Dynamical supersymmetry for strange quark and $ud$ antidiquark in hadron mass spectrum

Speculating that the $ud$ diquark with spin 0 has a similar mass to the constituent $s$ quark, we introduce a symmetry between the $s$ quark and the $\overline{ud}$ diquark. Constructing an algebra for this symmetry, we regard a triplet of the $s$ quarks with spin up and down and the $\overline{ud}$ diquark with spin 0 as a fundamental representation of this algebra. We further build higher representations constructed by direct products of the fundamental representations. We propose assignments of hadrons to the multiples of this algebra, in which we find in particular that $\{D_{s}, D_{s}^{*}, \Lambda_{c}\}$ and $\{\eta_{s}, \phi, \Lambda, f_{0}(1370)\}$ form a triplet and a nonet, respectively. We also find a mass relation between them by introducing the symmetry breaking due to the mass difference between the $s$ quark and the $\overline{ud}$ diquark.


Introduction
Symmetries play important roles in hadron physics. Hadrons can be classified into the representations of symmetry groups, and the hadron masses and interactions can be explained by the symmetry properties. In particular, the flavor SU(3) symmetry is one of the most successful examples to understand the hadron spectra. After having discovered strangeness, one collects hadrons having a similar mass and classifies into octets and decuplet of the SU(3) representation [1,2] according to the celebrated Gell-Mann Nishijima relation [3,4]. Behind this classification, the up, down and strange quarks are found as the fundamental representation of the symmetry [5]. The flavor SU(3) symmetry is not exact but is broken explicitly with the quark mass difference. The symmetry breaking pattern is also constrained by the symmetry properties. Treating the quark mass difference as a first order perturbation, one obtains the so-called Gell-Mann Okubo mass formulae [2,6] which relates the masses of the hadrons in the same multiplet. In this way, one finds the substantial objects which carry the fundamental properties of symmetry out of the hadron spectrum. In this paper, regarding the constituent strange quark and the ud diquark as a fundamental object of a symmetry, we find mass relations among the hadrons classified in the same multiplet of the symmetry and discuss the possibility of the existence of the ud diquark as an effective constituent of hadrons.
The diquark is a pair of two quarks and is considered as a strong candidate of the hadron consituent [7][8][9]. Because the diquark has color charge, it cannot be isolated and should exist inside hadrons. One expects a strong correlation particularly between up and down quarks with spin 0 and isospin 0 due to color magnetic interaction [10] and such strong correlations are also found in Lattice QCD studies [11][12][13][14]. The diquark has been investigated in a context of the quark models in Refs. [15][16][17][18][19][20][21], and recently it has been found in Refs. [22,23] that the color electric force between diquark and quark could be weaker than that between quark and antiquark. A QCD sum rule approach [24] has suggested also the ud diquark as a constituent of the ground states of Λ, Λ c and Λ b having a constituent diquark mass around 0.4 GeV. The mass of the ud diquark is not fixed yet. Considering the u and d constituent quark mass to be about 0.3 GeV, one expects that the diquark mass be 0.4 to 0.6 GeV depending on the attraction between the u and d quark. Such value of the diquark mass is very similar to the constituent strange mass, which may be 0.5 GeV.
In this paper, we introduce a symmetry in which the constituent s quark and the ud diquark form a fundamental representation thanks to their similar masses and classify hadrons according to the symmetry to discuss the breaking pattern of the symmetry in the mass spectrum of hadrons composed of the s quark and ud diquark. This is the same approach to find the flavor SU(3) symmetry in the hadron spectrum. While both the s quark and the ud diquark have the same color charge, they have different spins; the s quark is a fermion with spin 1/2 and the ud diquark is a boson with spin 0. Thus, the symmetry that we consider here is a supersymmetry which transforms fermions and bosons. This kind of supersymmetry was introduced first in hadron physics in Refs. [25,26]. There (p,n,Λ) and (K + ,K 0 ,η) were considered as flavor fundamental representations, and a supersymmetry between these two triplets were investigated using so-called V(3) algebra. A supersymmetry between quark and diquark was discussed also in Refs. [27][28][29]. Dynamical supersymmetry in nuclear physics was suggested in Ref. [30].
In this paper, in Sec. 2 we define the algebra in which the spin up and down s quark and the ud diquark with spin 0 form a triplet. In Sec. 3, we discuss the representation of the algebra introduced in Sec. 2, and show examples of the representations for hadrons in Sec. 4. Section 5 presents the symmetry breaking by the mass difference of the s quark and the ud diquark, and derives a Gell-Mann Okubo type mass formula for φ, Λ and f 0 . Section 6 is devoted to summary and conclusion.

