Chiral Lagrangians with decuplet baryons to one loop
aa r X i v : . [ h e p - ph ] J a n Chiral Lagrangians with decuplet baryons to one loop
Shao-Zhou Jiang , , ∗ Yan-Rui Liu , † Hong-Qian Wang , and Qin-He Yang Department of Physics and GXU-NAOC Center for Astrophysics and Space Sciences,Guangxi University, Nanning, Guangxi 530004, People’s Republic of China Guangxi Key Laboratory for the Relativistic Astrophysics,Nanning, Guangxi 530004, People’s Republic of China School of Physics and Key Laboratory of Particle Physics and Particle Irradiation (MOE),Shandong University, Jinan 250100, People’s Republic of China
We construct the relativistic chiral Lagrangians with decuplet baryons up to the order O ( p )(one loop). For the meson-decuplet-decuplet couplings, there are 1, 13, 55, and 548 terms in the O ( p )- O ( p ) order Lagrangians, respectively. For the meson-octet-decuplet Lagrangians, the numberof independent terms from O ( p ) to O ( p ) are 1, 5, 67, and 611, respectively. For convenience ofapplications, the π ∆∆ and πN ∆ chiral Lagrangians are picked out. This new form of ∆ Lagrangiansis equivalent to the original isovector-isospinor one and we establish relations between these twoforms. I. INTRODUCTION
Chiral perturbation theory (ChPT) is a useful tool to describe low-energy strong interactions of mesons [1–3] andbaryons [4]. This effective theory is based on the chiral symmetry of QCD and its spontaneous breaking. Theinteraction terms and various physical quantities in this theory are organized perturbatively by chiral dimension, theorder of p/ Λ χ where p represents the typical scale of momentum and Λ χ is the scale of chiral symmetry breaking.Theoretically, the higher chiral dimension terms are considered, the higher precisions of results would be obtained. Atpresent, the chiral Lagrangians containing the pseudoscalar mesons [2, 3, 5–12] and the ground state baryons [4, 13–20](both SU(2) and SU(3)) have been already constructed to the sixth and forth order, respectively. Recently, the chiralLagrangians with ∆(1232) are also considered up to the forth chiral order [21, 22]. For the purpose of applications,the current existent chiral Lagrangians are precise enough for theoretical studies on low energy interactions. However,the above investigations missed a kind of particles, the spin-3/2 hyperons.In reality, a lot of low-energy QCD problems are related to the chiral Lagrangians with decuplet states which aredegenerate with the octet baryons in the large N c limit. Such problems include: the masses of the octet/decupletbaryons and the mass relations between octet/decuplet baryons [23–26], the electromagnetic structures of octet anddecuplet baryons (magnetic moments, electric quadrupole moments, and electromagnetic form factors) [27–30], themeson-octet/decuplet scattering processes [31, 32], the transitions from decuplet states to octet states [33, 34], latticestudies of baryon properties [35–37], and so on. Especially, the studies of the transitions between decuplet and octetbaryons can shed light on the possible dibaryons [38]. The lowest order chiral Lagrangian with decuplet states isobtained easily [39], but we find only fragmentary results for high order terms in the literature (see the referencesmentioned above). Such Lagrangians are constructed just for special problems one focuses on. A complete andminimal set of Lagrangians with decuplet baryons is still needed. One purpose of this paper is to construct the chiralLagrangians with the decuplet baryons to one loop (the 4th chiral order) systematically.In the SU(2) case, we have obtained the chiral Lagrangians with ∆ up to the order O ( p ) [22], where we use theisovector-isospinor representation [40] in the isospin space for the Rarita-Schwinger (RS) fields. The application ofsuch Lagrangians is not so convenient in some cases. On the other hand, in the SU(3) case, the decuplet baryons arerepresented in the flavour space as a totally symmetric tensor T abc . Since the ∆ baryons are members of the decupletrepresentation, the Lagrangians with ∆ can also be expressed with the symmetric tensor. However, it is apparentlynot straightforward to make a relation between these two formalisms. Another purpose of the present study is to givenew chiral Lagrangians with ∆ in the form of T abc ( a, b, c = 1 ,
2) and establish the relations to the former formalism.This paper is organized as follows. In Sec. II, we review the building blocks for the construction of the chiralLagrangians with the mesons, the external sources, and a part of building blocks with baryon fields. In Sec. III, wepresent the structures of the chiral Lagrangians and give a full building blocks with baryon fields. In Sec. IV, theproperties of the building blocks, the linear relations of invariant monomials, and the relations between the originalchiral Lagrangians with ∆ and the new forms are given. In Sec. V, we list our results and present some discussions.Section VI is a short summary. ∗ [email protected] † [email protected] II. BUILDING BLOCKS IN CONSTRUCTING CHIRAL LAGRANGIANS
Generally speaking, the constructed Lagrangians in ChPT involve the pseudoscalar mesons, the external sources,the decuplet baryons, and the octet baryons. In this section, we present appropriate building blocks in constructingthe chiral Lagrangians. More detailed discussions about them can be found in Refs. [2–6, 9, 10, 12, 17, 18, 20–22].For the spin-3/2 baryon states, we consider both SU (3) and SU (2) cases. For convenience, we simply call the form ofchiral Lagrangians with ∆ in Ref. [22] “original” and that in this paper “new”. Needless to say, the new form SU (2)Lagrangians are just selected terms of the SU (3) Lagrangians with decuplet baryons. Hence, in the following parts,we treat them in the same way. A. Building blocks of the mesons and the external sources
The QCD Lagrangian L can be written as L = L + ¯ q ( /v + /aγ − s + ipγ ) q , (1)where L is the original QCD Lagrangian and q denotes the quark field. We use s , p , v µ , and a µ to denote scalar,pseudoscalar, vector, and axial-vector external sources, respectively. Conventionally, the tensor source and the θ termare ignored. As usual, we consider that only a µ is traceless in the two-flavour case, but both a µ and v µ are tracelessin the three-flavour case.In ChPT, the pseudoscalar mesons (Goldstone bosons) come from the spontaneous breaking of the global symmetry SU ( N f ) L × SU ( N f ) R into SU ( N f ) V . The resulting meson fields are collected in u and it transforms as u → g L uh † = hug † R (2)under the chiral rotation, where g L and g R represent elements in SU ( N f ) L and SU ( N f ) R , respectively, and h is acompensator field which is a function of the pion fields.To construct the chirally invariant Lagrangians involving only meson fields and external sources, the building blocksare usually chosen as u µ = i { u † ( ∂ µ − ir µ ) u − u ( ∂ µ − il µ ) u † } ,χ ± = u † χu † ± uχ † u,h µν = ∇ µ u ν + ∇ ν u µ ,f µν + = uF µνL u † + u † F µνR u,f µν − = uF µνL u † − u † F µνR u = −∇ µ u ν + ∇ ν u µ , (3)where r µ = v µ + a µ , l µ = v µ − a µ , χ = 2 B ( s + ip ), F µνR = ∂ µ r ν − ∂ ν r µ − i [ r µ , r ν ], F µνL = ∂ µ l ν − ∂ ν l µ − i [ l µ , l ν ], and B is a constant related to the quark condensate. The form of these building blocks, however, is not very useful inthe construction of chiral Lagrangians with decuplet baryons. For convenience, we write the flavour indices of thesebuilding blocks (or any other matrices in the flavour space) explicitly, X = X ab + X s I, X s = 1 N f h X i , (4)where X denotes any building block in Eq. (3) (or any matrix in the flavour space), X ab ( X s ) is the traceless(traceable) part of X , I is the N f × N f identity matrix in the N f -flavour space, and h· · ·i means the trace in theflavour space. We use a and b ( a, b = 1 , ,
3) to denote the row index and column index of the matrix X , respectively.In the following, we will treat the row index (or the first index) of X ab as the subscript and the column index (or thesecond index) as the superscript. According to these notations, we have u µs = f µν − ,s = h µνs = 0 in the two-flavour caseand an additional relation f µν + ,s = 0 in the three-flavour case. The chiral transformations ( R ) for these building blocksare X ab R −→ X ′ ab = h aa ′ X ′ b ′ a ′ h † bb ′ ,X s R −→ X ′ s = X s . (5)Here h aa ′ does not need to be traceless as the definition of X ab in Eq. (4). The row index of X ab is related to the h field, but the column index is related to the h † field.The covariant derivative ∇ µ acting on the building blocks in Eq. (5) are ∇ µ X ab = ∂ µ X ab + Γ ac,µ X cb − X ac Γ cb,µ , ∇ µ X s = ∂ µ X s , Γ µ = 12 { u † ( ∂ µ − ir µ ) u + u ( ∂ µ − il µ ) u † } . (6)In constructing the Lagrangian, the following two relations will be useful[ ∇ µ , ∇ ν ] X ab = Γ ac,µν X cb − X ac Γ cb,µν , (7)[ ∇ µ , ∇ ν ] X s = 0 , (8)Γ µν = ∇ µ Γ ν − ∇ ν Γ µ − [Γ µ , Γ ν ] = 14 [ u µ , u ν ] − i f µν + . (9) B. Building blocks of baryons
Besides the meson fields and external fields, we also need baryons belonging to SU (3) 8 and 10 representations.The octet baryons are represented by a matrix B ab , B ab = Σ √ + Λ √ Σ + p Σ − − Σ √ + Λ √ n Ξ − Ξ − √ . (10)In the two-flavour case, it is reduced to the nucleon doublet, ψ a = (cid:18) pn (cid:19) . (11)One may also use the symbol B a ( a = 1 ,
2) to denote this nucleon doublet. For the decuplet baryons, they aredenoted by a totally symmetrical tensor T abc with T = ∆ ++ , T = ∆ + √ , T = ∆ √ , T = ∆ − ,T = Σ ∗ + √ , T = Σ ∗ √ , T = Σ ∗− √ ,T = Ξ ∗ √ , T = Ξ ∗− √ , T = Ω − . In the SU (2) case, only the first four fields are needed.The chiral transformations for these baryon fields are B ab R −→ B ′ ba = h aa ′ B b ′ a ′ h † bb ′ ,ψ a R −→ ψ ′ a = h ab ψ b ,T abc R −→ T ′ abc = h aa ′ h bb ′ h cc ′ T a ′ b ′ c ′ , ¯ B ab R −→ ¯ B ′ ba = h aa ′ ¯ B b ′ a ′ h † bb ′ , ¯ ψ a R −→ ψ ′ a = ¯ ψ b h † ab , ¯ T abc R −→ ¯ T ′ abc = ¯ T a ′ b ′ c ′ h † aa ′ h † bb ′ h † cc ′ . (12)From the transformations, the indices of ψ a and T abc ( ¯ ψ a and ¯ T abc ) can be treated as row (column) indices and thoseof B ab and ¯ B ab are self-evident. From Eqs. (5) and (12), we can see that if a term is chirally invariant, all the rowindices must be contracted with the column indices and vice versa. This is the reason why we write the row andcolumn indices explicitly.The covariant derivative D µ acting on the baryon fields are [15, 18, 27] D µ ψ a = ∂ µ ψ a + Γ abµ ψ b ,D µ B ab = ∂ µ B ab + Γ ac,µ B cb − B ac Γ cb,µ ,D µ T abc = ∂ µ T abc + Γ adµ T dbc + Γ bdµ T adc + Γ cdµ T abd . (13)It seems that, in the three (two)-flavour case, we can choose T µabc , ¯ T abc,µ , B ab , ¯ B ab ( T µabc , ¯ T abc,µ , ψ a , ¯ ψ a ), and theircovariant derivatives as building blocks. But it is a bit more complex for the spin-3 / III. STRUCTURES OF CHIRAL LAGRANGIANS WITH DECUPLET BARYONS
A similar discussion in this section has been presented in Ref. [22]. Here we only list the necessary ingredients forthe Lagrangian construction. More details can be found in Refs. [17–21, 40–56].In this paper, we adopt the vector-spinor representation Ψ µ ( µ = 0 , , ,
3) [41] for the spin-3/2 fields. The generalLagrangian for a free RS field with mass m reads [42] L f = ¯Ψ µ Λ µνA Ψ ν , (14)Λ µνA = − (cid:2) ( i /∂ − m ) g µν + iA ( γ µ ∂ ν + γ ν ∂ µ )+ i A + 2 A + 1) γ µ /∂γ ν + m (3 A + 3 A + 1) γ µ γ ν (cid:3) , where A = − / i /∂ − m )Ψ µ = 0 , (15) γ µ Ψ µ = 0 , (16) ∂ µ Ψ µ = 0 . (17)The two unphysical spin- degrees of freedom in the vector-spinor representation can be eliminated with these twosubsidiary conditions.There exists a so-called “point” or “contact” transformation under which the above Lagrangian is invariant,Ψ µ → Ψ ′ µ = Ψ µ + 12 aγ µ γ ν Ψ ν , (18) A → A ′ = A − a a , a = − . (19)The choice for the value of A does not affect physical quantities [50, 55, 57]. Therefore, one may simplify the aboveLagrangian by a field redefinition [48] L f = ¯ ψ Aµ Λ µν ψ Aν , (20)Λ µν = − ( i /∂ − m ) g µν + 14 γ µ γ λ ( i /∂ − m ) γ λ γ ν , where ψ µA ≡ O µνA Ψ ν = ( g µν + Aγ µ γ ν )Ψ ν . Now, Λ µν is independent of A and the A dependence is implied in ψ µA .For the meson-decuplet-decuplet (MTT) interactions, the chiral Lagrangian has the form L MTT = ¯ T abcµ Λ def,µνA,abc T def,ν , (21)Λ def,µνA,abc = − (cid:2) ( i /D − m T ) g µν + iA ( γ µ D ν + γ ν D µ )+ i A + 2 A + 1) γ µ /Dγ ν + m T (3 A + 3 A + 1) γ µ γ ν (cid:3) δ ad δ be δ cf + O def,µν ,A,abc , (22)where m T is the decuplet mass in the SU (3) limit and O def,µν ,A,abc contains the meson fields and the external sources.Then the EOM and the subsidiary conditions in ChPT are( i /D − m T ) T µabc . = 0 , (23) D µ T µabc . = 0 , (24) γ µ T µabc . = 0 , (25)where the symbol “ . =” means that both sides are equal if high order terms are ignored. We may write the structureof any term in O def,µν ,abc as [17, 21, 22] ¯ T abc,µ O ······ Θ ······ T νdef + H . c ., (26)where · · · denotes suitable flavour and Lorentz indices, O ······ is the product of the building blocks with the mesonfields and the external sources in Sec. II A, and Θ ······ contains a Clifford algebra element Γ ∈ { , γ µ , γ , γ γ µ , σ µν } , theLevi-Civita tensors in Lorentz space ε µνλρ , and the covariant derivatives acting on T νdef . Up to the order O ( p ), thestructures of Θ ······ can be found below Eq. (49) in Ref. [22].With the structure in Eq. (26), the low-energy constants (LECs) in O def,µν ,abc are dependent on A . One can absorbthe parameter A into the redefined RS fields according to the point transformation (Eqs. (18) and (20)). Then theLagrangian (22) can be rewritten as L MTT = − ¯ T abcA,µ (cid:2) ( i /D − m T ) g µν − γ µ γ λ ( i /D − m T ) γ λ γ ν (cid:3) ¯ T A,abc,ν + ¯ T abcA,µ O def,µν ,abc T A,def,ν , (27)where T µA,abc = O µνA T abc,ν . Now, the LECs in Eq. (27) are independent of A , but the invariant monomials have thesame structures as those in Eq. (22), i.e. one may get Eq. (27) from Eq. (22) by changing T abc,µ to T A,abc,µ only.The LECs in these two equations are equal if A = 0. Physically, we can choose any value of A ( A = − /
2) ( A = − π ∆∆ Lagrangians are very similar to the MTT Lagrangians. The differences lie only in the baryonmass and the flavour indices. By changing m T to m ∆ (∆ mass in the chiral limit) and limiting all the flavour indicesto 1 and 2, the new form of π ∆∆ Lagrangians is obtained.For the meson-octet-decuplet and πN ∆ interactions, the chiral Lagrangians have the following structures, respec-tively, ǫ abc ¯ B de O ······ Θ ······ T µA,n,fgh + H . c ., (28) ǫ ab ¯ ψ c O ······ Θ ······ T µA,n,def + H . c ., (29)where O ······ and Θ ······ have the same meanings as those in Eq. (26). For the Levi-Civita tensor, we have column indicesin ǫ abc ( a, b, c = 1 , ,
3) and row indices in ǫ abc (in the H.c. part). Here, ǫ ab ≡ ǫ ab . The RS field depending on A isdefined through T A,n,fgh,µ = Θ
A,n,µν ( z n ) T νfgh , (30)Θ A,n,µν ( z n ) = g µν + [ z n + 12 (1 + 4 z n ) A ] γ µ γ ν ≡ Θ n,µα ( z n ) O αAν = O Aµα Θ n,αν ( z n ) , Θ n,µα ( z n ) ≡ g µα + z n γ µ γ α . (31)Some z n parameters are needed because of the point transformation [58]. They can be obtained from experiments.In Eqs. (28) and (29), the point-invariant structures have been implied and the LECs are already independent of A .To construct Lagrangians, for the baryon fields, we choose T µabc , ¯ T abc,µ , T µA,abc , ¯ T abc,µA , B ab , ¯ B ab , and their covariantderivatives as building blocks in the three-flavour case. In the two-flavour case, we adopt T µabc , ¯ T abc,µ , T µA,abc , ¯ T abc,µA , ψ a , ¯ ψ a , and their covariant derivatives. IV. PREPARATIONS FOR LAGRANGIAN CONSTRUCTION
In this section, we make preparations for the construction of chiral Lagrangians with decuplet baryons. The newform of chiral Lagrangians with ∆ is understood. The recipes are very similar to those in constructing Lagrangiansfor mesons, meson-baryon systems, and the π - N -∆ systems in Refs. [12, 20, 22]. A. Power counting and transformation properties
The chiral dimensions [2–4, 6, 17, 18] of the building blocks with the external sources are listed in the second columnof Table I and those of the Clifford algebra and the Levi-Civita tensors are given in the second column of Table II[17, 18, 56]. The baryon fields are chiral dimensionless and the information is not shown in these tables. The covariantderivatives acting on the meson fields and the external sources are counted as O ( p ), but those acting on the baryonfields are counted as O ( p ).The chiral Lagrangian should be invariant under the chiral rotation ( R ), parity transformation ( P ), charge con-jugation transformation ( C ), and Hermitian transformation (h.c.). The chiral rotations for the building blocks havebeen discussed in Eqs. (5) and (12). The P , C , and h.c. transformations are almost the same as those in Ref. [22]and we also present such properties in Tables I and II. Only different properties will be mentioned.Compared with Table I of Ref. [22], Table I here shows the flavour indices explicitly. The meanings of plus andminus signs in Table II are the same as those in Refs. [17, 20, 22]. One thing different is the ǫ ijk . This symbol in Ref.[22] is in the isovector space and it absorbs a minus sign in C transformations (Eq. (31) of Ref. [22]). But now ǫ abc and ǫ ab are the Levi-Civita tensors in the three (two)-flavour space. They do not need to absorb an extra minus sign. TABLE I. Chiral dimension (Dim), parity ( P ), charge conjugation ( C ), and Hermiticity (h.c.) of the building blocks with theexternal sources. Dim P C h.c. u b,µa − u ba µ u a,µb u b,µa h b,µνa − h ba µν h a,µνb h b,µνa χ b ± ,a ± χ b ± ,a χ a ± ,b ± χ b ± ,a χ ± ,s ± χ ± ,s χ ± ,s ± χ ± ,s f b,µν ± ,a ± f b ± ,a µν ∓ f a,µν ± ,b f b,µν ± ,a f µν + ,s f + ,s,µν − f µν + ,s f µν + ,s TABLE II. Chiral dimension (Dim), parity ( P ), charge conjugation ( C ), and Hermiticity (h.c.) of the Clifford algebra elements,the Levi-Civita tensors, and the covariant derivatives. The subscript ‘TT’ (‘BT’) denotes the meson-decuplet-decuplet (meson-octet-decuplet) interactions in the three flavours ( π ∆∆ ( πN ∆) interactions in the two-flavour case). Ψ denotes any baryonfield, decuplet baryon, ∆, octet baryon, or nucleon. ǫ abc ( ǫ ab ) is the Levi-Civita tensor in three (two)-flavour space. Themeaning of the plus or minus sign is explained in the text.Dim P TT C TT h.c. TT P BT C BT h.c. BT − + + γ − + − + + − γ µ − + − − + γ γ µ − + + + + + σ µν − + − − + ǫ µνλρ − + + − + + ǫ abc ǫ ab D µ Ψ 0 + − − + + +
B. Linear relations
Some linear relations exist in reducing the chiral-invariant terms to a minimal set. The relations coming frompartial integration, EOM, covariant derivatives, and Bianchi identity are the same as those in Ref. [22]. The relationscoming from the Cayley-Hamilton relation are the same as those in Ref. [6]. We will not discuss them any more andwe only focus on the different and new relations in the following parts.
