ ∗ c Q Ξ c → Ξ ∗ c + 1 (cid:19) M ( r ) , (35) G / → / (wf) M ( r ) = q √ (cid:18) Q Λ c → Σ ∗ c −Q Ξ c → Ξ ∗ c + 2 (cid:19) (cid:18) Q ( r ) + 1 I Q ( r ) −
12 1 I X ( r ) (cid:19) − p (cid:18) Q Λ c → Σ ∗ c − Q Ξ c → Ξ ∗ c + 4 (cid:19) (cid:18) Q ( r ) + 1 I Q ( r ) + 32 1 I X ( r ) (cid:19) , (36)in the basis of the [Λ c → Σ ∗ c , Ξ c → Ξ ∗ c ] for / → / G / → / (0) M ( r ) = 130 √ Q B → B ∗ − (cid:18) Q ( r ) + 1 I Q ( r ) + 13 1 I X ( r ) + 12 1 I X ( r ) (cid:19) , (37) G / → / (op) M ( r ) = − m s √ Q Σ c → Σ ∗ c − Q Ξ (cid:48) c → Ξ ∗ c − Q Ω c → Ω ∗ c + 3 (cid:18) K I X ( r ) − M ( r ) (cid:19) − m s √ Q Σ c → Σ ∗ c + 77 Q Ξ (cid:48) c → Ξ ∗ c − Q Ω c → Ω ∗ c + 3 (cid:18) K I X ( r ) − M ( r ) (cid:19) + m s √ − Q Σ c → Σ ∗ c + 7 − Q Ξ (cid:48) c → Ξ ∗ c + 11 − Q Ω c → Ω ∗ c + 15 M ( r ) , (38) G / → / (wf) M ( r ) = q Q Σ c → Σ ∗ c − Q Ξ (cid:48) c → Ξ ∗ c − (cid:18) Q ( r ) + 1 I Q ( r ) − I X ( r ) −
12 1 I X ( r ) (cid:19) − q √ Q Σ c → Σ ∗ c + 12 Q Ξ (cid:48) c → Ξ ∗ c + 43 Q Ω c → Ω ∗ c + 6 (cid:18) Q ( r ) + 1 I Q ( r ) − I X ( r ) − I X ( r ) (cid:19) , (39)in the basis of the [Σ c → Σ ∗ c , Ξ (cid:48) c → Ξ ∗ c , Ω c → Ω ∗ c ] for / → / . Similarly, the densities for the E2 form factors arewritten by G / → / (0) E ( r ) = − √ Q B → B ∗ −
2) 1 I I E , (40) G / → / (op) E ( r ) = − m s √ − Q Σ c → Σ ∗ c + 18 Q Ξ (cid:48) c → Ξ ∗ c − − Q Ω c → Ω ∗ c + 3 (cid:18) K I I E ( r ) − K E ( r ) (cid:19) , (41) G / → / (wf) E ( r ) = − q Q Σ c → Σ ∗ c − Q Ξ (cid:48) c → Ξ ∗ c − − q √ Q Σ c → Σ ∗ c + 12 Q Ξ (cid:48) c → Ξ ∗ c + 43 Q Ω c → Ω ∗ c + 6 I I E ( r ) , (42)in the basis of the [Σ c → Σ ∗ c , Ξ (cid:48) c → Ξ ∗ c , Ω c → Ω ∗ c ] for / → / . Q B → B ∗ stand for the charges of the correspondingheavy baryons. Note that the E2 and C2 transtion from factors ( / ( J = 0) → / ( J = 1)) are forbidden withinthis model. V. RESULTS AND DISCUSSIONA. Comparison with the lattice data
Since we use exactly the same set of the model parameters as in Refs. [25, 36, 37], we proceed to present thenumerical results and discuss them. To compare the present results with those from lattice QCD [19], we need toemploy the values of the unphysical pion mass. We refer to Ref. [38] for details. In the left and right panels of Fig. 1,we draw, respectively, the results for the M1 and E2 transition form factors with the pion mass varied from the chirallimit ( m π = 0) to m π = 550 MeV. We find that when the larger value of the pion mass is used, the results for boththe M1 and E2 form factors fall off more slowly, as Q increases. This feature is already known from the results forthe EM form factors of both the light and singly heavy baryons [36–38, 43]. There are only two lattice data at Q = 0and Q = 0 . and they indicate that the lattice data on the M1 transition form factor of the Ω c γ → Ω ∗ c processfalls off rather slowly, compared to the present results with the corresponding value of the pion mass, i.e., m π = 156MeV. As shown in the left panel of Fig. 1, the present results are overestimated approximately by 50 %. Consideringthe fact that the lattice data on the Ω c → Ω ∗ c E2 transition form factor show large numerical uncertainties, we arenot able to draw any definitive conclusions from the comparison with the lattice data. Actually, the lattice data onthe E2 form factors of Ω ∗ c contain similar uncertainties as shown in Ref. [46]. We anticipate future experimental andlattice data, which will allow one to make a quantitative comparison.It is also interesting to see that the magnitude of the E2 form factor of the Ω ∗ c → Ω c transition increases drasticallyas Q gets closer to zero. This is in line with what was found in Ref. [8], where the radiative decay Σ ∗ c → Λ c γ wasexamined. This indicates that the effects of the vacuum polarization or the sea quarks become dominant over those Q [GeV ] | G ∗ Ω c → Ω ∗ c M ( Q ) | χ QSM (chiral limit) χ QSM ( m π = 140 MeV) χ QSM ( m π = 156 MeV) χ QSM ( m π = 350 MeV) χ QSM ( m π = 550 MeV) LQCD ( m π = 156 MeV) Q [GeV ] | G ∗ Ω c → Ω ∗ c E ( Q ) | χ QSM (chiral limit) χ QSM ( m π = 140 MeV) χ QSM ( m π = 156 MeV) χ QSM ( m π = 350 MeV) χ QSM ( m π = 550 MeV) LQCD ( m π = 156 MeV) FIG. 1. Numerical results for the magnetic dipole and electric quadrupole transition form factors of the Ω c γ → Ω ∗ c transitionwith the pion mass varied from 0 to 550 MeV, drawn respectively in the left and right panels. The results are compared withthe lattice data taken from Ref.[19] of the valence quarks as Q decreases. We will later discuss each contribution of the valence and sea quarks to theE2 form factors in detail.It is of great importance to know the magnetic dipole transition form factors of the baryon sextet with spin 3/2,since they provide essential information on their radiative decays. As expressed in Eq. (11), the values of the M1 andE2 transition form factors at Q = 0 will determine the decay rates of the radiative decays of the baryon sextet withspin 3/2. However, since the values of the E2 transition form factors are known to be rather small in the case of thebaryon decuplet, we expect that they would be also small in the case of the baryon sextet with spin 3/2. As will beshown later, the magnitudes of the E2 transition form factors are indeed very small, compared with those of the M1transition form factors. B. Valence- and sea-quark contributions
