Strong decays of singly heavy baryons from a chiral effective theory of diquarks

A chiral effective theory of scalar and vector diquarks is formulated, which is based on $SU(3)_R\times SU(3)_L$ chiral symmetry and includes interactions between scalar and vector diquarks with one or two mesons. We find that the diquark interaction term with two mesons breaks the $U(1)_A$ and flavor $SU(3)$ symmetries. To determine the coupling constants of the interaction Lagrangians, we investigate one-pion emission decays of singly heavy baryons $Qqq$ ($Q=c, b$ and $q=u,d,s$), where baryons are regarded as diquark--heavy-quark two-body systems. Using this model, we present predictions of the unobserved decay widths of singly heavy baryons. We also study the change of masses and strong decay widths of singly heavy baryons under partial restoration of chiral symmetry.

Another important aspect of hadron spectroscopy is the chiral symmetry and its spontaneous symmetry breaking in the low-energy regime of QCD.In our previous works, an effective Lagrangian based on the chiral SU (3) R × SU (3) L symmetry for diquarks was constructed.In Refs.[35,36], the chiral effective model of scalar and pseudoscalar diquarks with spin 0 is proposed and is employed to investigate the mass spectra of singly heavy baryons with the diquark-heavy-quark potential model.Here we derived the mass formulas for the diquarks and discovered the inverse mass hierarchy of diquark masses: The nonstrange pseudoscalar diquark becomes heavier than the singly strange pseudoscalar diquark [M (ud, 0 − ) > M (ds/su, 0 − )] due to the U (1) A anomaly [42,43].Besides, in Ref. [40], we constructed the chiral effective model of vector and axial-vector diquarks with spin 1 and updated our numerical results of the spectrum of singly heavy baryons with a renewal of potential models.We obtained the mass formula for the axial-vector diquarks [M 2 (uu/ud/dd, 1 + )+M 2 (ss, 1 + ) = 2M 2 (ds/su, 1 + )], which is a generalization of the Gell-Mann-Okubo mass formula [44,45]. 1ne of the characteristic points of these works is applying mass formulas of diquarks to the spectrum of singly heavy baryons.Here scalar and vector diquarks are considered independently and interactions between these diquarks are neglected because they are expected to be irrelevant to the diquark mass formulas.Instead of this, however, such an interaction plays the main role in strong decays of singly heavy baryons [41,.
In this paper, we extend our approach [35,36,40] to interactions between the scalar and vector diquarks, which satisfies the chiral SU (3) R × SU (3) L symmetry.We determine the coupling constants for these interactions from the experimentally known strong decay widths of singly heavy baryons and also investigate the effect of chiral symmetry restoration.
This paper is organized as follows.In Sec.II, we formulate a chiral effective model including interactions between scalar and vector diquarks.In Sec.III, we investigate one-pion emission decays of singly heavy baryons based on a diquark-heavy-quark description and determine the coupling constants of the interaction Lagrangian.Besides, modifications of decay widths of the baryons triggered by the partial restoration of chiral symmetry are demonstrated.Finally, Sec.IV is for the con-clusion of the present work.

II. CHIRAL EFFECIVE MODEL OF DIQUARKS
Main purpose of this work is to study strong decays of singly heavy baryons from the diquark-heavy-quark description based on chiral symmetry.In this framework, interactions between singly heavy baryons and light mesons are determined by chiral dynamics of the diquarks and light mesons, where the remaining heavy quark simply serves as a spectator.To this end, in this section we introduce a chiral effective Lagrangian describing interactions between diquarks and light mesons.For the diquarks, in particular, we include the vector (V) and axial-vector (A) diquarks as well as the scalar (S) and pseudo-scalar (P) ones.

A. Diquark operators
Within a simple description, singly heavy baryons are composed of one heavy quark and one diquark belonging to the color 3 and 3 representations, respectively.Here, we briefly explain the properties of color 3 diquarks and their structures based on chiral symmetry.
In Table I, we summarize interpolating fields (or operators) of the diquarks studied in this work and their quantum numbers. 2 In this table, C = iγ 0 γ 2 is the charge conjugation Dirac matrix.The subscript "A" and "S" stand for the antisymmetric and symmetric combinations in flavor indices, respectively, which are shown in the rightmost column.The superscript 3 means that the diquarks are antisymmetric in color indices.Their spin and parities are represented by J P .
The diquarks listed in Table I are parity eigenstates which are useful to see connections with physical states of the singly heavy baryons.To see diquarks from the aspects of chiral symmetry, it is useful to classify them in terms of the left-handed and right-handed quarks defined by q a R,i = P R q a i and q L,i = P L q a i with the chiral projection operators P R,L ≡ (1 ± γ 5 )/2.Here, a and i denote the color and flavor indices, respectively.In

