Zero temperature properties of mesons and baryons from an extended linear sigma-model
aa r X i v : . [ h e p - ph ] F e b Zero temperature properties of mesons and baryonsfrom an extended linear sigma-model
P Kov´acs and Gy Wolf
Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, HungarianAcademy of Sciences, XII. Konkoly Thege Mikl´os ´ut 29-33, 1121 Budapest, HungaryE-mail: [email protected]
Abstract.
An extended linear sigma model with mesons ( q ¯ q states) and baryons ( qqq states) ispresented. The model contains a low energy multiplet for every hadronic particle type, namely ascalar, a pseudoscalar, a vector and an axialvector nonet, a baryon octet and a baryon decuplet.The model parameters are determined through a multiparametric minimalization with the helpof well known physical quantities. It is found that the considered zero temperature quantities(masses and decay widths) can be described well at tree-level and are in good agreement withthe experimental data.
1. Introduction
The vacuum properties of strong interaction are very hard to investigate within the frameworkof QCD – the fundamental theory of strong interaction – (see e.g.[1]), which is due to its subtletyat low energies. Consequently, instead of solving QCD one can set up an effective theory, whichreflects some properties of the original theory. The underlying principle in the construction ofsuch theories is that they share the same global symmetries as QCD.For zero masses of the u , d and s quarks, the global symmetry of QCD is U (3) R × U (3) L ,the so-called chiral symmetry. In the vacuum, this chiral symmetry is spontaneously brokendue to the existence of a quark-antiquark condensate. The chiral symmetry can be realizedin different ways, nonlinearly [2] and linearly [3], which we choose here. Accordingly, in thispaper we set up an extended linear sigma model, which contains mesonic and baryonic degreesof freedom. The previous versions of our model [4, 5] contained the scalar, pseudoscalar, vector-and axial-vector nonets. The vacuum phenomenology of mesons was described very well in thatmodel. As an improvement, in this paper we include additionally the nucleon-octet and theDelta-decuplet to extend the vacuum phenomenology for baryons as well. Another approach tobaryon phenomenology can be found in [6].Our paper is organized as follows. In the Sec. 2 we briefly present the model, while in Sec. 3we describe how to calculate various tree-level quantities. The Sec. 4 is dedicated to the resultsin the mesonic and baryonic sector and finally we conclude in Sec. 5.
2. The Model
The model can be defined through a Lagrangian consisting of a purely mesonic and a baryonic-mesonic part as L = L meson + L baryon . The terms in the meson part are limited by chiral anddilaton symmetry (for details see [5]), while in case of the baryon part we included all the SU (3) V nvariants which can produce baryon masses – with different masses for different particles in thegiven multiplet – and decuplet decays with the lowest possible dimension ( B − B − Φ − Φ,∆ − ∆ − Φ − Φ and ∆ − B − Φ terms). The mesonic part has the following form L meson = Tr[( D µ Φ) † ( D µ Φ)] − m Tr(Φ † Φ) − λ [Tr(Φ † Φ)] − λ Tr(Φ † Φ) −
