Simplified Bethe-Salpeter Description of Basic Pseudoscalar-Meson Features
aa r X i v : . [ h e p - ph ] J u l Simplified Bethe–Salpeter Description of BasicPseudoscalar-Meson Features
Wolfgang Lucha , ∗ Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18,A-1050 Vienna, Austria
Abstract.
We assess a description of pseudoscalar mesons as pseudo-Goldstonebosons by its compatibility with some Gell-Mann–Oakes–Renner-type relation.
Goldstone’s theorem necessitates the presence of a massless boson in the spectrum of physicalparticles for any spontaneously broken chiral symmetry of quantum chromodynamics (QCD);these hypothetical Goldstone bosons are identified with the ground-state pseudoscalar mesons(pions, kaons, η ). Their finite (but comparatively small) masses are attributed to the additionalexplicit breakdown of the chiral symmetries of QCD enforced by nonvanishing quark masses.We analyze the Goldstone-boson nature of the lightest pseudoscalar mesons by means of aformalism [1–4] situated somewhere between the fully relativistic Bethe–Salpeter descriptionof bound states [5], with several still to be resolved inherent obstacles, and the latter’s extremeinstantaneous limit, represented by its three-dimensional reduction devised by Salpeter [6]. Arather promising tool to judge the merits of such kind of intermediate framework proves to be,among others, the fulfilment of (generalized) Gell-Mann–Oakes–Renner-type relations [7] bycharacteristic features ( viz. , decay constants or condensates) of light pseudoscalar mesons [8]. The homogeneous Bethe–Salpeter equation describes, in Poincaré-covariant manner, a boundstate | B( P ) i of mass b M and momentum P , formed by two particles of relative momentum p , byits Bethe–Salpeter amplitude Φ ( p , P ). One ingredient are the constituents’ full propagators, inthe case of spin- fermions i given by mass M i ( p ) and wave-function renormalization Z i ( p ): S i ( p ) = i Z i ( p ) / p − M i ( p ) + i ε , / p ≡ p µ γ µ , ε ↓ , i = , . The other ingredient are the interactions responsible for bound-state formation. Ignoring theirdependence on time components of momenta and the above propagators’ dependence on zeromomentum components squared led to a bound-state equation [1] for a Salpeter amplitude [6] φ ( p ) ∝ Z d p Φ ( p , P ) , interpretable as equal-time bound-state wave function. An integral kernel K ( p , q ) captures theentirety of, by assumption instantaneous, e ff ective interactions of all bound-state constituents. ∗ e-mail: [email protected] he bound-state equation proposed in Ref. [1], formulated in terms of the kinetic energies, E i ( p ) ≡ q p + M i ( p ) , and the energy projectors onto positive and negative energies of the bound-state constituents i , Λ ± i ( p ) ≡ E i ( p ) ± γ [ γ · p + M i ( p )]2 E i ( p ) , reads, for the case of fermion–antifermion bound states in the center-of-momentum frame [1], φ ( p ) = Z ( p ) Z ( p ) Z d q (2 π ) Λ + ( p ) γ [ K ( p , q ) φ ( q )] Λ − ( p ) γ b M − E ( p ) − E ( p ) − Λ − ( p ) γ [ K ( p , q ) φ ( q )] Λ + ( p ) γ b M + E ( p ) + E ( p ) . For any one-particle states | B ( P ) i normalized according to the Lorentz-invariant condition h B ( P ) | B ( P ′ ) i = (2 π ) P δ (3) ( P − P ′ ) , the normalization condition for the corresponding Salpeter amplitudes φ ( p ) is given by [9–11] Z d p (2 π ) Tr " φ † ( p ) γ [ γ · p + M ( p )] E ( p ) φ ( p ) = P . Things simplify considerably if the propagator functions of involved fermion and antifermionhappen to be identical, M ( p ) = M ( p ) and Z ( p ) = Z ( p ); the Salpeter amplitude of everypseudoscalar bound state is then fully defined by just two Lorentz-scalar components, ϕ , ( p ): φ ( p ) = √ " ϕ ( p ) γ [ γ · p + M ( p )] E ( p ) + ϕ ( p ) γ . For K ( p , q ) compatible with spherical and rather specific Fierz symmetries of the e ff ectiveinteractions, our bound-state equation governing φ ( p ) collapses to an eigenvalue problem [12]fixing the radial parts ϕ , ( p ) , p ≡ | p | , of ϕ , ( p ) , with a single, spherically symmetric potential V ( r ) , r ≡ | x | , encoding the configuration-space interactions between bound-state constituents: E ( p ) ϕ ( p ) + Z ( p ) π p ∞ Z d q q d r sin( p r ) sin( q r ) V ( r ) ϕ ( q ) = b M ϕ ( p ) , E ( p ) ϕ ( p ) = b M ϕ ( p ) . (1)For the actual case of interest, mesonic bound states of quarks and antiquarks, the e ff ective interaction potential V ( r ) was extracted pointwise [2–4] by straightforward inversion [13, 14]of our Bethe–Salpeter-inspired bound-state equation [1], starting from that Salpeter amplitude φ ( p ) that represents massless pseudoscalar mesons. The latter, in turn, is connected [15–17] tothe chiral-quark propagator [18, 19] by a Ward–Takahashi identity of QCD [7]. The emergingconfining V ( r ) rises, from its slightly negative value at r = , rather steeply to infinity (Fig. 1). r @ GeV - D V H r L @ G e V D Figure 1. E ff ective interquark central potential V ( r ) pinned down by inverting the radial Bethe–Salpeterproblem (1) for a Salpeter-amplitude input derived from some quark-propagator model solution [18, 19].The almost flatness of the potential V ( r ) near the origin in combination with its extraordinarily steep riseto infinity renders the form of V ( r ) pretty close but clearly not identical to that of a square-well potential. In order to scrutinize pseudoscalar quark–antiquark bound states for physical ( i.e. , non-chiral)quark masses, we ought to find the corresponding solutions to our set (1) of coupled equationsfor the two independent radial Salpeter components ϕ , ( p ) . Our task is greatly facilitated by adefinitely obvious move enabled by the purely algebraic nature of one of the two relations (1):inserting any of the two relations (1) into the other leads to single explicit eigenvalue problemsfor either ϕ ( p ) or ϕ ( p ) with mass-squared eigenvalue b M [2–4, 12]; conversion to equivalentmatrix eigenvalue problems by expansion over a convenient function-space basis is among thestandard solution procedures [20–24]. With these solutions at hand, we may create trust in thereliability of our approach by assessing and exploring its predictions for hadronic observables.The spatial extension of the pion deduced, from the numerical ground-state solution to ourbound-state formalism inferred along the above lines, in form of the pion’s average interquarkdistance h r i = .
478 fm or root-mean-square radius p h r i = .
529 fm fits nicely to the pion’selectromagnetic charge radius, p h r π i = (0 . ± . V ( r ) , and the experimental u / d quark masses are pretty close to their chiral limit.onsequently, we have to look for di ff erent and more significant criteria which allow us toappraise the reasonableness of any inversion considerations. Fortunately, this proves to be nottoo di ffi cult: Equating the residues of pseudoscalar-meson pole terms in both axial-vector andpseudoscalar vertex functions entering into the axial-vector Ward–Takahashi identity of QCDexpressing the invariance of QCD under chiral transformations leads to a generalization [7] ofthe Gell-Mann–Oakes–Renner relation [26]; this innovation relates, for a pseudoscalar boundstate | B ( P ) i , its (weak) decay constant f B , defined in terms of the axial-vector quark current by h | : ¯ ψ (0) γ µ γ ψ (0) : | B ( P ) i = i f B P µ , — which, consequently, may be found by projection of φ ( p ) onto the axial-vector current, i.e. , f B ∝ Z d p Tr[ γ γ φ ( p )]— and its in-hadron condensate [7] (universalizing the notion of quark vacuum condensates) C B ≡ h | : ¯ ψ (0) γ ψ (0) : | B ( P ) i ∝ Z d p Tr[ γ φ ( p )]to the mass b M B of this pseudoscalar bound state and the two relevant quark mass parameters inthe QCD Lagrangian [7]. For the simpler case of equal quark masses m , this relationship reads f B b M B = m C B . (2)Compatibility with Eq. (2) may be inspected by solving our formalism with the previouslyestablished potential V ( r ) for bound states of chiral, u / d , and s quarks, taking advantage of theappropriate model propagator functions [18, 19]. Comparison of the quark masses m , fixed bythe thus determined values of b M B , f B , and C B , with the current-quark masses m ( µ ) in modifiedminimal subtraction at scale µ proves that our findings for m are in the right ballpark (Table 1). Table 1.