Chiral operator Spin Color Chiral
d a R,i = abc ijk (q bT R,j Cq c R,k ) 0 3 (3, 1) d a L,i = abc ijk (q bT L,j Cq c L,k ) 0 3 (1, 3) d a,µ ij = abc (q bT L,i Cγ µ q c R,j ) 1 3 (3,3) such chiral basis, the four diquark operators in Table I are decomposed to three chiral diquark operators given in Table II.From the definition, q a R,i and q a L,i belong to the (3,1) and (1,3) representations of SU (3) R × SU (3) L chiral symmetry, respectively, and accordingly the chiral representations of the diquarks are determined as in the table.
Below, we show the chiral and parity transformations of the diquarks in Table II.First, since q a R,i and q a L,i belong to (3,1) and (1,3), respectively, they transform under SU (3) R × SU (3) L chiral symmetry as q a R,i → (U R ) ij q a R,j and q a L,i → (U L ) ij q a L,j , with U R ∈ SU (3) R and U L ∈ SU (3) L .Thus, one can easily see that the chiral diquarks are transformed as [35] Ch : Next, it is well known that the spatial inversion of the left-handed or right-haded quark is given by q a R/L,i (t, x) → γ 0 q a L/R,i (t, − x).From this formula, the parity transformation of the chiral diquarks reads These transformation properties are important for constructing a chiral Lagrangian of the diquarks.Besides, the parity transformation law in Eq. ( 2) enables us to express the parity-eigenstate diquark operators in Table I in terms of the chiral operators in Table II as (q bT i Cγ µ γ 5 q c j ), ( 5) From these expressions, we find that S and P diquarks are chiral partners belonging to the chiral ( 3, 1)+(1, 3) representation, while V and A diquarks are chiral partners belonging to the chiral (3,3) < l a t e x i t s h a 1 _ b a s e 6 4 = " H s Q 8 w 6 P E j q v a P 4 G + V 0 T u b R U l t 4 c = " > A A A C Z n i c h V H L S s N A F D 2 N r 1 q 1 r Y o o u C m W i q s y k a L i q u j G Z R / SV [Eq.(10)] and L SV [Eq.( 11)], respectively.Arrows with "R" or "L" represent the right-handed or left-handed quark lines.

B. Chiral effective Lagrangian
In this subsection, we present interaction Lagrangian including the diquarks and light mesons based on According to Ref. [40], the chiral effective Lagrangian for diquarks in Table II and light mesons are expressed within the linear sigma model as In the first and second terms, Σ ij is the meson nonet composed of the scalar σ ij and pseudoscalar π ij mesons [97,98].The latter mesons are regarded as the Nambu-Goldstone (NG) bosons in association with the breakdown of SU (3) R × SU (3) L chiral symmetry triggered by the instability of the potential V (Σ) at Σ = 0, where V (Σ) is the potential terms of the meson nonet.Since the meson nonet Σ ij = qR,j q L,i belongs to chiral ( 3, 3) representation, its chiral transformation is given as The third and forth terms in Eq. ( 7) represent the kinetic and mass terms of the diquarks, where L S includes spin 0 (S and P) diquarks while L V includes spin 1 (V and A) diquarks.For their explicit expressions, see Refs.[35,40].
The last term of chiral effective Lagrangian in Eq. ( 7), L SV , represents couplings between the diquarks with different spins mediated by the meson nonet.In this work, we propose the following two terms: where The coefficients g 1 and g 2 are dimensionless coupling constants, and f π in Eq. ( 11) is the pion decay constant.All the color indices are implicitly contracted.In constructing the Lagrangian (9) we have taken into account the contributions up to O(Σ 2 ) so as to satisfy chiral symmetry in Eqs.(1) and (8).
Since the spatial inversion of Σ is expressed as P : Σ(t, x) → Σ † (t, − x), the Lagrangians in Eqs.(10) and (11) are invariant under parity transformation as well as chiral transformation.However, under the axial U (1) A transformation, only L SV keeps the symmetry while L (2) SV breaks it.That is, the latter is responsible for the U (1) A anomaly effects, which may be caused by instantons (a topologically nontrivial configuration of gluon fields) [99].
In order to diagrammatically understand the U (1) A anomaly effects, in Fig. 1, we draw schematic pictures of the vertices and the quark lines: In the left figure (a) for L (1) SV , all the right-handed or left-handed quark lines, denoted by "R" or "L", is conserved through the interaction vertex.On the other hand, in the right figure SV , the quark lines are not conserved.

C. Explicit chiral symmetry breaking
In the vacuum where SU (3) R × SU (3) L chiral symmetry is spontaneously broken, the vacuum expectation values (VEVs) of the meson nonet Σ is nonzero: Σ = 0.
As a result, the Σ with no derivatives in Eq. ( 11) can be replaced by its VEV Σ , and couplings describing one meson emission decays of the diquarks are obtained from Eq. (11) as well as Eq.(10).In such treatment, effects from the violation of SU (3) R × SU (3) L chiral symmetry due to a mass of strange quark cannot be ignored.In this subsection, we explain our method to incorporate such explicit chiral symmetry breaking (ECSB) effects into the Lagrangian (9).
When taking into account the current quark masses, within the linear sigma model, the mass matrix M eff of constituent quarks can be expressed as where  [100].As a result, the effective quark mass matrix can be expressed in a simple form as One of the most useful ways to incorporate the ECSB effects is to replace Σ in the Lagrangian (7) by the following shifted nonet Σ [35,40]: where its VEV is given by By substituting Eqs. ( 13)-( 15) into the effective Lagrangian (9), we obtain the interaction Lagrangian with the ECSB effects: with and In order to obtain Eqs. ( 17) and ( 18), we have rearranged the g 1 and g 2 terms in Eqs.(10) and (11) such that L ASπ and L V P π contain couplings of the S and A diquarks and those of the P and V diquarks, respectively.Note that we have omitted interactions mediated by scalar mesons σ ij in Eqs. ( 17) and (18).When we decompose the pseudoscalar nonet π ij into the singlet and octet parts as the flavor structures of Eq. ( 16) become more transparent.The Lagrangian written in terms of π ij and π ij is straightforwardly obtained by substituting Eq. ( 19) into Eq.( 16), but the resultant expression is lengthy.Hence, here we only comment on its flavor symmetric properties.The resultant Lagrangian indicates that the flavor-singlet pseudoscalar meson π (1) does not couple to A and S diquarks.This is because the A and S diquarks belong to flavor 6 and 3 representations, respectively, so that original flavor symmetry prohibits such couplings.Similarly, both the V and P diquarks belong to flavor 3, and thus they couple to both flavor singlet π (1) and octet π (8)  mesons. 3