1. Schouten identity
The Schouten identity in the Lorentz space is the same as that in Ref. [22], but some differences exist in the flavourspace. For the Levi-Civita tensor ǫ abc ( ǫ ab ) in the three (two)-flavour space, the Schouten identities for any operator A are 0 = ǫ abc A d − ǫ dbc A a − ǫ adc A b − ǫ abd A c , ǫ ab A c − ǫ cb A a − ǫ ac A b . (32)There are two types of indices in A (row or column). Eq. (32) works only for the case that the indices in theLevi-Civita tensor and the indices in A are the same type.
2. Fierz transformations
The basic Fierz transformation for the Pauli matrices is τ iab τ icd = 2 δ ad δ cb − δ ab δ cd . (33)With this equation, for any two 2 × X ab and Y ab in Table I, one may obtain [59] X da Y eb = 12 ( Y ea X db + X ea Y db + X fc Y cf δ ea δ db − X fc Y cf δ da δ eb + X ca Y ec δ db − δ ea X cb Y dc ) . (34)The basic Fierz transformation for the Gall-Mann matrices is λ iac λ ibd = 2 δ ad δ cb − δ ac δ bd . (35)With the relation in Ref. [60] and the properties of the structure constants of SU (3), one finds that the followingrelation exists for any two 3 × X ab and Y ab in Table I,0 = X ba Y dc − X da Y bc − X bc Y da + X dc Y ba + X ea Y be δ dc − X ea Y de δ bc − X ec Y be δ da + X ec Y de δ ba + δ ba Y ec X de − δ da Y ec X be − δ bc Y ea X de + δ dc Y ea X be − X fe Y ef δ ba δ dc + X fe Y ef δ da δ bc . (36)
3. Contact terms
The method to construct contact terms is the same as that in Ref. [22]. In the two (three)-flavour case, the totalnumber of the contact terms is six (five) and we list them in the end of Table V. The last term in Table V is at the O ( p ) order in the SU (3) case. C. Relations between the original chiral Lagrangians with ∆ and the new ones In Ref. [22], we have obtained the chiral Lagrangians with ∆ to one loop. There, the ∆ fields are represented by anisovector-isospinor RS field ψ µi ( i = 1 , , T µabc ( a, b, c = 1 ,
2) to representthem. The difference lies only in the flavour representations. By some calculations, one gets the following relationsbetween these two formalisms of interaction terms,¯ T abc OT abc = ¯ ψ i Oψ i , (37)¯ T abe O fe T abf = ¯ ψ i O j τ j ψ i , (38)¯ T abc X eb Y fc T aef = ¯ ψ i X j Y j ψ i − ¯ ψ i X i Y j ψ j − ¯ ψ i X j Y i ψ j , (39)¯ T abc X da Y eb Z fc T def = 16 ¯ ψ i X l Y j τ j Z l ψ i −
13 ¯ ψ i X i Y k τ k Z j ψ j + P ( X, Y, Z ) , (40) ǫ ab ¯ ψ c O fa T bcf = √ ψO i ψ i , (41) ǫ ab ¯ ψ c X ea Y fc T ebf = √ ψX i Y j τ j ψ i , (42) ǫ ab ¯ ψ c X ea Y fb T efc = √ iǫ ijk ¯ ψX i Y j ψ k , (43) ǫ ab ¯ ψ c X da Y eb Z fc T def = √ iǫ ijk ¯ ψX i Y j Z l τ l ψ k , (44)where P ( X, Y, Z ) means all permutations for the symbols X , Y , and Z . O , O i , X i , Y i and Z i are building blocks inRef. [22] or their products. The definitions of the symbols in the right-hand side can be found in Ref. [22].Alternatively, we may transform the original formalism to the new one. To do that, we define transition isospin I j through ψ j = I j φ with φ = (∆ ++ , ∆ + , ∆ , ∆ − ) T . Similarly, we define T abc = W iabc φ i . The matrix forms of I j andthe values of W iabc are easy to obtain from the definitions. We have two relations in connecting the original π ∆∆Lagrangians with the new ones,( I † i I j ) xy = 12 [ W abcx ( τ i τ j ) ad W ybcd − W abcx ( τ i ) ad ( τ j ) be W ycde ] , (45)( I † i τ l I j ) xy = 12 [ W abcx ( τ i τ j ) ad ( τ l ) be W ycde − W abcx ( τ i ) ad ( τ j ) be ( τ l ) cf W ydef ] . (46)For the special case j = i , one has ( I † i I i ) xy = W abcx W yabc , (47)( I † i τ l I i ) xy = W abcx ( τ l ) ad W ybcd . (48)To connect the original πN ∆ Lagrangians with the new ones, we may use( I i ) xy = 1 √ ǫ ab ( τ i ) ac W yxbc , (49)( τ i I j ) xy = 1 √ ǫ ab ( τ i ) xc ( τ j ) ad W ybcd . (50)Note ( τ i I j ) xy = √ ǫ ab ( τ i ) ac ( τ j ) xd W ybcd . Substituting these six equations into the right-hand sides of Eqs. (37)-(44),one may prove the equivalence of the two sets of relations by using the formula ǫ ab ( τ i τ j ) ac W yxbc = ǫ ab ( τ i ) ac ( τ j ) bd W yxcd . V. RESULTS AND DISCUSSIONS
Following the same steps from Sec. IV C to Sec. IV E in Ref. [22], we obtain the chiral Lagrangians with decupletbaryons up to the order O ( p ) and list them below. A. O ( p ) order In the three-flavour case, the lowest order meson-decuplet-decuplet chiral Lagrangian is L (1)MTT = · · · + C (1)1 ¯ T abcµ u adν γ γ ν T bcdµ , (51)where C (1)1 is the low-energy constant at this order and the ellipsis represents the terms coming from the first part inEq. (22). The lowest order meson-octet-decuplet chiral Lagrangian reads L (1)MBT = D (1)1 ǫ abc ¯ B ad u beµ T cdeµ + H . c .. (52)In the two-flavour case, the lowest order π ∆∆ chiral Lagrangian has the same form as Eq. (51), L (1) π ∆∆ = · · · + e (1)1 ¯ T abcµ u adν γ γ ν T bcdµ . (53)The difference lies only in the allowed numbers for the indices a , b , c , and d . Similarly, the lowest order πN ∆ chiralLagrangian can be written as L (1) πN ∆ = f (1)1 ( ǫ ab ¯ ψ c u adµ T A,n,bcdµ + H . c . ) . (54)We have confirmed the previous results in Ref. [22] with the newly constructed Lagrangians. With the relations inthe last section, we get the relations between these two kinds of LECs, e (1)1 = c (1)1 = 12 g , f (1)1 = 1 √ g πN ∆ (55) B. O ( p ) order The O ( p ) order meson-decuplet-decuplet chiral Lagrangian has the form L (2)MTT = X n =1 C (2) n O (3 , n , (56)where the operators O ( N f =3 , n are listed in Table III. The meson-octet-decuplet chiral Lagrangian at this order is L (2)MBT = D (2)1 ( ǫ abc ¯ B ad u beµ u cfν γ γ µ T A,n,defν + H . c . ) + D (2)2 ( ǫ abc ¯ B ad u beµ u dfν γ γ µ T A,n,cefν + H . c . )+ D (2)3 ( ǫ abc ¯ B ad u beµ u efν γ γ µ T A,n,cdfν + H . c . ) + D (2)4 ( ǫ abc ¯ B ad u beµ u efν γ γ ν T A,n,cdfµ + H . c . )+ D (2)5 ( iǫ abc ¯ B ad f + beµν γ γ µ T A,n,cdeν + H . c . ) . (57)This result is consistent with that in Ref. [34]. TABLE III. The order O ( p ) meson-decuplet-decuplet ( π ∆∆) chiral Lagrangians, and the relations between π ∆∆ LECs hereand those in Ref. [22]. O ( n f , n SU (2) SU (3) e (2) n ¯ T abcµ u adµ u beν T cdeν − c (2)1 / − c (2)2 / T abcµ u adν u beν T cdeµ − c (2)3 / T abcµ u adµ u deν T bceν c (2)1 / c (2)4 / T abcµ u adν u deµ T bceν c (2)2 / c (2)4 / T abcµ u adν u deν T bceµ c (2)3 / c (2)5 ¯ T abcµ u deµ u edν T abcν T abcµ u deν u edν T abcµ T abcµ u adν u beλ D νλ T cdeµ − c (2)6 / T abcµ u adν u deλ D νλ T bceµ c (2)6 / c (2)7 ¯ T abcµ u deν u edλ D νλ T abcµ i ¯ T abcµ f s, + µν T abcν c (2)8 i ¯ T abcµ f + adµν T bcdν c (2)9 ¯ T abcµ χ + ,s T abcµ
10 12 c (2)10 ¯ T abcµ χ + ad T bcdµ
11 13 c (2)11 The new form of the π ∆∆ chiral Lagrangian at the O ( p ) order is L (2) π ∆∆ = X n =1 e (2) n O (2 , n , (58)where the operators O ( N f =2 , n can also be found in Table III. The new form πN ∆ chiral Lagrangian reads L (2) πN ∆ = f (2)1 ( ǫ ab ¯ ψ c u adµ u beν γ γ µ T A,n,cdeν + H . c . ) + f (2)2 ( ǫ ab ¯ ψ c u adµ u ceν γ γ µ T A,n,bdeν + H . c . )+ f (2)3 ( iǫ ab ¯ ψ c f + adµν γ γ µ T A,n,bcdν + H . c . ) . (59)This result is consistent with the Lagrangian in Ref. [22]. We present the relations between these two kinds of π ∆∆LECs in the last column of Table III. The obtained relations for the πN ∆ LECs are f (2)1 = − √ d (2)1 , (60) f (2)2 = 1 √ d (2)1 + 1 √ d (2)2 , (61) f (2)3 = 1 √ d (2)3 . (62)0 C. O ( p ) and O ( p ) orders We define the O ( p ) and O ( p ) chiral Lagrangians as L ( m )MTT = X n C ( m ) n O (3 ,m ) n , (63) L ( m )MBT = X n D ( m ) n ( P (3 ,m ) n + H . c . ) , (64) L ( m ) π ∆∆ = X n e ( m ) n O (2 ,m ) n , (65) L ( m ) πN ∆ = X n f ( m ) n ( P (2 ,m ) n + H . c . ) , (66)where m = 3 or 4 denotes the chiral dimension, C ( m ) n , D ( m ) n , e ( m ) n , and f ( m ) n are the LECs, and O ( N f ,m ) n and P ( N f ,m ) n are the independent chiral-invariant terms in the N f -flavour case. The results are listed in Appendix A. At the O ( p )order, the meson-decuplet-decuplet ( π ∆∆) Lagrangians are presented in Table IV. There are 55 (38) independentterms in the SU (3) ( SU (2)) case. The meson-octet-decuplet ( πN ∆) Lagrangians are given in Table VI. There are67 (33) independent terms in the SU (3) ( SU (2)) case. At the O ( p ) order, the meson-decuplet-decuplet ( π ∆∆)Lagrangians are presented in Table V. There are 548 (318) independent terms in the SU (3) ( SU (2)) case. The meson-octet-decuplet ( πN ∆) Lagrangians are listed in Table VII. There are 611 (218) independent terms in the SU (3)( SU (2)) case. Note that the z n parameters should be different for the meson-octet-decuplet and πN ∆ Lagrangiansat the different orders, but we do not distinguish them explicitly in the results.To merge the meson-octet-decuplet and the πN ∆ results, similar to those for the meson-decuplet-decuplet and π ∆∆, we write them in a unified form. We have changed the SU (2) results with ǫ ab ¯ ψ c → ǫ dab ¯ B dc by setting d = 3but a, b, c = 1 , SU (2) results from corresponding terms in Table VI and Table VIIwith ǫ abc ¯ B ad · · · N f =2 −−−−→ ǫ bc ¯ ψ d · · · . (67)Because the number of LECs in O ( p ) and O ( p ) Lagrangians is large and only several LECs will be involved ina study, we here do not give the LEC relations between the new and original results at high orders. Each form ofLagrangians can be chosen to study low-energy processes. One may use relations in Sec. IV C to determine LECsfrom another form terms, if necessary.From the results, one can see that not only the total number of terms but also the number in each type of externalsources in the chiral Lagrangians with ∆ are the same as those in Ref. [22]. The equality in number is a strictcondition for consistency of Lagrangians in different forms. The violation of this condition means that the number ofterms in either or both forms is not minimal. This check confirms our previous results. VI. SUMMARY
In this paper, we construct the relativistic chiral Lagrangians with decuplet baryons and give a new form of thechiral Lagrangians with ∆(1232) to one loop. These chiral Lagrangians are for the meson-decuplet-decuplet, meson-octet-decuplet, π ∆∆, and πN ∆ interactions. The correspondence between the π ∆∆ and πN ∆ chiral Lagrangians inRef. [22] and those in the present form can be obtained with the relations we get in Sec. IV C. ACKNOWLEDGEMENTS
We thank Prof. Li-Sheng Geng for useful discussions. YRL also thanks the hospitality from Prof. M. Oka andother people at Tokyo Institute of Technology where the draft was finished. This work was supported by the NationalScience Foundation of China (NSFC) under Grants No. 11565004, No. 11775132, No. U1731239, and No. 11673006,the special funding for Guangxi distinguished professors (Bagui Yingcai and Bagui Xuezhe) and High Level InnovationTeam and Outstanding Scholar Program in Guangxi Colleges.1
Appendix A: Independent terms in O ( p ) and O ( p ) chiral Lagrangians with decuplet baryons TABLE IV: Terms in the O ( p ) meson-decuplet-decuplet and π ∆∆ chi-ral Lagrangians, where O ( N f , n is defined in Eqs. (63) and (65). O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3)¯ T abcµ u adµ u beν u cfλ γ γ ν T defλ T abcµ u adν f − deµλ D ν T bceλ + H . c .
15 30¯ T abcµ u adµ u beν u dfλ γ γ ν T cefλ T abcµ u adν f − deµλ D λ T bceν + H . c .
16 31¯ T abcµ u adµ u beν u dfλ γ γ λ T cefν + H . c . T abcµ u adν f − deνλ D λ T bceµ + H . c .
17 32¯ T abcµ u adµ u beν u efλ γ γ λ T cdfν + H . c . T abcµ u deµ f − edνλ D ν T abcλ + H . c . T abcµ u adν u beν u cfλ γ γ λ T defµ T abcµ u deν f − edµλ D ν T abcλ T abcµ u adµ u deν u efλ γ γ ν T bcfλ T abcµ u adµ h deνλ D ν T bceλ + H . c .
18 35¯ T abcµ u adν u beν u dfλ γ γ λ T cefµ + H . c . T abcµ u adν h deνλ D λ T bceµ + H . c .
19 36¯ T abcµ u adν u beλ u df µ γ γ λ T cefν T abcµ u adν h deλρ D νλρ T bceµ + H . c .
20 37¯ T abcµ u adµ u deν u efλ γ γ λ T bcfν + H . c . T abcµ ∇ ν f − adνλ γ γ λ T bcdµ
21 38¯ T abcµ u adν u beλ u df ν γ γ λ T cefµ i ¯ T abcµ f + adµν u beλ γ γ ν T cdeλ + H . c .
22 39¯ T abcµ u adµ u efν u feλ γ γ ν T bcdλ + H . c . i ¯ T abcµ f + adµν u beλ γ γ λ T cdeν
23 40¯ T abcµ u adν u deµ u efλ γ γ λ T bcfν + H . c . i ¯ T abcµ f + adµν u deλ γ γ ν T bceλ + H . c .
24 41¯ T abcµ u adν u deν u efλ γ γ λ T bcfµ + H . c . i ¯ T abcµ f + adµν u deλ γ γ λ T bceν + H . c .
25 42¯ T abcµ u adν u deλ u ef µ γ γ λ T bcfν i ¯ T abcµ f + adνλ u deµ γ γ ν T bceλ + H . c .
26 43¯ T abcµ u adν u deλ u ef ν γ γ λ T bcfµ i ¯ T abcµ f + adνλ u deν γ γ λ T bceµ + H . c .
27 44¯ T abcµ u adν u ef µ u feλ γ γ ν T bcdλ i ¯ T abcµ f + deµν u edλ γ γ ν T abcλ + H . c . T abcµ u adν u ef ν u feλ γ γ λ T bcdµ i ¯ T abcµ f + deµν u edλ γ γ λ T abcν T abcµ u adν u efλ u feλ γ γ ν T bcdµ i ¯ T abcµ f + adνλ u deρ γ γ ν D λρ T bceµ + H . c .
28 47¯ T abcµ u deµ u efν u fdλ γ γ ν T abcλ i ¯ T abcµ f s, + µν u adλ γ γ ν T bcdλ + H . c . T abcµ u adν u beλ u cfρ γ γ ν D λρ T defµ i ¯ T abcµ f s, + µν u adλ γ γ λ T bcdν T abcµ u adν u beλ u dfρ γ γ ν D λρ T cefµ + H . c .
10 21 i ¯ T abcµ ∇ ν f + adνλ D λ T bcdµ
31 48¯ T abcµ u adν u beλ u dfρ γ γ λ D νρ T cefµ
11 22 i ¯ T abcµ ∇ ν f s, + νλ D λ T abcµ T abcµ u adν u deλ u efρ γ γ ν D λρ T bcfµ + H . c .
23 ¯ T abcµ u adν χ + be γ γ ν T cdeµ
33 49¯ T abcµ u adν u deλ u efρ γ γ λ D νρ T bcfµ
24 ¯ T abcµ u adν χ + de γ γ ν T bceµ + H . c .
34 50¯ T abcµ u adν u efλ u feρ γ γ ν D λρ T bcdµ
25 ¯ T abcµ u deν χ + ed γ γ ν T abcµ T abcµ u adν u efλ u feρ γ γ λ D νρ T bcdµ
26 ¯ T abcµ u adν χ + ,s γ γ ν T bcdµ
35 52¯ T abcµ u adµ f − beνλ D ν T cdeλ + H . c .
12 27 i ¯ T abcµ u adν χ − de D ν T bceµ + H . c .
36 53¯ T abcµ u adν f − beµλ D ν T cdeλ
13 28 i ¯ T abcµ ∇ ν χ − ad γ γ ν T bcdµ
37 54¯ T abcµ u adµ f − deνλ D ν T bceλ + H . c .
14 29 i ¯ T abcµ ∇ ν χ − ,s γ γ ν T abcµ
38 55TABLE V: Terms in the O ( p ) meson-decuplet-decuplet and π ∆∆ chiralLagrangians, where O ( N f , n is defined in Eqs. (63) and (65). O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3)¯ T abcµ u adµ u beν u cf ν u dgλ T efgλ T abcµ h deνλ h edρσ D νλρσ T abcµ T abcµ u adµ u beν u cf ν u egλ T dfgλ + H . c . i ¯ T abcµ h adµν h deλρ σ νλ T bceρ
122 300¯ T abcµ u adµ u beν u cfλ u egν T dfgλ T abcµ u adµ ∇ ν f − beνλ T cdeλ + H . c .
123 301¯ T abcµ u adµ u beν u cfλ u egλ T dfgν + H . c . T abcµ u adν ∇ λ f − beνλ T cdeµ
124 302¯ T abcµ u adµ u beν u df ν u egλ T cfgλ T abcµ u adµ ∇ ν f − deνλ T bceλ + H . c .
125 303¯ T abcµ u adµ u beν u dfλ u egν T cfgλ T abcµ u adν ∇ λ f − deµλ T bceν + H . c .
126 304¯ T abcµ u adµ u beν u df ν u f gλ T cegλ T abcµ u adν ∇ λ f − deνλ T bceµ + H . c .
127 305¯ T abcµ u adµ u beν u dfλ u egλ T cfgν + H . c . T abcµ u deµ ∇ ν f − edνλ T abcλ + H . c . T abcµ u adµ u beν u dfλ u f gν T cegλ + H . c . T abcµ u deν ∇ λ f − edνλ T abcµ T abcµ u adµ u beν u ef ν u f gλ T cdgλ + H . c .
10 ¯ T abcµ u adν ∇ λ f − beλρ D νρ T cdeµ
128 308¯ T abcµ u adµ u beν u efλ u f gν T cdgλ + H . c .