In Fig. 2, we show the results for the M1 form factors of the EM transitions from the baryon antitriplet to thebaryon sextet with spin 3/2, drawing separately the valence- and sea-quark contributions. On the other hand, Figure 3depicts the results for those from the baryon sextet with spin 1/2 to the baryon sextet with spin 3/2, again with thevalence- and sea-quark contributions separated. In general, the valence-quark contributions dominate over those ofthe sea quarks. In the case of the radiative excitation Ξ c γ → Ξ ∗ c , the effect of the sea quarks is negligibly small.Note that the magnitude of this M1 form factor is approximately more than ten times smaller, compared with thosefor the Λ + c γ → Σ ∗ + c and Ξ + c γ → Ξ ∗ + c excitations. This is due to the U -spin symmetry, which will be discussedlater. Comparing the results in Fig. 2 with those in Fig. 3, we find that the M1 excitations for the baryon antitripletare larger than those for the baryon sextet except for the Ξ c γ → Ξ ∗ c excitation. While this can be understood bycomparing Eq. (34) with Eq. (37), the physical interpretation of these results is originated from the spin configurationof the light valence quarks inside heavy baryons. The spins of the light valence quarks for the baryon antitriplet arein the spin-singlet state ( S L = 0), whereas those for the baryon sextet are in the spin-triplet state ( S L = 1). The M1transitions occur more likely due to the spin flip of the light valence quarks based on the naive quark model. Thus,the M1 form factors for the radiative excitations from the baryon antitriplet to the baryon sextet with spin 3/2 turnout naturally larger than those from the baryon sextet with spin 1/2 to the baryon sextet with spin 3/2.When it comes to the E2 transitions, the situation is the other way around. The E2 transitions are forbiddenbetween the different spin states. Thus, we have only the finite results for the E2 form factors for the EM transitionsbetween the baryon sextet with spin 1/2 and with spin 3/2, as shown in Fig. 4. The sea-quark contributions to theE2 form factors are remarkably sizable, which was already shown in those of the baryon decuplet [42, 43]. In thecase of the Ξ (cid:48) c → Ξ ∗ c and Ω c → Ω ∗ c EM transitions, the sea-quark contributions dominate over those of the valencequarks, in particular, in the smaller Q region. This implies that the effects of the pion clouds play a significantrole in describing the E2 form factors. Knowing the fact that the E2 transition form factors reveal how a baryonwith spin 3/2 is deformed, the outer part of the charge distribution governs the E2 form factors. As shown inRefs. [44, 47], the sea-quark part constructs the outer place of the charge and mechanical distributions in a baryon,whereas the valence-quark part is responsible for the inner part of these distributions. In this sense, it is natural that0 Q [GeV ] G ∗ Λ + c → Σ ∗ + c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ c → Ξ ∗ c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea)
FIG. 2. Results for the magnetic dipole transition form factors from the baryon antitriplet to the baryon sextet with spin 3/2,with the valence- and sea-quark contributions separated. The dashed and short-dashed curves draw the valence- and sea-quarkcontributions, respectively. The solid ones depict the total results. the sea-quark contributions contribute significantly to the E2 transition form factors in the smaller Q region. Notethat as Q increases, the sea-quark contributions fall off much faster than those of the valence quarks, which is alsounderstandable.The magnitudes of the E2 transition form factors of a baryon are in general much smaller than those of the M1transition form factors. The leading contribution to the E2 form factors vanishes within the χ QSM, so that therotational 1 /N c correction takes the place of the leading contribution as shown in Eq. (40). This indicates that themagnitudes of the E2 form factors should be smaller than those of the M1 ones. Moreover, the E2 form factors aresuppressed by the mass of a decaying baryon. Considering the fact that the mass of a singly heavy baryon is