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 electromagnetism 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 consider 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 chiral 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 confine into hadronic composites. We shall consider the following 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 confinement 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). (29) whilst SU(2)R act as 7→ − Then the residual Lagrangian of Eq. (25) is altered Φ exp(iβaτa)Φ, (21) 7→ accordingly: L where τa = iσ2K, σ2K, 12 , with K the complex { } { − } µ 1 µν 1 µ 1 0 µ 0 conjugation operator. Notice that the phase rotation [σ, πa ,A ]= F Fµν + ∂µσ∂ σ + ∂µπ ∂ π L −4 2 2 Φ exp(iβ3)Φ is a subgroup of SU(2)R. µ 2 2 + D π†Dµπ + λf σ +h.o.t. (30) 7→One should note that the above form of the sigma π model is the same as the scalar sector of the Weinberg- with the covariant derivative Salam model. The SU(2)L is equivalent to weak isospin, µ µ µ whereas the SU(2)R is a global symmetry of the scalar D = ∂ + ieA , (31) sector but not a symmetry of the electroweak gauge sector [3]. and the electromagnetic field tensor Symmetry breaking F µν = ∂ν Aµ ∂µAν . (32) − SU(2)L SU(2)R SU(2)L+R, (22) × → Of course one should also include the pionic mass terms is achieved through minimisation of Eq. (17), with a typ- of Eq. (26). ical vacuum of the form f Φ = 1 π . (23) 0 √2 0 III. EFFECTIVE RHO-PION-ELECTROMAGNETIC The degenerate equivalent vacua collectively form the INTERACTIONS vacuum manifold 3 M = SU(2)L SU(2)R Φ = S . (24) Before postulating the form of the effective theory we × · 0 ∼ Then the residual theory is obtained upon expansion shall motivate it by discussing a couple of topics. Firstly, around the vacuum we shall discuss the possible form of the interaction terms between the rho mesons and the pseudo-scalars. These [σ, πa ]= [Φ0 +Φ] interaction terms must be such that they satisfy the rel- L L 1 µ 1 0 µ 0 1 µ 1 1 evant symmetries. Secondly, we discuss how to consis- = 2∂µσ∂ σ + 2 ∂µπ ∂ π + 2 ∂ π ∂µπ + 1 ∂µπ2∂ π2 + λf 2σ2 +h.o.t. (25) tently include the electromagnetic interactions of pions. 2 µ π By attempting to describe both of these simultaneously and has a global SU(2)L+R symmetry. we shall motivate our model. Because of the non-zero quark masses and hadronic effects, the pions are not to be massless, although their masses are small compared to typical hadronic mass A. Rho-pion interactions scales. Thus a pionic mass term must be included by hand into Eq. (25) Our first motivation is to consider the form of the possi- 1 2 i i ble interaction terms between rho mesons and pions that m = mππ π . (26) L 2 have the required symmetry properties.

3 We shall consider the pseudo-scalar field to be defined π0 π0 (42) 7→ as in Eq. (13) π eiθπ. (43) 7→ 1 1 Φ= 212σ+2iσaπa, (33) The problem is that since SU(2)L is a global symmetry then U(1) must also be global. This is incompatible with the transformation properties of Eq. (14) Q with introducing the electromagnetic interaction. Thus to include electromagnetic interactions it is nec- Φ gL† φ gR. (34) → · · essary to make SU(2)L a local symmetry: then the sym- Instead of the pseudo-vector field V µ we shall deal with metry breaking in Eq. (41) would be the Weinberg-Salam the rho meson component model with a naturally local U(1)Q. It is natural to as- sociate the left handed rho mesons as the gauge bosons µ 1 µ ρ = iσk ρ . (35) of this local symmetry. 2 · k Because of the symmetry between the left and right According to Eq. (12) these decomposes into two compo- chiral symmetry then one may also consider just the right nents handed rho meson. Then the counterpart to Eq. (41) is ρµ = ρµ + ρµ , (36) L R SU(2)R U(1)L U(1)Q. (44) × → which transform as However one may not incorporate electromagnetism suc- µ µ µ cessfully as a gauge theory into the linear sigma model if ρ g† ρ gL + g† ρ gR. (37) → L · L · R · R · one considers both the left and right sectors together. Now we shall consider possible interaction terms of the rho mesons with the pseudo-scalars. These split into two C. Construction of the Model possible types, a left and a right part

= L + R, (38) Guided by the discussions above we shall attempt to L L L motivate a theory of interacting pseudo-scalar and rho which are invariant under only the left chiral symmetry mesons, including electromagnetic eﬀects. The theory and the only right chiral symmetry, respectively should satisfy the following criteria:

µ µ µ (1) The particle spectrum corresponds to a pseudo-scalar L[Φ,ρL]=cL1(∂µΦ†ρLΦ+Φ†ρL∂ Φ) µ µ L doublet Φ, a rho triplet ρi and a U(1) gauge field B . µ µ + cL2(ρLΦ)†(ρLµΦ), (39) The photon A is associated with a linear combination µ µ µ of ρµ and Bµ. R[Φ,ρR]=cR1(∂µΦρRΦ†+ΦρR∂ Φ†) 3 L µ (2) It is based upon an SU(2) U(1) local symmetry, +cR2(Φρ )†(ΦρRµ). (40) R with the associated gauge bosons× the rho mesons and It is important to notice that the two interactions are the photon. Their dynamics are governed by of the same form, although their coefficients may differ. Rµν = ∂ν ρµ ∂µρν +˜g[ρµ,ρν], (45) However there are no interaction terms that are invariant − under both the left and right chiral symmetries. Bµν = ∂ν Bµ ∂µBν , (46) − the associated field tensors. B. Pion-Electromagnetic Interactions (3) The interaction of the rho mesons and electromagnetism with Φ is specified through minimal coupling in Our second motivation is to attempt including elec- the covariant derivative tromagnetic effects in the sigma model. The idea is to µ µ 1 µ 1 µ = ∂ + ieB˜ + igρ˜ σi, (47) incorporate a U(1) gauge symmetry such that the resid- D 2 2 i ual theory takes the form in Eq. (30). whereg ˜ is the gauge coupling associated with ρµ gauge As a first try consider just the left handed rho me- field ande ˜ is associated with Bµ. son. Then electromagnetism may be included through (4) Condensation of the pseudo-scalar field breaks the the symmetry breaking SU(2) U(1) symmetry to a residual electromagnetic × gauge symmetry U(1)Q. The electromagnetic gauge field SU(2)L U(1)R U(1)Q, (41) × → couples to charged pions through the covariant derivative with U(1)R a gauge symmetry representing a local phase Dµ = ∂µ + ieAµ, (48) rotation of Φ and SU(2)L the global ‘left’ strong isospin symmetry. The embedding of the U(1)Q is of the correct where e is the electromagnetic coupling constant. form taking (5) After symmetry breaking extra mass terms must be

4 introduced by hand to represent the fact that the con- associated with the corresponding generators stituent quarks are not massless. 1 αX cos θ sin θ 1igσ˜ One should note that the SU(2) symmetry is not the 2 0 s s 2 3 = − 1 . (59) eXQ sin θs cos θs ie˜12 SU(2)L or the SU(2)R symmetry but a new symmetry 2 of the Lagrangian (49) below. If one were to consider Here θs we shall call the ‘strong mixing angle’, with the the rho meson to consist of only ρL or ρR then it would relation coincide with a chiral symmetry. However, we are considering the rho meson to consist of both the left and e˜ tan θs = . (60) right components. g˜ Basing a theory upon the above points leads to the specifying Lagrangian 10 00 µ µ µ µ 1 µ X0 = i − ,XQ=i , (61) [Φ,ρi ,B ]= g[ρi,B ]+ 2 Φ† µΦ V[Φ] (49) 0cos2θs 01 L µ µ L1 µν D1 µνD − g[ρ ,B ]= trR Rµν B Bµν (50) L i −4 − 4 where α = g/ cos θ , e =˜ecos θ . 1 2 2 s s V [Φ] = λ(Φ†Φ f ) . (51) 4 − π It is also useful to express the rho meson gauge bosons in a charge eigenstate basis It has an invariance under SU(2) U(1) local transformations, such that a gauge transformations× exp(ieθ˜ (x)/2) µ µ µ µ µ ρ1 + iρ2 µ ρ1 iρ2 ∈ ρ = ,ρ†= − , (62) U(1)R acts as √2 √2 Φ exp(ieθ˜ (x)/2)Φ (52) with the charge neutral component ρ0 defined in Eq. (58). 7→ Bµ Bµ ie∂˜ µθ(x), (53) From the non-vanishing of the quark masses we now 7→ − include the following mass terms by hand whilst a gauge transformation l(x)=exp(igα˜ (x)σ /2) i i 1 2 0 0µ µ 1 2 0 0 ∈ = m (ρ ρ +2ρ†ρ )+ m (π π +2π†π). (63) SU(2)L takes Lm 2 0 µ µ 2 1 Φ l Φ (54) We are neglecting the small mass difference between the µ 7→ · µ 1 µ 1 charged and neutral pions. ρ l ρ l− ig∂˜ l l− , (55) 7→ · · − · Putting this all together we obtain a residual La- grangian of the form The potential is of the Landau form in (17), to break the SU(2) U(1) gauge symmetry. = g + + + + + , (64) × L L L1 L2 L3 Lint Lunphys with g as in Eq. (50) and D. Symmetry Breaking L µ 2 2 µ = D π†Dµπ + m π†π + m ρ† ρ , (65) L1 π ρ µ 1 µ 0 0 1 2 0 0µ 1 2 0 0 Symmetry breaking is achieved exactly as in the 2 = ∂ π ∂µπ + m ρ ρ + m π π , (66) L 2 2 ρ0 µ 2 π0 Weinberg-Salam model. The residual theory is obtained 1 µ 1 2 = ∂ σ∂µσ + m σσ. (67) upon expansion around the vacuum, with the rho mass L2 2 2 σ eigenstates representing a linear mixing of the SU(2) and The corresponding particle masses are, using Eq. (63) U(1) gauge fields. This mixing is described by an analo- m2 = 1 f 2g˜2 + m2,m2 =1f2α2+m2. (68) gous angle to the weak mixing angle. ρ 4 π 0 ρ0 4 π 0 A typical vacuum Interactions between the pseudo-scalar mesons, photons and rho mesons split into terms of different forms 1 fπ Φ(x)=Φ0 = √ (56) 2 0 A B C int = + + . (69) L Lint Lint Lint breaks [Φ,ρµ,Bµ] to a residual theory L a The explicit forms are given here. In Appendix B we present the specific associated Feynman diagrams. σ+iπ0 [σ, π, ρµ,Bµ]= Φ + 1 (57) (A) Single rho, two pseudo-scalar mesons and electro- L L 0 √2 π magnetism

