Baryons in the light-front approach: The three-quark picture

In this work, a three-quark picture is constructed using a bottom-up approach for baryons in light-front quark model. The shape parameters, which characterize the momentum distribution inside a baryon, are determined with the help of the pole residue of the baryon. The relation between the three-quark picture and the diquark picture is clarified. When building the model, we find that Lorentz boost plays a crucial role, and the bottom-up modeling approach can be generalized to multiquark states. Based on this, a unified theoretical framework for describing multiquark states may be established. As a by-product of model construction, we can easily obtain a newly improved definition of baryon interpolating current. The hadron interpolating currents are the starting point of Lattice QCD and QCD sum rules, and therefore are of great importance.


I. INTRODUCTION
Recently, there have been some big events in the field of heavy flavor physics, including the discovery of CP violation (CPV) in neutral charm mesons [1] and the discovery of doubly charmed baryons [2].On the one hand, CPV has been confirmed in many meson systems [3][4][5], however, CPV has never been observed in the baryon sector and to search for baryonic CPV is becoming particularly urgent [6].On the other hand, the discovery of doubly charmed baryon has attracted great interests both theoretically and experimentally.The study of the properties of heavy baryons plays an important role in accurately testing the standard model, searching for the origin of CP violation and new physics, and understanding strong interactions.
From a theoretical perspective, baryons are generally more complicated than mesons.In spite of this, there have been many methods to study the decay properties of heavy flavor baryons, including the light-front quark model, SU(3) flavor symmetry, effective field theory, QCD and light-cone sum rules, perturbative QCD, Lattice QCD, and so forth.Some recent progress can be found in Refs. .
In Ref. [7], light-front quark model (LFQM) is used to investigate the weak decays of doubly heavy baryons under the diquark picture.In this picture, the two quarks which do not participate in weak interactions are considered to form a loosely bound system -a diquark.In this way, the relatively complicated three-body problem inside a baryon is reduced to a relatively simple two-body problem.However, this diquark picture is criticized by some people.Take the weak decays of Ξ bc (bcq) as an example.When the b quark decays, (cq) is considered as a diquark, while when the c quark decays, (bq) is considered as a diquark.From this, one can see that the diquark picture is actually a matter of expediency.In addition, the diquark picture inevitably contains more parameters, such as the diquark masses.Even for the 0 + diquark [ud] and the 1 + diquark {ud} containing the same quark components, their masses are considered to be different [28].
Under the diquark picture, in Ref. [29], the baryonquark-diquark vertex is given by while in Ref. [30], it is where p 1,2 (m 1,2 ) are respectively the momenta (masses) of quark 1 and the diquark, λ 2 (ǫ) is the helicity (polarization vector) of the diquark, P ≡ p 1 +p 2 and M 2 0 ≡ P 2 .In fact, except for an unimportant negative sign, there is a Lorentz boost between the two expressions, and the latter have correctly considered this effect.Later, one can see that this Lorentz boost effect plays a crucial role in our model construction.
Early in 1998, the authors of Ref. [31] developed the three-quark picture of heavy baryons.Recently, the three-quark picture has been used to study the weak decays of heavy flavor baryons in Refs.[32][33][34][35][36][37], and this work aims to highlight the following points: • Spin wavefunction will be constructed in a bottomup approach.In this work, we are limited to considering ground state baryons.Several typical baryons include: Λ Q , Σ Q , and Σ * Q .This is due to the following coupling in spin space: One of our potentially important discoveries is that Lorentz boost may become more important when constructing the quark model of multiquark states, which have a certain size, and are multi-body QCD bound states.Since we may still lack a quark model that can describe all the multiquark states, this discovery may be a key to unlock the door.It is worth noting that LFQM has been applied to some relatively simple multiquark states in Refs.[38,39].In addition, as a by-product of model construction, we can easily obtain improved definitions of interpolating currents of baryons.
• A method for determining the shape parameters will be proposed.The shape parameters (see Eq. ( 15)) characterize the momentum distribution inside a hadron.For the meson case, the shape parameter is determined by the decay constant of the meson [40].The "decay constant" of a baryon is the pole residue, which is used to determine the shape parameters of the baryon, see below.
• The relation between the diquark picture and the three-quark picture will be clarified.In this work, we will consider three weak decays: For the first two processes, the spectator quarks are respectively a scalar and axial-vector diquark, while for the last process, a diquark is broken up in the initial state and a new diquark emerges by rearranging quarks in the final state.Here, for Ξ cc → Λ c , the two charmed quarks in Ξ cc are usually considered as an axial-vector diquark and the u, d quarks in Λ c form a scalar one.In the diquark picture, overlap factors are important quantities for obtaining the physical form factors [7].We will derive the overlap factors for Ξ cc → Λ c , through which the relation between the diquark picture and the three-quark picture can be illustrated.
The rest of this article is arranged as follows.In Sec.II, theoretical framework and some applications are introduced, including: the definitions of baryon states; the determination of the shape parameters; the form factors of Λ b → Λ c , Σ b → Σ c , and Ξ cc → Λ c ; the relation between the two pictures; improved definitions of interpolating currents of baryons.In Sec.III, numerical results of shape parameters, form factors, and semileptonic decay widths will be shown and will be compared with others in the literature.We conclude this article in the last section.