muchlarger than those of the baryon decuplet, one can expect that the E2 transition form factors of the baryon sextetwould turn out to be much smaller than those of the baryon decuplet. In addition, the matrix elements of the SU(3)Wigner D functions for the baryon sextet are smaller than those for the baryon decuplet. Thus, the magnitudes ofthe E2 transition form factors for the baryon sextet become approximately five to ten times smaller than those for thebaryon decuplet. Figure 5 presents the numerical results for the Coulomb quadrupole form factors from the baryonsextet with spin 1/2 to that with spin 3/2. The main conclusion is the same as in the case of the E2 transition formfactors. The sea-quark contributions are again dominant over those of the valence-quarks in the smaller Q region. C. Effects of explicit breaking of flavor SU(3) symmetry
In Fig. 6, we examine the effects of flavor SU(3) symmetry breaking on the M1 transition form factors. While thelinear m s corrections are very small to the Λ + c → Σ ∗ + c and Ξ + c → Ξ ∗ + c magnetic transitions, they become the leadingcontributions to the Ξ c → Ξ ∗ c . The reason is clear as mentioned previously. The U -spin symmetry forbids the EM1 Q [GeV ] G ∗ Σ ++ c → Σ ∗ ++ c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Σ + c → Σ ∗ + c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Σ c → Σ ∗ c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ c → Ξ ∗ c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ω c → Ω ∗ c M ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea)
FIG. 3. Results for the magnetic dipole transition form factors from the baryon sextet with spin 1/2 to the baryon sextet withspin 3/2, with the valence- and sea-quark contributions separated. The notations are the same as in Fig. 2. transition from Ξ c → Ξ ∗ c as shown in Eq. (34). Note that Ξ c belongs to the U -spin singlet, while Ξ ∗ c is the U -spintriplet. Only when m s (cid:54) = m u . d , it allows the Ξ c → Ξ ∗ c transition mode. As a result, the magnitude of the Ξ c → Ξ ∗ c M1 form factor is tiny, compared to those of the other M1 transition form factors. In Fig. 7, we depict the resultsfor the M1 transition form factors from the baryon sextet with spin 1/2 to the baryon sextet with spin 3/2 with thelinear m s considered. The effects of the flavor SU(3) symmetry breaking are again negligibly small. It is interestingto compare these results with those for the M1 form factors of the baryon decuplet presented in Ref. [48]. Whilethe linear m s corrections are also very small in the case of the M1 transition form factors for the baryon decuplet,they turn out to be much smaller for the baryon sextet than for the decuplet. In the case of the E2 transition form2 Q [GeV ] G ∗ Σ ++ c → Σ ∗ ++ c E ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Σ + c → Σ ∗ + c E ( Q ) × χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Σ c → Σ ∗ c E ( Q ) × χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c E ( Q ) × χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ c → Ξ ∗ c E ( Q ) × χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ω c → Ω ∗ c E ( Q ) × χ QSM (total) χ QSM (valence) χ QSM (sea)
FIG. 4. Results for the electric quadrupole transition form factors from the baryon sextet with spin 1/2 to the baryon sextetwith spin 3/2, with the valence- and sea-quark contributions separated. The notations are the same as in Fig. 2. factors, the linear m s corrections are sizable for the Σ + c → Σ ∗ + c and Ξ (cid:48) + c → Ξ ∗ + c E2 excitations as shown in Fig. 8.It is also interesting to see that the linear m s corrections suppress the E2 transition form factors for the Σ + c → Σ ∗ + c excitation. These can be understood by examining Eqs. (41) and (42). In Fig. 9, we draw the numerical results forthe C2 transition form factors from the baryon sextet with spin 1/2 to that with spin 3/2. Knowing that the densitiesfor them are the same as those for the E2 transition form factors, we can understand the sizable effects of the linear m s on the Σ + c → Σ ∗ + c and Ξ (cid:48) + c → Ξ ∗ + c transitions.3 Q [GeV ] G ∗ Σ ++ c → Σ ∗ ++ c C ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Σ + c → Σ ∗ + c C ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Σ c → Σ ∗ c C ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c C ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ξ c → Ξ ∗ c C ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea) Q [GeV ] G ∗ Ω c → Ω ∗ c C ( Q ) χ QSM (total) χ QSM (valence) χ QSM (sea)
FIG. 5. Results for the Coulomb quadrupole transition form factors from the baryon sextet with spin 1/2 to the baryon sextetwith spin 3/2, with the valence- and sea-quark contributions separated. The notations are the same as in Fig. 2.