11 ¯ T abcµ u adν ∇ λ f − deλρ D νρ T bceµ + H . c .
129 309¯ T abcµ u adµ u beν u efλ u f gλ T cdgν + H . c .
12 ¯ T abcµ u deν ∇ λ f − edλρ D νρ T abcµ T abcµ u adµ u beν u fgν u gf λ T cdeλ + H . c . i ¯ T abcµ f + adµν u beν u cfλ T defλ + H . c .
130 311¯ T abcµ u adν u beν u cfλ u f gµ T degλ i ¯ T abcµ f + adµν u beν u dfλ T cefλ + H . c .
131 312¯ T abcµ u adµ u beν u fgλ u gfλ T cdeν i ¯ T abcµ f + adµν u beν u efλ T cdfλ + H . c .
132 313¯ T abcµ u adν u beν u cfλ u dgλ T efgµ i ¯ T abcµ f + adµν u beλ u cf λ T defν
133 314¯ T abcµ u adν u beν u cfλ u f gλ T degµ i ¯ T abcµ f + adµν u beλ u df ν T cefλ + H . c .
134 315 O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3)¯ T abcµ u adν u beν u dfλ u f gµ T cegλ + H . c . i ¯ T abcµ f + adµν u beλ u df λ T cefν + H . c .
135 316¯ T abcµ u adµ u deν u ef ν u f gλ T bcgλ i ¯ T abcµ f + adµν u beλ u ef ν T cdfλ + H . c .
136 317¯ T abcµ u adν u beν u dfλ u egλ T cfgµ
10 20 i ¯ T abcµ f + adµν u beλ u ef λ T cdfν
137 318¯ T abcµ u adν u beλ u df µ u egν T cfgλ
11 21 i ¯ T abcµ f + adµν u deν u efλ T bcfλ + H . c . T abcµ u adν u beν u dfλ u f gλ T cegµ + H . c . i ¯ T abcµ f + adνλ u beµ u df ν T cefλ + H . c .
138 320¯ T abcµ u adµ u deν u efλ u f gν T bcgλ + H . c . i ¯ T abcµ f + adµν u deλ u ef ν T bcfλ + H . c . T abcµ u adν u beλ u df µ u egλ T cfgν i ¯ T abcµ f + adµν u deλ u ef λ T bcfν + H . c . T abcµ u adµ u deν u efλ u f gλ T bcgν + H . c . i ¯ T abcµ f + adνλ u beν u df µ T cefλ + H . c .
139 323¯ T abcµ u adν u beν u fgµ u gf λ T cdeλ i ¯ T abcµ f + adνλ u beν u df λ T cefµ + H . c .
140 324¯ T abcµ u adν u beλ u df ν u egλ T cfgµ
12 27 i ¯ T abcµ f + adµν u ef ν u feλ T bcdλ + H . c . T abcµ u adν u beλ u df λ u egν T cfgµ i ¯ T abcµ f + adνλ u beν u ef λ T cdfµ
141 326¯ T abcµ u adν u beλ u df λ u f gµ T cegν i ¯ T abcµ f + adνλ u deµ u ef ν T bcfλ + H . c . T abcµ u adν u beλ u df λ u f gν T cegµ i ¯ T abcµ f + adµν u efλ u feλ T bcdν T abcµ u adν u beν u fgλ u gfλ T cdeµ i ¯ T abcµ f + adνλ u deν u ef µ T bcfλ + H . c . T abcµ u adν u deµ u efλ u f gν T bcgλ i ¯ T abcµ f + adνλ u deν u ef λ T bcfµ + H . c . T abcµ u adν u deµ u efλ u f gλ T bcgν + H . c . i ¯ T abcµ f + deµν u adν u efλ T bcfλ + H . c . T abcµ u adν u deµ u fgν u gf λ T bceλ + H . c . i ¯ T abcµ f + deµν u adλ u ef ν T bcfλ + H . c . T abcµ u adν u beλ u fgν u gfλ T cdeµ i ¯ T abcµ f + deµν u adλ u ef λ T bcfν T abcµ u adν u deν u fgµ u gf λ T bceλ i ¯ T abcµ f + deµν u af ν u edλ T bcfλ + H . c . T abcµ u adν u deµ u fgλ u gfλ T bceν i ¯ T abcµ f + deµν u afλ u edν T bcfλ + H . c . T abcµ u adν u deν u efλ u f gλ T bcgµ i ¯ T abcµ f + deµν u afλ u edλ T bcfν T abcµ u adν u deλ u ef ν u f gλ T bcgµ i ¯ T abcµ f + deνλ u adν u ef λ T bcfµ T abcµ u adν u deν u fgλ u gfλ T bceµ i ¯ T abcµ f + deµν u ef ν u fdλ T abcλ + H . c . T abcµ u adν u ef µ u f gν u geλ T bcdλ i ¯ T abcµ f + deνλ u ef ν u fdλ T abcµ T abcµ u deµ u edν u fgν u gf λ T abcλ i ¯ T abcµ f + adµν u beλ u cfρ D νλ T defρ + H . c .
142 340¯ T abcµ u deµ u edν u fgλ u gfλ T abcν i ¯ T abcµ f + adµν u beλ u cfρ D λρ T defν
143 341¯ T abcµ u deν u edν u fgλ u gfλ T abcµ i ¯ T abcµ f + adµν u beλ u dfρ D νλ T cefρ + H . c .
144 342¯ T abcµ u deν u edλ u fgν u gfλ T abcµ i ¯ T abcµ f + adµν u beλ u dfρ D νρ T cefλ + H . c .
145 343¯ T abcµ u adµ u beν u cfλ u dgρ D νλ T efgρ
13 46 i ¯ T abcµ f + adµν u beλ u dfρ D λρ T cefν + H . c .
146 344¯ T abcµ u adµ u beν u cfλ u dgρ D νρ T efgλ + H . c .
14 47 i ¯ T abcµ f + adµν u beλ u efρ D νλ T cdfρ + H . c .
147 345¯ T abcµ u adµ u beν u cfλ u egρ D νρ T dfgλ
15 48 i ¯ T abcµ f + adµν u beλ u efρ D νρ T cdfλ + H . c .
148 346¯ T abcµ u adµ u beν u cfλ u egρ D λρ T dfgν + H . c .
16 49 i ¯ T abcµ f + adµν u beλ u efρ D λρ T cdfν
149 347¯ T abcµ u adµ u beν u dfλ u egρ D νλ T cfgρ
17 50 i ¯ T abcµ f + adµν u deλ u efρ D νλ T bcfρ + H . c . T abcµ u adµ u beν u dfλ u egρ D νρ T cfgλ i ¯ T abcµ f + adµν u deλ u efρ D νρ T bcfλ + H . c . T abcµ u adµ u beν u dfλ u egρ D λρ T cfgν + H . c .
18 52 i ¯ T abcµ f + adµν u deλ u efρ D λρ T bcfν + H . c . T abcµ u adµ u beν u dfλ u f gρ D νλ T cegρ i ¯ T abcµ f + adνλ u beµ u dfρ D νρ T cefλ + H . c .
150 351¯ T abcµ u adµ u beν u dfλ u f gρ D νρ T cegλ + H . c . i ¯ T abcµ f + adνλ u beν u dfρ D λρ T cefµ + H . c .
151 352¯ T abcµ u adµ u beν u dfλ u f gρ D λρ T cegν + H . c . i ¯ T abcµ f + adνλ u beν u efρ D λρ T cdfµ + H . c .
152 353¯ T abcµ u adµ u beν u efλ u f gρ D νρ T cdgλ + H . c . i ¯ T abcµ f + adνλ u beρ u df µ D νρ T cefλ + H . c .
153 354¯ T abcµ u adµ u beν u efλ u f gρ D λρ T cdgν + H . c . i ¯ T abcµ f + adµν u efλ u feρ D νλ T bcdρ + H . c . T abcµ u adµ u beν u fgλ u gf ρ D νλ T cdeρ + H . c . i ¯ T abcµ f + adµν u efλ u feρ D λρ T bcdν T abcµ u adµ u beν u fgλ u gf ρ D λρ T cdeν i ¯ T abcµ f + adνλ u beρ u df ν D λρ T cefµ + H . c .
154 357¯ T abcµ u adν u beν u cfλ u dgρ D λρ T efgµ + H . c .
19 60 i ¯ T abcµ f + adνλ u deµ u efρ D νρ T bcfλ + H . c . T abcµ u adν u beν u cfλ u f gρ D λρ T degµ
20 61 i ¯ T abcµ f + adνλ u deν u efρ D λρ T bcfµ + H . c . T abcµ u adµ u deν u efλ u f gρ D νλ T bcgρ i ¯ T abcµ f + adνλ u deρ u ef µ D νρ T bcfλ + H . c . T abcµ u adµ u deν u efλ u f gρ D νρ T bcgλ + H . c . i ¯ T abcµ f + adνλ u deρ u ef ν D λρ T bcfµ + H . c . T abcµ u adµ u deν u efλ u f gρ D λρ T bcgν + H . c . i ¯ T abcµ f + deµν u adλ u efρ D νλ T bcfρ + H . c . T abcµ u adν u beν u dfλ u egρ D λρ T cfgµ + H . c .
21 65 i ¯ T abcµ f + deµν u adλ u efρ D νρ T bcfλ + H . c . T abcµ u adν u beν u dfλ u f gρ D λρ T cegµ + H . c . i ¯ T abcµ f + deµν u adλ u efρ D λρ T bcfν T abcµ u adν u beλ u df µ u egρ D νρ T cfgλ
22 67 i ¯ T abcµ f + deµν u afλ u edρ D νλ T bcfρ + H . c . T abcµ u adν u beλ u df µ u egρ D λρ T cfgν
23 68 i ¯ T abcµ f + deµν u afλ u edρ D νρ T bcfλ + H . c . T abcµ u adν u beλ u df µ u f gρ D λρ T cegν + H . c . i ¯ T abcµ f + deµν u afλ u edρ D λρ T bcfν T abcµ u adν u beλ u cfρ u dgµ D λρ T efgν i ¯ T abcµ f + deνλ u adν u efρ D λρ T bcfµ + H . c . T abcµ u adν u beλ u cfρ u dgν D λρ T efgµ
24 71 i ¯ T abcµ f + deµν u efλ u fdρ D νλ T abcρ + H . c . T abcµ u adν u beλ u df ν u egρ D λρ T cfgµ
25 72 i ¯ T abcµ f + deνλ u ef ν u fdρ D λρ T abcµ + H . c . T abcµ u adν u beλ u df ν u f gρ D λρ T cegµ + H . c .
73 ¯ T abcµ f + adµν u beλ u cfρ σ νλ T defρ + H . c .
155 371¯ T abcµ u adν u beλ u df λ u egρ D νρ T cfgµ
74 ¯ T abcµ f + adµν u beλ u dfρ σ νλ T cefρ + H . c .
156 372 O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3)¯ T abcµ u adν u beν u fgλ u gf ρ D λρ T cdeµ
75 ¯ T abcµ f + adµν u beλ u dfρ σ νρ T cefλ + H . c .
157 373¯ T abcµ u adν u beλ u df λ u f gρ D νρ T cegµ
76 ¯ T abcµ f + adµν u beλ u efρ σ νλ T cdfρ + H . c .
158 374¯ T abcµ u adν u beλ u dfρ u f gµ D λρ T cegν
77 ¯ T abcµ f + adµν u beλ u efρ σ νρ T cdfλ + H . c .
159 375¯ T abcµ u adν u beλ u dfρ u f gν D λρ T cegµ i ¯ T abcµ f s, + µν u adν u beλ T cdeλ + H . c . T abcµ u adν u deµ u efλ u f gρ D νρ T bcgλ i ¯ T abcµ f s, + µν u adλ u beλ T cdeν T abcµ u adν u deµ u efλ u f gρ D λρ T bcgν + H . c . i ¯ T abcµ f s, + µν u adν u deλ T bceλ + H . c . T abcµ u adν u beλ u fgµ u gf ρ D νλ T cdeρ i ¯ T abcµ f s, + µν u adλ u deν T bceλ + H . c . T abcµ u adν u beλ u fgν u gf ρ D λρ T cdeµ i ¯ T abcµ f s, + µν u adλ u deλ T bceν T abcµ u adν u deµ u fgλ u gf ρ D νλ T bceρ + H . c . i ¯ T abcµ f s, + νλ u adν u deλ T bceµ T abcµ u adν u deµ u fgλ u gf ρ D λρ T bceν i ¯ T abcµ f s, + µν u adλ u beρ D νλ T cdeρ + H . c . T abcµ u adν u deν u efλ u f gρ D λρ T bcgµ + H . c . i ¯ T abcµ f s, + µν u adλ u beρ D λρ T cdeν T abcµ u adν u deλ u ef µ u f gρ D νρ T bcgλ i ¯ T abcµ f s, + µν u adλ u deρ D νλ T bceρ + H . c . T abcµ u adν u beλ u fgρ u gfρ D νλ T cdeµ i ¯ T abcµ f s, + µν u adλ u deρ D νρ T bceλ + H . c . T abcµ u adν u deν u fgλ u gf ρ D λρ T bceµ i ¯ T abcµ f s, + µν u adλ u deρ D λρ T bceν T abcµ u adν u deλ u ef ν u f gρ D λρ T bcgµ + H . c . i ¯ T abcµ f s, + νλ u adν u deρ D λρ T bceµ + H . c . T abcµ u adν u deλ u ef λ u f gρ D νρ T bcgµ
90 ¯ T abcµ f s, + µν u adλ u beρ σ νλ T cdeρ + H . c . T abcµ u adν u ef µ u f gλ u geρ D νλ T bcdρ
91 ¯ T abcµ f s, + µν u adλ u deρ σ νλ T bceρ + H . c . T abcµ u deµ u edν u fgλ u gf ρ D νλ T abcρ
92 ¯ T abcµ f s, + µν u adλ u deρ σ νρ T bceλ + H . c . T abcµ u deµ u edν u fgλ u gf ρ D λρ T abcν
93 ¯ T abcµ f + adµν u deλ u efρ σ νλ T bcfρ + H . c . T abcµ u deν u edν u fgλ u gf ρ D λρ T abcµ
94 ¯ T abcµ f + adµν u deλ u efρ σ νρ T bcfλ + H . c . T abcµ u deν u edλ u fgν u gf ρ D λρ T abcµ
95 ¯ T abcµ f + adµν u efλ u feρ σ νλ T bcdρ + H . c . T abcµ u adν u beλ u cfρ u dgσ D νλρσ T efgµ
26 96 ¯ T abcµ f + deµν u adλ u efρ σ νλ T bcfρ + H . c . T abcµ u adν u beλ u dfρ u egσ D νλρσ T cfgµ
27 97 ¯ T abcµ f + deµν u adλ u efρ σ νρ T bcfλ + H . c . T abcµ u adν u beλ u dfρ u f gσ D νλρσ T cegµ
98 ¯ T abcµ f + deµν u afλ u edρ σ νλ T bcfρ + H . c . T abcµ u adν u beλ u fgρ u gf σ D νλρσ T cdeµ
99 ¯ T abcµ f + deµν u efλ u fdρ σ νλ T abcρ + H . c . T abcµ u adν u deλ u efρ u f gσ D νλρσ T bcgµ i ¯ T abcµ f + adµν f − beλρ γ γ ν D λ T cdeρ + H . c .
175 383¯ T abcµ u deν u edλ u fgρ u gf σ D νλρσ T abcµ i ¯ T abcµ f + adµν f − beλρ γ γ λ D ν T cdeρ + H . c .
176 384 i ¯ T abcµ u adµ u beν u cfλ u dgρ σ νρ T efgλ + H . c .
28 102 i ¯ T abcµ f + adνλ f − beνρ γ γ λ D ρ T cdeµ
177 385 i ¯ T abcµ u adµ u beν u cfλ u egρ σ νρ T dfgλ
29 103 i ¯ T abcµ f + adνλ f − beνρ γ γ ρ D λ T cdeµ
178 386 i ¯ T abcµ u adµ u beν u dfλ u egρ σ νλ T cfgρ
30 104 i ¯ T abcµ f + adµν f − deλρ γ γ ν D λ T bceρ + H . c .
179 387 i ¯ T abcµ u adµ u beν u dfλ u egρ σ νρ T cfgλ i ¯ T abcµ f + adµν f − deλρ γ γ λ D ν T bceρ + H . c .
180 388 i ¯ T abcµ u adµ u beν u dfλ u f gρ σ νρ T cegλ + H . c . i ¯ T abcµ f + adµν f − deλρ γ γ λ D ρ T bceν + H . c .
181 389 i ¯ T abcµ u adµ u beν u efλ u f gρ σ νρ T cdgλ + H . c . i ¯ T abcµ f + adνλ f − deµρ γ γ ν D λ T bceρ + H . c .
182 390 i ¯ T abcµ u adµ u beν u fgλ u gf ρ σ νλ T cdeρ + H . c . i ¯ T abcµ f + adνλ f − deµρ γ γ ν D ρ T bceλ + H . c .
183 391 i ¯ T abcµ u adν u beλ u df µ u egρ σ νρ T cfgλ
31 109 i ¯ T abcµ f + adνλ f − deµρ γ γ ρ D ν T bceλ + H . c .
184 392 i ¯ T abcµ u adµ u deν u efλ u f gρ σ νλ T bcgρ i ¯ T abcµ f + adνλ f − deνρ γ γ λ D ρ T bceµ + H . c .
185 393 i ¯ T abcµ u adµ u deν u efλ u f gρ σ νρ T bcgλ + H . c . i ¯ T abcµ f + adνλ f − deνρ γ γ ρ D λ T bceµ + H . c .
186 394 i ¯ T abcµ u adν u deµ u efλ u f gρ σ νρ T bcgλ i ¯ T abcµ f s, + µν f − adλρ γ γ ν D λ T bcdρ + H . c . i ¯ T abcµ u adν u deµ u fgλ u gf ρ σ νλ T bceρ + H . c . i ¯ T abcµ f s, + µν f − adλρ γ γ λ D ν T bcdρ + H . c . i ¯ T abcµ u deµ u edν u fgλ u gf ρ σ νλ T abcρ i ¯ T abcµ f s, + νλ f − adνρ γ γ λ D ρ T bcdµ T abcµ u adµ u beν f − cfλρ γ γ ν D λ T defρ + H . c .
32 115 i ¯ T abcµ f s, + νλ f − adνρ γ γ ρ D λ T bcdµ T abcµ u adµ u beν f − cfλρ γ γ λ D ν T defρ + H . c .
33 116 i ¯ T abcµ f + deµν f − edλρ γ γ ν D λ T abcρ + H . c . T abcµ u adµ u beν f − dfλρ γ γ ν D λ T cefρ + H . c .
34 117 i ¯ T abcµ f + deµν f − edλρ γ γ λ D ν T abcρ + H . c . T abcµ u adµ u beν f − dfλρ γ γ λ D ν T cefρ + H . c .
35 118 i ¯ T abcµ f + deνλ f − edνρ γ γ λ D ρ T abcµ T abcµ u adµ u beν f − dfλρ γ γ λ D ρ T cefν + H . c .
36 119 i ¯ T abcµ f + deνλ f − edνρ γ γ ρ D λ T abcµ T abcµ u adµ u beν f − efλρ γ γ ν D λ T cdfρ + H . c .
37 120 i ¯ T abcµ f + adµν h beλρ γ γ ν D λ T cdeρ + H . c .
191 399¯ T abcµ u adµ u beν f − efλρ γ γ λ D ν T cdfρ + H . c .
38 121 i ¯ T abcµ f + adµν h beλρ γ γ λ D ν T cdeρ + H . c .
192 400¯ T abcµ u adµ u beν f − efλρ γ γ λ D ρ T cdfν + H . c .
39 122 i ¯ T abcµ f + adνλ h beµρ γ γ ν D λ T cdeρ
193 401¯ T abcµ u adµ u deν f − bfλρ γ γ ν D λ T cefρ + H . c .
40 123 i ¯ T abcµ f + adνλ h beνρ γ γ λ D ρ T cdeµ
194 402¯ T abcµ u adµ u deν f − bfλρ γ γ λ D ν T cefρ + H . c .
41 124 i ¯ T abcµ f + adνλ h beνρ γ γ ρ D λ T cdeµ
195 403¯ T abcµ u adν u beλ f − cf µρ γ γ ν D λ T defρ
42 125 i ¯ T abcµ f + adµν h deλρ γ γ ν D λ T bceρ + H . c .