II. THEORETICAL FRAMEWORK AND SOME APPLICATIONS
A. The baryon states In this section, we will consider three baryon states: Λ Q , Σ Q and Σ * Q .They all have the same quark components udQ and are all S-wave baryons, and their spins are respectively 1/2, 1/2, and 3/2.
Under the three-quark picture, the baryon state in LFQM is expressed as where p i (λ i ) is the light-front momentum (helicity) of the i-th quark, the color wavefunction C ijk = ǫ ijk / √ 6, and the spin and momentum wavefunctions are contained in Ψ SSz .The light-front momentum is decomposed into The intrinsic variables (x i , k i⊥ ) are introduced through where x i is the light-front momentum fraction constrained by 0 < x i < 1.The invariant mass M 0 is defined by M 2 0 ≡ P 2 with P = p 1 + p 2 + p 3 , and it can be shown that M 0 is in general different from the baryon mass M which obeys the condition M 2 = P 2 .This is due to the fact that the baryon and the constituent quarks cannot be on their mass shell simultaneously.However, γ + u( P ) = γ + u(P ) holds [30].The internal momenta are defined as then it is easy to obtain where e i denotes the energy of the i-th quark in the rest frame of P .The momenta k i⊥ and k iz constitute a momentum 3-vector k i = (k i⊥ , k iz ).For Λ Q , in which the u, d quarks are considered to form a 0 + diquark, Ψ in Eq. ( 3) can be shown as for Σ Q , in which the u, d quarks are considered to form a 1 + diquark, and for Σ * Q , in which the u, d quarks are also considered to form a 1 + diquark, where v µ ≡ P µ /M 0 , and Φ is the momentum wavefunction.The proofs of Eqs.(9-11) can be found in Appendix A.
With the normalization of the baryon state and one can obtain The momentum wavefunction can be given by where , and β 1 and β 23 are the shape parameters.Some important notes are given below.
• The definition of a baryon state is the most important part of LFQM, while the spin wavefunction is the most important part in the definition of a baryon state.It is worth pointing out that, when we arrive at Eqs. (9-11), we do not introduce any additional assumptions, for example, it does not assume heavy quark symmetry, nor does it depend on the coordinate system selection of LFQM (see below).Therefore, the spin wavefunctions in Eqs.(9)(10)(11) may not only apply to heavy flavor baryons, but should also apply to light flavor baryons.
• From the proof in Appendix A, one can clearly see that, the Lorentz boost between the rest frame of "diquark" and the rest frame of P plays a crucial role.The proof of Eq. ( 9) is relatively simple, this is because the first case involves a scalar diquark, whose Lorentz boost is trivial.The proofs of Eqs. ( 10) and ( 11) are relatively complicated, because the latter two cases involve an axis-vector diquark, whose Lorentz boost is non-trivial.
• If we only consider the spin coupling, in principle, we can choose any two quarks for spin coupling first.However, when we consider the flavor wavefunction, the two quarks that are coupled first are usually already determined.For example, for Λ Q , whose flavor wavefunction is (ud − du)Q/ √ 2, we couple the u and d quarks first; while for Ξ ++ cc , whose flavor wavefunction is just ccu, we couple the two charm quarks first.
• If identical quarks are contained in the baryon state, some additional factor should be added.For example, for Ξ ++ cc , when we calculate B(P ′ , S ′ , S ′ z )|B(P, S, S z ) to normalize the baryon state, a factor 2 appears because of two equivalent contractions.An additional factor 1/ √ 2 should be added in the definition of |Ξ ++ cc in order to keep Eq. ( 12) unchanged.

B. To determine the shape parameters
The shape parameters in Eq. ( 15) characterize the momentum distribution inside the baryon.In this work, we suggest that the shape parameters can be determined by the pole residue of the baryon, whose numerical result can be taken from, for example, Lattice QCD or QCD sum rules.
Taking Λ Q as an example, let us focus on the matrix element 0|J On the one hand, this matrix element can be calculated in LFQM On the other hand, the pole residue of baryon is defined by Respectively multiplying Eqs. ( 16) and (17) with Sz ū(P, S z )γ + from the left, also noting that γ + u(P ) = γ + u( P ), one can arrive at and Equating Eqs. ( 18) and ( 19), one can obtain the expression of pole residue in LFQM which can be used to determine the shape parameters in Φ provided the pole residue is known.Since there are two shape parameters in Φ, one more equation is desirable, at this time, use Sz ū(P, S z )γ + γ − to left multiply instead, finally we have The expressions of pole residues of Σ Q and Ξ cc can also be obtained in a similar way.
In addition, it should be noted that, the baryon mass M can in turn be extracted by equating Eqs. ( 20) and (21) once the shape parameters are fixed by, for example, global fitting.

C. Form factors of Λ b → Λc
On the one hand, the weak decay matrix element Λ c |cγ µ (1 − γ 5 )b|Λ b can be parameterized in terms of form factors where q = P − P ′ , and f i , g i are the form factors. On the other hand, the matrix element can also be calculated in LFQM Now we extract the form factors f 1,2 and g 1,2 in the following method.
Respectively multiplying the "+" component of the vector current part of Eq. ( 22) by and Respectively multiplying the "+" component of the axial-vector current part of Eq. ( 22) by Sz,S ′ z ū(P, S z )γ + γ 5 u(P ′ , S ′ z ) and iσ +j q j γ 5 )u(P ′ , S ′ z ) from the left, one can obtain and iσ +j q j γ 5 )( Then doing the same thing for Eq. ( 23), one can obtain with where we have used γ + u(P ) = γ + u( P ).Same expressions can be obtained for f 2 and g 1,2 , except that, • for g 1 , • for g 2 , In practice, we choose the frame that satisfies q + = 0, that is, When calculating the weak decay matrix element in Eq. ( 23), one can find that the momenta of quark 2 and quark 3 remain unchanged from the initial state to the final state, from which one can obtain and furthermore, One comment.As pointed out in Ref. [32], the form factors f 3 and g 3 cannot be extracted for we have imposed the condition q + = 0.However, these two form factors do not contribute to the 1/2 → 1/2 semileptonic decays if the electron mass is neglected.
where v µ = P µ /M 0 and v ′µ = P ′µ /M ′ 0 .It turns out that with f 2 and g 1,2 can also be obtained by the same assignments to Γ 1,2 as those in Subsec.II C.
)c|Ξ cc can also be obtained in LFQM as where the factor 2 comes from the two equivalent contractions, and the factor 1/ √ 2 comes from the normalization of Ξ ++ cc state, which has been pointed out in Subsec.II A. It turns out that with f 2 and g 1,2 can also be obtained by the same assignments to Γ 1,2 as those in Subsec.II C except for the only one difference for g 1,2 because we have performed a transpose in Eq. (39).