D. Decay widths of the radiative decay for the baryon sextet with spin 3/2
In Table I we list the results for the widths of the radiative decays from the baryon sextet with spin 3/2 to the baryonantitriplet in the first three lines and to the baryon sextet with spin 1/2 in the next lines. As written in Eq. (11), thedecay width for the B ∗ / → B / γ is proportional to | G ∗ M | + 3 | G ∗ E | . Since we have already shown that the valuesof G ∗ E are much smaller than those of G ∗ M , the decay widths are approximately proportional to | G ∗ M | . Table Iindicates that the baryon sextet with spin 3/2 decay more likely into the baryon antitriplet. As explained previously,the spin state of the decaying baryon is flipped in the M1 transition. This explains why the decay rates from the4 Q [GeV ] G ∗ Λ + c → Σ ∗ + c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ c → Ξ ∗ c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) FIG. 6. Results for the magnetic dipole transition form factors from the baryon antitriplet to the baryon sextet with spin 3/2.The dashed curves draw the results in flavor SU(3) symmetry, whereas the solid ones depic those with flavor SU(3) symmetrybreaking taken into account.TABLE I. Results for the radiative decay widths of Bγ → B ∗ with and without flavor SU(3) symmetry breaking. Γ( B c γ → B ∗ c ) χ QSM χ QSM χ SM [34] LQCD [19] Bag [49] χ PT [11] QCDSR [15, 16] QM [14]( m s = 0 MeV) ( m s = 180 MeV)Λ + c γ → Σ ∗ + c . ± .
15 – 126 161.8 130(45) 151(4)Ξ + c γ → Ξ ∗ + c . ± .
22 – 44.3 21.6 52(25) 54(3)Ξ c γ → Ξ ∗ c . ± .
42 – 0.908 1.84 0.66(32) 0.68(4)Σ ++ c γ → Σ ∗ ++ c . ± .
22 – 0.826 1.20 2.65(1.20) –Σ + c γ → Σ ∗ + c . ± .
02 – 0.004 0.04 0.40(16) 0.140(4)Σ c γ → Σ ∗ c . ± .
06 – 1.08 0.49 0.08(3) –Ξ (cid:48) + c γ → Ξ ∗ + c . ± .
02 – 0.011 0.07 0.274 –Ξ (cid:48) c γ → Ξ ∗ c . ± .
05 – 1.03 0.42 2.142 –Ω c γ → Ω ∗ c . ± .
08 0.074 1.07 0.32 0.932 – baryon sextet with spin 3/2 to the baryon antitriplet are much larger than the / → / γ decays. As we havealready seen from the results for the form factors, the effects of the flavor SU(3) symmetry are rather small. Thefourth column lists the results from the chiral quark-soliton model in a model-independent approach [34] where alldynamical variables were determined by experimental data on the light baryons . The present results seem overall Note that in Ref. [34] the formula for the decay width contains an error. The present results listed in the fourth column are the correctedones. Q [GeV ] G ∗ Σ ++ c → Σ ∗ ++ c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Σ + c → Σ ∗ + c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Σ c → Σ ∗ c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ c → Ξ ∗ c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ω c → Ω ∗ c M ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) FIG. 7. Results for the magnetic dipole transition form factors from the baryon sextet with spin 1/2 to the baryon sextet withspin 3/2. The notations are the same as in Fig. 6. underestimated, compared with those from Ref. [34]. What is interesting is that the present results are in agreementwith those from chiral perturbation theory [11]. On the other hand, lattice QCD yields a very small value of thedecay width for Ω ∗ c → Ω c γ , compared with the results from all other works.We already observed that the E2 transition form factors are very small. This means that the ratios R EM as definedin Eq. (10) will turn out to be also very small. As listed in Table II, the values of R EM for the radiative transitionsof the baryon sextet with spin 3/2 are indeed very small. Except for the Ξ (cid:48) + c γ → Ξ ∗ + c excitation, the values of theratios R EM and R SM are approximately two times smaller than those for the baryon decuplet.6 Q [GeV ] G ∗ Σ ++ c → Σ ∗ ++ c E ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Σ + c → Σ ∗ + c E ( Q ) × χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Σ c → Σ ∗ c E ( Q ) × χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c E ( Q ) × χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ c → Ξ ∗ c E ( Q ) × χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ω c → Ω ∗ c E ( Q ) × χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) FIG. 8. Results for the electric qudrupole transition form factors from the baryon sextet with spin 1/2 to the baryon sextetwith spin 3/2. The notations are the same as in Fig. 6.