196 404¯ T abcµ u adµ u deν f − efλρ γ γ ν D λ T bcfρ + H . c . i ¯ T abcµ f + adµν h deλρ γ γ λ D ν T bceρ + H . c .
197 405¯ T abcµ u adµ u deν f − efλρ γ γ λ D ν T bcfρ + H . c . i ¯ T abcµ f + adµν h deλρ γ γ λ D ρ T bceν + H . c .
198 406¯ T abcµ u adµ u deν f − efλρ γ γ λ D ρ T bcfν + H . c . i ¯ T abcµ f + adνλ h deµρ γ γ ν D λ T bceρ + H . c .
199 407¯ T abcµ u adν u beλ f − df µρ γ γ ν D λ T cefρ + H . c .
43 129 i ¯ T abcµ f + adνλ h deµρ γ γ ν D ρ T bceλ + H . c .
200 408¯ T abcµ u adν u beλ f − df µρ γ γ ν D ρ T cefλ + H . c .
44 130 i ¯ T abcµ f + adνλ h deνρ γ γ λ D ρ T bceµ + H . c .
201 409¯ T abcµ u adν u deµ f − bfλρ γ γ ν D λ T cefρ + H . c .
45 131 i ¯ T abcµ f + adνλ h deνρ γ γ ρ D λ T bceµ + H . c .
202 410 O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3)¯ T abcµ u adν u beν f − dfλρ γ γ λ D ρ T cefµ + H . c .
46 132 i ¯ T abcµ f + deµν h edλρ γ γ ν D λ T abcρ + H . c . T abcµ u adµ u efν f − beλρ γ γ λ D ν T cdfρ + H . c .
47 133 i ¯ T abcµ f + deµν h edλρ γ γ λ D ν T abcρ + H . c . T abcµ u adν u beλ f − df µρ γ γ λ D ν T cefρ + H . c .
48 134 i ¯ T abcµ f + deνλ h edµρ γ γ ν D λ T abcρ T abcµ u adν u beλ f − df µρ γ γ λ D ρ T cefν + H . c .
49 135 i ¯ T abcµ f + deνλ h edνρ γ γ λ D ρ T abcµ T abcµ u adν u deµ f − bfλρ γ γ λ D ν T cefρ + H . c .
50 136 i ¯ T abcµ f + deνλ h edνρ γ γ ρ D λ T abcµ T abcµ u adν u beλ f − df µρ γ γ ρ D λ T cefν + H . c .
51 137 i ¯ T abcµ f + adνλ h beρσ γ γ ν D λρσ T cdeµ
203 416¯ T abcµ u adν u beλ f − df νρ γ γ λ D ρ T cefµ + H . c .
52 138 i ¯ T abcµ f + adνλ h deρσ γ γ ν D λρσ T bceµ + H . c .
204 417¯ T abcµ u adµ u efν f − deλρ γ γ ν D λ T bcfρ + H . c . i ¯ T abcµ f s, + µν h adλρ γ γ ν D λ T bcdρ + H . c . T abcµ u adν u beλ f − df νρ γ γ ρ D λ T cefµ + H . c .
53 140 i ¯ T abcµ f s, + µν h adλρ γ γ λ D ν T bcdρ + H . c . T abcµ u adν u beλ f − df λρ γ γ ν D ρ T cefµ + H . c .
54 141 i ¯ T abcµ f s, + νλ h adµρ γ γ ν D λ T bcdρ T abcµ u adµ u efν f − deλρ γ γ λ D ν T bcfρ + H . c . i ¯ T abcµ f s, + νλ h adνρ γ γ λ D ρ T bcdµ T abcµ u adν u deλ f − bf µρ γ γ ν D λ T cefρ + H . c .
55 143 i ¯ T abcµ f s, + νλ h adνρ γ γ ρ D λ T bcdµ T abcµ u adν u deµ f − efλρ γ γ ν D λ T bcfρ + H . c . i ¯ T abcµ f s, + νλ h adρσ γ γ ν D λρσ T bcdµ T abcµ u adν u beλ f − df λρ γ γ ρ D ν T cefµ + H . c .
56 145 i ¯ T abcµ f + deνλ h edρσ γ γ ν D λρσ T abcµ T abcµ u adµ u efν f − feλρ γ γ ν D λ T bcdρ + H . c . i ¯ T abcµ ∇ ν f + adνλ u beρ γ γ λ D ρ T cdeµ
211 419¯ T abcµ u adµ u efν f − feλρ γ γ λ D ν T bcdρ + H . c . i ¯ T abcµ ∇ ν f + adνλ u beρ γ γ ρ D λ T cdeµ
212 420¯ T abcµ u adµ u efν f − feλρ γ γ λ D ρ T bcdν + H . c . i ¯ T abcµ ∇ ν f + adλρ u beν γ γ λ D ρ T cdeµ
213 421¯ T abcµ u adν u deµ f − efλρ γ γ λ D ν T bcfρ + H . c . i ¯ T abcµ ∇ ν f + adνλ u deρ γ γ λ D ρ T bceµ + H . c .
214 422¯ T abcµ u adν u deµ f − efλρ γ γ λ D ρ T bcfν + H . c . i ¯ T abcµ ∇ ν f + adνλ u deρ γ γ ρ D λ T bceµ + H . c .
215 423¯ T abcµ u adν u deλ f − bf νρ γ γ λ D ρ T cefµ + H . c .
57 151 i ¯ T abcµ ∇ ν f + adλρ u deν γ γ λ D ρ T bceµ + H . c .
216 424¯ T abcµ u adν u deλ f − bf νρ γ γ ρ D λ T cefµ + H . c .
58 152 i ¯ T abcµ ∇ ν f s, + νλ u adρ γ γ λ D ρ T bcdµ T abcµ u adν u ef µ f − deλρ γ γ ν D λ T bcfρ + H . c . i ¯ T abcµ ∇ ν f s, + νλ u adρ γ γ ρ D λ T bcdµ T abcµ u adν u deλ f − ef µρ γ γ ν D λ T bcfρ + H . c . i ¯ T abcµ ∇ ν f s, + λρ u adν γ γ λ D ρ T bcdµ T abcµ u adν u deλ f − ef µρ γ γ ν D ρ T bcfλ + H . c . i ¯ T abcµ ∇ ν f + deνλ u edρ γ γ λ D ρ T abcµ T abcµ u adν u deν f − efλρ γ γ λ D ρ T bcfµ + H . c . i ¯ T abcµ ∇ ν f + deνλ u edρ γ γ ρ D λ T abcµ T abcµ u adν u deλ f − ef µρ γ γ λ D ν T bcfρ + H . c . i ¯ T abcµ ∇ ν f + deλρ u edν γ γ λ D ρ T abcµ T abcµ u adν u deλ f − ef µρ γ γ λ D ρ T bcfν + H . c . iε µνλρ ¯ T abcσ f + adµσ f − beνλ T cdeρ + H . c .
220 428¯ T abcµ u adν u ef µ f − deλρ γ γ λ D ν T bcfρ + H . c . iε µνλρ ¯ T abcσ f + adµσ f − deνλ T bceρ + H . c .
221 429¯ T abcµ u adν u deλ f − ef µρ γ γ ρ D ν T bcfλ + H . c . iε µνλρ ¯ T abcσ f + adµν f − deλσ T bceρ + H . c .
222 430¯ T abcµ u adν u deλ f − ef µρ γ γ ρ D λ T bcfν + H . c . iε µνλρ ¯ T abcµ f + adνσ f − deλσ T bceρ + H . c .
223 431¯ T abcµ u adν u deλ f − ef νρ γ γ λ D ρ T bcfµ + H . c . iε µνλρ ¯ T abcσ f s, + µσ f − adνλ T bcdρ + H . c . T abcµ u adν u ef µ f − feλρ γ γ ν D λ T bcdρ + H . c . iε µνλρ ¯ T abcσ f + deµσ f − edνλ T abcρ + H . c . T abcµ u adν u deλ f − ef νρ γ γ ρ D λ T bcfµ + H . c . iε µνλρ ¯ T abcσ f + adµν h beλσ T cdeρ + H . c .
225 433¯ T abcµ u adν u deλ f − ef λρ γ γ ν D ρ T bcfµ + H . c . iε µνλρ ¯ T abcσ f + adµν h deλσ T bceρ + H . c .
226 434¯ T abcµ u adν u ef µ f − feλρ γ γ λ D ν T bcdρ + H . c . iε µνλρ ¯ T abcµ f + adνσ h deλσ T bceρ + H . c .
227 435¯ T abcµ u adν u deλ f − ef λρ γ γ ρ D ν T bcfµ + H . c . iε µνλρ ¯ T abcσ f s, + µν h adλσ T bcdρ + H . c . T abcµ u adν u efλ f − deµρ γ γ ν D λ T bcfρ + H . c . iε µνλρ ¯ T abcσ f + deµν h edλσ T abcρ + H . c . T abcµ u adν u efλ f − deνρ γ γ λ D ρ T bcfµ + H . c . iε µνλρ ¯ T abcµ ∇ σ f + adνσ u beλ T cdeρ
229 437¯ T abcµ u adν u efλ f − deνρ γ γ ρ D λ T bcfµ + H . c . iε µνλρ ¯ T abcµ ∇ σ f + adνσ u deλ T bceρ + H . c .
230 438¯ T abcµ u adν u efλ f − feµρ γ γ ν D λ T bcdρ iε µνλρ ¯ T abcµ ∇ σ f s, + νσ u adλ T bcdρ T abcµ u adν u efλ f − feµρ γ γ λ D ν T bcdρ iε µνλρ ¯ T abcµ ∇ σ f + deνσ u edλ T abcρ T abcµ u deµ u edν f − afλρ γ γ ν D λ T bcfρ + H . c .
173 ¯ T abcµ f + adµν f + beνλ T cdeλ
232 440¯ T abcµ u deµ u edν f − afλρ γ γ λ D ν T bcfρ + H . c .
174 ¯ T abcµ f + adνλ f + beνλ T cdeµ
233 441¯ T abcµ u deµ u f dν f − ef λρ γ γ ν D λ T abcρ + H . c .
175 ¯ T abcµ f + adµν f + deνλ T bceλ
234 442¯ T abcµ u deµ u f dν f − ef λρ γ γ λ D ν T abcρ + H . c .
176 ¯ T abcµ f + adνλ f + deµν T bceλ T abcµ u deν u edλ f − af µρ γ γ ν D λ T bcfρ
177 ¯ T abcµ f + adνλ f + deνλ T bceµ T abcµ u deν u efλ f − fdνρ γ γ λ D ρ T abcµ + H . c .
178 ¯ T abcµ f + adµν f + beλρ D νλ T cdeρ
235 445¯ T abcµ u deν u efλ f − fdνρ γ γ ρ D λ T abcµ + H . c .
179 ¯ T abcµ f + adνλ f + beνρ D λρ T cdeµ
236 446¯ T abcµ u adν u beλ f − dfρσ γ γ ρ D νλσ T cefµ + H . c .
59 180 ¯ T abcµ f + adµν f + deλρ D νλ T bceρ
237 447¯ T abcµ u adν u deλ f − efρσ γ γ ρ D νλσ T bcfµ + H . c .
181 ¯ T abcµ f + adνλ f + deµρ D νρ T bceλ T abcµ u adµ u beν h cfλρ γ γ ν D λ T defρ + H . c .
60 182 ¯ T abcµ f + adνλ f + deνρ D λρ T bceµ T abcµ u adµ u beν h dfλρ γ γ ν D λ T cefρ + H . c .
61 183 i ¯ T abcµ f + adµν f + beλρ σ νλ T cdeρ
238 450¯ T abcµ u adµ u beν h dfλρ γ γ λ D ν T cefρ + H . c .
62 184 i ¯ T abcµ f + adµν f + deλρ σ νλ T bceρ
239 451¯ T abcµ u adµ u beν h dfλρ γ γ λ D ρ T cefν + H . c .
63 185 ¯ T abcµ f + adµν f s, + νλ T bcdλ + H . c . T abcµ u adµ u beν h efλρ γ γ ν D λ T cdfρ + H . c .
64 186 ¯ T abcµ f + adνλ f s, + νλ T bcdµ T abcµ u adµ u beν h efλρ γ γ λ D ν T cdfρ + H . c .
65 187 ¯ T abcµ f + adµν f s, + λρ D νλ T bcdρ + H . c . O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3)¯ T abcµ u adµ u beν h efλρ γ γ λ D ρ T cdfν + H . c .
66 188 ¯ T abcµ f + adνλ f s, + νρ D λρ T bcdµ T abcµ u adµ u deν h bfλρ γ γ λ D ν T cefρ + H . c .
67 189 i ¯ T abcµ f + adµν f s, + λρ σ νλ T bcdρ + H . c . T abcµ u adµ u deν h efλρ γ γ ν D λ T bcfρ + H . c .
190 ¯ T abcµ f s, + µν f s, + νλ T abcλ T abcµ u adµ u deν h efλρ γ γ λ D ν T bcfρ + H . c .
191 ¯ T abcµ f s, + νλ f s, + νλ T abcµ T abcµ u adµ u deν h efλρ γ γ λ D ρ T bcfν + H . c .
192 ¯ T abcµ f s, + µν f s, + λρ D νλ T abcρ T abcµ u adν u beλ h df µρ γ γ ν D ρ T cefλ + H . c .
68 193 ¯ T abcµ f s, + νλ f s, + νρ D λρ T abcµ T abcµ u adν u beν h dfλρ γ γ λ D ρ T cefµ + H . c .
69 194 i ¯ T abcµ f s, + µν f s, + λρ σ νλ T abcρ T abcµ u adν u beλ h df µρ γ γ λ D ρ T cefν + H . c .
70 195 i ¯ T abcµ f + deµν f + edλρ σ νλ T abcρ T abcµ u adν u beλ h df νρ γ γ λ D ρ T cefµ + H . c .
71 196 ¯ T abcµ u adµ u beν χ + cf T defν
250 453¯ T abcµ u adµ u efν h deλρ γ γ ν D λ T bcfρ + H . c .
197 ¯ T abcµ u adν u beν χ + cf T defµ
251 454¯ T abcµ u adν u beλ h df νρ γ γ ρ D λ T cefµ + H . c .
72 198 ¯ T abcµ u adµ u beν χ + df T cefν + H . c .
252 455¯ T abcµ u adν u beλ h df λρ γ γ ν D ρ T cefµ + H . c .
73 199 ¯ T abcµ u adµ u beν χ + ef T cdfν + H . c .
253 456¯ T abcµ u adµ u efν h deλρ γ γ λ D ν T bcfρ + H . c .
200 ¯ T abcµ u adν u beν χ + df T cefµ + H . c .
254 457¯ T abcµ u adν u beλ h df λρ γ γ ρ D ν T cefµ + H . c .
74 201 ¯ T abcµ u adµ u deν χ + bf T cefν
255 458¯ T abcµ u adµ u efν h feλρ γ γ ν D λ T bcdρ + H . c .
202 ¯ T abcµ u adν u deµ χ + bf T cefν
256 459¯ T abcµ u adµ u efν h feλρ γ γ λ D ρ T bcdν + H . c .
203 ¯ T abcµ u adν u deν χ + bf T cefµ
257 460¯ T abcµ u adν u deµ h efλρ γ γ λ D ν T bcfρ + H . c .
204 ¯ T abcµ u adµ u deν χ + ef T bcfν + H . c . T abcµ u adν u deµ h efλρ γ γ λ D ρ T bcfν + H . c .
205 ¯ T abcµ u adν u deµ χ + ef T bcfν + H . c . T abcµ u adν u deλ h bf νρ γ γ ρ D λ T cefµ + H . c .
75 206 ¯ T abcµ u adν u deν χ + ef T bcfµ + H . c . T abcµ u adν u ef µ h deλρ γ γ ν D λ T bcfρ + H . c .
207 ¯ T abcµ u adµ u efν χ + de T bcfν T abcµ u adν u deν h efλρ γ γ λ D ρ T bcfµ + H . c .
208 ¯ T abcµ u adν u ef µ χ + de T bcfν T abcµ u adν u deλ h ef µρ γ γ λ D ρ T bcfν + H . c .
209 ¯ T abcµ u adν u ef ν χ + de T bcfµ T abcµ u adν u deλ h ef νρ γ γ λ D ρ T bcfµ + H . c .
210 ¯ T abcµ u adµ u efν χ + fe T bcdν + H . c . T abcµ u adν u ef µ h feλρ γ γ ν D λ T bcdρ + H . c .
211 ¯ T abcµ u adν u ef ν χ + fe T bcdµ T abcµ u adν u deλ h ef νρ γ γ ρ D λ T bcfµ + H . c .
212 ¯ T abcµ u deµ u edν χ + af T bcfν T abcµ u adν u deλ h ef λρ γ γ ν D ρ T bcfµ + H . c .
213 ¯ T abcµ u deν u edν χ + af T bcfµ T abcµ u adν u deλ h ef λρ γ γ ρ D ν T bcfµ + H . c .
214 ¯ T abcµ u deµ u efν χ + fd T abcν T abcµ u adν u efλ h deνρ γ γ ρ D λ T bcfµ + H . c .
215 ¯ T abcµ u adν u beλ χ + cf D νλ T defµ
258 472¯ T abcµ u deµ u f dν h ef λρ γ γ ν D λ T abcρ + H . c .
216 ¯ T abcµ u adν u beλ χ + df D νλ T cefµ + H . c .
259 473¯ T abcµ u deµ u f dν h ef λρ γ γ λ D ν T abcρ + H . c .
217 ¯ T abcµ u adν u deλ χ + bf D νλ T cefµ
260 474¯ T abcµ u deν u efλ h fdνρ γ γ λ D ρ T abcµ + H . c .
218 ¯ T abcµ u adν u deλ χ + ef D νλ T bcfµ + H . c . T abcµ u deν u efλ h fdνρ γ γ ρ D λ T abcµ + H . c .
219 ¯ T abcµ u adν u efλ χ + de D νλ T bcfµ T abcµ u adν u beλ h dfρσ γ γ ν D λρσ T cefµ + H . c .
76 220 ¯ T abcµ u adν u efλ χ + fe D νλ T bcdµ T abcµ u adν u beλ h dfρσ γ γ λ D νρσ T cefµ + H . c .
77 221 ¯ T abcµ u deν u edλ χ + af D νλ T bcfµ T abcµ u adν u beλ h dfρσ γ γ ρ D νλσ T cefµ + H . c .
78 222 ¯ T abcµ u adµ u beν χ + ,s T cdeν
261 479¯ T abcµ u adν u deλ h efρσ γ γ ν D λρσ T bcfµ + H . c .
223 ¯ T abcµ u adν u beν χ + ,s T cdeµ
262 480¯ T abcµ u adν u deλ h efρσ γ γ λ D νρσ T bcfµ + H . c .
224 ¯ T abcµ u adµ u deν χ + ,s T bceν
263 481¯ T abcµ u adν u deλ h efρσ γ γ ρ D νλσ T bcfµ + H . c .
225 ¯ T abcµ u adν u deµ χ + ,s T bceν
264 482¯ T abcµ u deν u efλ h fdρσ γ γ ν D λρσ T abcµ + H . c .
226 ¯ T abcµ u adν u deν χ + ,s T bceµ
265 483 ε µνλρ ¯ T abcµ u adν u beλ f − df ρσ T cefσ + H . c .
79 227 ¯ T abcµ u deµ u edν χ + ,s T abcν ε µνλρ ¯ T abcµ u adν u deλ f − bf ρσ T cefσ + H . c .
80 228 ¯ T abcµ u deν u edν χ + ,s T abcµ ε µνλρ ¯ T abcµ u adν u deλ f − ef ρσ T bcfσ + H . c .
229 ¯ T abcµ u adν u beλ χ + ,s D νλ T cdeµ
266 486 ε µνλρ ¯ T abcµ u adν u ef λ f − beρσ T cdfσ + H . c .
81 230 ¯ T abcµ u adν u deλ χ + ,s D νλ T bceµ
267 487 ε µνλρ ¯ T abcµ u adν u beσ f − cf λρ T defσ + H . c .
82 231 ¯ T abcµ u deν u edλ χ + ,s D νλ T abcµ ε µνλρ ¯ T abcµ u adν u ef λ f − deρσ T bcfσ + H . c .
232 ¯ T abcµ f − adνλ χ + de γ γ ν D λ T bceµ + H . c .
268 489 ε µνλρ ¯ T abcµ u adν u beσ f − df λρ T cefσ + H . c .
83 233 ¯ T abcµ h adνλ χ + de γ γ ν D λ T bceµ + H . c .
269 490 ε µνλρ ¯ T abcµ u adν u beσ f − ef λρ T cdfσ + H . c .