F. The relation between the two pictures
Define the spin wavefunction in Eq. ( 9) as ψ 0 (321) and that in Eq. ( 10) as ψ 1 (321), i.e., ψ 0,1 (321) have the same normalization factor as that in Eq. ( 14).Moreover, ψ 0,1 (321) are orthogonal where quark 1 ′ does not need to be the same as quark 1.Now consider the relation between the three-quark picture and the diquark picture.The key observation is that in the three-quark picture, the first two quarks form a diquark in the diquark picture.Specifically, in ψ 0 (321)/ψ 1 (321), quark 3 and quark 2 are considered to form a scalar/axial-vector diquark.In fact, ψ 0,1 (321) constitute a diquark basis.
One can easily check that In addition, it can be shown that with the transition matrix which satisfies T −1 = T , as expected.Now we are ready to determine the overlap factors in the diquark picture.Take the process of Ξ + bc (cbu) → Λ b (dbu) as an example, where the b, c quarks in the initial state are considered to form an axial-vector diquark, while the u, d quarks in the final state are considered to form a scalar diquark.The initial and final states can be rewritten as Then it is easy to write the transition matrix element as from which, one can read the two overlap factors They are the same as those in the diquark picture in Ref. [7].It's time to go one step further to derive the overlap factors of Ξ ++ cc (ccu) → Λ c (dcu).To this end, notice that there is only one difference between this process and Ξ + bc (cbu) → Λ b (dbu), that is, Ξ ++ cc contains two identical quarks.At this time, one can obtain the overlap factors for where the factor 2/ √ 2 can be found in Eq. (39).These factors are also the same as those in the diquark picture in Ref. [7].

G. Improved definitions of interpolating currents
The definitions of interpolating currents are the starting point of Lattice QCD and QCD sum rules.The following definitions are usually adopted for Λ Q and Σ Q in the literature [41][42][43][44] These interpolating currents of baryons were first given in Ref. [45], and then widely used to study the properties of baryons.They are obtained with the help of symmetry analysis, however, to our knowledge, there is no literature that provides a rigorous proof starting from quark spinors and Dirac matrices.This work may fill this gap.Hermite conjugating Eq. ( 43), one can obtain the following improved definitions of interpolating currents for Λ Q and Σ Q where v µ ≡ p µ / p 2 with p the four momentum of baryon.Some comments are in order.
• It can be seen that, we can even let v → 0 in Eq. ( 53) to get the definitions in Eq. ( 52) if we temporarily forget the coefficient 1/ √ 3.However, we cannot do that, because v µ , according to its definition, is O(1).
• It would be interesting to compare the shape parameters with those used in the diquark picture [8], and the latter are in fact the shape parameters of mesons.For example, numerically, our β b,[ud] , β c,[ud] and β [ud] are respectively close to (in units of GeV) β bs = 0.623, β cs = 0.535 and β ds = 0.393 in Ref. [8].A significant difference is found between β {cc} = 0.400 GeV in this work and β cc = 0.753 GeV in Ref. [8].It seems that the charm quark and anti-charm quark in η c are more energetic than the two charm quarks in Ξ cc .It is worth noting that, when deriving the pole residue expressions for Ξ cc , a factor of 2/ √ 2 also appears.Using the pole residue λ Ξcc in Eq. ( 56), and having considered this factor, one can obtain the much smaller shape parameter β {cc} together with β u,{cc} .
To access the q 2 dependence of the form factors, we calculate the form factors in an interval q 2 ∈ [−5, 0] GeV 2 for Λ b → Λ c and Σ b → Σ c , and q 2 ∈ [−0.5, 0] GeV 2 for Ξ cc → Λ c , and fit the results with the following simplified z-expansion [50]: where m pole = m Bc for Λ b → Λ c and Σ b → Σ c , and The fitted results of (a, b) for the three processes are given in Table I.
The obtained form factors are then applied to semileptonic decays, it turns out that the central values of decay widths and branching ratios are where τ Λ b = 1.471 × 10 −12 s, |V cb | = 0.0408, and 221 have been used [49].Considering the uncertainties of form factors in Eq. (58-60), there are respectively about 13%, 7%, and 29% uncertainties in these phenomenological predictions.