To exhibit the residual theory one must change the A 1 µ µ 0 = iα cos 2θs(π†D π D π†π)ρ vector meson and photon basis to a basis of mass eigen- Lint 2 − µ 1 µ 0 0 µ 1 µ 0 0 µ states + g˜(π†∂ π π D π†)ρµ + g˜(π∂ π π D π)ρ† 2 − 2 − µ 1 µ µ 1 µ µ 0µ µ + ig˜(σD π† π†∂ σ)ρµ + ig˜(σD π π∂ σ)ρµ† ρ cos θs sin θs ρ3 2 − 2 − µ = − µ , (58) 1 µ 0 0 µ 0 A sin θs cos θs B + α(σ∂ π π ∂ σ)ρ . (70) 2 − µ

5 (B) Two rho’s and a pseudo-scalar meson This implies that the neutral rho coupling α is approximately coincident with the charged rho couplingg ˜. B 1 2 µ µ 0 int = iαgf˜ π sin θs(πρ† + π†ρ )ρµ Although such a value,g ˜ 12.2, of the strong isospin L − 2 ∼ 1 2 µ 1 2 0µ 0 gauge coupling seems rather high when comparing to the + g˜ fπσρ ρµ† + α fπσρ ρµ. (71) 2 2 standard model weak coupling g 0.6, it is typical of ∼ (C) Two rho’s and two pseudo-scalar mesons the strong interaction. Examination of pion-proton scattering yields a typical coupling in the region of fifteen or C 1 2 2 0 0 0 0µ = α (2 cos 2θsπ†π + π π + σσ)ρ ρ so. This may be simply seen by comparing the photon- Lint 8 µ 1 2 0 0 µ proton Thompson scattering cross section + 4 g˜ (2π†π + π π + σσ)ρµ† ρ + 1 g˜2 sin2 θ (π ρµ + ρµ π)σρ . (72) e2 2 4 s † † µ σ(eγ)= 2 (4πR2) (78) 3 4π p Finally there are some extra terms that are unphysical, since they represent a mixing between the rho mesons and with an analogous formula for the pion-proton cross sec- the pions tion 2 2 1 µ µ 0 0µ g˜ 2 unphys = igf˜ π(Dµπ†ρ ρµ† D π)+αfπ∂µπ ρ (73) σ(πp)= (4πR ), (79) L 2 − 4π p In Weinberg-Salam theory they are eliminated through where Rp is the Compton wavelengths of the proton. taking the unitary gauge. Here we may not perform such Comparison of the experimental results implies thatg ˜ a unitary transformation because of the pion mass terms. has a value in the region of 10 to 15. Instead we shall not include the above terms in the following analysis. However, it should be noted that the term may be eliminated through a redefinition of the IV. RHO MESON DECAYS pion and rho field variables. This will affect some of the pseudo-scalar self interactions and induces a finite From the above Lagrangian we calculate decay widths renormalisation of the pion fields. The issue is rather of the rho mesons into various products. The specific complicated and will be addressed in further work [4]. processes we shall examine are:

0 + ρ π π− E. Discussion → 0 ρ± π±π → ρ0 π+π γ Masses of the pions and the rho mesons are given by − → 0 ρ± π±π γ their experimental values → which we find to be in reasonable agreement with exper- mπ 140 MeV,mρ 770 MeV, (74) ∼ ∼ iment where data is available. The last decay has not been seen and represents a prediction of the model. and fπ is about 92 GeV. We estimate the value of the strong isospin gauge couplingg ˜ by assuming that most of the rho mass originates A. Decays of the form ρ ππ from symmetry breaking. There will be also be a small → amount of mass from strong interaction effects, and we estimate this to be of order the pion mass. This estima- Only the decays 0 + tion is very natural when one considers that the pion and ρ π π− rho have exactly the same quark content. Taking → 0 ρ± π±π → 1 mρ gf˜ π + mπ, (75) ∼ 2 are allowed. In the context of our model this is seen directly in the by the absence of any term describing ρ0 implies a valueg ˜ 13. Later we shall determine the π0π0. However this may also be seen directly through→ value of the gauge∼ couplingg ˜ to be isospin conservation. g˜ 12.2, (76) Isospin symmetry also constrains the above decays to ∼ have near equal widths. The small amount of isospin determined from the ρ ππ decay in sec. (IV A). breaking due to electromagnetic effects will produce a 2 ˜2 From the above we immediately→ infer that correction of the order ofe ˜ /g . By sec. (III E) such a correction is negligible. e2 g˜2. (77) At tree level the above decays relate to processes of the form

6 π although there is an appreciable correction from the two ρ bremsstrahlung processes π+ π+ π γ γ From A in Eq. (70), referring in particular to Appendix int 0 0 B, theL associated matrix elements are are ρ ρ

1 µ r π− π− ρ+ π+π0 = ig˜(q1 q2) µ, (80) M → − 2 − 1 µ r Associated with the single vertex interaction is the ma- ρ0 π+π = iα(q1 q2) µ, (81) M → − − 2 − trix element µ µ where q1 =(mρ/2,q)andq2 =(mρ/2, q) are the ﬁnal µ ν r − I = iegg˜ µν r s0 , (87) state momenta and µ refers to the polarisation vector M of the initial rho meson. Four momentum conservation µ with r the polarisation vector corresponding to the ini- leads to the centre of mass frame value ν tial rho meson and s0 the polarisation vector of the outgoing photon. m2 ρ 2 The two bremsstrahlung contributions are respectively q = q = mπ 358 MeV. (82) | | r 4 − ∼ of the form Averaging over the initial polarisation, and determining 1 ieg˜ µ ν µ ν the decay width from the standard two body decay width IIa = (q1 k2) (k2 + q2) r s0 . (88) M 2 k2 m2 − 2 − π q 1 Γ= 2, (83) and 8πm2 3 ρ × r |M| X 1 ieg˜ µ ν µ ν = (k q ) (k + q ) 0 , (89) which gives the result MIIb − 2 k2 m2 1 − 2 1 1 r s 1 − π 2 3 µ µ g˜ q Here q1 and q2 are the four-momenta of the outing pions Γ . (84) µ µ ρ ππ 2 → ∼ 24πmρ and k1 and k2 are the four momenta of the two internal pions such that Comparison with the measured value µ µ µ µ µ µ k1 = p q2 ,k2=pq1, (90) exp − − Γρ+ ππ = 151 1MeV. (85) → ± µ where p =(mρ,0) is the four momentum of the initial yields the determination rho meson. To determine the decay width we use the standard g˜ 12.2, (86) ∼ three body decay width formula consistent with the estimation in Eq. (75). 1 1 Γ= dE dE + + 2. (91) 64π3m 1 2 3 |MI MIIa MIIb | ρ Z 0 + B. The Decay ρ π π−γ The relevant phase space limits are determined in Ap- → pendix A. We used a combination of FORM to determine Next we shall consider the decay the matrix element and MAPLE to carry out the phase space integration. The results yield 0 + ρ π π−γ → Γ 0 + (2.02 + ∆)MeV, (92) ρ π π−γ which represents the second most dominant decay of the → ∼ rho meson. forg ˜ 12.2. Due to soft photon processes in the ∼ The main contribution is through a single vertex inter- bremsstrahlung diagrams there is an infrared divergent action contribution ∆. We neglect this, regarding that in anal- π+ ogous quantum electrodynamic processes the one loop contributions cancel the tree level infrared divergences. This result is a good estimation of the measured value ρ0 γ exp Γρ0 π+π γ 1.5 0.2MeV. (93) → − ∼ ± π−