VI. SUMMARY AND CONCLUSION
We aimed in the present work at investigating the electromagnetic transition form factors for the singly heavybaryons with spin 3/2, based on the pion mean-field approach or the chiral quark-soliton model. Having taken intoaccount the rotational 1 /N c and linear m s corrections, we computed the magnetic dipole, electric quadrupole andCoulomb quadrupole transition form factors for the radiative excitations from the baryon antitriplet and sextet withspin 1/2 to the baryon sextet with spin 3/2. Since the model parameters were already fixed in producing propertiesof the light baryons, we use exactly the same set of the parameters for the present investigation. We compared the7 Q [GeV ] G ∗ Σ ++ c → Σ ∗ ++ c C ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Σ + c → Σ ∗ + c C ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Σ c → Σ ∗ c C ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ + c → Ξ ∗ + c C ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ξ c → Ξ ∗ c C ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) Q [GeV ] G ∗ Ω c → Ω ∗ c C ( Q ) χ QSM ( m s = 180 MeV) χ QSM ( m s = 0 MeV) FIG. 9. Results for the Coulomb quadrupole transition form factors from the baryon sextet with spin 1/2 to the baryon sextetwith spin 3/2. The notations are the same as in Fig. 6. results for the magnetic dipole (M1) and electric quadrupole (E2) transition form factors with those from a latticecalculation. However, the lattice results for the form factor for the Ω c → Ω ∗ c M1 transition seems to be underestimatedin comparison with those from other works. Since the lattice calculation suffers from large uncertainties in the resultsfor the corresponding E2 transition form factor, we were not able to draw any conclusion from the comparison of thepresent results with the lattice ones. We then examined the valence- and sea-quark contributions to the M1, E2 andC2 transition form factors. While the sea-quark contributions are marginal to the M1 form factors, they dominateover the valence-quark contributions in the smaller Q region. On the other hand, the sea-quark contributions falloff faster than the valence-quark ones as Q increases. The magnitudes of the M1 transitions form factors from the8 TABLE II. Results for the R EM and R SM on Bγ → B ∗ with and without flavor SU(3) symmetry breaking. χ QSM( m s = 0 MeV) χ QSM( m s = 180 MeV)[%] R EM R SM R EM R SM Σ ++ c γ → Σ ∗ ++ c -0.87 -0.88 -0.75 -0.76Σ + c γ → Σ ∗ + c -0.87 -0.88 -0.58 -0.59Σ c γ → Σ ∗ c -0.87 -0.88 -0.89 -0.91Ξ (cid:48) + c γ → Ξ ∗ + c -0.93 -0.95 -1.63 -1.69Ξ (cid:48) c γ → Ξ ∗ c -0.93 -0.95 -1.04 -1.06Ω c γ → Ω ∗ c -0.96 -0.98 -1.19 -1.22 baryon antitriplet to the baryon sextet with spin 3/2 are in general larger than those from the baryon sextet withspin 1/2 to the sextet with spin 3/2. This indicates that the M1 transitions occur more naturally between the stateswith the total spin flipped. Since the E2 and C2 transitions take place in the transitions without any spin flip, wehave null results for the transitions from the baryon antitriplet to the sextet with spin 3/2.We also examined the effects of flavor SU(3) symmetry breaking by considering the linear m s corrections. Exceptfor the Ξ c → Ξ ∗ c transition that is forbidden by the U -spin symmetry, we found that the effects of the flavor SU(3)symmetry breaking are negligibly small. Since the U -spin symmetry is broken by the finite value of the strangecurrent quark mass, the M1 transition form factor for the Ξ c → Ξ ∗ c radiative excitation is finite but tiny, comparedwith those for other transition modes. Similarly, we found that the linear m s corrections are also very small to theE2 transition form factors except for the Σ + c → Σ ∗ + c and Ξ (cid:48) + c → Ξ ∗ + c transitions. Similar features