Definition of algebra
In this section, we introduce a supersymmetry for a Dirac fermion with spin 1/2 and a charged scalar boson in the flavor space according to Ref. [25].

Field definition
Let us first define the fermion and boson fields. We write the fermion and boson fields as ψ and ϕ, respectively. The fermion field ψ has four components, two of them are so-called upper components in the Dirac representation, the others are the lower components, while the scalar field ϕ is composed of two independent real fields for a charged boson.

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The Lagrangians for the free Dirac field and the scalar boson field are written as respectively. Defining the conjugate momenta as we have the corresponding Hamiltonians as Quantization is performed by introducing the equal-time commutation relations for the fermion and boson fields. The field commutation relations are given as for fermion, where α and β stand for the Dirac components, and for boson. These expressions are not symmetric in terms of the fermion and boson fields. In the following we redefine the fields in a symmetric form. Let us introduce two-component fields, ψ (+) and ψ (−) , as the eigenfunction of γ 0 with eigenvalue ±1, respectively. In the Dirac representation, ψ (+) and ψ (−) are the upper and lower components of the Dirac field, respectively, as Their conjugate fields are denoted bŷ Using theψ (±) field, the anti-commutation relation of the fermion field is written as for i, j = 1, 2, and the ψ (+) and ψ (−) are anti-commuting. The pseudoscalar and vector feilds,ψγ 5 ψ andψγ i ψ, are decomposed intoψ (±) and ψ (±) asψ where σ i is the Pauli matrix in the spin space.

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For the boson field, we introduce the following two independent fields and their conjugate fields are denoted bȳ Now, in the similar way to the fermion field, we introducê where γ 0 for the boson field is a 2 by 2 matrix and ϕ (±) is the eigenvector of γ 0 with eigenvalue ±1. It is easy to check that the boson fields ϕ (±) andφ (±) satisfy the following commutation relation: The ϕ (±) andφ (±) fields are commuting. Now we have the commutation relations (10) and (16) in a symmetric form.
Writing the mass term of the Hamiltonians in the redefined fields, we obtain if one assumes the same mass m for the fermion and boson.

V(3) algebra
2.2.1. Generators of V(3). Now let us consider the fermion and boson fields as a triplet for each (±) component: Hereafter we indicate the (+) and (−) components by the superscript and subscript, respectively: We introduce transformation among the triplet for each component. We call this algebra by V(3) accordingly to Ref. [26]. This algebra has SU(2) as a subalgebra. The fermion field is transformed as a doublet of SU(2), while the boson field is transformed as a singlet of SU (2). Regarding the fermion field as quark and the boson field as antidiquark, we also introduce baryon number. The fermion field has baryon number 1/3, while the boson field has baryon number −2/3. We can label each component of the V(3) representation by the The anticommutation relations for the fermionic generators are and the other commutation relations vanish: It is also notable that the relations implies that the commutation relations can only provide the combinations of the generators, . This is true also for the generators with the subscript.
If we write the bosonic and fermionic generators as B AB and F AB , respectively, that is, the commutation relations are written as We have the same relations for the generators with the subscript. As seen in the above commutation relations, G AB and G AB satisfy the same commutation relations. This implies that G AB and G AB are algebraically equivalent. According to Lorentz symmetry, both Ψ i and Ψ i fields should participate in the theory. Thus, in order to incorporate both Ψ i and Ψ i fields into the theory, we would consider the V(3)⊗V(3) symmetry. However, because the spin operator of the Dirac fermion in the standard notation is expressed as which implies that the spin operation for the fermion acts on both ψ i and ψ i in the same direction, we should consider only the subalgebra of V(3)⊗V(3) which is generated by G AB = G AB + G AB . As a result, G AB satisfies the same commutation relations of G AB and G AB , that is, G AB generates the V(3) algebra.

Fundamental representation:
The triplets, Ψ i and Ψ i , given in Eq. (19) are the fundamental representations of the V(3) algebra. These fields are transformed by the generators G AB and G AB , respectively. Similarly the conjugate fieldsΨ i andΨ i are the complex representation of the fundamental representation.