III. DECAYS OF ΣQ AND Ξ Q BARYONS
In Sec.II, we have formulated the chiral effective model of diquarks describing their one-pion emission decays.In this section, from the chiral model and the diquarkheavy-quark description, we investigate strong decays of singly heavy baryons and quantify the model using the experimental data.In Table III, we summarize the experimental data of singly heavy baryons from the PDG [100], which are expected to be in the ground states.
The Λ Q and Ξ Q are composed of one heavy quark and the S diquark, while Σ Q and Ξ Q includes the A diquark.Thus, in order to study Σ Q → Λ Q π and Ξ Q → Ξ Q π decays, we need to focus on the couplings between the S diquark, the A diquark, and the pion in Lagrangian (16).The interaction Lagrangian is given by In this Lagrangian, we have rewritten the numbers in the subscripts of diquarks as the flavors: the original A µ ij and S † i denote where {• • • } and [• • • ] stand for the symmetric and antisymmetric ordering of quarks, respectively.The new coupling constants G 1 and G 2 are given as From Eq. ( 23), it is obvious that the difference between G 1 and G 2 comes from the violation of SU (3) flavor symmetry characterized by α = 1.
From now on, in order to study the decays of singly heavy baryons, we employ a diquark-heavy-quark picture: While the dynamics of pion decays is determined by the Lagrangian (20) for diquarks, the property as the bound state composed of a diquark and a heavy quark is given by a quark model calculation.In this picture, the heavy-baryon state |d Q with the energy E d Q is represented as the product of the diquark state |d with E d and a heavy-quark state |Q with E Q , superposed by the relative wave function φ d ( p) between them.It is explicitly given in the momentum space as where the integral is over the relative momentum, p, keeping the total momentum conserved as Here, with m d and m Q as the diquark and heavy-quark masses, respectively) are the momenta of the heavy baryon, diquark, and heavy quark, respectively.(See Fig. 2 for the definitions.)The energies of diquark and heavy quark also depend on the relative momentum as Q .In Eq. ( 24), J d Q , s d , and s Q represent the spins of the corresponding states.The conservation law of spins leads to for the S diquark (d = S) and < l a t e x i t s h a 1 _ b a s e 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " T Z I / m X Z T A f Z e q z b U z z m e b U Z s T a v q z X 9 U G 3 y K R Z 3 j 5 U D G K I H u q I X u q d r e q L 3 X 2 s F Y Y 2 q l z L P R k 0 r 3 F z n U e / K 2 7 + q A s 8 K u 5 + q P z 0 r 7 G A q 9 C r Z u x s y 1 V u Y N X 3 p 4 P h l Z X p 5 K B i m M 3 p m / 6 f 0 S H d 8 A 7 v 0 a p 4 v i e U T x P k D 0 t + f + y d Y G 0 u l J 1 L j S + P J 2 b n o K 2 L o w y B G + L 0 n M Y s F L C L D 5 x 7 i E j e 4 1 e q 1 U Y 2 0 s V q q V h d p e v A l t J k P A 9 m V y g = = < / l a t e x i t > p⇡ < l a t e x i t s h a 1 _ b a s e 6 4 = " U Q

