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Modelling Rho-Pion-Electromagnetic Interactions with an Effective View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Modelling Rho-Pion-Electromagnetic Interactions with an Effective Weinberg-Salam Theory Nathan F. Lepora Department of Applied Mathematics and Theoretical Physics, Cambridge University, England (October 28, 1999) We propose an effective gauge theory of pions, rho mesons and electromagnetism. The theory is analogous to the Weinberg-Salam model of electroweak interaction, with the pions associated with the Goldstone bosons, the rho mesons corresponding to the W and Z gauge bosons, and electro- magnetism treated similarly in both cases. To test the model we examine two distinct experimental situations: firstly, decays of the rho meson, and secondly, properties of the pion-pion interaction. Our results are compatible with experiment. PACS numbers: 12.38.Aw, 12.90.+b, 13.20.Jf, 13.75.Lb I. INTRODUCTION where data is available. The particular reactions we con- sider are: At low energies quantum chromodynamics is an empir- 0 + ically non-perturbative description of the strong interac- ρ π π− → 0 tions. Consequently it is difficult to calculate implica- ρ± π±π → tions of the theory. To obtain results one method is to ρ± π±γ construct effective theories so that approximate results 0 → + ρ π π−γ may be obtained in certain circumstances. In their con- → 0 ρ± π±π γ struction some general properties of interaction, such as → symmetry and local invariance, are sufficiently restrictive The last decay has not yet been measured and represents to specify the form of the theory. Theories constructed a prediction of the model. The second experimental sit- along these lines often give very good approximations uation we discuss is pion-pion interactions. Specifically of the experimental data, although their predictions are processes of the form valid only for a limited range of phenomena. Chiral perturbation theory [1] represents a good low π+π+ π+π+ energy effective theory for the interaction of pions, giving → π+π π+π an accurate description pion-pion scattering near thresh- − − + 0 → + 0 old [2]. However, its results are less accurate at higher π π π π → energies. In particular it is unable to describe the rho π0π0 π0π0 resonance, which occurs at about 770 MeV. + → 0 0 π π− π π . In this paper we present an effective theory that de- → scribes the interactions of rho mesons, pions and electro- We find that the general form of the partial wave ampli- magnetism. It differs from chiral perturbation theory, in tudes correspond well to the experimental results. that it has a local gauge theoretic structure based upon The plan of this paper is as follows. Firstly we dis- the symmetry breaking cuss chiral symmetry breaking and the construction of the linear sigma model. This introduces our construc- SU(2) U(1) U(1)Q, × → tion of an effective theory for rho-pion-electromagnetic interactions. Using this description we examine two dis- with SU(2) a gauge symmetry relating to the rho de- tinct experimental situations: firstly, the decays of the grees of freedom, U(1) a phase rotation and U(1) the Q rho meson, and secondly, the interaction of two pions. Fi- electromagnetic gauge symmetry. All of the symmetries nally we discuss some implications and further prospects are local. Essentially the above symmetry breaking is the of this work. Weinberg-Salam model applied to hadronic interactions. Pions correspond to the Goldstone bosons, whilst the rho meson triplet correspond to the W and Z gauge bosons. II. CHIRAL SYMMETRY BREAKING Electromagnetism is embedded similarly in both cases. To test the validity of such a scheme we evaluate tree It will prove useful to quickly review chiral symmetry level processes corresponding to two distinct experimen- breaking in the linear sigma model [1], and the construc- tal situations. Firstly we examine decays of the rho tion of fermionic composites from the constituent u and d meson into different combinations of pions and photons. quarks. To stress an analogy with electroweak theory we Our results are in reasonable agreement with experiment shall re-express the analysis in terms of a pseudo-scalar 1 µ 1 µ 1 µ complex doublet. This is completely equivalent to the V = 2 12ω + 2 ρa σa, (10) usual approach, where the pseudo-scalars are defined in a terms of a complex two-by-two matrix. where σ are the Pauli spin matrices. We complete this section with a discussion of