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The commutation relations of these fields for the spin G 3 and the baryon number G 8 read This implies that ψ 1 , ψ 2 and ϕ 3 fields have the quantum number (S 3 , B) as (1/2, 1/3), (−1/2, 1/3) and (0, −2/3), respectively. The same relations are satisfied for the fields with the subscript. Regarding Ψ i ⊕ Ψ i as a color triplet "quark" field andΨ i ⊕Ψ i as a "antiquark" field with a color anti-triplet, we construct the representation of "hadron" as a composites of the quark fields.
With the commutation relations derived in the previous section, we find thatΨ i Ψ i and Ψ i Ψ i are invariant under any transformations G AB and G AB , respectively, as If we take a linear combination of these terms asΨ Next, we consider composite fields made of two fundamental representations, Ψ i ⊗ Ψ i and Ψ i ⊗Ψ i , which can be regarded as "mesonic" fields. We have two kinds of the combinations: The former is favored by the ground state in the nonrelativistic limit, because Ψ i andΨ i contain the large components of the Dirac spinor. First, we consider the former combination. Let us introduce These fields are algebraically independent and belong to the same representations, as we shall see below. For the Lorentz symmetry we need both fields in an appropriate combination.
Let us see the irreducible representation for M j i and M i j . We start with the M 2 1 field, which has S 3 = −1 for the 3rd component of spin and B = 0 for the baryon number, and therefore M 2 1 is a boson field. The generator G 21 raises 1/2 for the 3rd component of spin and does not change the baryon number. Having the commutation relations , M 1 2 } form a spin triplet. Thus, these fields have total spin 1 with B = 0. We also find that the 1 2 ) has total spin 0. 8/21 Next let us consider the transformation of M 2 1 by G 23 which raises the 3rd component of spin by 1/2 and change the baryon number by −1: This implies that M 3 1 has S 3 = −1/2 and B = −1, and thus, it is a fermion field. We consider the transformation of the field M 3 1 by G 21 changing S 3 by +1 as shows that the field M 3 3 , which has total spin S = 0 and baryon number B = 0, is also within this multiplet. This implies that we need all of three components, 1 is the spin singlet with B = 0. We also confirm in the same way thatΨ i Ψ j ⊕Ψ i Ψ j forms a nonet of V(3).
In this way, we find a nonet representation of V(3) and write 3 ⊗3 We consider higher dimensional representations composed of two fundamental representa- , which can be regarded as "diquark" fields. Again, Ψ i ⊗ Ψ j is favored by the ground state in the nonrelativistic limit than Ψ i ⊗ Ψ j . Here we consider the decomposition of Ψ i ⊗ Ψ j into the irreducible representations of V(3). The other combinations are also decomposed in the same way.
We introduce nine fields, Ψ i a Ψ j b , where i and j are indices of the V(3) fundamental representation running 1 to 3, and a and b are fixed labels representing other quantum numbers such as color. The product Ψ i Ψ j has 9 components. Here we consider the decomposition of the 9 components into the irreducible representations of V(3). We will see that Ψ i Ψ j is decomposed into a quintet and a quartet representations, that is written as 3 ⊗ 3 = 5 ⊕ 4.

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3.3.1. Quintet. Let us start a highest field ψ 1 a ψ 1 b which has S 3 = +1 and baryon number B = 2/3. We lower spin quantum number by take the commutation relations and then we find that {ψ 2 } forms a spin triplet with total spin 1 and baryon number 2/3. Next we consider the transformation of the field ψ 1 a ψ 1 b by G 13 , and consider the transformation of the field ϕ 3 Therefore, we find that {ϕ 3 } form a spin doublet with total spin 1/2 and baryon number −1/3.
Taking the transformation of the field ϕ 3 we find that there are no components with baryon number −4/3 for the multiplet starting with ψ 1 a ψ 1 b . Consequently, there are five components, forming a quintet representation 5. This representation is anti-symmetric under the exchange of indices a and b. (Note that ψ i is a fermion field having {ψ i a , ψ j b } = 0.)