X p T + S I 3 z h w v x B E u x I P p + g n Q I E g h j y Y x f I I 9 t m O A o o w Q B A y 5 h H S o c a p t I g 8 E i b g s e c
T Y h G e w L + I i S t k x Z g j J U Y v d o 3 K X V Z s g a t K 7 W d A I 1 p 1 N 0 6 j Y p + 5 B k D + y S v b B 7 d s W e 2 P u v t b y g R t X L P s 1 a T S u s Q u y w J / v 2 r 6 p E s 4 v i p + p P z y 5 2 M B F 4 l e T d C p j q L X h N X z k 4 f s l O r S S 9 A X b G n s n / K X t k d 3 Q D o / L K z 5 f F y g m i 9 A H p 7 8 / 9 E 6 y N p t J j q c x y J j E 7 F 3 5 F B L 3 o x x C 9 9 z h m s Y g l 5 O j c I 1 z j B r d K m z K q T C r T t V S l L t R 0 4 0 s o C x 9 Q P p h w < / l a t e x i t > S ( p0 = ~ + p) FIG. 2. Kinematics employed for AQ → SQπ decays with pion momentm pπ.PA Q and PS Q denote the momenta of initial-and final-state baryons, respectively, and p and p are the relative momenta between a diquark and a heavy quark in the initial and final states, respectively.φA,J and φS are the relative wave functions.m d (d = A, S) and mQ are the diquark mass and the heavy-quark mass, respectively, and G is the coupling constant.
for the A diquark (d = A), if there is no orbital excitation.Note that the normalizations of the state kets are defined in the relativistic way in Eq. ( 24), such as with Similar normalizations are also applied to the state kets of the diquark and the heavy quark.Hence, we need to include the factors of 2E d Q , √ 2E d , and 2E Q .The normalization of the relative wave function is given by For the pionic decays where the heavy quark is treated as a spectator, we only need to take into account the overlaps of the diquark states |d (and pion states |π ) together with the wave function φ d ( p).Meanwhile, the heavy-quark part simply leads to their momentum conservation law.Keeping the above in mind, we focus on decays of Using the interaction between diquarks and pions written as Eq. ( 20), the decay amplitude is given by (29) From Eq. ( 20), the Lagrangian L A→Sπ are expressed as the sum of the terms with operators A µ {qq} (∂ µ π)S † [qq] .Substituting Eq. ( 24) into Eq.( 29), they are calculated for each term as Here the factor G denotes the coupling constant, in which its absolute value is given for each decay process as Also, (p π ) µ and µ A (p A , s A ) denote the pion momentum and the polarization vector of the A diquark, respectively.φ S ( p) and φ A,J ( p) show the S-wave relative wave functions between the diquark and the heavy quark for S Q and A Q baryons, respectively.As shown in Eqs.(25) and (26), while S Q baryon belongs to heavy-quark spin singlet, A Q belongs to the doublet, so that the total spin of A Q is labeled as the subscript J = 1/2, 3/2 in φ A,J .The delta function in Eq. ( 30) represents the momentum and energy conservation laws of diquarks and an emitted pion.In addition, the momenta carried by the heavy quarks preserve during the interaction.These conservation laws determine the recoil momentum χ ≡ p − p in the relative motion φ S ( p ).In particular, at the rest frame of the A Q baryon ( P A Q = 0), χ is written as In this frame, one can easily confirm P S Q = − p π , and the momentum of A diquark coincides with the relative momentum as p A = p.The kinematics employed in the present analysis is depicted in Fig. 2. For the amplitude in Eq. ( 30), we use an approximation for the momentum of A diquark, p A .That is, in our present analysis, p A is replaced by the expectation value: In this way, the momentum squared of the S diquark is evaluated as p 2 + | p π | 2 from the momentum conservation.Then, by defining a velocity υ 2 A using p 2 ≡ m 2 A υ 2 A , the energies of the A and S diquarks are written as respectively.This approximation is also applied to p A in the sum of the polarization vector µ A (p A , s A ) as As a result, we find the decay amplitude M as where the inner product φ S |φ A,J is defined in the momentum space as Finally, from the square of the amplitude (37), we obtain the decay width: where | p π | is given as (with the masses of the baryons and pion) The relative wave functions φ S ( p) and φ A,J ( p) are obtained by solving the Schrödinger equation Hφ d = Eφ d describing a bound state of the heavy quark and diquark.The non-relativistic diquark-heavy-quark Hamiltonian is where the relative momentum p and the reduced mass µ are defined as respectively.For the potential between the heavy quark and the diquark, we use the potential in Ref. [40] which is called the Y-potential: where we include the Coulomb term with the coefficient α coul , the linear confinement term with λ, the constant shift term C, and the spin-spin potential term with κ Q and a cutoff parameter Λ.These model parameters are summarized in Table IV together with the masses of diquarks and heavy quarks.In Eq. ( 44), we neglect other terms such as the spin-orbit potential term and the tensor term because the wave functions φ S ( p) and φ A,J ( p) are in the S-wave states.By using the Gaussian expansion method [101,102], we solve the Schrödinger equation and obtain the wave functions φ S and φ A,J .

B. Determination of coupling constants
By using the observed decay widths of singly heavy baryons, we determine the coupling constant between the diquarks and the pions in the interaction Lagrangian (20) via the formula (39) and Eq.(31).More concretely, G 1 and G 2 are determined as follows: (i) The value of G 1 can be determined by the experimental data of the π ± emission decays of Σ ( * ) Q baryons.On the other hand, the π 0 emission decays only give the upper limit of the decay width.
(ii) The value of G 2 is determined by the experimental data of the decays of Ξ Q baryons with J = 3/2.This is because there are no available data of Ξ Q with J = 1/2 in PDG.
In Tables V and VI, we show the obtained G 1 and G 2 for each decay.From these tables, one sees that both the values of G 1 and G 2 range from 20 to 30, where 21 < G 1 < 27 and 24 < G 2 < 30, so that G 2 tends to be a bit larger than G 1 .We note that, depending on used experimental data, both G 1 and G 2 have the indeterminacy of approximately 6.
We comment on the errors of experimental data.The values of G 1 and G 2 shown in Tables V and VI are evaluated from the averages of the experimental data.If TABLE VI.The coupling constant G2 determined from the one-pion decay widths of Ξ Q baryons.The experimental data of decay widths are also shown in units of MeV.Note that the values of partial decay widths with the asterisk (*) are the predicted widths in our present work [40].
Baryon J P Decay mode Decay width (MeV) G2 we take into account the errors of these data, we can also estimate the errors of G 1 and G 2 .From all of the eight results of G 1 and the four results of G 2 , we have confirmed the margin of errors of G 1 and G 2 are from 2 to 7. For example, according to PDG [100], the decay width and baryon masses of Σ ++ In the following analysis, for simplicity, we neglect these errors.
We take the isospin and spin averages of G 1 's in Table V, and obtain 21.04 , and For G 2 's in Table VI, by taking only the isospin average, we obtain The values of G 1 and G 2 in Eqs. ( 46) and ( 47) indicate that the violation of SU (3) flavor symmetry characterized by the difference between G 1 and G 2 are considerably small.In addition, the violation of heavy-quark flavor symmetry characterized by the difference between G c 1,2 and G b 1,2 is also small.Using Eq. ( 23), we determine the g 1 and g 2 parameters in the original effective Lagrangian (9).Since the g 2 term breaks the U (1) A symmetry while the g 1 term does not, the influence of the U (1) A anomaly on the diquarks are quantified by translating the values of G 1 and G 2 into those of g 1 and g 2 .When we take g s = 3 for the quarkmeson coupling constant, the dimensionless quantity α in Eq. ( 13) is estimated to be α 1.73.Thus, from the relations in Eq. ( 23), the magnitudes of g 1 and g 2 are estimated as from Eqs. ( 46) and ( 47) for charmed and bottom sectors, respectively.From these values, one immediately sees that the magnitude of g 1 is much larger than g 2 : The ratio of g 2 to g 1 is evaluated to be |g c 2 /g c 1 | 0.11 for the charm sector and |g b 2 /g b 1 | 0.17 for the bottom one.Therefore, we can conclude that the U (1) A anomaly effects to interactions between the diquarks and pions are small.Note that the same conclusion is obtained from a chiral effective model for singly heavy baryons [41,67,72].(In Appendix A, we compare the results from our diquark model and from the model for singly heavy baryons.) Next, in Tables VII and VIII, we show our predictions for decay widths unknown in PDG, or Table III.First, we tabulate the predictions of π 0 emission decays of Σ Q TABLE VII.Our predictions of π 0 emission decay widths of ΣQ baryons in units of MeV.The asterisk (*) indicates that the mass is taken from [40].For the decay width, (cal.) and (exp.)represent our predictions and experimental data [100], respectively.