14 Tr( L µν + R µν ) + Tr (cid:20)(cid:18) m (cid:19) ( L µ + R µ ) (cid:21) + Tr[ H (Φ + Φ † )]+ c (det Φ − det Φ † ) + i g { L µν [ L µ , L ν ] } + Tr { R µν [ R µ , R ν ] } )+ h † Φ) Tr( L µ + R µ ) + h Tr[( L µ Φ) + (Φ R µ ) ] + 2 h Tr( L µ Φ R µ Φ † ) . + g [Tr( L µ L ν L µ L ν ) + Tr( R µ R ν R µ R ν )] + g [Tr ( L µ L µ L ν L ν ) + Tr ( R µ R µ R ν R ν )]+ g Tr ( L µ L µ ) Tr ( R ν R ν ) + g [Tr( L µ L µ ) Tr( L ν L ν ) + Tr( R µ R µ ) Tr( R ν R ν )] , (1)where D µ Φ ≡ ∂ µ Φ − ig ( L µ Φ − Φ R µ ) − ieA e µ [ T , Φ] ,L µν ≡ ∂ µ L ν − ieA e µ [ T , L ν ] − { ∂ ν L µ − ieA e ν [ T , L µ ] } ,R µν ≡ ∂ µ R ν − ieA e µ [ T , R ν ] − { ∂ ν R µ − ieA e ν [ T , R µ ] } , The quantities Φ = P i =0 ( S i + iP i ) T i , L µ /R µ = P i =0 ( V µi ± A µi ) T i represent the scalar-pseudoscalar nonets and the left-/right-handed vector nonets. T i ( i = 0 , . . . ,
8) denote thegenerators of U (3), while S i represents the scalar, P i the pseudoscalar, V µi the vector, and A µi the axial-vector meson fields, and A e µ is the electromagnetic field. H and ∆ are some constantexternal fields. It should be noted that in the (0 −
8) sector of the scalars and pseudoscalarsthere is a mixing and it is more suitable to use the non strange – strange basis defined as ϕ N = 1 / √ √ ϕ + ϕ ), ϕ S = 1 / √ ϕ − √ ϕ ) for ϕ i ∈ ( S i , P i , V µi , A µi ).Moving on to the baryonic-mesonic part, the Lagrangian is given by L baryon = Tr (cid:2) ¯ B (cid:0) iD/ − M (8) (cid:1) B (cid:3) − Tr (cid:8) ¯∆ µ · (cid:2)(cid:0) iD/ − M (10) (cid:1) g µν − i ( γ µ D ν + γ ν D µ ) + γ µ (cid:0) iD/ + M (10) (cid:1) γ ν (cid:3) ∆ ν (cid:9) + C Tr (cid:20) ¯∆ µ · (cid:18) − f ( ∂ µ − ieA eµ [ T , Φ]) − f [Φ , V µ ] + A µ (cid:19) B (cid:21) + h. c. − ξ Tr (cid:0) ¯ BB (cid:1) Tr (cid:16) Φ † Φ (cid:17) − ξ Tr (cid:16) ¯ B {{ Φ , Φ † } , B } (cid:17) − ξ Tr (cid:16) ¯ B [ { Φ , Φ † } , B ] (cid:17) − ξ (cid:16) Tr (cid:0) ¯ B Φ (cid:1) Tr (cid:16) Φ † B (cid:17) + Tr (cid:16) ¯ B Φ † (cid:17) Tr (Φ B ) (cid:17) − ξ Tr (cid:16) ¯ B { [Φ , Φ † ] , B } (cid:17) (2) − ξ Tr (cid:16) ¯ B [[Φ , Φ † ] , B ] (cid:17) − ξ (cid:16) Tr (cid:0) ¯ B Φ (cid:1) Tr (cid:16) Φ † B (cid:17) − Tr (cid:16) ¯ B Φ † (cid:17) Tr (Φ B ) (cid:17) − ξ (cid:16) Tr (cid:16) ¯ B Φ B Φ † (cid:17) − Tr (cid:16) ¯ B Φ † B Φ (cid:17)(cid:17) + χ Tr (cid:0) ¯∆ · ∆ (cid:1) Tr (cid:16) Φ † Φ (cid:17) + χ Tr (cid:16) ( ¯∆ · ∆) { Φ , Φ † } (cid:17) + χ Tr (cid:16) ( ¯∆ · Φ)(Φ † · ∆) + ( ¯∆ · Φ † )(Φ · ∆) (cid:17) + χ Tr (cid:16) ( ¯∆ · ∆)[Φ , Φ † ] (cid:17) , where B = √ P i =1 B a T a and ∆ µ stands for the baryon octet and decuplet. M (8) and M (10) are the bare masses of the baryon octet and decuplet. f is the pion decay constant, while [ , ] A 2 by 2 segment of the mass matrix consisting of the components: 00 , , , nd { , } denote the commutator and the anticommutator. Here the meson-baryon interactionterms are all the possible SU (3) V invariants that can be written down with the given numberof fields [7, 8]. The covariant derivatives are defined as D µ B = ∂ µ B + i [ B, V µ ] + 1 f { [ A µ , Φ] , B } ,D µ ∆ ijkν = ∂ ∆ ijkν + (cid:18) f [ A µ , Φ] il − iV iµ l (cid:19) ∆ ljkν + (cid:18) f [ A µ , Φ] jl − iV jµ l (cid:19) ∆ ilkν + (cid:18) f [ A µ , Φ] kl − iV kµ l (cid:19) ∆ ijlν , and the following dot notation is used:( ¯∆ · ∆) mk ≡ ¯∆ ijk ∆ ijm , ( ¯∆ · Φ) mk ≡ ¯∆ ijk Φ il ǫ jlm , (Φ · ∆) mk ≡ ∆ ijm Φ li ǫ jlm . (3)From the given Lagrangian various tree-level quantities such as masses and decay widths canbe calculated and can be used to determine the unknown parameters of the model. During thisparametrization process we can check how well the physical spectrum is reproduced. In the nextsection calculation of tree-level quantities and the parametrization are discussed.