Quartet.
To find further representation, we start with 1 and has spin 0. Calculating the commutation relations we find that four components, We decompose the fields composed of three fundamental representations, Ψ i a Ψ j b Ψ k c , into irreducible representations of V(3), where again a, b and c label other quantum numbers. This field configuration is favored by the ground state in the nonrelativistic limit and corresponds to baryons for color singlet. Finally we will find that Ψ i a Ψ j b Ψ k c is decomposed into a septet, a quartet and two octet representations, namely we write 3 ⊗ 3 ⊗ 3 = 7 ⊕ 4 ⊕ 8 ⊕ 8. Other configurations, Ψ i Ψ j Ψ k , Ψ i Ψ j Ψ k and Ψ i Ψ j Ψ k forms the same multiplets.
3.4.1. Septet. We start with a highest component ψ 1 a ψ 1 b ψ 1 c with spin S 3 = 3/2 and baryon number B = 1. Taking the commutation relations we find a spin quartet {ψ 1 c } with total spin 3/2 and baryon number 1. Next, we consider the transformation of ψ 1 a ψ 1 b ψ 1 c by G 13 and the spin partners of its product: These form spin triplet with baryon number 0. Decreasing the baryon number of these terms further, we find This implies that we have a septet representation 7 as where the first four terms have total spin 3/2 and baryon number 1 and the last three terms have total spin 1 and baryon number 0. These terms are totally anti-symmetric under the exchange of indices a, b and c.

Quartet.
To find further representations, we start with another highest component, ϕ 3 a ϕ 3 b ϕ 3 c , which has spin 0 and baryon number −2. Increasing its baryon number by G 31 , we which has spin S 3 = +1/2 and baryon number B = −1, and its spin partner can be found by applying G 12 on it as which has spin S 3 = −1/2 and baryon number B = −1. Calculating we have a further component with spin 0 and baryon number 0 in this multiplet. Thus the second multiplet of three fundamental representations is a quartet representation 4: This representation is symmetric under the exchange of indices a, b and c.

Octet symmetric.
Next, we see another representation by starting with a spin double 1 c with baryon number 1, which are symmetric under the exchange of indices a and b. We decrease the baryon number of the former term: This term has spin S 3 = +1 and baryon number B = 0. Its spin partners are found by decreasing its spin with G 12 sequently as 1 . Thus, these components form a spin triplet with baryon number 0. We further apply G 13 , and we have This has spin S 3 = +1/2 and baryon number B = −1. Its spin partner is found as In addition, we have which has spin 0 and baryon number 0. This can be written as a linear combination of two components as is also a component of the multiplet. Consequently we have eight components in this multiplet forming a 12/21 octet 8: 3.4.4. Octet asymmetric. Next we start with another spin 1 2 doublet 1 with baryon number 1, which is asymmetric under the exchange of indices a and b. We decrease the baryon number of the former component by considering the transformation of G 23 : which has spin S 3 = +1 and baryon number B = 0. Its spin partners are found by operating . We further decrease the baryon number by using G 13 : and its spin partner is found as 1 . These components have spin 1/2 and baryon number −1. We also calculate This component can be written as 13/21 Thus, we have the following 8 components in this multiplet: Then we find that the baryonic representation is where subscrips A and S mean totally asymmetry and symmetry under the exchange of indices a, b and c, respectively, while subscript ρ and λ stand for asymmetry and symmetry under the exchange of indices a and b, respectively.

Representations of hadrons
Here we show examples of the V(3) representations for hadrons. Regarding that the strange constituent quark and the ud-diquark have a very similar mass, such as 500 MeV, we assign the fundamental representation of V(3), 3, into a strange quark with spin up, s ↑ , a strange Table 1 Possible hadrons in the same multiplet of V(3). The lowest states in each category are considered. The wavefunction of the orbital motion is assumed to be symmetric, while for the multiplets with asterisks, their symmetry property makes the orbital wavefunction asymmetric with orbital or radial excitation. The number in the parenthesis denotes the considerable spin-parity J p of the state. For the excited state, the possible spin S not total spin J is written. One understands that c is a charm quark as a representative of quarks in the outside of the V(3) multiplets. One can replace the charm quark into another quark such as a bottom quark. quark with spin down, s ↓ , and an ud antidiquark, ud, as which has color triplet. We compose hadrons out of the Ψ fields. The possible hadrons are summarized in Table 1. We consider nonrelativisic favor configurations made of Ψ i andΨ i . It is easy to recover the relativistic covariance by making up other components composed of Ψ i andΨ i appropriately.