Baryon (J
baryons in Table VII.In this evaluation, we have used the isospin average coupling G 1 in Eq. ( 45).For the unknown masses of Σ b baryons, we have used the values predicted by the diquark-heavy-quark model in Ref. [40].From this table, the predicted widths of Σ + c (1/2 + ) and Σ * + c (3/2 + ) are below the experimental upper limits.In this sense, our predictions are consistent with the experiments.Also, in Table VIII, we show our predictions of one pion decay widths of Ξ b baryons with J P = 1/2 + .Since there is no experimental value for the mass of Ξ 0 b (1/2 + ), again we have adopted that from Ref. [40].Using the isospin averaged value of coupling G 2 = 27.16 for Ξ b (3/2 + ) , however, the predicted decay widths are quite different from the upper limit of the decay width of Ξ − b → Ξ b π: Γ max = 0.08 MeV.To satisfy this condition, the coupling constant G 2 needs to be less than 15.61.

C. Decays under chiral symmetry restoration
In this subsection, we discuss a chiral symmetry restoration effect for decays.Chiral symmetry tends to be restored in an extreme environment such as high temperature and/or high baryon density.In chiral effective models, such chiral symmetry restoration can be described as a decrease of the absolute value of VEV toward zero, i.e., Σ → 0. In order to incorporate the change of VEV, in Refs.[40,103], the authors introduced a parameter x characterizing chiral symmetry breaking: Σ = x Σ | vac .In this treatment, the range of x is limited to 0 ≤ x ≤ 1. x = 0 corresponds to chiral symmetry restored phase while x = 1 corresponds to the ordinary chiral-symmetry broken vacuum.Then, the mass formulas of nonstrange S and A diquarks are given as a function of x [40]: where the diquark mass parameters are [36,40] 4 53) in units of MeV 2 .The mass formulas ( 50) and ( 51) indicates that, at the chiral restoration point x = 0, the S diquark mass becomes heavier than the A diquark one, i.e., m [ud] > m {uu/ud/dd} , which is contrast to the normal mass ordering m {uu/ud/dd} > m [ud] realized in the vacuum at x = 1.In particular, within our parameters, the mass inversion occurs at x 0.6 [40].
When we take into account the chiral symmetry restoration, not only the diquark masses in Eqs.