84 234 ¯ T abcµ u adν ∇ λ χ + de γ γ λ D ν T bceµ + H . c .
270 491 ε µνλρ ¯ T abcµ u adν u ef λ f − feρσ T bcdσ + H . c . ε µνλρ ¯ T abcµ f − adνλ χ + de T bceρ + H . c .
271 492 ε µνλρ ¯ T abcµ u adν u beσ f − df λσ T cefρ + H . c .
85 236 ε µνλρ ¯ T abcµ u adν ∇ λ χ + de T bceρ + H . c .
272 493 ε µνλρ ¯ T abcµ u adν u deσ f − bf λρ T cefσ + H . c .
86 237 ¯ T abcµ ∇ ν ∇ ν χ + ad T bcdµ
273 494 ε µνλρ ¯ T abcµ u adν u deσ f − ef λρ T bcfσ + H . c .
238 ¯ T abcµ ∇ ν ∇ ν χ + ,s T abcµ
274 495 ε µνλρ ¯ T abcµ u adν u deσ f − ef λσ T bcfρ + H . c . i ¯ T abcµ f + adµν χ + be T cdeν
275 496 ε µνλρ ¯ T abcµ u adν u efσ f − beλρ T cdfσ + H . c .
87 240 i ¯ T abcµ f + adµν χ + de T bceν + H . c .
276 497 ε µνλρ ¯ T abcµ u adν u efσ f − deλρ T bcfσ + H . c . i ¯ T abcµ f s, + µν χ + ad T bcdν ε µνλρ ¯ T abcµ u adν u efσ f − feλρ T bcdσ + H . c . i ¯ T abcµ f + deµν χ + ed T abcν ε µνλρ ¯ T abcµ u deν u ef λ f − adρσ T bcfσ + H . c . i ¯ T abcµ f + adµν χ + ,s T bcdν
278 499 ε µνλρ ¯ T abcµ u deν u ef λ f − fdρσ T abcσ + H . c . i ¯ T abcµ f s, + µν χ + ,s T abcν ε µνλρ ¯ T abcµ u adσ u deν f − ef λρ T bcfσ + H . c .
245 ¯ T abcµ χ + ad χ + be T cdeµ
280 500 O ( N f , n SU (2) SU (3) O ( N f , n SU (2) SU (3) ε µνλρ ¯ T abcµ u deν u edσ f − af λρ T bcfσ + H . c .
246 ¯ T abcµ χ + ad χ + de T bceµ
281 501 ε µνλρ ¯ T abcµ u deν u efσ f − adλρ T bcfσ + H . c .
247 ¯ T abcµ χ + de χ + ed T abcµ ε µνλρ ¯ T abcµ u adσ u ef ν f − feλρ T bcdσ + H . c .
248 ¯ T abcµ χ + ad χ + ,s T bcdµ
282 503 ε µνλρ ¯ T abcµ u adν u beλ h df ρσ T cefσ + H . c .
88 249 ¯ T abcµ χ + ,s χ + ,s T abcµ
283 504 ε µνλρ ¯ T abcµ u adν u deλ h bf ρσ T cefσ + H . c .
89 250 i ¯ T abcµ u adν u beλ χ − df γ γ ν D λ T cefµ + H . c .
284 505 ε µνλρ ¯ T abcµ u adν u deλ h ef ρσ T bcfσ + H . c . i ¯ T abcµ u adν u beλ χ − df γ γ λ D ν T cefµ + H . c .
285 506 ε µνλρ ¯ T abcµ u adν u ef λ h deρσ T bcfσ + H . c . i ¯ T abcµ u adν u deλ χ − bf γ γ ν D λ T cefµ + H . c .
286 507 ε µνλρ ¯ T abcµ u adν u ef λ h feρσ T bcdσ + H . c . i ¯ T abcµ u adν u deλ χ − ef γ γ ν D λ T bcfµ + H . c . ε µνλρ ¯ T abcµ u adν u beσ h df λσ T cefρ + H . c .
90 254 i ¯ T abcµ u adν u deλ χ − ef γ γ λ D ν T bcfµ + H . c . ε µνλρ ¯ T abcµ u adν u deσ h ef λσ T bcfρ + H . c . i ¯ T abcµ u adν u efλ χ − de γ γ ν D λ T bcfµ + H . c . T abcµ f − adµν f − beνλ T cdeλ
91 256 i ¯ T abcµ u deν u efλ χ − fd γ γ ν D λ T abcµ + H . c . T abcµ f − adνλ f − beνλ T cdeµ
92 257 i ¯ T abcµ u adν u deλ χ − ,s γ γ ν D λ T bceµ + H . c .
287 512¯ T abcµ f − adµν f − deνλ T bceλ
93 258 iε µνλρ ¯ T abcµ u adν u beλ χ − df T cefρ + H . c .
288 513¯ T abcµ f − adνλ f − deµν T bceλ
94 259 iε µνλρ ¯ T abcµ u adν u deλ χ − bf T cefρ
289 514¯ T abcµ f − adνλ f − deνλ T bceµ
95 260 iε µνλρ ¯ T abcµ u adν u deλ χ − ef T bcfρ + H . c . T abcµ f − deµν f − edν λ T abcλ iε µνλρ ¯ T abcµ u adν u ef λ χ − de T bcfρ T abcµ f − deνλ f − edνλ T abcµ iε µνλρ ¯ T abcµ u deν u ef λ χ − fd T abcρ T abcµ f − adµν f − beλρ D νλ T cdeρ
96 263 iε µνλρ ¯ T abcµ u adν u deλ χ − ,s T bceρ
290 518¯ T abcµ f − adνλ f − beνρ D λρ T cdeµ
97 264 i ¯ T abcµ f − adµν χ − de T bceν + H . c .
291 519¯ T abcµ f − adµν f − deλρ D νλ T bceρ
98 265 i ¯ T abcµ h adµν χ − be T cdeν
292 520¯ T abcµ f − adνλ f − deµρ D νρ T bceλ
99 266 i ¯ T abcµ h adµν χ − de T bceν + H . c .
293 521¯ T abcµ f − adνλ f − deνρ D λρ T bceµ
100 267 i ¯ T abcµ h deµν χ − ed T abcν T abcµ f − deµν f − edλρ D νλ T abcρ i ¯ T abcµ h adνλ χ − be D νλ T cdeµ
294 523¯ T abcµ f − deνλ f − edν ρ D λρ T abcµ i ¯ T abcµ h adνλ χ − de D νλ T bceµ + H . c .
295 524 i ¯ T abcµ f − adµν f − beλρ σ νλ T cdeρ
101 270 i ¯ T abcµ h deνλ χ − ed D νλ T abcµ i ¯ T abcµ f − adµν f − deλρ σ νλ T bceρ
102 271 i ¯ T abcµ h adµν χ − ,s T bcdν
296 526 i ¯ T abcµ f − deµν f − edλρ σ νλ T abcρ i ¯ T abcµ h adνλ χ − ,s D νλ T bcdµ
297 527¯ T abcµ h adµν f − beνλ T cdeλ + H . c .
103 273 i ¯ T abcµ u adν ∇ ν χ − be T cdeµ
298 528¯ T abcµ h adµν f − deνλ T bceλ + H . c .
104 274 i ¯ T abcµ u adν ∇ ν χ − de T bceµ + H . c .
299 529¯ T abcµ h adνλ f − deµν T bceλ + H . c .
105 275 i ¯ T abcµ u deν ∇ ν χ − ed T abcµ T abcµ h deµν f − edνλ T abcλ + H . c . i ¯ T abcµ u adν ∇ ν χ − ,s T bcdµ
300 531¯ T abcµ h adµν f − beλρ D νλ T cdeρ + H . c .
106 277 ¯ T abcµ f + adνλ χ − be γ γ ν D λ T cdeµ
301 532¯ T abcµ h adνλ f − beνρ D λρ T cdeµ
107 278 ¯ T abcµ f + adνλ χ − de γ γ ν D λ T bceµ + H . c .
302 533¯ T abcµ h adµν f − deλρ D νλ T bceρ + H . c .
108 279 ¯ T abcµ f s, + νλ χ − ad γ γ ν D λ T bcdµ T abcµ h adνλ f − deµρ D νλ T bceρ + H . c .
109 280 ¯ T abcµ f + deνλ χ − ed γ γ ν D λ T abcµ T abcµ h adνλ f − deνρ D λρ T bceµ + H . c .
110 281 ¯ T abcµ f + adνλ χ − ,s γ γ ν D λ T bcdµ
304 535¯ T abcµ h deµν f − edλρ D νλ T abcρ + H . c .
282 ¯ T abcµ f s, + νλ χ − ,s γ γ ν D λ T abcµ T abcµ h deνλ f − edνρ D λρ T abcµ ε µνλρ ¯ T abcµ f + adνλ χ − be T cdeρ
306 536 i ¯ T abcµ h adµν f − deλρ σ νλ T bceρ + H . c .
111 284 ε µνλρ ¯ T abcµ f + adνλ χ − de T bceρ + H . c .
307 537¯ T abcµ h adµν h beνλ T cdeλ
112 285 ε µνλρ ¯ T abcµ f s, + νλ χ − ad T bcdρ T abcµ h adνλ h beνλ T cdeµ
113 286 ε µνλρ ¯ T abcµ f + deνλ χ − ed T abcρ T abcµ h adµν h deνλ T bceλ
114 287 ε µνλρ ¯ T abcµ f + adνλ χ − ,s T bcdρ
309 539¯ T abcµ h adνλ h deνλ T bceµ
115 288 ε µνλρ ¯ T abcµ f s, + νλ χ − ,s T abcρ T abcµ h deµν h edνλ T abcλ
289 ¯ T abcµ χ − ad χ − be T cdeµ
311 540¯ T abcµ h deνλ h edνλ T abcµ
290 ¯ T abcµ χ − ad χ − de T bceµ T abcµ h adµν h beλρ D νλ T cdeρ
116 291 ¯ T abcµ χ − de χ − ed T abcµ T abcµ h adνλ h beνρ D λρ T cdeµ
117 292 ¯ T abcµ χ − ad χ − ,s T bcdµ
312 543¯ T abcµ h adµν h deλρ D νλ T bceρ
118 293 ¯ T abcµ h F Lµν F Lνλ i T abcλ + H . c .
313 544¯ T abcµ h adνλ h deνρ D λρ T bceµ
119 294 ¯ T abcµ h F Lνλ F Lνλ i T abcµ + H . c .
314 545¯ T abcµ h deµν h edλρ D νλ T abcρ
295 ¯ T abcµ h F Lµν F Lλρ i D νλ T abcρ + H . c .
315 546¯ T abcµ h deνλ h edνρ D λρ T abcµ
296 ¯ T abcµ h F Lνλ F Lνρ i D λρ T abcµ + H . c .
316 547¯ T abcµ h adνλ h beρσ D νλρσ T cdeµ
120 297 ¯ T abcµ h χχ † i T abcµ
317 548¯ T abcµ h adνλ h deρσ D νλρσ T bceµ
121 298 ¯ T abcµ det χT abcµ + H . c . TABLE VI: Terms in the O ( p ) meson-octet-decuplet and πN ∆ chiralLagrangians, where P ( N f , n is defined in Eqs. (64) and (66). For the SU (2) case, the form needs to be changed, see the sentences around Eq.(67). P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ǫ abc ¯ B ad u beµ u cf µ u egν T A,n,dfgν ǫ abc ¯ B ad u beµ f − efνλ γ γ ν D λ T A,n,cdfµ ǫ abc ¯ B ad u beµ u cfν u dgµ T A,n,efgν ǫ abc ¯ B ad u efµ f − beνλ γ γ µ D ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u egµ T A,n,dfgν ǫ abc ¯ B ad u efµ f − beνλ γ γ ν D µ T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u egν T A,n,dfgµ ǫ abc ¯ B ad u efµ f − beνλ γ γ ν D λ T A,n,cdfµ ǫ abc ¯ B ad u beµ u df µ u egν T A,n,cfgν ǫ abc ¯ B ad u beµ h cfνλ γ γ µ D ν T A,n,defλ
15 40 ǫ abc ¯ B ad u beµ u dfν u egµ T A,n,cfgν ǫ abc ¯ B ad u beµ h dfνλ γ γ µ D ν T A,n,cefλ
16 41 ǫ abc ¯ B ad u beµ u dfν u egν T A,n,cfgµ ǫ abc ¯ B ad u beµ h efνλ γ γ µ D ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u ef µ u f gν T A,n,cdgν ǫ abc ¯ B ad u beµ h efνλ γ γ ν D µ T A,n,cdfλ ǫ abc ¯ B ad u beµ u efν u f gµ T A,n,cdgν ε µνλρ ǫ abc ¯ B ad u beµ f − cf νλ T A,n,defρ
17 44 ǫ abc ¯ B ad u beµ u efν u f gν T A,n,cdgµ ε µνλρ ǫ abc ¯ B ad u beµ f − df νλ T A,n,cefρ
18 45 ǫ abc ¯ B ad u beµ u cfν u dgλ D µλ T A,n,efgν ε µνλρ ǫ abc ¯ B ad u beµ f − ef νλ T A,n,cdfρ ǫ abc ¯ B ad u beµ u cfν u egλ D µν T A,n,dfgλ ε µνλρ ǫ abc ¯ B ad u ef µ f − beνλ T A,n,cdfρ ǫ abc ¯ B ad u beµ u cfν u egλ D µλ T A,n,dfgν ǫ abc ¯ B ad ∇ µ f − beµν T A,n,cdeν
19 48 ǫ abc ¯ B ad u beµ u cfν u egλ D νλ T A,n,dfgµ iǫ abc ¯ B ad ∇ µ f − beνλ σ µν T A,n,cdeλ
20 49 ǫ abc ¯ B ad u beµ u dfν u egλ D µν T A,n,cfgλ iǫ abc ¯ B ad f + beµν u cf µ T A,n,defν
21 50 ǫ abc ¯ B ad u beµ u dfν u egλ D µλ T A,n,cfgν iǫ abc ¯ B ad f + beµν u df µ T A,n,cefν
22 51 ǫ abc ¯ B ad u beµ u dfν u egλ D νλ T A,n,cfgµ iǫ abc ¯ B ad f + beµν u ef µ T A,n,cdfν ǫ abc ¯ B ad u beµ u efν u f gλ D µν T A,n,cdgλ iǫ abc ¯ B ad f + efµν u beµ T A,n,cdfν ǫ abc ¯ B ad u beµ u efν u f gλ D µλ T A,n,cdgν iǫ abc ¯ B ad f + beµν u cfλ D µλ T A,n,defν
23 54 ǫ abc ¯ B ad u beµ u efν u f gλ D νλ T A,n,cdgµ iǫ abc ¯ B ad f + beµν u dfλ D µλ T A,n,cefν
24 55 iǫ abc ¯ B ad u beµ u cfν u dgλ σ µν T A,n,efgλ iǫ abc ¯ B ad f + beµν u efλ D µλ T A,n,cdfν iǫ abc ¯ B ad u beµ u cfν u egλ σ µν T A,n,dfgλ iǫ abc ¯ B ad f + efµν u beλ D µλ T A,n,cdfν iǫ abc ¯ B ad u beµ u cfν u egλ σ µλ T A,n,dfgν ǫ abc ¯ B ad f + beµν u cfλ σ µν T A,n,defλ
25 58 iǫ abc ¯ B ad u beµ u dfν u egλ σ µν T A,n,cfgλ ǫ abc ¯ B ad f + beµν u dfλ σ µν T A,n,cefλ
26 59 iǫ abc ¯ B ad u beµ u dfν u egλ σ µλ T A,n,cfgν iǫ abc ¯ B ad f s, + µν u beµ T A,n,cdeν iǫ abc ¯ B ad u beµ u efν u f gλ σ µν T A,n,cdgλ iǫ abc ¯ B ad f s, + µν u beλ D µλ T A,n,cdeν iǫ abc ¯ B ad u beµ u efν u f gλ σ µλ T A,n,cdgν ǫ abc ¯ B ad f s, + µν u beλ σ µν T A,n,cdeλ ǫ abc ¯ B ad u beµ f − cfνλ γ γ µ D ν T A,n,defλ ǫ abc ¯ B ad f + beµν u efλ σ µν T A,n,cdfλ ǫ abc ¯ B ad u beµ f − cfνλ γ γ ν D µ T A,n,defλ
10 29 ǫ abc ¯ B ad f + efµν u beλ σ µν T A,n,cdfλ ǫ abc ¯ B ad u beµ f − cfνλ γ γ ν D λ T A,n,defµ
11 30 ǫ abc ¯ B ad u beµ χ + cf T A,n,defµ
30 62 ǫ abc ¯ B ad u beµ f − dfνλ γ γ µ D ν T A,n,cefλ
12 31 ǫ abc ¯ B ad u beµ χ + df T A,n,cefµ
31 63 ǫ abc ¯ B ad u beµ f − dfνλ γ γ ν D µ T A,n,cefλ
13 32 ǫ abc ¯ B ad u beµ χ + ef T A,n,cdfµ ǫ abc ¯ B ad u beµ f − dfνλ γ γ ν D λ T A,n,cefµ
14 33 ǫ abc ¯ B ad u efµ χ + be T A,n,cdfµ ǫ abc ¯ B ad u beµ f − efνλ γ γ µ D ν T A,n,cdfλ ǫ abc ¯ B ad u beµ χ + ,s T A,n,cdeµ
32 66 ǫ abc ¯ B ad u beµ f − efνλ γ γ ν D µ T A,n,cdfλ iǫ abc ¯ B ad ∇ µ χ − be T A,n,cdeµ
33 67TABLE VII: Terms in the O ( p ) meson-octet-decuplet and πN ∆ chiralLagrangians, where P ( N f , n is defined in Eqs. (64) and (66). For the SU (2) case, the form needs to be changed, see the sentences around Eq.(67). P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ǫ abc ¯ B ad u beµ u cf µ u dgν u ehλ γ γ ν T A,n,fghλ ǫ abc ¯ B ad u beµ ∇ ν f − cf µλ γ γ ν T A,n,defλ
78 322 ǫ abc ¯ B ad u beµ u cf µ u dgν u ehλ γ γ λ T A,n,fghν ǫ abc ¯ B ad u beµ ∇ ν f − cf νλ γ γ µ T A,n,defλ
79 323 ǫ abc ¯ B ad u beµ u cf µ u egν u f hλ γ γ ν T A,n,dghλ ǫ abc ¯ B ad u beµ ∇ ν f − cf νλ γ γ λ T A,n,defµ
80 324 ǫ abc ¯ B ad u beµ u cf µ u egν u ghλ γ γ ν T A,n,dfhλ ǫ abc ¯ B ad u beµ ∇ µ f − dfνλ γ γ ν T A,n,cefλ
81 325 ǫ abc ¯ B ad u beµ u cf µ u egν u ghλ γ γ λ T A,n,dfhν ǫ abc ¯ B ad u beµ ∇ ν f − df µλ γ γ ν T A,n,cefλ
82 326 ǫ abc ¯ B ad u beµ u cfν u dgµ u ehλ γ γ ν T A,n,fghλ ǫ abc ¯ B ad u beµ ∇ ν f − df νλ γ γ µ T A,n,cefλ
83 327 ǫ abc ¯ B ad u beµ u cfν u dgµ u ehλ γ γ λ T A,n,fghν ǫ abc ¯ B ad u beµ ∇ ν f − df νλ γ γ λ T A,n,cefµ
84 328 P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ǫ abc ¯ B ad u beµ u cfν u dgµ u f hλ γ γ ν T A,n,eghλ ǫ abc ¯ B ad u beµ ∇ µ f − efνλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u dgµ u f hλ γ γ λ T A,n,eghν ǫ abc ¯ B ad u beµ ∇ ν f − ef µλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehµ γ γ ν T A,n,fghλ ǫ abc ¯ B ad u beµ ∇ ν f − ef νλ γ γ µ T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehµ γ γ λ T A,n,fghν ǫ abc ¯ B ad u beµ ∇ ν f − ef νλ γ γ λ T A,n,cdfµ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehν γ γ µ T A,n,fghλ ǫ abc ¯ B ad u efµ ∇ µ f − beνλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehν γ γ λ T A,n,fghµ ǫ abc ¯ B ad u efµ ∇ ν f − beµλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehλ γ γ µ T A,n,fghν ǫ abc ¯ B ad u efµ ∇ ν f − beν λ γ γ µ T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehλ γ γ ν T A,n,fghµ ǫ abc ¯ B ad u efµ ∇ ν f − beν λ γ γ λ T A,n,cdfµ ǫ abc ¯ B ad u beµ u cfν u egµ u f hλ γ γ ν T A,n,dghλ ǫ abc ¯ B ad u beµ ∇ µ h cfνλ γ γ ν T A,n,defλ