Triplet Representation
Since the triplet field Ψ has color, a single Ψ cannot form hadrons. We consider color singlet hadrons made of an heavy-quark and the anti-triplet field. First we take the charm quark as one of the heavy quarks and consider hadrons composed of c i andΨ j . The charm quark has spin 1/2 and i runs from 1 to 2. Here we have six components,ψ 1 c 1 ,ψ 2 c 1 ,φ 3 c 1 ,ψ 1 c 2 , ψ 2 c 2 ,φ 3 c 2 , which are two triplets of V(3). For the spin eigenstates, we have a spin triplet, a spin singlet and a spin doublet as Here we assign possible hadrons which have the appropriate quantum number. In this way, in the V(3) symmetry, D * s , D s and Λ c are in the same multiplet. We Introduce the conjugate fields like Eqs. (9) and (15) as, (109) Here we have used (ψ 2 c 1 ) † = (c 1 ) † (ψ 2 ) † =ĉ 1 ψ 2 and (φ 3 The mass term for the V(3) triplet hadron can be written with a common mass m 0 as

Mesonic nonet representation
Next let us consider the mesonic nonet representation,Ψ i Ψ j ⊕Ψ i Ψ j . We introduce the matrix representation like Eq. (57) as Using these values we find the spin averaged mass forss as ms s = 0.925 GeV (132) for m φ = 1.019 GeV. With the Λ mass m Λ = 1.116 GeV, the mass formula (127) suggests that a scalar meson f 0 composed of two diquarks udud has a mass m f0 = 1.320 GeV.
The corresponding particle can be found as f 0 (1370) in the particle listing of Particle Data Group [31],  3) breaking found in these hadrons is attributed to the mass difference between the s quark and the ud diquark, the mass of the ud diquark may be 700 MeV if one assumes the constituent strange quark mass to be 500 MeV. Nevertheless, it should be worth mentioning that the V(3) breaking could stem from asymmetry of the interaction ofs-Q and (ud)-Q as pointed out in Refs. [22,23]. There they found that the string tension in the color electric confinement potential between quark and diquark is as weak as half of that between quark and anti-quark, even though these two systems have the same color configuration. Further investigation on the symmetry between the quark and diquark should be necessary.

Summary
We have introduced a symmetry among the constituent strange quark and the ud diquark, supposing that their masses be very similar to each other, say 500 MeV. To investigate the properties of this symmetry, we have constructed an algebra which transforms a fermion with spin 1/2 and a boson with spin 0. Regarding these fermion and boson as a fundamental representation of this algebra, we have built higher representations for mesonic, diquark and baryonic configurations. We have proposed possible assignments of these irreducible representations to existing hadrons, which is summarized in Table 1. Particularly investigating the triplet and nonet representations, we have found that (Λ c , D s , D * s ) and (η s , φ, Λ, f 0 (1370)) could form multiplets, respectively. Introducing a symmetry breaking coming from the mass difference the s quark and the ud diquark, we have derived a mass relation among each multiplet. In the nonet representation, we have the mass relation among φ, η s , Λ, f 0 . In our formulation, both φ and η s are composed of the s ands quarks, while the physical η meson is known to be expressed almost as the octet. Thus, estimating the strength of the spin-spin interaction from the mass difference of D s and D * s , we have found the spin averaged mass φ and η s to be 920 MeV. Using the mass relation with this mass and the observed Λ mass, 20/21 we have found the mass of f 0 in the multiplet to be 1320 MeV. This may correspond to the observed f 0 (1370) meson. Thus, our model suggests f 0 (1370) to be a tetraquark composed of ud and ud diquarks. In addition, finding the mass differences among the nonet to be 200 MeV, and the difference between the spin averaged mass of D s and D * s (B s and B * s ) and Λ c (Λ b ) also to be 200 MeV, we have suggested the mass difference between the constituent s quark and the ud diquark to be 200 MeV. Thus, if we regard the strange quark mass as 500 MeV, the mass of ud diquark may be 700 MeV. Nevertheless, we could have another possibility for the source of the mass difference to be a perspective suggesting that the symmetry braking comes from the difference of interactions betweens-Q and (ud)-Q. It is an open question that the origin of the symmetry breaking between the s quark and the ud diquark, and further investigations on this issue are necessary.