C. Comparison with other results in the literature
In Table II and Table III, we respectively compare our form factors and semileptonic decay widths with those in the literature.It can be seen that our predictions are comparable with other results; in addition, it seems that the diquark picture tends to give larger predictions.It is likely that larger shape parameters are used in the diquark picture, as pointed out in Subsec.III A.

IV. CONCLUSIONS AND DISCUSSIONS
In this work, a three-quark picture is constructed using a bottom-up approach for baryons in light-front quark model, where quark spinors and Dirac matrices act as building blocks.The shape parameters, which characterize the momentum distribution inside a baryon, are determined with the help of the pole residue of the baryon.Some semileptonic decays are investigated under this three-quark picture.The relation between the three-quark picture and the diquark picture is clarified.There is still a small flaw worth pointing out, that is, when determining the shape parameters, we demand that λ 1 ≈ λ 2 ≈ λ QCDSR , some uncertainty can still be introduced.A better prescription we can think of is to do global fitting.Given that our main goal in this article is to develop a set of methods, such a more detailed consideration is left for our future work.Here are some prospects.
• At this point, we have constructed a relatively complete three-quark picture for baryons, which can be applied to study semileptonic, nonleptonic, strong, and electromagnetic decay processes of heavy flavor baryons in the future.
• When building the model, we found that Lorentz boost plays a crucial role.As can be seen in Appendix A, in spin space, we first couple quark 3 and quark 2 to form a "diquark", then couple this "diquark" to quark 1. Obviously, this bottom-up modeling approach can be generalized to multiquark states.We may establish a unified theoretical framework for describing multiquark states.
• As a by-product of model construction, we can easily obtain an improved definition of baryon interpolating current.The hadron interpolating currents are the starting point of Lattice QCD and QCD sum rules, and therefore are of great importance.These new interpolating currents can be applied to study more detailed problems, such as the Ξ Q − Ξ ′ Q mixing [53].Dirac matrices, the instant spinors can then be transformed into the light-front ones using Eq.(A2).Therefore, in the following, we will focus on rewriting the CG coefficients in Eq. (A1) into the product of instant spinors and Dirac matrices.Note that the spinors appearing below are all instant spinors (we have omitted their subscript D), except u( P , S z ) and u µ ( P , S z ).When one of these two spinors is involved, we always take the rest frame of P , where its instant form and light-front form coincide.
It is worth pointing out that the proof given here does not introduce any additional assumptions, for example, it does not assume heavy quark symmetry, nor does it depend on the coordinate system selection of LFQM.In addition, one can clearly see that, the Lorentz boost between the rest frame of "diquark" and the rest frame of P plays a crucial role for the case involving an axial-vector "diquark".(Of course, for the case involving a scalar "diquark", the Lorentz boost is trivial.)a.To derive the spin wavefunction of ΛQ Λ Q has quark components udQ, in which ud are considered to form a 0 + diquark.
• Step 1, couple the spins of quark 3 and quark 2 to form a scalar "diquark" • Step 2, calculate the trivial coupling Therefore, for Λ Q , the CG coefficients in Eq. (A1) can be rewritten into  Σ Q also has quark components udQ, in which ud are considered to form a 1 + diquark.Technically, it is much more complicated to arrive at the spin wavefunction of Σ Q .

TABLE II .
[19]form factors are compared with other results in the literature.The asterisk on Ref.[19]indicates that, in this literature, we made a mistake in the sign for the axial-vector form factors, and here we have corrected it.

TABLE III .
[19]decay widths (in units of 10 −14 GeV) are compared with other results in the literature.The asterisk on Ref.[19]indicates that, in this literature, although we made a mistake in the sign for the axial-vector form factors, the prediction for the decay width is not affected.Λ b → Λce − νeDecay width Σ b → Σce − νe Decay width Ξcc → Λce + νe Decay width This work 2.54 ± 0.33 This work 0.870 ± 0.061 This work 0.755 ± 0.219 (e 1 + m 1 )(e 2 + m 2 )(e 3 + m 3 ) b.To derive the spin wavefunction of ΣQ