elec- The above mesons transform as multiplets under an tromagnetism within the pion spectrum and interaction, element (gL,gR) of the chiral symmetry such that the which will prove useful in the next section of this paper. pseudo-scalar mesons transform as ˜ Φ=¯qL>γ5qR> +¯qR>γ5qL> A. Construction of the Fermion Composites g† q¯>γq> gR+g† q¯>γq> gL. (11) → L· L 5 R· R· R 5 L· We shall start by considering the u and d quark doublet whilst the pseudo-vector mesons transform as µ µ µ u V =¯qL>γγ5qL>+¯qR>γ γ5qR> q = . (1) µ µ d g† q¯>γ γ q> gL+g† q¯>γ γ q> gR. (12) → L· L 5 L · R· R 5 R · This is, to good approximation, massless when compared One should note that the components of the pseudo- to the QCD energy scale Λ 200 300 MeV. There- vector and pseudo-scalar mesons transform differently QCD ∼ − fore it can be represented by the Lagrangian under the chiral symmetry. It is conventional to express the above pseudo-scalar µ = iqγ¯ ∂µq. (2) L multiplet in a slightly different form so as to simplify its chiral transformation properties. Considering instead This is symmetric under the global flavour symmetry Φ= 11 σ+1iπ σ , (13) q SU(2) q. (3) 2 2 2 a a → · this transforms in a simpler fashion Associated with the fermion doublet q are the two chi- ral eigenstates Φ g† Φ gR, (14) → L · · 1 1 qL = (1 + γ )q, qR = (1 γ )q, (4) 2 5 2 − 5 under the SU(2)L SU(2)R chiral symmetry. We should mention× that under the flavour symmetry such that q = q + q . Then the Lagrangian (1) splits L R SU(2) an element (g,g)actsonΦandVµ as into two parts L+R µ µ Φ g† Φ g, (15) = iq¯Lγ ∂µqL + iq¯Rγ ∂µqR. (5) µ → · µ· L V g† V g, (16) → · · Thus is in fact invariant under a larger global SU(2)L L × i.e. as the adjoint action on each. This symmetry decom- SU(2)R chiral symmetry group, poses the multiplets into their mass eigenstates with σ qL SU(2)L qL,qR SU(2)R qR, (6) and ω transforming as singlets under SU(2)L+R, whereas → · → · µ πa and ρa transform as triplets. Thus the flavour sym- that contains the flavour symmetry as SU(2)L+R. metry SU(2)L+R is associated with strong isospin. Due to the strong interaction the u and d quarks con- fine into hadronic composites. We shall consider the fol- lowing matrix of pseudo-scalar mesons, which arises nat- B. Linear Sigma Model urally from combinations of fermions Interactions of the pseudo-scalar mesons may be mod- ˜ uγ¯ 5u uγ¯ 5d elled through the Lagrangian [1] Φ=(γ5)ij q¯i qj =¯q>γ5q>= ¯ ¯ , (7) ⊗ dγ5u dγ5d µ 1 2 2 [Φ] = tr ∂ Φ†∂µΦ λ(tr(Φ†Φ) f ) . (17) and the analagous object for the pseudo-vector mesons L − − 2 π It is straightforward to verify from the above that this µ µ µ µ uγ¯ γ5u uγ¯ γ5d theory is symmetric under the action of the SU(2) V =¯q>γ γ5q>= ¯ µ ¯ µ . (8) L dγ γ5u dγ γ5d × SU(2)R chiral symmetry. From these one identifies the pion and sigma pseudo- The above theory models the effects of quark confine- ment through breaking the chiral symmetry SU(2)L scalar fields × SU(2)R to its diagonal subgroup SU(2)L+R.Belowthe ˜ 1 1 scale of symmetry breaking the pseudo-scalar mesons Φ= 12σ+ πaσa, (9) 2 2 split into mass eigenstates transforming under the bro- and the rho and omega pseudo-vector fields ken symmetry. The sigma meson become massive whilst 2 the pions correspond to massless Goldstone bosons. This C. Inclusion of Electromagnetism seems to hold approximately, as the pion has a small mass mπ 140 MeV compared to the mass of the sigma me- The linear sigma model only relates to the strong ∼ son, estimated to be around 400 1200 MeV. isospin symmetry within the strong interaction. How- − For later sections of this paper it will be convenient ever within the pion triplet the combinations to write the above Lagrangian in a slightly different, but equivalent form. Defining instead Φ to be a complex π1 + iπ2 π1 iπ2 π = ,π†= − (27) doublet √2 √2 σ+iπ Φ= 0 (18) have electromagnetic charge. Thus to describe the prop- π + iπ 1 2 erties of pions one should also include electromagnetic we may rewrite the above Lagrangian as effects within the linear sigma model. µ 2 2 Electromagnetism may be introduced by introduc- [Φ] = ∂ Φ†∂µΦ λ(Φ†Φ f ) . (19) L − − π ing an electromagnetic gauge field Aµ associated with Then the chiral symmetries act in an associated way upon an invariance under the local transformations g(x)= Φ such that the Lagrangian is still invariant. The left exp(ieθ(x)) U(1)Q such that SU(2)L acts upon Φ as ∈ π exp(ieθ(x))π, (28) Φ exp(iαaσa)Φ, (20) 7→ 7→ Aµ Aµ ie∂µθ(x).
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