were seen in theresults for the Coulomb qudrupole form factors. We also computed the widths of the radiative decays for the baryonsextet with spin 3/2. The results for the transitions within the baryon sextet are in agreement with those from chiralperturbation theory. Finally, we presented the results for the ratios of the E2 over M1 and C2 over M1. They turn outto be approximately two times smaller than those for the baryon decuplet. As the electromagnetic (transition) formfactors provide the most valuable information on the structure of the hadrons, it is extremely important to measurethem, experimentally and/or by lattice QCD computations. So far very limited information is given for the charmedbaryons, and further experimental and computational efforts are highly desired. ACKNOWLEDGMENTS
The authors are grateful to K. U. Can for valuable discussion related to the lattice results. The present work wassupported by Basic Science Research Program through the National Research Foundation of Korea funded by theMinistry of Education, Science and Technology (Grant-No. 2018R1A2B2001752 and 2018R1A5A1025563). J.-Y.Kis supported by the Deutscher Akademischer Austauschdienst(DAAD) doctoral scholarship. The work of Gh.-S.-Y.is supported by NRF-2019R1A2C1010443. The work of M.O. is supported in part by JSPS KAKENHI Grants No.JP19H05159 and JP20K03959.9
Appendix A: Densities for the EM transition form factors
The densities of the magnetic dipole transition form factors are expressed explicitly as follows: Q ( r ) =( N c − (cid:104) val | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | val (cid:105) + N c (cid:88) n R ( E n ) (cid:104) n | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105) , Q ( r ) = i N c − (cid:88) n (cid:54) =val sign( E n ) E n − E val (cid:104) n | r (cid:105) γ [ { ˆ r × σ } × τ ] (cid:104) r | val (cid:105) · (cid:104) val | τ | n (cid:105) + i N c (cid:88) n,m R ( E n , E m ) (cid:104) m | r (cid:105) γ [ { ˆ r × σ } × τ ] (cid:104) r | n (cid:105) · (cid:104) m | τ | n (cid:105) , X ( r ) =( N c − (cid:88) n (cid:54) =val E n − E val (cid:104) val | r (cid:105) γ { ˆ r × σ }(cid:104) r | val (cid:105) · (cid:104) n | τ | val (cid:105) + 12 N c (cid:88) n,m R ( E n , E m ) (cid:104) n | r (cid:105) γ { ˆ r × σ }(cid:104) r | m (cid:105) · (cid:104) m | τ | n (cid:105) , X ( r ) =( N c − (cid:88) n E n − E val (cid:104) val | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105)(cid:104) n | val (cid:105) + N c (cid:88) n ,m R ( E m , E n ) (cid:104) m | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105)(cid:104) n | m (cid:105) , M ( r ) =( N c − (cid:88) n (cid:54) =val E n − E val (cid:104) val | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105)(cid:104) n | γ | val (cid:105)− N c (cid:88) n,m R ( E n , E m ) (cid:104) m | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105)(cid:104) n | γ | m (cid:105) , M ( r ) =( N c − (cid:88) n (cid:54) =val E n − E val (cid:104) val | r (cid:105) γ { ˆ r × σ }(cid:104) r | n (cid:105) · (cid:104) n | γ τ | val (cid:105)− N c (cid:88) n,m R ( E n , E m ) (cid:104) m | r (cid:105) γ { ˆ r × σ }(cid:104) r | n (cid:105) · (cid:104) n | γ τ | m (cid:105) , M ( r ) =( N c − (cid:88) n E n − E val (cid:104) val | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105)(cid:104) n | γ | val (cid:105)− N c (cid:88) n ,m R ( E n , E m ) (cid:104) m | r (cid:105) γ { ˆ r × σ } · τ (cid:104) r | n (cid:105)(cid:104) n | γ | m (cid:105) . (A1)The densities of the electric quadrupole transition form factors are given as − √ I E ( r ) = ( N c − (cid:88) n (cid:54) =val E n − E val (cid:104) val | τ | n (cid:105) · (cid:104) n | r (cid:105){√ πY ⊗ τ } (cid:104) r | val (cid:105) + 12 N c (cid:88) n,m R ( E n , E m ) (cid:104) n | τ | m (cid:105) · (cid:104) m | r (cid:105){√ πY ⊗ τ } (cid:104) r | n (cid:105) , − √ K E ( r ) = ( N c − (cid:88) n (cid:54) =val E n − E val (cid:104) val | γ τ | n (cid:105) · (cid:104) n | r (cid:105){√ πY ⊗ τ } (cid:104) r | val (cid:105) + 12 N c (cid:88) n,m R ( E n , E m ) (cid:104) n | γ τ | m (cid:105) · (cid:104) m | r (cid:105){√ πY ⊗ τ } (cid:104) r | n (cid:105) . (A2)0The regularization functions in Eqs. (A1) and (A2) are defined by R ( E n ) = − √ π E n (cid:90) ∞ φ ( u ) duu e − uE n , R ( E n , E m ) = 12 √ π (cid:90) ∞ φ ( u ) du √ u E m e − uE m − E n e − uE n E n − E m , R ( E n , E m ) = 12 √ π (cid:90) ∞ φ ( u ) du √ u (cid:34) e − uE m − e − uE n u ( E n − E m ) − E m e − uE m + E n e − uE n E n + E m (cid:35) , R ( E n , E m ) = 12 π (cid:90) ∞ φ ( u ) du (cid:90) dαe − uE n (1 − α ) − uE m α E n (1 − α ) − αE m (cid:112) α (1 − α ) , R ( E n , E m ) = sign( E n ) − sign( E m )2( E n − E m ) , (A3)with proper-time regulator φ ( u ) = cθ ( u − Λ − )+(1 − c ) θ ( u − Λ − ). The cutoff parameters c, Λ and Λ were determinedin Ref. [41]. | val (cid:105) and | n (cid:105) denote the states of the valence and sea quarks with the corresponding eigenenergies E val and E n of the single-quark Hamiltonian h ( U c ), respectively [41]. Appendix B: Collective matrix elements of electromagnetic form factor
In Tables III, IV, V and VI, we present the results for all relevant matrix elements of the SU(3) Wigner D functions. TABLE III. The matrix elements of the collective operators for the leading-order contributions and the 1 /N c rotational correc-tions to the electrimagnetic transition form factors. B γ ∗ → B Λ c γ ∗ → Σ ∗ c Ξ c γ ∗ → Ξ ∗ c B γ ∗ → B Σ c γ ∗ → Σ ∗ c Ξ (cid:48) c γ ∗ → Ξ ∗ c Ω c γ ∗ → Ω ∗ c (cid:104) B | D (8)33 | B (cid:105) √ − √ T (cid:104) B | D (8)33 | B (cid:105) √ T √ T (cid:104) B | D (8)83 | B (cid:105) − (cid:104) B | D (8)83 | B (cid:105) √ − √ − (cid:113) (cid:104) B | D (8)38 J | B (cid:105) (cid:104) B | D (8)38 J | B (cid:105) − √ T − √ T (cid:104) B | D (8)88 J | B (cid:105) (cid:104) B | D (8)88 J | B (cid:105) − √ √ √ (cid:104) B | d ab D (8)3 a J b | B (cid:105) − √ √ T (cid:104) B | d ab D (8)3 a J b | B (cid:105) − √ T − √ T (cid:104) B | d ab D (8)8 a J b | B (cid:105) (cid:104) B | d ab D (8)8 a J b | B (cid:105) − √ √ √ (cid:104) B | D (8)3 i J i | B (cid:105) (cid:104) B | D (8)3 i J i | B (cid:105) (cid:104) B | D (8)8 i J i | B (cid:105) (cid:104) B | D (8)8 i J i | B (cid:105) , 132 (1988).[2] M. N. Butler, M. J. Savage and R. P. Springer, Phys. Lett. B , 353 (1993) [hep-ph/9302214].[3] C. P. Jessop et al. [CLEO Collaboration], Phys. Rev. Lett. , 492 (1999) [hep-ex/9810036].[4] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. , 232001 (2006) [hep-ex/0608055].[5] E. Solovieva et al. , Phys. Lett. B , 1 (2009) [arXiv:0808.3677 [hep-ex]].[6] J. Yelton et al. [Belle Collaboration], Phys. Rev. D , no. 5, 052011 (2016) [arXiv:1607.07123 [hep-ex]].[7] H. Y. Cheng, C. Y. Cheung, G. L. Lin, Y. C. Lin, T. M. Yan and H. L. Yu, Phys. Rev. D , 1030 (1993) [hep-ph/9209262].[8] M. J. Savage, Phys. Lett. B , 61 (1995) [hep-ph/9408294].[9] S. L. Zhu and Y. B. Dai, Phys. Rev. D , 114015 (1999) [hep-ph/9810243]. TABLE IV. The matrix elements of the collective operators for the m s corrections to the electromagnetic transition formfactors. B γ ∗ → B Λ c γ ∗ → Σ ∗ c Ξ c γ ∗ → Ξ ∗ c B γ ∗ → B Σ c γ ∗ → Σ ∗ c Ξ (cid:48) c γ ∗ → Ξ ∗ c Ω c γ ∗ → Ω ∗ c (cid:104) B | D (8)88 D (8)33 | B (cid:105) √ (cid:104) B | D (8)88 D (8)33 | B (cid:105) √ T √ T (cid:104) B | D (8)88 D (8)83 | B (cid:105) (cid:104) B | D (8)88 D (8)83 | B (cid:105) − √ √ (cid:104) B | D (8)83 D (8)38 | B (cid:105) √ − √ T (cid:104) B | D (8)83 D (8)38 | B (cid:105) √ T √ T (cid:104) B | D (8)83 D (8)88 | B (cid:105) (cid:104) B | D (8)83 D (8)88 | B (cid:105) − √ √ (cid:104) B | d ab D (8)8 a D (8)8 b | B (cid:105) √ (cid:104) B | d ab D (8)8 a D (8)8 b | B (cid:105) − √
