Journal of Physics: Conference Series PAPER OPEN ACCESS A study of π and ρ mesons with a nonperturbative approach To cite this article: Rocio Bermudez et al 2015 J. Phys.: Conf. Ser. 651 012002 Related content - X (3872) production from reactions involving D and D* mesons A Martínez Torres, K P Khemchandani, F S Navarra et al. - Non-perturbative QCD and hadron physics J J Cobos-Martínez - STAR D0 meson 2 measurement Liang He View the article online for updates and enhancements. This content was downloaded from IP address 46.3.204.127 on 24/02/2018 at 19:53
A study of π and ρ mesons with a non-perturbative approach Rocio Bermudez Departamento de Investigación en Fsica, Universidad de Sonora, Blvd. Transversal S/N,83000, Hermosillo, Mexico E-mail: rabermudezr@gmail.com J.J. Cobos-Martnez Departamento de Física, Universidad de Sonora, Blvd. Transversal S/N,83000, Hermosillo, Mexico E-mail: j.j.cobos.martinez@gmail.com M.E. Tejeda-Yeomans Departamento de Física, Universidad de Sonora, Blvd. Transversal S/N,83000, Hermosillo, Mexico E-mail: elena.tejeda@correo.fisica.uson.mx Abstract. In this poster, we give a summary of work in progress regarding the study of meson s PDAs. We explore the consequences of a momentum-independent interaction as an ad-hoc tool to rebuild the pion distribution amplitude (PDA). The PDA is obtained through its moments using the Schwinger-Dyson formalism within QCD in the rainbow-ladder approximation. 1. Introduction We can say that and mesons are the simplest bound-states to study in QCD. This can be achieved by describing light-quark confinement and dynamical chiral symmetry breaking (DCSB), and admits a symmetry-preserving truncation scheme. QCDs Dyson-Schwinger equations (DSEs) provide a picture of these mesons in hadron physics. We assume that u/dmesons are produced by a vector-vector current-current interaction that is mediated by a momentum-independent boson propagator, i.e., by the symmetry preserving regularisation of a contact interaction. 2. Pion Distribution Amplitude Consider S the dressed-quark propagator, which is obtained from the gap equation: S 1 (p) = iγ p + m + (2π) 4 g2 D µν (p q) λa 2 γ µs(q) λa 2 Γ ν(q, p) (1) Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1
where m is the Lagrangian current-quark mass, D µν is the vector-boson propagator, and Γ ν is the quark vector-boson vertex which contains the non-perturbative information. In order to build a stock of material that can be used to identify unambiguous signals in experiment for the pointwise behavior of interaction between light-quarks, their mass-functions and other quantities, we use a description through a contact interaction. We thus define g 2 D µν (p q) = 1 m 2 δ µν (2) G where m G is a gluon mass-scale, and preceed by embedding this interaction in a rainbow-ladder truncation of the DSEs, which implies Γ ν (k, p) = γ ν. This leads to the quark propagator equation: S 1 (p) = iγ p + M (3) where M is momentum independent and is defined as: M = m + 4 3 The regularisation procedure used is 1 m 2 G (2π) 4 γ µs(q)γ µ (4) 1 s + M 2 = dτe τ(s+m 2) 0 τ 2 ir τ 2 uv dτe τ(s+m 2 ) (5) where τ uv,ir are, respectively, infrared and ultraviolet regulators. A nonzero value of τ ir = 1/Λ ir implements confinement by ensuring the absence of quark production thresholds. τ ir = 1/Λ ir plays a dynamical role and sets the scale of all dimensioned quantities. The homogeneous Bethe-Salpeter Equation (BSE) for the pseudoscalar meson for this interaction is Γ π (P ) = 4 1 3 (2π) 4 γ µχ π (q +, q )γ µ (6) m 2 G where q + = q + P and q = q are the gluons relative momentum. With a symmetrypreserving regularisation of the interaction, the Bethe-Salpeter amplitude cannot depend on relative momentum. We use Γ π (P ) = γ 5 [ie π (P ) + 1 M γ P F π(p )] (7) We want to build the pion distribution amplitude (PDA) from a moments calculation of it. The PDA is f π ϕ π (x) = Z 2 N c (2π) 4 δ(n q x n P ) tr[γ 5γ ns(q + )Γ π (q; P )S(q )] (8) The PDAs moments are So, we need to solve f π x m = Z 2 N c x m = 1 0 dx x m ϕ π (x) (9) (n q) m (2π) 4 (n P ) m+1 tr[γ 5γ ns(q + )Γ π (q; P )S(q )] (10) where f π is the pions leptonic decay constant define as the zero-moment of the PDA. 2
2.1. Results Weve obtained the following expression for the PDAs moments: f π x m = 1 4π 2 Z 1 1 2N c dα( α) [E m π M 2 C 1 (M 2 ) + 1 [ ] ] M 0 2 F π (M 2 4M 2 )C 1 (M 2 ) + C(M 2 ) (11) where M 2 = α(1 α)p 2 + M 2 and the constants C 1 (M 2 ; τ 2 uv, τ 2 ir) = Γ(0; M 2 τ 2 uv) Γ(0; M 2 τ 2 ir) C(M 2 ; τ 2 uv, τ 2 ir) = M 2 [Γ( 1; M 2 τ 2 uv) Γ( 1; M 2 τ 2 ir)] (12) The f π, define as m = 0, is f π = 1 4π 2 Z 1 1 [ 2N c dα E π M 2 C 1 (M 2 ) + 1 M 0 2 F π This means x 0 = 1, i.e. PDAs moments describe a point-like particle. [ (M 2 4M 2 )C 1 (M 2 ) + C(M 2 )] ] (13) Figure 1. Results for the Pions moments and the point-like behavior of the PDA. 3. Pion Distribution Amplitude We want to build a distribution amplitude for the ρ-meson by getting a procedure analogous to the PDAs moments calculation. In this case, we need to change the Bethe-Salpeter amplitude 3
in order to have a description of a vector rather than a pseudoscalar meson. The vector BSA takes the form Γ ρ µ(p ) = γ T µ E ρ (P ) (14) with P µ γ T µ = 0 γ L µ + γ T µ = γ µ (15) The first attempt for a ρ distribution amplitude (ρda) from a moment calculation is f ρ ϕ ρ (x) = 1 3 Z 2N c The moments are given by f ρ x m L = 1 3 Z 2N c f ρ x m T = 1 3 Z 2N c (2π) 4 f ρ x m i = 3 Z 2N c (2π) 4 δ(n q x n P )tr[γ µs(q + )Γ ρ µ(q; P )S(q )] (16) (n q) m (2π) 4 (n P ) m+1 tr[γ µs(q + )Γ ρ µ(q; P )S(q )] (17) (2π) 4 (n q) m (n P ) m+1 tr[σ µνn ν S(q + )Γ ρ µ(q; P )S(q )] (18) (n q) m (n P ) m+1 tr[n γγ µs(q + )Γ ρ µ(q; P )S(q )] (19) where f ρ is the ρ s leptonic decay constant define as the zero-moment of the ρda. The test moments x m L and xm T are a longitudinal-like and a transverse-like, respectively, forms inspired from [4]. On the other hand, test moment x m it s a fit to the point-like form that we expect. The results of these test expressions are presented in the following graph (20) 4
Figure 2. Results for the ρ s moments and the point-like behavior of the ρda. 4. Summary Weve obtained expressions for the DAs moments using the contact interaction. This represent an easy way to describe the behavior of the π- and ρ- mesons as point-like particles. Its important to note that weve only tested expressions that still need to be explored and fixed for a better description. It is also necessary to find a way to reconstruct ρda using the moments for the complete picture. References [1] Pion form factor from a contact interaction. Phys. Rev. C 81 065202 [2] pi- and rho-mesons, and their diquark partners, from a contact interaction. Phys. Rev. C 83 065206 [3] Pion and kaon valence-quark parton distribution functions. Phys. Rev. C 83 062201 [4] The rho- meson light-cone distribution amplitudes of leading twist revisited. Phys. Rev. D 54 2182 [5] Bethe-Salpeter study of vector meson masses and decay constants. Phys. Rev. C 60 055214 5