3. Tree-level quantities and parametrization
As a standard procedure in a spontaneously broken theory we assume non-zero vacuumexpectation values (vev) to certain fields, in our case to the σ N ≡ / √ √ σ + σ ) and σ S ≡ / √ σ − √ σ ) scalar fields, and denote their vev by φ N and φ S . After that the σ N and σ S fields are shifted with their non zero vev’s φ N and φ S . Consequently, the quadraticand three-coupling terms of the Lagrangian can be determined from which the masses and thedecay widths originate. However, it should be noted that as a technical difficulty – due to the σ N and σ S field shifts – different particle mixings emerge. In detail, there will be mixings inthe N − S (or 0 −
8) sector of the scalar and pseudoscalar octets and between the vector-scalarand axialvector-pseudoscalar nonets. The N − S mixings can be resolved by some orthogonaltransformation, while the other mixings by redefinition of certain (axial-)vector fields. Thedetails can be found in [5] together with explicit expressions for the meson masses and variousdecay widths.Regarding the baryon sector the tree-level octet and decuplet masses – from the terms of theLagrangian quadratic in the fields B and ∆ µ – are found to be m p = m n = M (8) + 12 ξ (Φ N + 2Φ S ) + 12 ξ (Φ N − S ) ,m Ξ = M (8) + 12 ξ (Φ N + 2Φ S ) − ξ (Φ N − S ) ,m Σ = M (8) + ξ Φ N ,m Λ = M (8) + 13 ξ (Φ N + 4Φ S ) + 13 ξ (Φ N − √ S ) , m ∆ = M (10) + 12 χ Φ N ,m Σ ⋆ = M (10) + 13 χ (Φ N + Φ S ) + 16 χ (Φ N − √ S ) ,m Ξ ⋆ = M (10) + 16 χ (Φ N + 4Φ S ) + 16 χ (Φ N − √ S ) ,m Ω = M (10) + χ Φ N . (4)Beside the masses one can consider two-body decays of the decuplet baryons. According to PDG[9] there are four such physically allowed decays,∆ → pπ, Σ ⋆ → Λ π, Ξ ⋆ → Ξ π, Σ ⋆ → Σ π. (5)nd the decay widths are given byΓ ∆ → πp = k m ∆ ( m p + E p ) G , Γ Σ ⋆ → π Λ = k ⋆ m Σ ⋆ ( m Λ + E Λ ) G , Γ Ξ ⋆ → π Ξ = k ⋆ m Ξ ⋆ ( m Ξ + E Ξ ) G , Γ Σ ⋆ → π Σ = k ⋆ m Σ ⋆ ( m Σ + E Σ ) G , (6)with G = C Z π (cid:18) w a + 1 f (cid:19) . As it can be seen from the Lagrangian there are 30 unknown parameters of the model, 14 in themeson sector and 16 in the baryon sector. However some of them can be set to zero withoutthe loss of generality, some of them not even appear in the formulas of the physical quantitiesconsidered here, while some of them appear only in certain combinations. All in all there is19 parameters which should be determined. These parameters are determined through thecomparison of the calculated tree-level expressions – from which we have 23 – of the spectrumand decay widths with their experimental value taken from [9] with artificially increased errors(5% for the masses and 10% for the decay widths, since we do not expect from a tree-levelmodel to be more precise). Our strategy is that first we set the parameters of the meson sector[5] and then we fit the remaining parameters of the meson-baryon interaction terms. For thiswe used a χ -minimalization, which was realized with a multiparametric minimalization code(MINUIT [10]).