245 130 √ √ (cid:104) B | d ab D (8)3 a D (8)8 b | B (cid:105) − T (cid:104) B | d ab D (8)3 a D (8)8 b | B (cid:105) √ T √ T (cid:104) B | D (8)83 D (8)33 | B (cid:105) (cid:104) B | D (8)83 D (8)33 | B (cid:105) − (cid:113) T (cid:113) T (cid:104) B | D (8)83 D (8)83 | B (cid:105) (cid:104) B | D (8)83 D (8)83 | B (cid:105) − √ √ − √ (cid:104) B | D (8)8 i D (8)3 i | B (cid:105) (cid:104) B | D (8)8 i D (8)3 i | B (cid:105) (cid:113) T − (cid:113) T (cid:104) B | D (8)8 i D (8)8 i | B (cid:105) (cid:104) B | D (8)8 i D (8)8 i | B (cid:105) √ − √ √ TABLE V. The relevant transition matrix elements of the collective operators coming from the 15-plet component of the baryonwave functions. B γ ∗ → B Λ c γ ∗ → Σ ∗ c Ξ c γ ∗ → Ξ ∗ c B γ ∗ → B Σ c γ ∗ → Σ ∗ c Ξ (cid:48) c γ ∗ → Ξ ∗ c Ω c γ ∗ → Ω ∗ c (cid:104) B | D (8)33 | B (cid:105) √ − √ T (cid:104) B | D (8)33 | B (cid:105) √ T (cid:113) T (cid:104) B | D (8)83 | B (cid:105) √ (cid:104) B | D (8)83 | B (cid:105) − √ − √ (cid:104) B | D (8)38 J | B (cid:105) (cid:104) B | D (8)38 J | B (cid:105) √ T (cid:113) T (cid:104) B | D (8)88 J | B (cid:105) (cid:104) B | D (8)88 J | B (cid:105) − √ − √ (cid:104) B | d ab D (8)3 a J b | B (cid:105) √ − √ T (cid:104) B | d ab D (8)3 a J b | B (cid:105) √ T (cid:113) T (cid:104) B | d ab D (8)8 a J b | B (cid:105) (cid:113) (cid:104) B | d ab D (8)8 a J b | B (cid:105) − √ − √ (cid:104) B | D (8)3 i J i | B (cid:105) (cid:104) B | D (8)3 i J i | B (cid:105) (cid:104) B | D (8)8 i J i | B (cid:105) (cid:104) B | D (8)8 i J i | B (cid:105) , 094009 (2000) [hep-ph/9911502].[11] G. J. Wang, L. Meng and S. L. Zhu, Phys. Rev. D , no. 3, 034021 (2019) [arXiv:1811.06208 [hep-ph]].[12] J. Dey, V. Shevchenko, P. Volkovitsky and M. Dey, Phys. Lett. B , 185 (1994). doi:10.1016/0370-2693(94)91466-4[13] S. Tawfiq, J. G. Korner and P. J. O’Donnell, Phys. Rev. D , 034005 (2001) [hep-ph/9909444].[14] M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and A. G. Rusetsky, Phys. Rev. D , 094002 (1999) [hep-ph/9904421].[15] T. M. Aliev, K. Azizi and A. Ozpineci, Phys. Rev. D , 056005 (2009) [arXiv:0901.0076 [hep-ph]].[16] T. M. Aliev, K. Azizi and H. Sundu, Eur. Phys. J. C , no. 1, 14 (2015) [arXiv:1409.7577 [hep-ph]].[17] T. M. Aliev, T. Barakat and M. Savcı, Phys. Rev. D , no. 5, 056007 (2016) [arXiv:1603.04762 [hep-ph]].[18] H. Y. Cheng, Front. Phys. (Beijing) , no. 6, 101406 (2015).[19] H. Bahtiyar, K. U. Can, G. Erkol and M. Oka, Phys. Lett. B , 281 (2015). [arXiv:1503.07361 [hep-lat]].[20] H. Bahtiyar, K. U. Can, G. Erkol, M. Oka and T. T. Takahashi, JPS Conf. Proc. , 022027 (2019).[21] E. Witten, Nucl. Phys. B , 57 (1979).[22] D. Diakonov, V. Y. Petrov and P. V. Pobylitsa, Nucl. Phys. B , 809 (1988).[23] A. Blotz, D. Diakonov, K. Goeke, N. W. Park, V. Petrov and P. V. Pobylitsa, Nucl. Phys. A , 765 (1993). TABLE VI. The relevant transition matrix elements of the collective operators coming from the 15- and 24-plet components ofthe baryon wave functions. B γ ∗ → B Λ c γ ∗ → Σ ∗ c Ξ c γ ∗ → Ξ ∗ c B γ ∗ → B Σ c γ ∗ → Σ ∗ c Ξ (cid:48) c γ ∗ → Ξ ∗ c Ω c γ ∗ → Ω ∗ c (cid:104) B | D (8)33 | B (cid:105) √ − √ T (cid:104) B | D (8)33 | B (cid:105) √ T √ T (cid:104) B | D (8)83 | B (cid:105) (cid:113) (cid:104) B | D (8)83 | B (cid:105) √ (cid:104) B | D (8)38 J | B (cid:105) (cid:104) B | D (8)38 J | B (cid:105) − √ T − T (cid:104) B | D (8)88 J | B (cid:105) (cid:104) B | D (8)88 J | B (cid:105) − √ − √ − √ (cid:104) B | d ab D (8)3 a J b | B (cid:105) √ − √ T (cid:104) B | d ab D (8)3 a J b | B (cid:105) √ T √ T (cid:104) B | d ab D (8)8 a J b | B (cid:105) (cid:113) (cid:104) B | d ab D (8)8 a J b | B (cid:105) (cid:113)
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