85 337 ǫ abc ¯ B ad u beµ u cfν u egµ u f hλ γ γ λ T A,n,dghν ǫ abc ¯ B ad u beµ ∇ µ h dfνλ γ γ ν T A,n,cefλ
86 338 ǫ abc ¯ B ad u beµ u cfν u egµ u ghλ γ γ ν T A,n,dfhλ ǫ abc ¯ B ad u beµ ∇ µ h efνλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u egµ u ghλ γ γ λ T A,n,dfhν ǫ abc ¯ B ad u efµ ∇ µ h beνλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u egν u f hλ γ γ µ T A,n,dghλ ε µνλρ ǫ abc ¯ B ad f − beµν f − cf λσ D ρ T A,n,defσ
87 341 ǫ abc ¯ B ad u beµ u cfν u egν u f hλ γ γ λ T A,n,dghµ ε µνλρ ǫ abc ¯ B ad f − beµν f − ef λσ D ρ T A,n,cdfσ ǫ abc ¯ B ad u beµ u cfν u egν u ghλ γ γ µ T A,n,dfhλ ε µνλρ ǫ abc ¯ B ad h beµσ f − cf νλ D ρ T A,n,defσ
88 343 ǫ abc ¯ B ad u beµ u cfν u egν u ghλ γ γ λ T A,n,dfhµ ε µνλρ ǫ abc ¯ B ad h beµσ f − df νλ D ρ T A,n,cefσ
89 344 ǫ abc ¯ B ad u beµ u cfν u egλ u f hλ γ γ µ T A,n,dghν ε µνλρ ǫ abc ¯ B ad h beµσ f − ef νλ D ρ T A,n,cdfσ ǫ abc ¯ B ad u beµ u cfν u egλ u ghµ γ γ ν T A,n,dfhλ ε µνλρ ǫ abc ¯ B ad h ef µσ f − beνλ D ρ T A,n,cdfσ ǫ abc ¯ B ad u beµ u cfν u egλ u ghµ γ γ λ T A,n,dfhν iǫ abc ¯ B ad f + beµν u cf µ u dgλ γ γ ν T A,n,efgλ
90 347 ǫ abc ¯ B ad u beµ u cfν u egλ u ghν γ γ µ T A,n,dfhλ iǫ abc ¯ B ad f + beµν u cf µ u dgλ γ γ λ T A,n,efgν
91 348 ǫ abc ¯ B ad u beµ u cfν u egλ u ghν γ γ λ T A,n,dfhµ iǫ abc ¯ B ad f + beµν u cf µ u egλ γ γ ν T A,n,dfgλ
92 349 ǫ abc ¯ B ad u beµ u cfν u egλ u ghλ γ γ µ T A,n,dfhν iǫ abc ¯ B ad f + beµν u cf µ u egλ γ γ λ T A,n,dfgν
93 350 ǫ abc ¯ B ad u beµ u cfν u egλ u ghλ γ γ ν T A,n,dfhµ iǫ abc ¯ B ad f + beµν u cf µ u f gλ γ γ ν T A,n,degλ
94 351 ǫ abc ¯ B ad u beµ u df µ u egν u f hλ γ γ ν T A,n,cghλ iǫ abc ¯ B ad f + beµν u cf µ u f gλ γ γ λ T A,n,degν
95 352 ǫ abc ¯ B ad u beµ u df µ u egν u ghλ γ γ ν T A,n,cfhλ iǫ abc ¯ B ad f + beµν u cfλ u dgµ γ γ ν T A,n,efgλ
96 353 ǫ abc ¯ B ad u beµ u df µ u egν u ghλ γ γ λ T A,n,cfhν iǫ abc ¯ B ad f + beµν u cfλ u dgµ γ γ λ T A,n,efgν
97 354 ǫ abc ¯ B ad u beµ u df µ u ghν u hgλ γ γ ν T A,n,cefλ iǫ abc ¯ B ad f + beµν u cfλ u dgλ γ γ µ T A,n,efgν
98 355 ǫ abc ¯ B ad u beµ u dfν u egµ u f hλ γ γ ν T A,n,cghλ iǫ abc ¯ B ad f + beµν u cfλ u egµ γ γ ν T A,n,dfgλ ǫ abc ¯ B ad u beµ u dfν u egµ u f hλ γ γ λ T A,n,cghν iǫ abc ¯ B ad f + beµν u cfλ u egµ γ γ λ T A,n,dfgν ǫ abc ¯ B ad u beµ u dfν u egµ u ghλ γ γ ν T A,n,cfhλ iǫ abc ¯ B ad f + beµν u cfλ u egλ γ γ µ T A,n,dfgν
99 358 ǫ abc ¯ B ad u beµ u dfν u egν u f hλ γ γ µ T A,n,cghλ iǫ abc ¯ B ad f + beµν u cfλ u f gµ γ γ ν T A,n,degλ
100 359 ǫ abc ¯ B ad u beµ u dfν u egν u f hλ γ γ λ T A,n,cghµ iǫ abc ¯ B ad f + beµν u cfλ u f gµ γ γ λ T A,n,degν
101 360 ǫ abc ¯ B ad u beµ u dfν u egν u ghλ γ γ µ T A,n,cfhλ iǫ abc ¯ B ad f + beµν u cfλ u f gλ γ γ µ T A,n,degν
102 361 ǫ abc ¯ B ad u beµ u dfν u egν u ghλ γ γ λ T A,n,cfhµ iǫ abc ¯ B ad f + beµν u df µ u egλ γ γ ν T A,n,cfgλ ǫ abc ¯ B ad u beµ u dfν u egλ u f hλ γ γ µ T A,n,cghν iǫ abc ¯ B ad f + beµν u df µ u egλ γ γ λ T A,n,cfgν ǫ abc ¯ B ad u beµ u dfν u egλ u ghν γ γ µ T A,n,cfhλ iǫ abc ¯ B ad f + beµν u df µ u f gλ γ γ ν T A,n,cegλ ǫ abc ¯ B ad u beµ u ef µ u f gν u ghλ γ γ ν T A,n,cdhλ iǫ abc ¯ B ad f + beµν u df µ u f gλ γ γ λ T A,n,cegν ǫ abc ¯ B ad u beµ u ef µ u f gν u ghλ γ γ λ T A,n,cdhν iǫ abc ¯ B ad f + beµν u dfλ u egµ γ γ ν T A,n,cfgλ ǫ abc ¯ B ad u beµ u ef µ u ghν u hgλ γ γ ν T A,n,cdfλ iǫ abc ¯ B ad f + beµν u dfλ u egµ γ γ λ T A,n,cfgν ǫ abc ¯ B ad u beµ u efν u f gµ u ghλ γ γ ν T A,n,cdhλ iǫ abc ¯ B ad f + beµν u dfλ u egλ γ γ µ T A,n,cfgν ǫ abc ¯ B ad u beµ u efν u f gµ u ghλ γ γ λ T A,n,cdhν iǫ abc ¯ B ad f + beµν u dfλ u f gµ γ γ ν T A,n,cegλ ǫ abc ¯ B ad u beµ u efν u f gλ u ghµ γ γ ν T A,n,cdhλ iǫ abc ¯ B ad f + beµν u dfλ u f gµ γ γ λ T A,n,cegν ǫ abc ¯ B ad u beµ u efν u f gλ u ghν γ γ µ T A,n,cdhλ iǫ abc ¯ B ad f + beµν u dfλ u f gλ γ γ µ T A,n,cegν ǫ abc ¯ B ad u beµ u efν u f gλ u ghν γ γ λ T A,n,cdhµ iǫ abc ¯ B ad f + beµν u ef µ u f gλ γ γ ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ µ D νλ T A,n,fghρ iǫ abc ¯ B ad f + beµν u ef µ u f gλ γ γ λ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ µ D νρ T A,n,fghλ iǫ abc ¯ B ad f + beµν u efλ u f gµ γ γ ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ µ D λρ T A,n,fghν iǫ abc ¯ B ad f + beµν u efλ u f gµ γ γ λ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ ν D µλ T A,n,fghρ
10 55 iǫ abc ¯ B ad f + beµν u efλ u f gλ γ γ µ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ ν D µρ T A,n,fghλ
11 56 iǫ abc ¯ B ad f + beµν u fgµ u gf λ γ γ ν T A,n,cdeλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ ν D λρ T A,n,fghµ
12 57 iǫ abc ¯ B ad f + beµν u fgµ u gf λ γ γ λ T A,n,cdeν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ λ D µν T A,n,fghρ
13 58 iǫ abc ¯ B ad f + beµν u fgλ u gfλ γ γ µ T A,n,cdeν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ λ D µρ T A,n,fghν iǫ abc ¯ B ad f + efµν u beµ u cgλ γ γ ν T A,n,dfgλ P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ λ D νρ T A,n,fghµ
14 60 iǫ abc ¯ B ad f + efµν u beµ u cgλ γ γ λ T A,n,dfgν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ ρ D µν T A,n,fghλ iǫ abc ¯ B ad f + efµν u beµ u dgλ γ γ ν T A,n,cfgλ ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ ρ D µλ T A,n,fghν iǫ abc ¯ B ad f + efµν u beµ u dgλ γ γ λ T A,n,cfgν ǫ abc ¯ B ad u beµ u cfν u dgλ u ehρ γ γ ρ D νλ T A,n,fghµ iǫ abc ¯ B ad f + efµν u beµ u f gλ γ γ ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u cfν u egλ u f hρ γ γ µ D νλ T A,n,dghρ iǫ abc ¯ B ad f + efµν u beµ u f gλ γ γ λ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν u egλ u f hρ γ γ µ D νρ T A,n,dghλ iǫ abc ¯ B ad f + efµν u beλ u cgµ γ γ ν T A,n,dfgλ ǫ abc ¯ B ad u beµ u cfν u egλ u f hρ γ γ µ D λρ T A,n,dghν iǫ abc ¯ B ad f + efµν u beλ u cgµ γ γ λ T A,n,dfgν ǫ abc ¯ B ad u beµ u cfν u egλ u f hρ γ γ λ D µν T A,n,dghρ iǫ abc ¯ B ad f + efµν u beλ u cgλ γ γ µ T A,n,dfgν ǫ abc ¯ B ad u beµ u cfν u egλ u f hρ γ γ λ D µρ T A,n,dghν iǫ abc ¯ B ad f + efµν u beλ u dgµ γ γ ν T A,n,cfgλ ǫ abc ¯ B ad u beµ u cfν u egλ u f hρ γ γ λ D νρ T A,n,dghµ iǫ abc ¯ B ad f + efµν u beλ u dgµ γ γ λ T A,n,cfgν ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ µ D νλ T A,n,dfhρ iǫ abc ¯ B ad f + efµν u beλ u dgλ γ γ µ T A,n,cfgν ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ µ D νρ T A,n,dfhλ iǫ abc ¯ B ad f + efµν u beλ u f gµ γ γ ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ µ D λρ T A,n,dfhν iǫ abc ¯ B ad f + efµν u beλ u f gµ γ γ λ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ ν D µλ T A,n,dfhρ iǫ abc ¯ B ad f + efµν u beλ u f gλ γ γ µ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ ν D µρ T A,n,dfhλ iǫ abc ¯ B ad f + efµν u bgµ u geλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ ν D λρ T A,n,dfhµ iǫ abc ¯ B ad f + efµν u bgµ u geλ γ γ λ T A,n,cdfν ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ λ D µν T A,n,dfhρ iǫ abc ¯ B ad f + beµν u cfλ u dgρ γ γ µ D νλ T A,n,efgρ
103 397 ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ λ D µρ T A,n,dfhν iǫ abc ¯ B ad f + beµν u cfλ u dgρ γ γ µ D νρ T A,n,efgλ
104 398 ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ λ D νρ T A,n,dfhµ iǫ abc ¯ B ad f + beµν u cfλ u dgρ γ γ µ D λρ T A,n,efgν
105 399 ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ ρ D µν T A,n,dfhλ iǫ abc ¯ B ad f + beµν u cfλ u dgρ γ γ λ D µρ T A,n,efgν
106 400 ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ ρ D µλ T A,n,dfhν iǫ abc ¯ B ad f + beµν u cfλ u dgρ γ γ ρ D µλ T A,n,efgν
107 401 ǫ abc ¯ B ad u beµ u cfν u egλ u ghρ γ γ ρ D νλ T A,n,dfhµ iǫ abc ¯ B ad f + beµν u cfλ u egρ γ γ µ D νλ T A,n,dfgρ
108 402 ǫ abc ¯ B ad u beµ u dfν u egλ u f hρ γ γ µ D νλ T A,n,cghρ iǫ abc ¯ B ad f + beµν u cfλ u egρ γ γ µ D νρ T A,n,dfgλ ǫ abc ¯ B ad u beµ u dfν u egλ u f hρ γ γ µ D νρ T A,n,cghλ iǫ abc ¯ B ad f + beµν u cfλ u egρ γ γ µ D λρ T A,n,dfgν
109 404 ǫ abc ¯ B ad u beµ u dfν u egλ u f hρ γ γ µ D λρ T A,n,cghν iǫ abc ¯ B ad f + beµν u cfλ u egρ γ γ λ D µρ T A,n,dfgν
110 405 ǫ abc ¯ B ad u beµ u dfν u egλ u f hρ γ γ λ D µν T A,n,cghρ iǫ abc ¯ B ad f + beµν u cfλ u egρ γ γ ρ D µλ T A,n,dfgν ǫ abc ¯ B ad u beµ u dfν u egλ u f hρ γ γ λ D µρ T A,n,cghν iǫ abc ¯ B ad f + beµν u cfλ u f gρ γ γ µ D νλ T A,n,degρ
111 407 ǫ abc ¯ B ad u beµ u dfν u egλ u f hρ γ γ λ D νρ T A,n,cghµ iǫ abc ¯ B ad f + beµν u cfλ u f gρ γ γ µ D νρ T A,n,degλ
112 408 ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ µ D νλ T A,n,cfhρ iǫ abc ¯ B ad f + beµν u cfλ u f gρ γ γ µ D λρ T A,n,degν
113 409 ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ µ D νρ T A,n,cfhλ iǫ abc ¯ B ad f + beµν u cfλ u f gρ γ γ λ D µρ T A,n,degν
114 410 ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ ν D µλ T A,n,cfhρ iǫ abc ¯ B ad f + beµν u cfλ u f gρ γ γ ρ D µλ T A,n,degν
115 411 ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ ν D µρ T A,n,cfhλ iǫ abc ¯ B ad f s, + µν u beµ u cfλ γ γ ν T A,n,defλ ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ ν D λρ T A,n,cfhµ iǫ abc ¯ B ad f s, + µν u beµ u cfλ γ γ λ T A,n,defν ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ λ D µν T A,n,cfhρ iǫ abc ¯ B ad f s, + µν u beµ u dfλ γ γ ν T A,n,cefλ ǫ abc ¯ B ad u beµ u dfν u egλ u ghρ γ γ λ D νρ T A,n,cfhµ iǫ abc ¯ B ad f s, + µν u beµ u dfλ γ γ λ T A,n,cefν ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ µ D νλ T A,n,cdhρ iǫ abc ¯ B ad f s, + µν u beλ u df λ γ γ µ T A,n,cefν ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ µ D νρ T A,n,cdhλ iǫ abc ¯ B ad f s, + µν u beλ u cfρ γ γ µ D νλ T A,n,defρ ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ µ D λρ T A,n,cdhν iǫ abc ¯ B ad f s, + µν u beλ u cfρ γ γ λ D µρ T A,n,defν ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ ν D µλ T A,n,cdhρ iǫ abc ¯ B ad f s, + µν u beλ u dfρ γ γ µ D νλ T A,n,cefρ ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ ν D µρ T A,n,cdhλ iǫ abc ¯ B ad f s, + µν u beλ u dfρ γ γ µ D λρ T A,n,cefν ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ ν D λρ T A,n,cdhµ iǫ abc ¯ B ad f s, + µν u beλ u dfρ γ γ λ D µρ T A,n,cefν ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ ρ D µν T A,n,cdhλ iǫ abc ¯ B ad f + beµν u dfλ u egρ γ γ µ D νλ T A,n,cfgρ ǫ abc ¯ B ad u beµ u efν u f gλ u ghρ γ γ ρ D µλ T A,n,cdhν iǫ abc ¯ B ad f + beµν u dfλ u egρ γ γ µ D νρ T A,n,cfgλ ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u dgλ u ehσ D ρ T A,n,fghσ
15 103 iǫ abc ¯ B ad f + beµν u dfλ u egρ γ γ µ D λρ T A,n,cfgν ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u dgσ u ehλ D ρ T A,n,fghσ iǫ abc ¯ B ad f + beµν u dfλ u egρ γ γ λ D µρ T A,n,cfgν ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u dgσ u ghλ D ρ T A,n,efhσ iǫ abc ¯ B ad f + beµν u dfλ u egρ γ γ ρ D µλ T A,n,cfgν ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u egλ u f hσ D ρ T A,n,dghσ iǫ abc ¯ B ad f + beµν u dfλ u f gρ γ γ µ D νλ T A,n,cegρ ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u egλ u ghσ D ρ T A,n,dfhσ iǫ abc ¯ B ad f + beµν u dfλ u f gρ γ γ µ D νρ T A,n,cegλ ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u egσ u ghλ D ρ T A,n,dfhσ iǫ abc ¯ B ad f + beµν u dfλ u f gρ γ γ µ D λρ T A,n,cegν ε µνλρ ǫ abc ¯ B ad u beµ u cf ν u ghλ u hgσ D ρ T A,n,defσ iǫ abc ¯ B ad f + beµν u dfλ u f gρ γ γ λ D µρ T A,n,cegν ε µνλρ ǫ abc ¯ B ad u beµ u cfσ u egν u f hλ D ρ T A,n,dghσ iǫ abc ¯ B ad f + beµν u dfλ u f gρ γ γ ρ D µλ T A,n,cegν ε µνλρ ǫ abc ¯ B ad u beµ u df ν u egλ u f hσ D ρ T A,n,cghσ iǫ abc ¯ B ad f + beµν u efλ u f gρ γ γ µ D νλ T A,n,cdgρ ε µνλρ ǫ abc ¯ B ad u beµ u df ν u egλ u ghσ D ρ T A,n,cfhσ iǫ abc ¯ B ad f + beµν u efλ u f gρ γ γ µ D νρ T A,n,cdgλ P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ε µνλρ ǫ abc ¯ B ad u beµ u ef ν u f gλ u ghσ D ρ T A,n,cdhσ iǫ abc ¯ B ad f + beµν u efλ u f gρ γ γ µ D λρ T A,n,cdgν ε µνλρ ǫ abc ¯ B ad u beµ u ef ν u f gσ u ghλ D ρ T A,n,cdhσ iǫ abc ¯ B ad f + beµν u efλ u f gρ γ γ λ D µρ T A,n,cdgν ǫ abc ¯ B ad u beµ u cf µ f − egνλ D ν T A,n,dfgλ