7 0 C. The Decay ρ± π±π γ V. PION-PION INTERACTIONS → Next we shall consider the decay In this section we investigate the form of pion-pion scattering due to rho meson exchange between the two 0 ρ± π±π γ. particles. In this sense we regard the rho meson as trans- → mitting the strong force, specifically the force due to This decay has not been observed experimentally, and gauging the strong isospin symmetry, in a completely represents a prediction of our model. We find that its analogous way to W and Z boson exchange between decay width is close to present experimental limits. weakly interacting particles. As in the previous decay the main contribution is We shall consider only the pion interaction due to rho through a single vertex interaction exchange. In further work [4] we will calculate the cor- π+ rection from pseudoscalar interactions. In this sense the following calculation should be interpreted as a modelling ρ+ γ of the rho resonance, although it transpires that it also gives a good description at lower energies, right down to threshold. π0 although there is an appreciable correction from one A. Amplitudes for Specific Processes bremsstrahlung type interaction + π For the interaction of two pions there are five distinct interactions: four corresponding to scattering ρ+ γ π+π+ π+π+ + → + π π− π π− 0 → π π+π0 π+π0 → Associated with the single vertex interaction is the ma- π0π0 π0π0 trix element → and one corresponding to the conversion of two charged 1 µ ν = egg˜ µν 0 , (94) pions into two neutral ones MI 2 r s + 0 0 µ π π− π π with r the polarisation vector corresponding to the ini- → ν tial rho meson and s0 the polarisation vector of the out- The charge conjugated processes have the same ampli- going photon. tudes by charge conjugation invariance of the effective The bremsstrahlung contributions is of the form theory. We find that the non-trivial rho-interaction processes 1 eg˜ µ ν µ ν = (k q ) (k + q ) 0 , (95) take one of two distinct forms. Firstly, a single exchange MII −2 k2 m2 − 2 1 r s − π of a rho meson corresponding to a process of the form µ µ π π Here q1 and q2 are the four-momenta of the outing pions and k the four momenta of the respective internal pions such that ρ kµ = pµ qµ (96) − 2 where pµ =(m,0) is the four momentum of the initial ρ π π rho meson. To determine the decay width we use the standard The second interaction consists of two contributions, one three body decay width formula above in Eq. (91), with from the exchange of a rho meson, the other from the the relevant phase space limits determined in Appendix annihilation and then creation of two pions B. The results yield π π π π Γ 0 (0.11+∆)MeV, (97) ρ± π±π γ → ∼ ρ forg ˜ 12.2. As for the previous decay there is an in- ρ frared∼ divergent contribution ∆ that is due to soft photon π π processes in the bremsstrahlung diagram. π π

8 Before we calculate the interaction amplitudes we shall 3. Scattering of π0π+ π0π+ ﬁrstly deﬁne some conventions we shall use. We shall ex- → press all amplitudes in terms of the invariant Mandelstam This is associated with the matrix element variables u t s u 0+ = 1 ig˜2 − + − (108) s = E2, (98) M 4 s m2 t m2 − ρ − ρ t =(2m2 1E2)(1 cos θ), (99) π −2 − and corresponds to both an exchange process and an 2 1 2 u =(2mπ E )(1 + cos θ). (100) annihilation-creation process. −2 Here E represents the total centre of mass energy of the two incoming (outgoing) pions, and θ represents the an- 4. Scattering of π0π0 π0π0 gle between the incoming and outgoing pions. These vari- → ables are related to the following invariant combinations This is associated with the matrix element of the incoming four-momenta p1, p2 and the outgoing 00 four-momenta q1, q2 = 0 (109) M 2 s =(p1+p2) , (101) as there does not exist a rho meson interaction term for t =(p q )2, (102) such a scattering, 1− 1 u =(p q )2, (103) 1− 2 + 0 0 5. Exchange of π π− π π and are constrained by →

2 This is associated with the matrix element s + t + u =4mπ. (104) s u Later we shall require some expressions in terms of the E = 1 ig˜2 − (110) M 2 t m2 centre of mass three-momentum of one of the pions. For − ρ reference the conversion is and corresponds to a single exchange process. q2 = E2/4 m2 , (105) − π B. A Qualitative Fit of the Rho Resonance with the factor of four relating to E being the total centre of mass energy. To illustrate the general form of the pion-pion interactions we shall calculate the scattering cross section for 1. Scattering of + + + + π π π π + + → π π− π π− → This process is associated with the matrix element from tree level rho meson exchange. The component of the cross section associated with s u ++ = 1 ig˜2 − (106) tree level rho meson exchange is obtained from the stan- M 2 t m2 − ρ dard two body scattering expression and corresponds to a single exchange process, of the ﬁrst 1 1 σ(π+π )= d(cos θ) + 2. (111) form above. A factor of two is included from the identi- − 2 − 32πE 1 |M | cality of the ﬁnal states. Z− This cross section is usefully compared to the tree level chiral perturbation theory result [2], derived from the + + 2. Scattering of π π− π π− → corresponding matrix element