4. Results
Using the above mentioned χ -minimalization method in the meson sector we obtain resultssummarized in Table 1, which can also be found in [5].In the baryon sector at first we only investigated the decuplet decays. The results are givenin Table 2.It can be seen that the meson observables are reproduced very well in general, while thedecuplet decays show a fair correspondence. It is worth to note that in case of the the decupletdecays their tree-level expressions Eq. (6) only differ in their kinematic parts and we have onlyone parameter to fit for the four observables, which can cause the deviation from the experimentalvalue.
5. Conclusion
We have presented an extended linear sigma model with meson and baryon degrees of freedom.This is a possible extension with baryon octet and decuplet of our previous meson model [5].We included interaction terms, such as ∆ − B − P and B − B − Φ − Φ to describe ∆ decaysand baryon masses. We calculated the tree-level masses and physically relevant decuplet decaywidths and we found that in general they are in good agreement with the experimental datataken from the PDG [9].As a continuation we plan to include the baryon masses to the fit and to add other (higherdimension) interaction terms containing derivatives which are important in case of scatteringprocesses [6]. Our further aim is to go to finite temperature and/or densities with all these fieldsincluded in our model.
Acknowledgments
P. Kov´acs and Gy. Wolf were partially supported by the Hungarian OTKA funds NK101438and K109462. The isospin violation in some cases (e.g. for pion) has the order of 5% able 1.
Calculated and experimental values of meson observablesObservable Fit [MeV] PDG [MeV] Error [MeV] m π . ± . . ± . m K . ± . . ± . m η . ± . . ± . m η ′ . ± . . ± . m ρ . ± . . ± . m K ⋆ . ± . . ± . m φ . ± . . ± . m a ± ± m f (1420) . ± . . ± . m a ± ± m K ⋆ ± ± ρ → ππ . ± . . ± . K ⋆ → Kπ . ± . . ± . φ → ¯ KK . ± .
14 3 . ± . a → ρπ ±
43 425 ± a → πγ . ± .
01 0 . ± . f (1420) → K ⋆ K . ± . . ± . a ±
12 265 ± K ⋆ → Kπ ±
12 270 ± Table 2.
Calculated and experimental values of baryon observablesObservable Fit [MeV] PDG [MeV] Error [MeV]Γ ∆ → pπ . . ± . Σ ⋆ → Λ π . . ± . Σ ⋆ → Σ π . . ± . Ξ ⋆ → Ξ π . . ± . References [1] Peskin M E and Schroeder D V 1995
An Introduction To Quantum Field Theory (Westview Press)[2] Weinberg S 1968
Phys. Rev.
Rev. Mod. Phys. Phys. Rev. D Phys. Rev. D Commun. Theor. Phys. Commun. Math. Phys. On the quark-mass dependence of baryon ground-state masses (Ph.D. Thesis, TU Darmstadt)http://tuprints.ulb.tu-darmstadt.de/2360/[9] Beringer J et al (Particle Data Group) 2012
Phys. Rev. D Comput. Phys. Commun.10