Singly charmed baryon mass [MeV]
< l a t e x i t s h a 1 _ b a s e 6 4 = " 2 v H r Y y I h f S u q E / O U I M z 9 w e r v j 9 0 = " > < l a t e x i t s h a 1 _ b a s e 6 4 = " h W b Q I X 4 f d z v A v h U u W q + L 7 j G S x h k = " > A A A C g X i c S y r I y S w u M T C 4 y c j E z M L K x s 7 B y c X N w 8 v H L y A o F F a c X 1 q U n B q a n J + T X x S R l F i c m p O Z l x p a k l m S k x p R U J S a m J u U k x q e l O 0 M k g 8 v S y 0 q z s z P C y m p L E i N z U 1 M z 8 t M y 0 ⌃b(1/2 + ) mass < l a t e x i t s h a 1 _ b a s e 6 4 = " t Y M 2 n o z G G N U 9 e H 1 l I R e P k A q W x q g = "  → Λ b π ± on the chiral-symmetry breaking parameter x in the range of 0.80 ≤ x ≤ 1.00.and (51) but also the coupling constant are modified due to Σ in the g 2 term in Eq. (11).That is, now the couplings G 1 and G 2 defined in Eq. ( 23) are replaced by In Figs. 3 and 4, we show the x dependence of the masses of Λ Q , Σ Q , and Σ * Q and the decay widths of Σ ( * ) Q → Λ Q π ± .We find that, as x decreases, the Λ Q becomes heavier while Σ Q and Σ * Q become lighter, which is naively expected from the x dependence of diquark mass formulas (50) and (51).Due to the changes of baryon masses and coupling constants, the decay widths monotonically decrease and finally vanish at a certain x.In other words, a decay channel is prohibited below a threshold of x.Such thresholds, x th , are estimated to be x th 0.95, 0.84, 0.91, and 0.87 for Σ c , Σ * c , Σ b , and Σ * b , respectively.
The chiral effective Lagrangian is constructed in the form of linear sigma model, which describes the interaction between scalar and vector diquarks with the coupling constants g 1 and g 2 .The term with the coupling g 1 represents the interaction between a diquark and one meson, while the term with g 2 does the interaction between a diquark and two mesons.We have found that the g 2 term breaks the U (1) A symmetry, and it also leads to violation of the flavor SU (3) symmetry due to explicit chiral symmetry breaking.
These coupling constants are determined from the decay width formula (39) and the experimental data of the one-pion emission decays of singly heavy baryons, where we have regarded these baryons as the two-body systems of a scalar or an axial-vector diquark and a heavy quark.Our findings from our model are as follows: (i) Our main finding is that the magnitude of g 2 is much smaller than that of g 1 , as shown in Eqs. ( 48) and ( 49), which indicates that the effects originated from the g 2 term, such as the U (1) A anomaly and the flavor SU (3) symmetry breaking, should be strongly suppressed.
(ii) We have predicted experimentally unknown decay widths, such as Σ Q baryons by changing a parameter x characterizing the magnitude of chiral symmetry breaking.As shown in Figs. 3 and  4, as x decreases, Λ Q and Σ ( * ) Q baryon masses become closer to each other due to the change of nonstrange S and A diquark masses [40].As a result, the decay widths of Σ ( * ) Q → Λ Q π ± decrease and finally vanish below a threshold of x.
In this work, we have focused on only the chiral effective model with the spin 0 and 1 diquarks with color 3.It may be interesting to improve our model by introducing other interactions or other diquark degrees of freedom.An example is the one-pion interactions between vector and axial-vector diquarks.Also, the tensor diquarks with color 3 and flavor 6 [104][105][106] could be important for the improvement of the diquark model, which is useful to describe the negative-parity excited states of flavor sextet singly heavy baryons (Σ Q , Ξ Q , and Ω Q ).
It is also important to examine the properties of diquarks inside hadrons under chiral symmetry restored environments such as finite temperature [107][108][109] and/or density.As predicted in this work, the suppression (and also prohibition) of Σ ( * ) Q → Λ Q π ± decay widths would be a good signal indicating chiral symmetry restoration via diquarks.The numerical results of G1 and G2 are summarized in Tables IX and X.As the same as the diquark model, the coupling constants of the SHB model, g1 and g2 , are given by the spin average of G1 and the isospin average of G2 with the parameter α 1.73 from the assumption g s = 3.Their results are ( Gc 1 , Gc 2 ; g1 , g2 ) = (0.69, 0.65; 0.61, 0.05), (A7) ( Gb 1 , Gb 2 ; g1 , g2 ) = (0.63, 0.74; 0.89, −0.15), (A8) whose values are much different from those in the diquark model, Eqs. ( 48) and (49).However, the ratio |g 1 /g 2 | is estimated to be 0.07 from the charm sector (A7) and 0.17 from the bottom sector (A8), which are similar to those obtained from the diquark model.This result means that the effects of the U (1) A anomaly and the SU (3) flavor symmetry breaking, included in the g 2 and g2 terms, from singly bottom baryons are larger than that from singly charmed baryons.In Fig. 5, we show the ratios of coupling constants, 2 A b W U J I 4 1 N E 1 C k h Z q 8 Q c E t 3 b h S k F E / A w 3 / o C L / o H i s o I b F 9 6 m A d G i 3 m F m z p y 5 5 8 6 Z G d n U V N t h r O s T R k b H x i f 8 k 4 G p 6 Z l g K D w 7 l 7 e N h q X k d e n o R 3 6 A i e x u o y 2 u u 8 v u K p j 0 B 4 K u e e h U E B H 9 j C 7 9 g Q 7 + g 6 K j Q Z c O v a 4 L U V K 9 w 8 w 8 8 8 z 7 v P P M j G S o i m U z 1 v M I Y + M T k 1 P e a d / M 7 J w

g 2 FIG. 1 .
FIG. 1. Schematic pictures for the effective interaction Lagrangian LSV [Eq.(9)] between diquarks and mesons.The left figure (a) and right figure (b) describe L 5 5 5 n 3 e e m Z E M V b F s x r o e Y W R 0 b H z C O + m b 8 k / P B I K z c 3 l L b 5 o y z 8 m 6 q p t F S b S 4 q m g 8 Z y u 2 y o u G y c W G 5 5 5 n 3 e e m Z E M V b F s x r o e Y W R 0 b H z C O + m b 8 k / P B I K z c 3 l L b 5 o y z 8 m 6 q p t F S b S 4 q m g 8 Z y u 2 y o u G y c W G d L b e q 0 d P Z / P u / K p 1 m F w d f q j 8 9 u 6 h h z f c q y L v l M 9 4 t 1 K 6 + e d z u 5 N e 3 k 6 0 F d s V e y f 8 l e 2 L 3 d A O j + a Z e 5 / j 2 B c L 0 A e m f z 9 0 9 l m f O P X P P n T M z h m e Z g W T s Z k J 5 8 H D y 0 d T 0 j D o 7 N / 9 4 I b P 4 5 C h w + z 4 X d e 5 a r t 8 0 9 E B Y p i P q 0 p S W a H q + 0 G 3 D E g 3 j b C 9 e b w y E H 5 i u c y j P P X B b o v 8 e r f I + a 9 C 6 U d P x 1 B q d o t O w S R l B j D 2 w G / b K 7 t k t e 2 I f v 9 a q e z U a X m o 0 q 0 0 t t 4 o D R 6 M b 7 / + q y j S 7 2 P 9 S / e n Z x S 7 m P a + C v F s e 0 7 i F 1 t R X D 0 5 e N x b W Y / U p d s m e y f 8 F e 2 R 3 d A O j + q Z d p f n 6 K U L 0 A P L P d r e C r Z m E P J t I p p P R x W X / K Y I Y w y T i 1 O 8 5 L G I V K W S 8 j h 3 j D O e B F 2 l E G p c m m q l S w N c M 4 1 t I 8 U 8 q N Y 2 4 < / l a t e x i t > A,J < l a t e x i t s h a 1 _ b a s e 6 4 = " z n g