This is associated with the matrix element + 1 − = (s + t u), (112) Mcpt 2f 2 − u t s u π + = 1 ig˜2 (107) − 4 − 2 − 2 M s mρ − t mρ and obtained using Eqs. (113) to (122) of the next sec- − − tion. and corresponds to both an exchange process and an The results were obtained using MAPLE for a gauge cou- annihilation-creation process, of the second form above. plings ofg ˜ =12.2: Our results are in reasonable qualitative agreement with the data, especially near the rho resonance. The

9 π 120

rho−pion interactions π chiral perturbation theory π π 90

60 The results are conventionally expressed in the isospin basis of pions rather than the electromagnetic charge eigenstate basis that we have been using. Representing cross section (mb) the three isospin eigenstates by I =0,1,2, the matrix 30 elements of the two bases are related by the associated Clebsch-Gordan coefficients: ++ = 2, (113) M+ M1 0 1 1 1 2 0 − = + + , (114) 200 400 600 800 1000 3 2 6 E (MeV) M M M M +0 = 1 1 + 1 2, (115) M 2 M 2 M + 00 0 2 FIG. 1. Theoretical π π− scattering cross section from rho 1 2 = 3 + 3 , (116) exchange M M M E = 1 2 1 0. (117) M 3 M − 3 M important point is that the width of the resonance is These yield the two constraints pretty much the correct value. Since the width relates 00 + E = ++, (118) to the strength of the coupling this is a good consistency M+ ME M+0 − + = (119) check on the value of the strong isospin gauge coupling. M M M It is likely that the skew in the data at the top the left which is trivially verified from Eqs. (113) to (117). Then side of the resonance is due to soft photon effects, since the relations may be inverted to yield similar effects happen with electron-positron scattering u s near the Z-resonance. The low energy tail of the reso- 0 = ig˜2 − , (120) nance does take significantly larger values than all three M t m2 − ρ of the theoretical predictions. Although all theoretical u t 1 = 1 ig˜2 , (121) prediction are roughly of the right order of magnitude. 2 − 2 M s mρ The chiral perturbation theory prediction and rho res- − 2 s u onance prediction are similar near threshold, E 2mπ. = 1 ig˜2 − . (122) ∼ M 2 t m2 Although we have not included the first order correc- − ρ tion to the chiral perturbation theory prediction, it will be similar near threshold since it relates to an energy expansion around the threshold energy. The first order cor- D. Comparison of the Phase Shifts rection to the chiral perturbation theory result is liable to give a better agreement than the tree level prediction. To test the effective gauge theory of rho-pion interac- However, it is impossible to obtain a full resonance type tions we derive the associated phase shifts for the first structure, most significantly the high energy tail, using three partial wave amplitudes of each isospin process. such methods. These are compared to the experimental results. We also derive the corresponding results from tree level chiral perturbation theory for a useful comparison. MAPLE was used C. The Isospin Basis to calculate all results. The partial wave amplitudes are extracted from the Experimental data for the pion-pion interaction is ob- matrix elements tained by extrapolating from pion-nucleon processes of 1 the form I 1 I l (s)= d(cos θ)Pl(cos θ)M (s, t, u), (123) M 64π 1 πN ππN, Z− → I Then the phase shifts δl are defined from through the following picture: s 4m2 1/2 − π I =sinδI. (124) s |Ml | l

10 I This formula generally holds whilst l <1, with de- |MI | rho−pion interactions viations developing in the region of δ 90∼◦. chiral perturbation theory l ∼ Matrix elements for the eﬀective rho-pion interaction 150 are obtained from Eqs. (120) to (122). Chiral perturbation theory results are obtained from the following tree level matrix elements [2] 100 2s m2 0 = π , (125) chiral −2 M fπ

t u phase shift (degrees) 1 = , (126) chiral −2 50 M fπ 2m2 s 2 = π . (127) chiral 2− M fπ

0 Our results are: 200 400 600 800 1000 E (MeV)

1 FIG. 4. Phase Shift δ1 from Rho Exchange

100 0

rho−pion interactions chiral perturbation theory

80 −20

60 −40

40 −60 phase shift (degrees) phase shift (degrees)

20 −80

rho−pion interactions chiral perturbation theory

0 −100 200 400 600 800 1000 200 400 600 800 1000 E (MeV) E (MeV)