16 115 iǫ abc ¯ B ad f + beµν u efλ u f gρ γ γ ρ D µλ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν f − dgµλ D ν T A,n,efgλ
17 116 iǫ abc ¯ B ad f + beµν u fgλ u gf ρ γ γ µ D νλ T A,n,cdeρ ǫ abc ¯ B ad u beµ u cfν f − dgµλ D λ T A,n,efgν
18 117 iǫ abc ¯ B ad f + beµν u fgλ u gf ρ γ γ µ D λρ T A,n,cdeν ǫ abc ¯ B ad u beµ u cfν f − egµλ D ν T A,n,dfgλ
19 118 iǫ abc ¯ B ad f + beµν u fgλ u gf ρ γ γ λ D µρ T A,n,cdeν ǫ abc ¯ B ad u beµ u cfν f − egµλ D λ T A,n,dfgν
20 119 iǫ abc ¯ B ad f + efµν u beλ u cgρ γ γ µ D νλ T A,n,dfgρ ǫ abc ¯ B ad u beµ u cfν f − egνλ D µ T A,n,dfgλ
21 120 iǫ abc ¯ B ad f + efµν u beλ u cgρ γ γ µ D νρ T A,n,dfgλ ǫ abc ¯ B ad u beµ u cfν f − egνλ D λ T A,n,dfgµ
22 121 iǫ abc ¯ B ad f + efµν u beλ u cgρ γ γ µ D λρ T A,n,dfgν ǫ abc ¯ B ad u beµ u df µ f − cgνλ D ν T A,n,efgλ
23 122 iǫ abc ¯ B ad f + efµν u beλ u cgρ γ γ λ D µρ T A,n,dfgν ǫ abc ¯ B ad u beµ u df µ f − egνλ D ν T A,n,cfgλ iǫ abc ¯ B ad f + efµν u beλ u cgρ γ γ ρ D µλ T A,n,dfgν ǫ abc ¯ B ad u beµ u dfν f − cgµλ D ν T A,n,efgλ
24 124 iǫ abc ¯ B ad f + efµν u beλ u dgρ γ γ µ D νλ T A,n,cfgρ ǫ abc ¯ B ad u beµ u dfν f − cgµλ D λ T A,n,efgν
25 125 iǫ abc ¯ B ad f + efµν u beλ u dgρ γ γ µ D νρ T A,n,cfgλ ǫ abc ¯ B ad u beµ u dfν f − egµλ D ν T A,n,cfgλ iǫ abc ¯ B ad f + efµν u beλ u dgρ γ γ µ D λρ T A,n,cfgν ǫ abc ¯ B ad u beµ u dfν f − egµλ D λ T A,n,cfgν iǫ abc ¯ B ad f + efµν u beλ u dgρ γ γ λ D µρ T A,n,cfgν ǫ abc ¯ B ad u beµ u dfν f − egνλ D µ T A,n,cfgλ iǫ abc ¯ B ad f + efµν u beλ u dgρ γ γ ρ D µλ T A,n,cfgν ǫ abc ¯ B ad u beµ u dfν f − egνλ D λ T A,n,cfgµ iǫ abc ¯ B ad f + efµν u beλ u f gρ γ γ µ D νλ T A,n,cdgρ ǫ abc ¯ B ad u beµ u ef µ f − cgνλ D ν T A,n,dfgλ iǫ abc ¯ B ad f + efµν u beλ u f gρ γ γ µ D νρ T A,n,cdgλ ǫ abc ¯ B ad u beµ u ef µ f − dgνλ D ν T A,n,cfgλ iǫ abc ¯ B ad f + efµν u beλ u f gρ γ γ µ D λρ T A,n,cdgν ǫ abc ¯ B ad u beµ u ef µ f − f gνλ D ν T A,n,cdgλ iǫ abc ¯ B ad f + efµν u beλ u f gρ γ γ λ D µρ T A,n,cdgν ǫ abc ¯ B ad u beµ u efν f − cgµλ D ν T A,n,dfgλ iǫ abc ¯ B ad f + efµν u beλ u f gρ γ γ ρ D µλ T A,n,cdgν ǫ abc ¯ B ad u beµ u efν f − cgµλ D λ T A,n,dfgν iǫ abc ¯ B ad f + efµν u bgλ u geρ γ γ µ D νλ T A,n,cdfρ ǫ abc ¯ B ad u beµ u efν f − cgνλ D µ T A,n,dfgλ iǫ abc ¯ B ad f + efµν u bgλ u geρ γ γ λ D µρ T A,n,cdfν ǫ abc ¯ B ad u beµ u efν f − cgνλ D λ T A,n,dfgµ iε µνλρ ǫ abc ¯ B ad f + beµν u cf λ u dgσ D ρ T A,n,efgσ
126 447 ǫ abc ¯ B ad u beµ u efν f − dgµλ D ν T A,n,cfgλ iε µνλρ ǫ abc ¯ B ad f + beµν u cf λ u egσ D ρ T A,n,dfgσ
127 448 ǫ abc ¯ B ad u beµ u efν f − dgµλ D λ T A,n,cfgν iε µνλρ ǫ abc ¯ B ad f + beµν u cf λ u f gσ D ρ T A,n,degσ
128 449 ǫ abc ¯ B ad u beµ u efν f − dgνλ D µ T A,n,cfgλ iε µνλρ ǫ abc ¯ B ad f + beµν u cfσ u dgλ D ρ T A,n,efgσ
129 450 ǫ abc ¯ B ad u beµ u efν f − dgνλ D λ T A,n,cfgµ iε µνλρ ǫ abc ¯ B ad f + beµν u cfσ u egλ D ρ T A,n,dfgσ ǫ abc ¯ B ad u beµ u efν f − f gµλ D ν T A,n,cdgλ iε µνλρ ǫ abc ¯ B ad f + beµν u cfσ u f gλ D ρ T A,n,degσ
130 452 ǫ abc ¯ B ad u beµ u efν f − f gµλ D λ T A,n,cdgν iε µνλρ ǫ abc ¯ B ad f s, + µν u beλ u cfσ D ρ T A,n,defσ ǫ abc ¯ B ad u beµ u efν f − f gνλ D µ T A,n,cdgλ iε µνλρ ǫ abc ¯ B ad f s, + µν u beλ u dfσ D ρ T A,n,cefσ ǫ abc ¯ B ad u beµ u efν f − f gνλ D λ T A,n,cdgµ iε µνλρ ǫ abc ¯ B ad f + beµν u df λ u egσ D ρ T A,n,cfgσ ǫ abc ¯ B ad u beµ u fgµ f − cf νλ D ν T A,n,degλ
26 145 iε µνλρ ǫ abc ¯ B ad f + beµν u df λ u f gσ D ρ T A,n,cegσ ǫ abc ¯ B ad u beµ u fgµ f − df νλ D ν T A,n,cegλ iε µνλρ ǫ abc ¯ B ad f + beµν u dfσ u egλ D ρ T A,n,cfgσ ǫ abc ¯ B ad u beµ u fgµ f − ef νλ D ν T A,n,cdgλ iε µνλρ ǫ abc ¯ B ad f + beµν u dfσ u f gλ D ρ T A,n,cegσ ǫ abc ¯ B ad u beµ u fgµ f − gf νλ D ν T A,n,cdeλ iε µνλρ ǫ abc ¯ B ad f + beµν u ef λ u f gσ D ρ T A,n,cdgσ ǫ abc ¯ B ad u beµ u fgν f − cfµλ D ν T A,n,degλ
27 149 iε µνλρ ǫ abc ¯ B ad f + beµν u efσ u f gλ D ρ T A,n,cdgσ ǫ abc ¯ B ad u beµ u fgν f − cfµλ D λ T A,n,degν
28 150 iε µνλρ ǫ abc ¯ B ad f + beµν u fgλ u gf σ D ρ T A,n,cdeσ ǫ abc ¯ B ad u beµ u fgν f − cfνλ D µ T A,n,degλ iε µνλρ ǫ abc ¯ B ad f + ef µν u beλ u cgσ D ρ T A,n,dfgσ ǫ abc ¯ B ad u beµ u fgν f − cfνλ D λ T A,n,degµ iε µνλρ ǫ abc ¯ B ad f + ef µν u beλ u dgσ D ρ T A,n,cfgσ ǫ abc ¯ B ad u beµ u fgν f − dfµλ D ν T A,n,cegλ iε µνλρ ǫ abc ¯ B ad f + ef µν u beλ u f gσ D ρ T A,n,cdgσ ǫ abc ¯ B ad u beµ u fgν f − dfµλ D λ T A,n,cegν iε µνλρ ǫ abc ¯ B ad f + ef µν u beσ u cgλ D ρ T A,n,dfgσ ǫ abc ¯ B ad u beµ u fgν f − dfνλ D µ T A,n,cegλ iε µνλρ ǫ abc ¯ B ad f + ef µν u beσ u dgλ D ρ T A,n,cfgσ ǫ abc ¯ B ad u beµ u fgν f − dfνλ D λ T A,n,cegµ iε µνλρ ǫ abc ¯ B ad f + ef µν u beσ u f gλ D ρ T A,n,cdgσ ǫ abc ¯ B ad u beµ u fgν f − efµλ D ν T A,n,cdgλ iε µνλρ ǫ abc ¯ B ad f + ef µν u bgλ u geσ D ρ T A,n,cdfσ ǫ abc ¯ B ad u beµ u fgν f − efµλ D λ T A,n,cdgν iǫ abc ¯ B ad f + beµν f − cf µλ D ν T A,n,defλ
133 467 ǫ abc ¯ B ad u beµ u fgν f − efνλ D µ T A,n,cdgλ iǫ abc ¯ B ad f + beµν f − cf µλ D λ T A,n,defν
134 468 ǫ abc ¯ B ad u beµ u fgν f − efνλ D λ T A,n,cdgµ iǫ abc ¯ B ad f + beµν f − df µλ D ν T A,n,cefλ
135 469 ǫ abc ¯ B ad u beµ u fgν f − gfµλ D ν T A,n,cdeλ iǫ abc ¯ B ad f + beµν f − df µλ D λ T A,n,cefν
136 470 ǫ abc ¯ B ad u beµ u fgν f − gfµλ D λ T A,n,cdeν iǫ abc ¯ B ad f + beµν f − ef µλ D ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u fgν f − gfνλ D µ T A,n,cdeλ iǫ abc ¯ B ad f + beµν f − ef µλ D λ T A,n,cdfν ǫ abc ¯ B ad u beµ u fgν f − gfνλ D λ T A,n,cdeµ iǫ abc ¯ B ad f + efµν f − beµλ D ν T A,n,cdfλ P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ǫ abc ¯ B ad u beµ u cfν f − egλρ D µνλ T A,n,dfgρ
29 165 iǫ abc ¯ B ad f + efµν f − beµλ D λ T A,n,cdfν ǫ abc ¯ B ad u beµ u dfν f − cgλρ D µνλ T A,n,efgρ
30 166 ǫ abc ¯ B ad f + beµν f − cfλρ σ µν D λ T A,n,defρ
137 475 ǫ abc ¯ B ad u beµ u dfν f − egλρ D µνλ T A,n,cfgρ ǫ abc ¯ B ad f + beµν f − cfλρ σ µλ D ν T A,n,defρ
138 476 ǫ abc ¯ B ad u beµ u efν f − cgλρ D µνλ T A,n,dfgρ ǫ abc ¯ B ad f + beµν f − dfλρ σ µν D λ T A,n,cefρ
139 477 ǫ abc ¯ B ad u beµ u efν f − dgλρ D µνλ T A,n,cfgρ ǫ abc ¯ B ad f + beµν f − dfλρ σ µλ D ν T A,n,cefρ
140 478 ǫ abc ¯ B ad u beµ u efν f − f gλρ D µνλ T A,n,cdgρ iǫ abc ¯ B ad f s, + µν f − beµλ D ν T A,n,cdeλ ǫ abc ¯ B ad u beµ u fgν f − cf λρ D µνλ T A,n,degρ
31 171 iǫ abc ¯ B ad f s, + µν f − beµλ D λ T A,n,cdeν ǫ abc ¯ B ad u beµ u fgν f − df λρ D µνλ T A,n,cegρ ǫ abc ¯ B ad f s, + µν f − beλρ σ µλ D ν T A,n,cdeρ ǫ abc ¯ B ad u beµ u fgν f − ef λρ D µνλ T A,n,cdgρ ǫ abc ¯ B ad f + beµν f − efλρ σ µν D λ T A,n,cdfρ ǫ abc ¯ B ad u beµ u fgν f − gf λρ D µνλ T A,n,cdeρ ǫ abc ¯ B ad f + beµν f − efλρ σ µλ D ν T A,n,cdfρ iǫ abc ¯ B ad u beµ u cfν f − dgλρ σ µν D λ T A,n,efgρ
32 175 ǫ abc ¯ B ad f + efµν f − beλρ σ µν D λ T A,n,cdfρ iǫ abc ¯ B ad u beµ u cfν f − dgλρ σ µλ D ν T A,n,efgρ
33 176 ǫ abc ¯ B ad f + efµν f − beλρ σ µλ D ν T A,n,cdfρ iǫ abc ¯ B ad u beµ u cfν f − egλρ σ µν D λ T A,n,dfgρ
34 177 iǫ abc ¯ B ad f + beµν h cf µλ D ν T A,n,defλ
144 483 iǫ abc ¯ B ad u beµ u cfν f − egλρ σ µλ D ν T A,n,dfgρ
35 178 iǫ abc ¯ B ad f + beµν h cf µλ D λ T A,n,defν
145 484 iǫ abc ¯ B ad u beµ u cfν f − egλρ σ µλ D ρ T A,n,dfgν
36 179 iǫ abc ¯ B ad f + beµν h df µλ D ν T A,n,cefλ
146 485 iǫ abc ¯ B ad u beµ u cfν f − egλρ σ νλ D µ T A,n,dfgρ
37 180 iǫ abc ¯ B ad f + beµν h df µλ D λ T A,n,cefν
147 486 iǫ abc ¯ B ad u beµ u dfν f − cgλρ σ µλ D ν T A,n,efgρ
38 181 iǫ abc ¯ B ad f + beµν h ef µλ D ν T A,n,cdfλ iǫ abc ¯ B ad u beµ u dfν f − cgλρ σ µλ D ρ T A,n,efgν
39 182 iǫ abc ¯ B ad f + beµν h ef µλ D λ T A,n,cdfν iǫ abc ¯ B ad u beµ u dfν f − egλρ σ µν D λ T A,n,cfgρ iǫ abc ¯ B ad f + efµν h beµλ D ν T A,n,cdfλ iǫ abc ¯ B ad u beµ u dfν f − egλρ σ µλ D ν T A,n,cfgρ iǫ abc ¯ B ad f + efµν h beµλ D λ T A,n,cdfν iǫ abc ¯ B ad u beµ u dfν f − egλρ σ µλ D ρ T A,n,cfgν iǫ abc ¯ B ad f + beµν h cfλρ D µλρ T A,n,defν
148 491 iǫ abc ¯ B ad u beµ u dfν f − egλρ σ νλ D µ T A,n,cfgρ iǫ abc ¯ B ad f + beµν h dfλρ D µλρ T A,n,cefν
149 492 iǫ abc ¯ B ad u beµ u efν f − cgλρ σ µν D λ T A,n,dfgρ iǫ abc ¯ B ad f + beµν h efλρ D µλρ T A,n,cdfν iǫ abc ¯ B ad u beµ u efν f − cgλρ σ µλ D ν T A,n,dfgρ iǫ abc ¯ B ad f + efµν h beλρ D µλρ T A,n,cdfν iǫ abc ¯ B ad u beµ u efν f − cgλρ σ µλ D ρ T A,n,dfgν ǫ abc ¯ B ad f + beµν h cfλρ σ µν D λ T A,n,defρ
150 495 iǫ abc ¯ B ad u beµ u efν f − cgλρ σ νλ D µ T A,n,dfgρ ǫ abc ¯ B ad f + beµν h cfλρ σ µλ D ν T A,n,defρ
151 496 iǫ abc ¯ B ad u beµ u efν f − dgλρ σ µν D λ T A,n,cfgρ ǫ abc ¯ B ad f + beµν h dfλρ σ µν D λ T A,n,cefρ
152 497 iǫ abc ¯ B ad u beµ u efν f − dgλρ σ µλ D ν T A,n,cfgρ ǫ abc ¯ B ad f + beµν h dfλρ σ µλ D ν T A,n,cefρ
153 498 iǫ abc ¯ B ad u beµ u efν f − dgλρ σ µλ D ρ T A,n,cfgν ǫ abc ¯ B ad f + beµν h efλρ σ µν D λ T A,n,cdfρ iǫ abc ¯ B ad u beµ u efν f − dgλρ σ νλ D µ T A,n,cfgρ ǫ abc ¯ B ad f + beµν h efλρ σ µλ D ν T A,n,cdfρ iǫ abc ¯ B ad u beµ u efν f − f gλρ σ µν D λ T A,n,cdgρ ǫ abc ¯ B ad f + efµν h beλρ σ µν D λ T A,n,cdfρ iǫ abc ¯ B ad u beµ u efν f − f gλρ σ µλ D ν T A,n,cdgρ ǫ abc ¯ B ad f + efµν h beλρ σ µλ D ν T A,n,cdfρ iǫ abc ¯ B ad u beµ u efν f − f gλρ σ µλ D ρ T A,n,cdgν iǫ abc ¯ B ad ∇ µ f + beµν u cfλ D ν T A,n,defλ
154 503 iǫ abc ¯ B ad u beµ u efν f − f gλρ σ νλ D µ T A,n,cdgρ iǫ abc ¯ B ad ∇ µ f + beµν u cfλ D λ T A,n,defν
155 504 iǫ abc ¯ B ad u beµ u fgν f − cf λρ σ µν D λ T A,n,degρ iǫ abc ¯ B ad ∇ µ f + beµν u dfλ D ν T A,n,cefλ
156 505 iǫ abc ¯ B ad u beµ u fgν f − cf λρ σ µλ D ν T A,n,degρ
40 200 iǫ abc ¯ B ad ∇ µ f + beµν u dfλ D λ T A,n,cefν
157 506 iǫ abc ¯ B ad u beµ u fgν f − cf λρ σ µλ D ρ T A,n,degν
41 201 iǫ abc ¯ B ad ∇ µ f s, + µν u beλ D ν T A,n,cdeλ iǫ abc ¯ B ad u beµ u fgν f − cf λρ σ νλ D µ T A,n,degρ iǫ abc ¯ B ad ∇ µ f s, + µν u beλ D λ T A,n,cdeν iǫ abc ¯ B ad u beµ u fgν f − df λρ σ µν D λ T A,n,cegρ iǫ abc ¯ B ad ∇ µ f s, + νλ u beµ D ν T A,n,cdeλ iǫ abc ¯ B ad u beµ u fgν f − df λρ σ µλ D ν T A,n,cegρ iǫ abc ¯ B ad ∇ µ f s, + νλ u beν D µ T A,n,cdeλ iǫ abc ¯ B ad u beµ u fgν f − df λρ σ µλ D ρ T A,n,cegν iǫ abc ¯ B ad ∇ µ f s, + νλ u beρ D µνρ T A,n,cdeλ iǫ abc ¯ B ad u beµ u fgν f − df λρ σ νλ D µ T A,n,cegρ ǫ abc ¯ B ad ∇ µ f s, + νλ u beρ σ µν D λ T A,n,cdeρ iǫ abc ¯ B ad u beµ u fgν f − ef λρ σ µν D λ T A,n,cdgρ ǫ abc ¯ B ad ∇ µ f s, + νλ u beρ σ µν D ρ T A,n,cdeλ iǫ abc ¯ B ad u beµ u fgν f − ef λρ σ µλ D ν T A,n,cdgρ ǫ abc ¯ B ad ∇ µ f s, + νλ u beρ σ µρ D ν T A,n,cdeλ iǫ abc ¯ B ad u beµ u fgν f − ef λρ σ µλ D ρ T A,n,cdgν iǫ abc ¯ B ad ∇ µ f + beµν u efλ D ν T A,n,cdfλ iǫ abc ¯ B ad u beµ u fgν f − ef λρ σ νλ D µ T A,n,cdgρ iǫ abc ¯ B ad ∇ µ f + beµν u efλ D λ T A,n,cdfν iǫ abc ¯ B ad u beµ u fgν f − gf λρ σ µν D λ T A,n,cdeρ iǫ abc ¯ B ad ∇ µ f + ef µν u beλ D ν T A,n,cdfλ iǫ abc ¯ B ad u beµ u fgν f − gf λρ σ µλ D ν T A,n,cdeρ iǫ abc ¯ B ad ∇ µ f + ef µν u beλ D λ T A,n,cdfν iǫ abc ¯ B ad u beµ u fgν f − gf λρ σ µλ D ρ T A,n,cdeν iǫ abc ¯ B ad ∇ µ ∇ µ f + beνλ γ γ ν T A,n,cdeλ