1 u m 8 Z
B W u z 6 r 0 7 p d 0 / b U K p 2 i U b d I G U W c P b J b 1 m I P 7 I 4 9 s f d f a 7 l e j b a X O s 1 K R 8 v N 0 s j J + N b r v 6 o K z Q 6 O P l V / e n Z w g E X P q y D v p s e 0 b 6 F 2 9 L X j 0 9 b W 0 m b c n W K X 7 J n 8 X 7 A m u 6 c b 6 L U X 9 S r L N 8 8 Q o g 9 I f n / u n y C f S i T n E + l s O p Z Z 8 b 8 i i A l M Y o b e e w E Z r G E D O T q 3 j n P c 4 D b w

Λ
c (1/2 + ) mass Σ c (1/2 + ) mass Σ c * (3/2 + ) mass Σ c -> Λ c π decay width Σ c * -> Λ c π decay width < l a t e x i t s h a 1 _ b a s e 6 4 = " t s k D w F B W u z c 8 T o r 5 y 6 N 9 T O 4 f L q 8 k z 9 9 w 5 M y N b m u q 4 j D 0 G h K 7 u n t 6 + / o H g 4 N D w S C g 8 O r b u m H V b 4 U X F 1 E x 7 U 5 Y c r q k G L 7 q q q / F N y + a S L m t 8 Q 6 4 t t / c 3 G t x 2 V N N Y c / c t v q 1 L V U O t q I r k E l U O R 0 q r l L w r l Z W 4 m E z t z M 4 c N U u 2 H t U l x z k s h 2 M s w b y I / g S i D 2 L w I 2 u G W y h h F y Y U 1 K G D w 4 B L W I M E h 9 o W R D B Y x G 2 j S Z x N S P X 2 O Q 4 R J G 2 d s j h l S M T W a K z S a s t n D V q 3 a z q e W q F T N O o 2 K a O Y Y g / s i r 2 w e 3 b N n t j 7 r 7 W a X o 2 2 l 3 2 a 5 Y 6 W W + X Q y U T h 7 V + V T r O L v U / V n 5 5 d V L D g e V X J u + U x 7 V s o H X 3 j 4 P S l s J i f a k 6 z C / Z M / s / Z I 7 u j G x i N V + U y x / N n C N I H i N + f + y d Y T y X E u U Q 6 l 4 5 l l v y v 6 E c E k 4 j T e 8 8 j g x V k U a R z j 9 H C D W 4 F Q Y g L S U H s p A o B X z O O L y E s f g B O b Z M P < / l a t e x i t > ⇤c(1/2 + ) mass < l a t e x i t s h a 1 _ b a s e 6 4 = " h 9 v 1 1 A / G k d s i n e G m W + W g h E d T u r Y = " > A A A C f 3 i c S y r I y S w u M T C 4 y c j E z M L K x s 7 B y c X N w 8 v H L y A o F F a c X 1 q U n B q a n J + T X x S R l F i c m p O Z l x p a k l m S k x p R U J S a m J u U k x q e l O 0 M k g 8 v S y 0 q z s z P C y m p L E i N z U 1 M z 8 t M y 0 x O L A E K x Q t I x Q R n p u c m x i d r G O o b x W l r 1 l X H F O U q 5 C Y W F 9 f G C y g b 6 B m A g Q I m w x D K U G a A g o B 8 g e U M M Q w p D P k M y Q y l D L k M q Q x 5 D C V A d g 5 D I k M x E E Y z G D I Y M B Q A x W I Z q o F i R U B W J l g + l a G W g Q u o t x S o K h W o I h E o m g 0 k 0 4 G 8 a K h o H p A P M r M Y r D s Z a E s O E B c B d S o w q B p c N V h p 8 N n g h M F q g 5 c G f 3 C a V Q 0 2 A + S W S i C d B N G b W h D P 3 y U R / J 2 g r l w g X c K Q g d C F 1 8 0 l D G k M F m C 3 Z g L d X g A W A f k i G a K / r G r 6 5 2 C r I N V q N Y N F B q + B 7 l 9 o c N P g M N A H e W V f k p c G p g b N Z u A C R o A h e n B j M s K M 9 A z N 9 E w C T Z Q d n K B R w c E g z a D E o A E M b 3 M G B w Y P h g C G U K C 9 D Q z L G N Y z b G B i Z F J n 0 m M y g C h l Y o T q E W Z A A U y W A I B k k r U = < / l a t e x i t > ⌃c(1/2 + ) mass < l a t e x i t s h a 1 _ b a s e 6 4 = " p I l Q F 8 b t I k 4 L s Z w b u L h 7 J c 1 d y V k = " > A A A C g X i c S y r I y S w u M T C 4 y c j E z M L K x s 7 B y c X N w 8 v H L y A o F F a c X 1 q U n B q a n J + T X x S R l F i c m p O Z l x p a k l m S k x p R U J S a m J u U k x q e l O 0 M k g 8 v S y 0 q z s z P C y m p L

FIG. 3 .Λ
FIG. 3. Dependence of the singly charmed baryon masses and decay widths of Σ ( * ) c→ Λcπ ± on the chiral-symmetry breaking parameter x in the range of 0.80 ≤ x ≤ 1.00.