0 2 FIG. 2. Phase Shift δ0 from Rho Exchange FIG. 5. Phase Shift δ0 from Rho Exchange

30 2

rho−pion interactions chiral perturbation theory

20 0

10 −2 phase shift (degrees) phase shift (degrees)

0 −4

rho−pion interactions chiral perturbation theory

−10 −6 200 400 600 800 1000 200 400 600 800 1000 E (MeV) E (MeV)

0 2 FIG. 3. Phase Shift δ2 from Rho Exchange FIG. 6. Phase Shift δ2 from Rho Exchange

11 All of the results for rho meson exchange are in good Values not included are all zero at tree level. qualitative agreement with measurements, particularly We think it fair to compare our rho exchange results when one appreciates that deviation from the data when to the tree level chiral perturbation theory results, since δ 90◦ is expected owing to the limitations of for- both are at the tree level. However, we also include the ∼ 1 mula (124). Bearing this in mind the δ1 results give a one loop corrected results. Since the chiral Lagrangian is 2 very good qualitative agreement with the data. The δ0 non renormalisable it transpires that two extra parame- results are poorest, although they approximate the data ters are required to calculate the one loop corrections. reasonably upto about 550 MeV or so. Again the results are reasonable, as should be expected We shall now also give the scattering lengths and slopes since we are really comparing the same data. The (2, 0) at threshold. However it should be noted that the above mode is the poorest, although much of this discrepancy is graphs represent the true comparison of the theories, due to the inflection point at threshold in our theoretical since the experimental values of the scattering lengths prediction. For the (2, 2) mode we comment that the and slopes are extrapolated from that data. experimental data in fig. (6) does seem to be consistent 2 The partial wave amplitudes, defined in Eq. (123), with a negative value for a2. specify the scattering lengths and slopes from a low mo- We should comment that it is surprising how two seem- mentum expansion ingly unrelated methods both give a reasonable description of pion scattering. Especially so when one considers l q2 q2 that chiral perturbation theory relates to a completely I = aI + bI + . (128) l 2 l l 2 M mπ mπ ··· scalar approach, whereas rho exchange relates to a gauge approach. Our results produce the following table of values:

Rho Exchange VI. DISCUSSION I I (I,l) al bl (0, 0) 0.19 0.28 In conclusion, the effective SU(2) U(1) gauge theory (0, 1) 0.036 0.004 description of the interactions between× pions, rho mesons (0, 2) 10 10 4 3 10 5 − − and electromagnetism appears to give a good approxima- (1, 0) ×0 × 0 tion of the actual physics. The results are not exact, but (1, 1) 0.037 0.006 agree reasonably with experiment. Considering that the (1, 2) 0 0 theory is effective, rather than fundamental, this is en- (2, 0) -0.096 -0.14 tirely reasonable. (2, 1) -0.018 -0.002 4 5 As a theoretical guidance we used the symmetry prop- (2, 2) 5 10− 2 10− − × − × erties of the pseudo-scalar and pseudo-vector mesons. These should be compared to the experimental results, Both transform under a global SU(2)L SU(2)R chi- × where we list only those measured lengths and slopes [5]: ral symmetry, with the natural representations of the pseudo-scalars and rho mesons corresponding to the fun- Experiment damental and adjoint representations. This is exactly as I I (I,l) al bl required to construct a gauge theory description of their (0, 0) 0.26 0.05 0.25 0.03 interaction. Furthermore the symmetry corresponding ± 4 ± (0, 2) (17 3) 10− — to electromagnetism is exactly as required to form an ± × (1, 1) 0.038 0.002 — SU(2) U(1) unified description of the interactions. ± (2, 0) 0.028 0.012 0.082 0.008 Although× symmetry guided the construction of the − ± 4 − ± (2, 2) (1.3 3) 10− — model, it is the gauge principle that specifies the form ± × of interaction. The gauge principle rests fundamentally Finally we give the results obtained using chiral pertur- on the locality of symmetry, and hence interaction. This bation theory [2]: is a very generic situation. Thus it might be expected Chiral Perturbation Theory to happen in a variety of cases in addition to the funda- Tree Level One loop mental interactions. As the theory presented here is an I I I I effective theory arising from the fundamental theory of (I,l) al bl al bl (0, 0) 0.16 0.18 0.20 0.26 QCD, it seems sensible that the effective theory might 4 inherit a general feature of the QCD interaction such as (0, 2) 0 0 0 20 10− (1, 1) 0.030 0 0.036 ×0 the gauge principle. (2, 0) -0.045 -0.089 -0.041 -0.070 A feature of the model presented here, which it in- 4 herits from the sigma model, is the effective description (2, 2) 0 0 3.5 10− 0 × of confinement through symmetry breaking. The strong

