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ization dependencies wipe out and no net effect can be extracted from experimental data. We will see that a combined analysis of some characteristic angles together with the dilepton invariant mass constitutes the appropriate framework to investigate the possible polarization asymmetry predicted by LPB.

This chapter is organised as follows: in Section 5.2, we introduce the Vector Meson Dominance model approach to LPB and derive in detail the distorted dispersion relation that affects the lightest meson states ρ and ω. In Section 5.3 their spectral functions are presented together with some considerations of a possible partial explanation of the abnormal dilepton production in the vicinity of their resonance peak position. An experimentally oriented study of the polarization asymmetry found in the spectral functions is discussed in Section 5.4. Two different analysis of the dilepton angular distribution are presented in order to reflect how the LPB effect may be experimentally detected.

5.1 Summary of the Vector Meson Dominance model

Before the formulation of QCD, hadron physics was deeply investigated through a variety of models incorporating approximate symmetries. The interaction between photons and hadronic matter is a subject of a particular interest in this work that the Vector Meson Dominance model (VMD) can address satisfactorily (for reviews, see [144]). The main idea behind this model is assuming that the quark loops appearing in the vacuum polarization of the photon can be described exclusively by vector mesons.

This approximation results to be very accurate in the regions around the vector meson masses. In QED the photon self-energy can be reasonably well approximated using bare propagators and vertices for electron-positron loops due to higher order suppressions.

However, the strong coupling of hadronic constituents requires considering the dressing of quark loops in this context.

The success of the principles of quantum field theories in QED led to a formulation of strong interactions based in the same principles. In particular, the notion of con-served currents, the gauge principle and the universality of couplings should be applied to strong-interaction physics. The VMD model was introduced as the theory of the strong interaction mediated by vector mesons and based on a non-abelian Yang-Mills theory [141]. Its simplest realization conceives the hadronic contribution to the photon polarization as a propagating vector meson that replaces the quark loop. This assump-tion is equivalent to considering the hadronic electromagnetic current operator (in the neutral isotriplet channel) given by the current-field identity

j_µ^em(x)  

I=1= m²_ρ gρ

ρ⁰_µ(x) (5.2)

is proportional to the vector meson field operator. As the electromagnetic current has null divergence, so will the field ρµ, both its neutral and its charged components so

∂_µ~ρ^µ = 0. The breaking of gauge invariance due to vector meson masses was ignored

since local gauge transformations were considered as the way to create interactions with universal couplings among nucleons and vector mesons, say gρ≡ g_{ρN N} = gρππ = gρρρ.

VMD was particularly successful in the description of the electromagnetic form factors of the nucleons. The nucleon was regarded as a core surrounded by a pion cloud and the form factor measurements were interpreted as an evidence for an isoscalar vector meson ω and for an isovector meson ρ⁰ decaying into 3 and 2 pions, respectively.

This picture of the nucleon form factors was subsequently generalised to all photon-hadron interactions. Based on the previous considerations, the electromagnetic current was associated to a linear combination of vector meson fields. A complete realization of the VMD model involves other light resonances such as ω and φ to be incorporated in the game equivalently to the ρ meson in Eq. (5.2) through the extension of the current-field identity

j_µ^em = m²_ρ

g_ρρ⁰_µ+m²_ω

g_ω ωµ+m²_φ

g_φφµ, (5.3)

which generalises Eq. (5.2). With this notation the SU (3) ratios gρ : gω : gφ = 1 : 3 : −3/√

2 hold. This treatment implies that the amplitudes of photon interactions are proportional to the corresponding vector meson processes. In this sense, a virtual photon is understood to fluctuate to a vector meson which is the responsible for in-teractions with hadronic matter. A parallel derivation of the ρ meson coupling can be obtained without any recurrence to gauge invariance.

Another way to derive Eq. (5.3) is from the hadronic electromagnetic current of the SM in terms of the quark fields in the SU (3) representation [145] given by

j_µ^em= 2

3uγ¯ µu −1

3dγ¯ µd − 1

3¯sγµs, (5.4)

which can be rewritten identically as j_µ^em = 1

√2j_µ^ρ+ 1 3√

2j_µ^ω− 1

3j_µ^φ, (5.5)

where the vector meson currents can be expressed in terms of their quark content via j_µ^ρ= 1

√

2 uγ¯ µu − ¯dγµd
, j_µ^ω= 1

√2 uγ¯ _µu + ¯dγ_µd
,

j^φ_µ= ¯sγµs. (5.6)

With this definition the matrix element can be expressed as h0|j_µ^em|V i = M_V²

gV

^{(V )}_µ , (5.7)

where ^{(V )}µ is a polarization vector and gω ' g_ρ ≡ g ' 6 < g_φ ' 7.8 and M_V² = 2g_V²f_π². We will use this notation for the couplings throughout this analysis.

5.1 Summary of the Vector Meson Dominance model

The fact that the electromagnetic current is made of vector meson fields suggests an underlying lagrangian with a mixing term proportional to ρµA^µ but this piece breaks gauge invariance. One of the consistent ways to introduce both gauge invariance and the photon-vector meson mixing is given by the current-mixing form [146] with an explicit mixing of vector mesons and photons both in the mass and kinetic terms. In the limit of universality g_ρ = g_ρππ = g_{ρN N} = . . . , an alternative description of VMD can be equivalently used with a mixing of vector mesons with photons in the mass term and a non-trivial photon mass (mass-mixing form) [141]. In any of these formulations the photon remains massless when one computes its propagator to all orders. Despite the second formulation is not as elegant as the first due to a photon mass term, it has been established as the most popular representation of VMD [141, 142]. From now on we will use this representation. Quark-meson interactions are described by [95]

Lint= ¯qγ_µV^µq; V_µ≡ −eA_µQ + 1

2g_ωω_µI_q+1

2g_ρρ_µλ₃+ 1

√2g_φφ_µI_s, (5.8)

where Q = ^λ₂³ + ¹₆I_q−¹₃I_s, I_q and I_s are the identities in the non-strange and strange sector, respectively; and λ3 is the corresponding Gell-Mann matrix. Then the Maxwell and mass terms of the VMD lagrangian are

Lkin= −¹₄(F_µνF^µν+ ω_µνω^µν+ ρ_µνρ^µν+ φ_µνφ^µν) +¹₂V_µ,amˆ²_abV_b^µ, reflects the VMD relations at the quark level [141, 142].

The pion form factor Fπ(q²) represents the contribution from the intermediate vec-tor mesons connecting the photon to pions in a process like γ → π⁺π⁻. The form factor is the multiplicative deviation from a point-like behaviour of photon-pion cou-pling. In the limit of zero momentum transfer (q² → 0), the photon is only affected by the charge of the pions and therefore one expects F_π(0) = 1. This relation is auto-matically satisfied by the pion form factor derived from the current-mixing Lagrangian [146] while for the mass-mixing scenario [141], it is necessary to impose universality via g_ρ= g_ρππ. This is the basis of the argument for universality given in [141], where the photon couples to the ρ due to the imposition of local gauge symmetry. Therefore g_ρππ must equal gρ. This is a direct consequence of assuming complete ρ dominance of the form factor.

5.2 Vector Meson Dominance Lagrangian in the presence