Predictions of the Bethe–Salpeter-inspired bound-state equation (1), with e ff ective interactionpotential V ( r ) as depicted in Fig. 1, for masses b M B , decay constants f B and in-meson condensates C B ofthe lightest pseudoscalar mesons, and confrontation of the quark-mass parameters m resulting from themore general Gell-Mann–Oakes–Renner relation (2) with their minimal-subtraction counterparts m ( µ ) . Constituents b M B f B C B m m (2 GeV)[MeV] [MeV] [GeV ] [MeV] [MeV] [25]chiral quarks 6.8 151 0.585 0 . u / d quarks 148.6 155 0.598 2 .
85 3 . + . − . s quarks 620.7 211 0.799 51 . + − In order to establish whether or not it is justified to lend trust to the outcomes of an approach tobound states proposed some time ago [1] and residing, as far as its compatibility with Poincarécovariance is concerned, somewhere in the vast range between bound-state descriptions alongthe ideas of Salpeter and Bethe, on the one hand, and static approximations, on the other hand,we evaluated the performance of the investigated framework’s predictions for those propertiesof the lightest pseudoscalar mesons that happen to be related by an advancement of the insightgained by Gell-Mann, Oakes, and Renner [26]: In brief, the formalism of Ref. [1] is still alive. A recent, comprehensive, Bethe–Salpeter-rooted evaluation of in-hadron condensates may be found in Ref. [27]. eferences [1] W. Lucha and F. F. Schöberl, J. Phys. G (2005) 1133, arXiv:hep-th / (2016) 1650202, arXiv:1606.04781[hep-ph].[3] W. Lucha, EPJ Web Conf. (2016) 00047, arXiv:1607.02426 [hep-ph].[4] W. Lucha, EPJ Web Conf. (2017) 13009, arXiv:1609.01474 [hep-ph].[5] E. E. Salpeter and H. A. Bethe, Phys. Rev. (1951) 1232.[6] E. E. Salpeter, Phys. Rev. (1952) 328.[7] P. Maris, C. D. Roberts, and P. C. Tandy, Phys. Lett. B (1998) 267,arXiv:nucl-th / (2018) 1850047, arXiv:1801.00264[hep-ph].[9] J.-F. Lagaë, Phys. Rev. D (1992) 305.[10] J. Resag, C. R. Münz, B. C. Metsch, and H. R. Petry, Nucl. Phys. A (1994) 397,arXiv:nucl-th / (1995) 5141,arXiv:hep-ph / (2007) 125028, arXiv:0707.3202[hep-ph].[13] W. Lucha and F. F. Schöberl, Phys. Rev. D (2013) 016009, arXiv:1211.4716 [hep-ph].[14] W. Lucha, Proc. Sci., EPS-HEP 2013 (2013) 007, arXiv:1308.3130 [hep-ph].[15] W. Lucha and F. F. Schöberl, Phys. Rev. D (2015) 076005, arXiv:1508.02951 [hep-ph].[16] W. Lucha and F. F. Schöberl, Phys. Rev. D (2016) 056006, arXiv:1602.02356 [hep-ph].[17] W. Lucha and F. F. Schöberl, Phys. Rev. D (2016) 096005, arXiv:1603.08745 [hep-ph].[18] P. Maris and P. C. Tandy, Phys. Rev. C (1999) 055214, arXiv:nucl-th / Proceedings of the International Conference on Quark Confinement andthe Hadron Spectrum IV , editors W. Lucha and K. Maung Maung (World Scientific,Singapore, 2002), p. 163, arXiv:nucl-th / (1997) 139, arXiv:hep-ph / (1999) 2309,arXiv:hep-ph / (2001) 056002,arXiv:hep-ph / (2001) 036007,arXiv:hep-ph / (2004) 1423,arXiv:hep-ph / et al. (Particle Data Group), Chin. Phys. C (2016) 100001.[26] M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. (1968) 2195.[27] T. Hilger, M. Gómez-Rocha, A. Krassnigg, and W. Lucha, Eur. Phys. J. A53