The magnetic moment of the \rho-meson

The magnetic moment of the \rho-meson is calculated in the framework of a low-energy effective field theory of the strong interactions. We find that the complex-valued strong interaction corrections to the gyromagnetic ratio are small leading to a value close to the real leading tree level result, g_\rho = 2. This is in a reasonably good agreement with the available lattice QCD calculations for this quantity.

In a covariant formalism, massive vector bosons are described by Lagrangians with constraints. The self-consistency of a system with constraints imposes non-trivial conditions on the form of the Lagrangian. In Ref. [15], the effective Lagrangian of Ref. [3] describing the interaction among ρ-mesons, pions and nucleons was considered. Requiring perturbative renormalizability in the sense of effective field theory [16], the universality of the vectormeson couplings was derived. The crucial ingredient of any effective field theory (EFT) is power counting. It is possible to consistently include virtual (axial-) vector mesons in EFT [11,17,18] provided they appear only as internal lines in Feynman diagrams involving soft external pions and nucleons with small three-momenta. The issue becomes highly non-trivial for energies when the intermediate resonant states can be generated. The problem is that vector mesons decay in light modes and therefore large imaginary parts appear [13]. First attempts have been made to handle this problem by applying the complex mass scheme [19][20][21][22][23][24][25][26].
In this work we calculate the magnetic moment of the ρ-meson as a function of quark masses in the framework of a low-energy effective theory of the strong interactions. As for any unstable particle, this quantity is a complex number. For a detailed discussion on this issue, see Ref. [27]. We start with the most general effective Lagrangian of vector mesons interacting with pions in the presence of external fields. It contains an infinite number of interaction terms which respect the underlying symmetries of QCD. In this work, we make the assumption that the interaction terms with a higher number of derivatives and/or more fields are suppressed by powers of some large hadronic scale. Therefore, only a finite number of terms of the effective Lagrangian is required to achieve a given accuracy. We apply the complex mass renormalization scheme [19,20] and calculate the magnetic moment of the vector meson and its pion mass dependence at one-loop order.

II. LAGRANGIAN
We start with the most general chiral effective Lagrangian for ρ-and ω-mesons, pions and external sources in the parametrization of the model III of Ref.
Below we specify only those terms of the Lagrangian which are relevant for the calculation of the magnetic moment of the ρ-meson presented in this work: where Here, F denotes the pion decay constant in the chiral limit, M 2 is the lowest order expression for the squared pion mass, M ρ and M ω refer to the ρ and ω masses in the chiral limit, respectively. Further, g, c x and g ωρπ are coupling constants and v µ is the external vector field. Notice that we do not show the counterterms explicitly. For the electromagnetic interaction we have v µ = −e τ 3 A µ /2. Demanding that couplings with different mass dimensions are independent, the consistency condition for the ρππ coupling [15] leads to the KSFR relation [28,29]

III. MAGNETIC MOMENT OF THE VECTOR MESON
As the ρ-meson is an unstable particle it does not appear as an asymptotic state in the effective field theory. Therefore, to define the magnetic moment of the ρ-meson, we follow the strategy of Ref. [27] and consider an amplitude of a process in which the γρρ vertex contributes as a sub-diagram. For the sake of definiteness, we take the process ππ → γππ shown in Fig. 1. We parameterize the amplitude of this process as where "Rest" denotes the non-resonant contributions. Here, M α 1 and M β 2 are the ρππ vertex functions, ǫ λ is the photon polarization, and −iD µν (p) with is the dressed propagator of the vector meson. Further, Z V is the (complex) residue at the pole z and R denotes the non-pole part. The γρρ vertex function can be written as where t λµν j denote the possible tensor structures which depend on the momenta p i , p f and the metric tensor, and V i (q 2 , p 2 f , p 2 i ) are the corresponding scalar functions.
Here and in what follows, we do not show the isospin indices (unless stated otherwise) for the sake of compactness. Expanding the V j about the z-pole and substituting, together with the expression for the propagator in Eq. (5), into Eq. (4) we obtain for the leading double-pole contribution In order to properly renormalize the γρρ vertex function we rewrite Eq. (7) in the form Noting that p µ D µν does not have a pole, we drop structures containing p ν f and p µ i and parameterize the "on-mass-shell" Γ as follows where the ellipsis refer to structures which do not involve the metric tensor.
The charge and magnetic moment e and µ ρ of the ρ-meson are defined in terms of the corresponding form factors f 1 (0) and f 2 (0) as There are both tree-level and loop contributions to these quantities. Loop diagrams are suppressed by powers of ξ = g 2 i /(16π 2 ), where g i stands for coupling constants in general.
Even for a sizeable coupling like g ρππ , this expansion parameter is small, ξ ≃ 0.2. Vertices generated by the c x -term of the Lagrangian are only included at tree order in our calculations as their contributions are suppressed at the one-loop order by two additional powers of the pion mass. At tree order, we obtain in agreement with the findings of Refs. [30,31]. One-loop diagrams contributing to the γρρ vertex function are shown in Fig. 2. Their full contributions to f 2 (0) are given in the appendix. Taking into account the wave function renormalization we obtain where the loop functions A 0 (m 2 ) and B 0 (p 2 , m 2 1 , m 2 2 ) are also defined in the appendix. Note that in Eq. (13), the quantity f loop 1 (0) only vanishes when the complex residue of the dressed propagator at the pole is used as the wave function renormalization constant for the vector field.
Comparing Eqs. (14) and (15) with Eq. (12) we see that the loop contributions are clearly suppressed in comparison with the tree-level result and also that the imaginary part of the vector meson masses has a little impact on the value of the loop correction. We thus conclude that the leading quantum corrections to the classical value of the g-factor, g ρ = 2, are suppressed. This gives a strong indication that the strong corrections to this observable are small. This conjecture is further supported by the available lattice QCD calculations for this quantity, namely (g ρ ) quenched ∼ 2.3 of Ref. [33] and (g ρ ) unquenched = 1.6(1) of Ref. [34] (see also Ref. [35] for an early study). Finally, we also plot in Fig. 3 the real and imaginary parts of the g-factor g ρ as functions of the pion mass. Both real and imaginary part show very little pion mass dependence, the cusp appears at the value of the pion mass, at which the ρ pole moves from the second to the first Riemann sheet.

IV. SUMMARY
In this paper, we have calculated the complex-valued magnetic moment of the ρ-meson in a chiral effective field theory utilizing the complex-mass renormalization scheme. Assuming that the interaction terms with a higher number of derivatives and/or more fields are suppressed by powers of some large hadronic scale, we perform a one-loop calculation in terms of the expansion parameter ξ = (g ρππ /4π) 2 ≃ 0.2. The pertinent results of our investigation can be summarized as follows: • At tree level (leading order), the magnetic moment of the ρ is real and its gyromagnetic ratio is g ρ = 2.
• At one-loop order, the magnetic moment picks up an imaginary part. We find that the one-loop corrections to g ρ are of the order of 10%, cf. Eqs. (14,15), and the imaginary part is about 0.04 (in units of the charge). The results are in agreement with recent lattice QCD determinations.
• We find that the pion mass dependence of the gyromagnetic ratio is very weak. This could be tested on the lattice for sufficiently small pion masses, say M π 0.3 GeV, that also allow for the ρ-meson to decay.