12 coupling increases as the energy scale is decreased, be- We work in the centre of mass frame, so that the initial coming sufficiently large such that confinement occurs. rho mesons has four-momentum The quark and gluon degrees of freedom then become µ an inappropriate description of the physics, at least in p =(mρ,0), (A1) a perturbative, calculational sense. We would say that symmetry breaking gives an effective description of how and the four momenta of the final two pions and photon these degrees of freedom are transferred, such that the are of the form: pseudo-vector and pseudo-scalar quark composites inter- µ p =(E1,p1) act precisely so in order for the effective theory to un- 1 dergoes symmetry breaking. Above the phase transition µ the composites are massless, and are inappropriate de- k =(E3,k) grees of freedom. When symmetry breaks, the composites gain mass and the physical degrees of freedom shift µ p =(E2,p2) accordingly. 2 As an extension to this work it would be interesting to see if the effective gauge interaction can be extended to with the values constrained by four momentum conser- cover particles other than pions and rho mesons. For vation instance the interaction of rho mesons with nucleons. µ µ µ µ p = p1 + p2 + k . (A2) Since the nucleon doublet N =(n, p)> transforms under the fundamental representation it may be possible to Also there are the constraints describe its interaction with rho mesons and electromagnetism through the Lagrangian E2 p 2 =E2 p 2 =m2, (A3) 1 −| 1| 2 −| 2| π 2 2 µ µ ¯ µ E3 k =0. (A4) [N,ρ ,A ]=iNγµD N, (129) −| | L i We determine the constraints on E and E . with Dµ the covariant derivative of Eq. (47). One should 1 3 (1) Bounds on E : note that this has precisely the correct electromagnetic 1 It is clear that the bounds are form. We shall investigate the experimental validity of such scheme in a further publication [4]. min E1 = mπ, (A5) We shall finish by indicating that the use of the Emax = 1 m . (A6) Weinberg-Salam model for an effective theory of the 1 2 ρ quark-hadron phase transition may imply additional The minimum value occurring when the associated pion non-perturbative effects for QCD. The non-trivial vac- is stationary, and the maximum value occurring when uum structure of electroweak theory implies certain non- E3 =0. perturbative effects, for instance, the electroweak strings (2) Bounds on E3: and sphalerons. By analogy one should expect similar Given a value of E1 we wish to determine the range of solutions, and maybe similar cosmological consequences, possible E3. To do this we observe that the balance of for the quark-hadron phase transition. These features four momenta is equivalent to: are investigated in [6]. µ µ p1 k2 ACKNOWLEDGMENTS µ µ µ k = k1 + k2 I am grateful to B. Allanach, P. Saffin, H. Shanahan and T. Kibble for invaluable discussions related to this µ µ p2 k1 work. I acknowledge King’s College, Cambridge, for a junior research fellowship. Then we may balance the two components of the photon contribution separately. Doing this yields the range APPENDIX A: PHASE SPACE OF A THREE 2 2 2 E m mρ m BODY DECAY 1 − π + π , 2 2 ( 2 − 2 2(mρ E mπ) p − 1 − Here we determine the phase space limits, i.e. the 2 2 2 E m mρ p m limits of integration, associated with decays of the form 1 − π + π (A7) 2 2 2 2 − 2(mρ E mπ)) p − 1 − ρ ππγ. p → for E3 as a function of E1.

13 APPENDIX B: ASSOCIATED FEYNMAN TERMS (C) Single rho, two pions and a photon ρ0 µ π+ We list here some Feynman diagrams that will be use- µν ful in this paper. These contain interactions between the ieα cos 2θsg rho mesons, pions and photons; not including any that relate to the sigma. π+ ν γ (A) Single rho and two pions + + π+ p π ρ µ

1 µν 0 1 µ egg˜ ρ iα cos 2θs(p + q) 2 µ − 2 q π0 ν γ π− + π p π ρ− µ −

1 µ 1 µν ρ+ g˜(p + q) 2 egg˜ µ 2 − q 0 ν γ π0 π

p π0

1 µ ρ− g˜(p + q) µ − 2 q π− [1] M.Gell-Mann and M.Levy, Nuovo Cim. 16 (1960) 705. (B) Photon and two charged pions [2] S.Weinberg, Phys. Rev. Lett. 17 616 (1966). [3] N.F. Lepora and T.W.B.Kibble, JHEP 04 (1999) 027 hep- π+ p th/9904178. [4] N.F.Lepora, In preparation. γ ie(p + q)µ [5] B.R. Martin, D. Morgan and G. Shaw, London 1976, 460p. − µ [6] N.F.Lepora, hep-ph/9910518. q π−