166 511 iǫ abc ¯ B ad u beµ u fgν f − gf λρ σ νλ D µ T A,n,cdeρ iǫ abc ¯ B ad ∇ µ ∇ ν f + beµλ γ γ ν T A,n,cdeλ
167 512 ǫ abc ¯ B ad u beµ u cf µ h egνλ D ν T A,n,dfgλ
42 215 ǫ abc ¯ B ad f + beµν f + cf µλ γ γ ν T A,n,defλ
168 513 ǫ abc ¯ B ad u beµ u cfν h dgµλ D ν T A,n,efgλ
43 216 ǫ abc ¯ B ad f + beµν f + df µλ γ γ ν T A,n,cefλ
169 514 P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) ǫ abc ¯ B ad u beµ u cfν h dgµλ D λ T A,n,efgν
44 217 ǫ abc ¯ B ad f + beµν f + ef µλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad u beµ u cfν h egµλ D ν T A,n,dfgλ
45 218 ǫ abc ¯ B ad f + beµν f + ef µλ γ γ λ T A,n,cdfν ǫ abc ¯ B ad u beµ u cfν h egµλ D λ T A,n,dfgν
46 219 ǫ abc ¯ B ad f + beµν f + cfλρ γ γ µ D νλ T A,n,defρ
170 517 ǫ abc ¯ B ad u beµ u cfν h egνλ D µ T A,n,dfgλ
47 220 ǫ abc ¯ B ad f + beµν f + dfλρ γ γ µ D νλ T A,n,cefρ
171 518 ǫ abc ¯ B ad u beµ u cfν h egνλ D λ T A,n,dfgµ ǫ abc ¯ B ad f + beµν f s, + µλ γ γ ν T A,n,cdeλ ǫ abc ¯ B ad u beµ u df µ h cgνλ D ν T A,n,efgλ
48 222 ǫ abc ¯ B ad f + beµν f s, + µλ γ γ λ T A,n,cdeν ǫ abc ¯ B ad u beµ u df µ h egνλ D ν T A,n,cfgλ ǫ abc ¯ B ad f + beµν f s, + λρ γ γ µ D νλ T A,n,cdeρ ǫ abc ¯ B ad u beµ u dfν h egµλ D ν T A,n,cfgλ ǫ abc ¯ B ad f + beµν f s, + λρ γ γ λ D µρ T A,n,cdeν ǫ abc ¯ B ad u beµ u dfν h egµλ D λ T A,n,cfgν ǫ abc ¯ B ad f + beµν f + efλρ γ γ µ D νλ T A,n,cdfρ ǫ abc ¯ B ad u beµ u dfν h egνλ D µ T A,n,cfgλ ǫ abc ¯ B ad f + beµν f + efλρ γ γ λ D µρ T A,n,cdfν ǫ abc ¯ B ad u beµ u dfν h egνλ D λ T A,n,cfgµ ε µνλρ ǫ abc ¯ B ad f + beµν f + cf λσ D ρ T A,n,defσ
176 521 ǫ abc ¯ B ad u beµ u ef µ h cgνλ D ν T A,n,dfgλ ε µνλρ ǫ abc ¯ B ad f + beµν f s, + λσ D ρ T A,n,cdeσ ǫ abc ¯ B ad u beµ u ef µ h dgνλ D ν T A,n,cfgλ ε µνλρ ǫ abc ¯ B ad f + beµν f + ef λσ D ρ T A,n,cdfσ ǫ abc ¯ B ad u beµ u ef µ h f gνλ D ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u cfν χ + dg γ γ µ T A,n,efgν
178 523 ǫ abc ¯ B ad u beµ u efν h cgµλ D ν T A,n,dfgλ ǫ abc ¯ B ad u beµ u cfν χ + eg γ γ µ T A,n,dfgν
179 524 ǫ abc ¯ B ad u beµ u efν h cgµλ D λ T A,n,dfgν ǫ abc ¯ B ad u beµ u cfν χ + eg γ γ ν T A,n,dfgµ
180 525 ǫ abc ¯ B ad u beµ u efν h cgνλ D µ T A,n,dfgλ ǫ abc ¯ B ad u beµ u dfν χ + cg γ γ µ T A,n,efgν
181 526 ǫ abc ¯ B ad u beµ u efν h dgµλ D ν T A,n,cfgλ ǫ abc ¯ B ad u beµ u dfν χ + eg γ γ µ T A,n,cfgν ǫ abc ¯ B ad u beµ u efν h dgµλ D λ T A,n,cfgν ǫ abc ¯ B ad u beµ u dfν χ + eg γ γ ν T A,n,cfgµ ǫ abc ¯ B ad u beµ u efν h dgνλ D µ T A,n,cfgλ ǫ abc ¯ B ad u beµ u efν χ + cg γ γ µ T A,n,dfgν ǫ abc ¯ B ad u beµ u efν h f gµλ D ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u efν χ + cg γ γ ν T A,n,dfgµ ǫ abc ¯ B ad u beµ u efν h f gµλ D λ T A,n,cdgν ǫ abc ¯ B ad u beµ u efν χ + dg γ γ µ T A,n,cfgν ǫ abc ¯ B ad u beµ u efν h f gνλ D µ T A,n,cdgλ ǫ abc ¯ B ad u beµ u efν χ + dg γ γ ν T A,n,cfgµ ǫ abc ¯ B ad u beµ u efν h f gνλ D λ T A,n,cdgµ ǫ abc ¯ B ad u beµ u efν χ + f g γ γ µ T A,n,cdgν ǫ abc ¯ B ad u beµ u fgµ h ef νλ D ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u efν χ + f g γ γ ν T A,n,cdgµ ǫ abc ¯ B ad u beµ u fgµ h gf νλ D ν T A,n,cdeλ ǫ abc ¯ B ad u beµ u fgν χ + cf γ γ µ T A,n,degν
182 535 ǫ abc ¯ B ad u beµ u fgν h efµλ D ν T A,n,cdgλ ǫ abc ¯ B ad u beµ u fgν χ + cf γ γ ν T A,n,degµ ǫ abc ¯ B ad u beµ u fgν h efµλ D λ T A,n,cdgν ǫ abc ¯ B ad u beµ u fgν χ + df γ γ µ T A,n,cegν ǫ abc ¯ B ad u beµ u cfν h dgλρ D µλρ T A,n,efgν
49 245 ǫ abc ¯ B ad u beµ u fgν χ + df γ γ ν T A,n,cegµ ǫ abc ¯ B ad u beµ u cfν h egλρ D µνλ T A,n,dfgρ
50 246 ǫ abc ¯ B ad u beµ u fgν χ + ef γ γ µ T A,n,cdgν ǫ abc ¯ B ad u beµ u cfν h egλρ D µλρ T A,n,dfgν
51 247 ǫ abc ¯ B ad u beµ u fgν χ + ef γ γ ν T A,n,cdgµ ǫ abc ¯ B ad u beµ u cfν h egλρ D νλρ T A,n,dfgµ ǫ abc ¯ B ad u beµ u fgν χ + gf γ γ µ T A,n,cdeν ǫ abc ¯ B ad u beµ u dfν h cgλρ D µνλ T A,n,efgρ
52 249 ǫ abc ¯ B ad u beµ u fgν χ + gf γ γ ν T A,n,cdeµ ǫ abc ¯ B ad u beµ u dfν h egλρ D µνλ T A,n,cfgρ ǫ abc ¯ B ad u beµ u cfν χ + ,s γ γ µ T A,n,defν
183 543 ǫ abc ¯ B ad u beµ u dfν h egλρ D µλρ T A,n,cfgν ǫ abc ¯ B ad u beµ u dfν χ + ,s γ γ µ T A,n,cefν
184 544 ǫ abc ¯ B ad u beµ u dfν h egλρ D νλρ T A,n,cfgµ ǫ abc ¯ B ad u beµ u efν χ + ,s γ γ µ T A,n,cdfν ǫ abc ¯ B ad u beµ u efν h cgλρ D µνλ T A,n,dfgρ ǫ abc ¯ B ad u beµ u efν χ + ,s γ γ ν T A,n,cdfµ ǫ abc ¯ B ad u beµ u efν h cgλρ D µλρ T A,n,dfgν ǫ abc ¯ B ad f − beµν χ + cf D µ T A,n,defν
185 547 ǫ abc ¯ B ad u beµ u efν h dgλρ D µνλ T A,n,cfgρ ǫ abc ¯ B ad f − beµν χ + df D µ T A,n,cefν
186 548 ǫ abc ¯ B ad u beµ u efν h dgλρ D µλρ T A,n,cfgν ǫ abc ¯ B ad f − beµν χ + ef D µ T A,n,cdfν ǫ abc ¯ B ad u beµ u efν h f gλρ D µνλ T A,n,cdgρ ǫ abc ¯ B ad f − efµν χ + be D µ T A,n,cdfν ǫ abc ¯ B ad u beµ u efν h f gλρ D µλρ T A,n,cdgν ǫ abc ¯ B ad h beµν χ + cf D µ T A,n,defν
187 551 ǫ abc ¯ B ad u beµ u efν h f gλρ D νλρ T A,n,cdgµ ǫ abc ¯ B ad h beµν χ + df D µ T A,n,cefν
188 552 ǫ abc ¯ B ad u beµ u fgν h ef λρ D µνλ T A,n,cdgρ ǫ abc ¯ B ad h beµν χ + ef D µ T A,n,cdfν ǫ abc ¯ B ad u beµ u fgν h ef λρ D µλρ T A,n,cdgν ǫ abc ¯ B ad h efµν χ + be D µ T A,n,cdfν iǫ abc ¯ B ad u beµ u cfν h dgλρ σ µν D λ T A,n,efgρ
53 262 ǫ abc ¯ B ad u beµ ∇ ν χ + ,s D µ T A,n,cdeν
189 555 iǫ abc ¯ B ad u beµ u cfν h dgλρ σ µλ D ν T A,n,efgρ
54 263 ǫ abc ¯ B ad u beµ ∇ ν χ + ,s D ν T A,n,cdeµ
190 556 iǫ abc ¯ B ad u beµ u cfν h egλρ σ µν D λ T A,n,dfgρ
55 264 ǫ abc ¯ B ad ∇ µ ∇ ν χ + be γ γ µ T A,n,cdeν
191 557 iǫ abc ¯ B ad u beµ u cfν h egλρ σ µλ D ν T A,n,dfgρ
56 265 iǫ abc ¯ B ad f + beµν χ + cf γ γ µ T A,n,defν
192 558 iǫ abc ¯ B ad u beµ u cfν h egλρ σ µλ D ρ T A,n,dfgν
57 266 iǫ abc ¯ B ad f + beµν χ + df γ γ µ T A,n,cefν
193 559 iǫ abc ¯ B ad u beµ u cfν h egλρ σ νλ D µ T A,n,dfgρ
58 267 iǫ abc ¯ B ad f s, + µν χ + be γ γ µ T A,n,cdeν iǫ abc ¯ B ad u beµ u dfν h egλρ σ µν D λ T A,n,cfgρ iǫ abc ¯ B ad f + beµν χ + ef γ γ µ T A,n,cdfν iǫ abc ¯ B ad u beµ u dfν h egλρ σ µλ D ν T A,n,cfgρ iǫ abc ¯ B ad f + efµν χ + be γ γ µ T A,n,cdfν P ( N f , n SU (2) SU (3) P ( N f , n SU (2) SU (3) iǫ abc ¯ B ad u beµ u dfν h egλρ σ µλ D ρ T A,n,cfgν iǫ abc ¯ B ad f + beµν χ + ,s γ γ µ T A,n,cdeν
195 562 iǫ abc ¯ B ad u beµ u dfν h egλρ σ νλ D µ T A,n,cfgρ iǫ abc ¯ B ad u beµ u cfν χ − dg D µ T A,n,efgν
196 563 iǫ abc ¯ B ad u beµ u efν h cgλρ σ µν D λ T A,n,dfgρ iǫ abc ¯ B ad u beµ u cfν χ − eg D µ T A,n,dfgν
197 564 iǫ abc ¯ B ad u beµ u efν h cgλρ σ µλ D ν T A,n,dfgρ iǫ abc ¯ B ad u beµ u cfν χ − eg D ν T A,n,dfgµ
198 565 iǫ abc ¯ B ad u beµ u efν h cgλρ σ µλ D ρ T A,n,dfgν iǫ abc ¯ B ad u beµ u dfν χ − cg D µ T A,n,efgν
199 566 iǫ abc ¯ B ad u beµ u efν h dgλρ σ µν D λ T A,n,cfgρ iǫ abc ¯ B ad u beµ u dfν χ − eg D µ T A,n,cfgν iǫ abc ¯ B ad u beµ u efν h dgλρ σ µλ D ν T A,n,cfgρ iǫ abc ¯ B ad u beµ u dfν χ − eg D ν T A,n,cfgµ iǫ abc ¯ B ad u beµ u efν h dgλρ σ µλ D ρ T A,n,cfgν iǫ abc ¯ B ad u beµ u efν χ − cg D µ T A,n,dfgν iǫ abc ¯ B ad u beµ u efν h f gλρ σ µν D λ T A,n,cdgρ iǫ abc ¯ B ad u beµ u efν χ − cg D ν T A,n,dfgµ iǫ abc ¯ B ad u beµ u efν h f gλρ σ µλ D ν T A,n,cdgρ iǫ abc ¯ B ad u beµ u efν χ − dg D µ T A,n,cfgν iǫ abc ¯ B ad u beµ u efν h f gλρ σ µλ D ρ T A,n,cdgν iǫ abc ¯ B ad u beµ u efν χ − dg D ν T A,n,cfgµ iǫ abc ¯ B ad u beµ u efν h f gλρ σ νλ D µ T A,n,cdgρ iǫ abc ¯ B ad u beµ u efν χ − f g D µ T A,n,cdgν iǫ abc ¯ B ad u beµ u fgν h ef λρ σ µν D λ T A,n,cdgρ iǫ abc ¯ B ad u beµ u efν χ − f g D ν T A,n,cdgµ iǫ abc ¯ B ad u beµ u fgν h ef λρ σ µλ D ν T A,n,cdgρ iǫ abc ¯ B ad u beµ u fgν χ − cf D µ T A,n,degν
200 575 iǫ abc ¯ B ad u beµ u fgν h ef λρ σ µλ D ρ T A,n,cdgν iǫ abc ¯ B ad u beµ u fgν χ − cf D ν T A,n,degµ ǫ abc ¯ B ad f − beµν f − cf µλ γ γ ν T A,n,defλ
59 285 iǫ abc ¯ B ad u beµ u fgν χ − df D µ T A,n,cegν ǫ abc ¯ B ad f − beµν f − df µλ γ γ ν T A,n,cefλ
60 286 iǫ abc ¯ B ad u beµ u fgν χ − df D ν T A,n,cegµ ǫ abc ¯ B ad f − beµν f − ef µλ γ γ ν T A,n,cdfλ iǫ abc ¯ B ad u beµ u fgν χ − ef D µ T A,n,cdgν ǫ abc ¯ B ad f − beµν f − ef µλ γ γ λ T A,n,cdfν iǫ abc ¯ B ad u beµ u fgν χ − ef D ν T A,n,cdgµ ǫ abc ¯ B ad f − beµν f − cfλρ γ γ µ D νλ T A,n,defρ
61 289 iǫ abc ¯ B ad u beµ u fgν χ − gf D µ T A,n,cdeν ǫ abc ¯ B ad f − beµν f − dfλρ γ γ µ D νλ T A,n,cefρ
62 290 iǫ abc ¯ B ad u beµ u fgν χ − gf D ν T A,n,cdeµ ǫ abc ¯ B ad f − beµν f − efλρ γ γ µ D νλ T A,n,cdfρ iǫ abc ¯ B ad u beµ u cfν χ − ,s D µ T A,n,defν
201 583 ǫ abc ¯ B ad f − beµν f − efλρ γ γ λ D µρ T A,n,cdfν iǫ abc ¯ B ad u beµ u dfν χ − ,s D µ T A,n,cefν
202 584 ǫ abc ¯ B ad h beµν f − cf µλ γ γ ν T A,n,defλ
63 293 iǫ abc ¯ B ad u beµ u efν χ − ,s D µ T A,n,cdfν ǫ abc ¯ B ad h beµν f − cf µλ γ γ λ T A,n,defν
64 294 iǫ abc ¯ B ad u beµ u efν χ − ,s D ν T A,n,cdfµ ǫ abc ¯ B ad h beµν f − df µλ γ γ ν T A,n,cefλ
65 295 iǫ abc ¯ B ad f − beµν χ − cf γ γ µ T A,n,defν
203 587 ǫ abc ¯ B ad h beµν f − df µλ γ γ λ T A,n,cefν
66 296 iǫ abc ¯ B ad f − beµν χ − df γ γ µ T A,n,cefν
204 588 ǫ abc ¯ B ad h beµν f − ef µλ γ γ ν T A,n,cdfλ iǫ abc ¯ B ad f − beµν χ − ef γ γ µ T A,n,cdfν ǫ abc ¯ B ad h beµν f − ef µλ γ γ λ T A,n,cdfν iǫ abc ¯ B ad f − efµν χ − be γ γ µ T A,n,cdfν ǫ abc ¯ B ad h efµν f − beµλ γ γ ν T A,n,cdfλ iǫ abc ¯ B ad h beµν χ − cf γ γ µ T A,n,defν
205 591 ǫ abc ¯ B ad h efµν f − beµλ γ γ λ T A,n,cdfν iǫ abc ¯ B ad h beµν χ − df γ γ µ T A,n,cefν
206 592 ǫ abc ¯ B ad h beµν f − cfλρ γ γ µ D νλ T A,n,defρ
67 301 iǫ abc ¯ B ad h beµν χ − ef γ γ µ T A,n,cdfν ǫ abc ¯ B ad h beµν f − cfλρ γ γ λ D µν T A,n,defρ
68 302 iǫ abc ¯ B ad h efµν χ − be γ γ µ T A,n,cdfν ǫ abc ¯ B ad h beµν f − cfλρ γ γ λ D µρ T A,n,defν
69 303 iǫ abc ¯ B ad f − beµν χ − ,s γ γ µ T A,n,cdeν
207 595 ǫ abc ¯ B ad h beµν f − dfλρ γ γ µ D νλ T A,n,cefρ
70 304 iǫ abc ¯ B ad h beµν χ − ,s γ γ µ T A,n,cdeν
208 596 ǫ abc ¯ B ad h beµν f − dfλρ γ γ λ D µν T A,n,cefρ
71 305 iǫ abc ¯ B ad u beµ ∇ ν χ − cf γ γ µ T A,n,defν
209 597 ǫ abc ¯ B ad h beµν f − dfλρ γ γ λ D µρ T A,n,cefν
72 306 iǫ abc ¯ B ad u beµ ∇ ν χ − cf γ γ ν T A,n,defµ
210 598 ǫ abc ¯ B ad h beµν f − efλρ γ γ µ D νλ T A,n,cdfρ iǫ abc ¯ B ad u beµ ∇ ν χ − df γ γ µ T A,n,cefν
211 599 ǫ abc ¯ B ad h beµν f − efλρ γ γ λ D µν T A,n,cdfρ iǫ abc ¯ B ad u beµ ∇ ν χ − df γ γ ν T A,n,cefµ
212 600 ǫ abc ¯ B ad h beµν f − efλρ γ γ λ D µρ T A,n,cdfν iǫ abc ¯ B ad u beµ ∇ ν χ − ef γ γ µ T A,n,cdfν ǫ abc ¯ B ad h efµν f − beλρ γ γ µ D νλ T A,n,cdfρ iǫ abc ¯ B ad u beµ ∇ ν χ − ef γ γ ν T A,n,cdfµ ǫ abc ¯ B ad h efµν f − beλρ γ γ λ D µν T A,n,cdfρ iǫ abc ¯ B ad u efµ ∇ ν χ − be γ γ µ T A,n,cdfν ǫ abc ¯ B ad h efµν f − beλρ γ γ λ D µρ T A,n,cdfν iǫ abc ¯ B ad u efµ ∇ ν χ − be γ γ ν T A,n,cdfµ ǫ abc ¯ B ad h beµν h cf µλ γ γ ν T A,n,defλ
73 313 iǫ abc ¯ B ad u beµ ∇ ν χ − ,s γ γ µ T A,n,cdeν
213 605 ǫ abc ¯ B ad h beµν h df µλ γ γ ν T A,n,cefλ
74 314 iǫ abc ¯ B ad u beµ ∇ ν χ − ,s γ γ ν T A,n,cdeµ
214 606 ǫ abc ¯ B ad h beµν h ef µλ γ γ ν T A,n,cdfλ ǫ abc ¯ B ad f + beµν χ − cf D µ T A,n,defν
215 607 ǫ abc ¯ B ad h beµν h ef µλ γ γ λ T A,n,cdfν ǫ abc ¯ B ad f + beµν χ − df D µ T A,n,cefν
216 608 ǫ abc ¯ B ad h beµν h cfλρ γ γ µ D νλ T A,n,defρ
75 317 ǫ abc ¯ B ad f s, + µν χ − be D µ T A,n,cdeν ǫ abc ¯ B ad h beµν h dfλρ γ γ µ D νλ T A,n,cefρ
76 318 ǫ abc ¯ B ad f + beµν χ − ef D µ T A,n,cdfν ǫ abc ¯ B ad h beµν h efλρ γ γ µ D νλ T A,n,cdfρ ǫ abc ¯ B ad f + efµν χ − be D µ T A,n,cdfν ǫ abc ¯ B ad h beµν h efλρ γ γ λ D µν T A,n,cdfρ ǫ abc ¯ B ad f + beµν χ − ,s D µ T A,n,cdeν
218 611 ǫ abc ¯ B ad u beµ ∇ µ f − cfνλ γ γ ν T A,n,defλ
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