⌃±
7 b 9 2 e l Z z v L 8 Q r 7 x E 7 J / 0 c 2 Y s d 0 A r / 3 y / m 8 L X Y + I E M P U L h 8 3 V d B 5 X G + 8 C S / v r 2 e 2 3 y e P s U s H u A h V u m + n 2 I T L 1 F E m f Y 9 w j f 8 w M h Y M 4 p G 1 d i b l B p T q e Y e L o R h / w V h X 6 T Q < / l a t e x i t > Decay width [MeV]< l a t e x i t s h a 1 _ b a s e 6 4 = " 4 u R y D o A l m T j F W i 0 P d G h t r 9 M e 8 / Y = " e G c 6 I C r q D U l O T u 6 5 O U l 0 z z J 9 y d h D Q m l p b W v v S H a m u r p 7 e v v S / Z l V 3 6 0 I g 5 c M 1 3 L F u q 7 5 3 D I d X p K m t P i 6 J 7 h m 6 x Z f 0 / f n w / 2 1 K h e + 6 T o r s u b x L V v b d c w d 0 9 A k U e V 0 J t g U d r a o C c 3 m k o t 6 o 3

d l 0 y y x 9 6 n 6 0 7 P
E D m Y i r y Z 5 9 y I m v I X R 1 F c P j 5 + X Z 5 d G g z F 2 z p 7 I / x l 7 Y L d 0 A 6 f 6 Y l w s 8 q U T p O g D 1 O / a D S T T g b x o q G g e k A 8 y s x i s O x l o S w 4Q F w F 1 K j C o G l w 1 W G n w 2 e C E w W q D l w Z / c J p V D T Y D 5 J Z K I J 0 E 0 Z t a E M / f J R H 8 n a C u X C B d w p C B 0 I X X z S U M a Q w W Y L d m A t 1 e A B Y B + S I Z o r + s a v r n Y K s g 1 W o 1 g 0 U G r 4 H u X 2 h w 0 + A w 0 A d 5 Z V + S l w a m B s 1 m 4 A J G g C F 6 c G M y w o z 0 D M 3 0 T A J N l B 2 c o F H B w S D N o M S g A Q x v c w Y H B g + G A I Z Q o L 1 N D C s Y N j J s Y m J m 0 m Q y Y D K C K G V i h O o R Z k A B T N Y A 7 e + T U g = = < / l a t e x i t > ⌃ ⇤ b (3/2 + ) mass< l a t e x i t s h a 1 _ b a s e 6 4 = " v 6 P 1 + p n r y p 8 v I 7 v T S Q 0 6 U + 8 t E Y I = " > A A A C f 3 i c S y r I y S w u M T C 4 y c j E z M L K x s 7 B y c X N w 8 v H L y A o F F a c X 1 q U n B q a n J + T X x S R l F i c m p O Z l x p a k l m S k x p R U J S a m J u U k x q e l O 0 M k g 8 v S y 0 q z s z P C y m p L h 7 2 a d Z 7 m i 5 V Q 6 e T G 6 8 / a v S a X a x 9 6 n 6 0 7 O L C h Y 8 r y p 5 t z y m f Q u l o 2 8 c n L 5 s L K 7 H m t P s g j 2 T / 3 P 2 y O 7 o B k b j V b l c 4 + t n C N A H i N + f + y f I p x L i X C K 9 l o 5 m l v y v G E A Y U 4 j T e 8 8 j g x V k k a N z j 9 H C D W 4 F Q Y g L S U H s p A p d v m Y C X 0 J Y / A B M W 5 M O < / l a t e x i t > ⇤b(1/2 + ) mass < l a t e x i t s h a _ b a s e = " B F e o S p l

FIG. 4 .
FIG. 4. Dependence of the singly bottom baryon masses and decay widths of Σ ( * ) b as summarized in Tables VII and VIII.(iii) We have investigated the modification of masses and decay widths of Λ Q and Σ ( * ) Numerical results of the coupling constant G2 from the SHB model.In each Ξ Q → ΞQπ decay mode, the experimental data of decay widths are also summarized in units of MeV.Note that the values of partial decay widths with the asterisk (*) are the calculated values in this work.Baryon J P Decay mode Decay width (MeV) G2 Ξ

TABLE I .
Local diquark operators belonging to color 3.
m d , m s ) denotes the current quark mass matrix, and Σ is the VEV of Σ with g s the quark-meson coupling constant.As for the current quark masses, the lattice QCD simulations suggest that m s = 93.1 MeV while m u ≈ m d ≈ 2 − 5 MeV, and hence we take m u = m d = 0 as a good approximation and assume SU (2) I isospin symmetry throughout the symmetry breaking.In this case, the VEV of Σ must be diagonal as Σ = diag( σ 11 , σ 22 , σ 33 ) with σ 11 = σ 22 .These values are determined by the pion and kaon decay constants f π and f K .That is, by evaluating the axial currents from Eq. (7), one can find σ 11 = σ 22 = f π = 92.1 MeV and σ 33 = f s ≡ 2f K − f π = 128.1 MeV, where f π = 92.1 MeV and f K = 110.1 MeV are from the Particle Data Group (PDG)

TABLE III .
[100]imental values of masses and decay widths of singly heavy baryons in units of MeV, taken from the PDG compilation[100].We also use the pion masses as m π ± = 139.57MeV and m π 0 = 134.97MeV.

TABLE IV .
(43)meters of the Y-potential model and masses of diquarks and heavy quarks[40].µ in the parameter α coul is the reduced mass of diquark-heavy-quark two-body system given in Eq.(43).The parameters κQ are determined separately for charm (Q = c) and bottom (Q = b) quarks.

TABLE V .
The coupling constant G1 determined from the one-pion decay widths of ΣQ baryons.The experimental data of decay widths are also shown in units of MeV.

TABLE IX .
Numerical results of the coupling constant G1 from the SHB model.In each ΣQ → ΛQπ decay mode, the experimental data of decay widths are also summarized in units of MeV.