Covariant Spectator Theory of heavy-light and heavy mesons and the predictive power of covariant interaction kernels
CCovariant Spectator Theory of heavy-light and heavy mesons and the predictivepower of covariant interaction kernels
Sofia Leit˜ao a , Alfred Stadler b,a, ∗ , M. T. Pe˜na c,a , Elmar P. Biernat a a CFTP, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal b Departamento de F´ısica, Universidade de ´Evora, 7000-671 ´Evora, Portugal c Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Abstract
The Covariant Spectator Theory (CST) is used to calculate the mass spectrum and vertex functions of heavy-light andheavy mesons in Minkowski space. The covariant kernel contains Lorentz scalar, pseudoscalar, and vector contributions.The numerical calculations are performed in momentum space, where special care is taken to treat the strong singularitiespresent in the confining kernel. The observed meson spectrum is very well reproduced after fitting a small number ofmodel parameters. Remarkably, a fit to a few pseudoscalar meson states only, which are insensitive to spin-orbit andtensor forces and do not allow to separate the spin-spin from the central interaction, leads to essentially the same modelparameters as a more general fit. This demonstrates that the covariance of the chosen interaction kernel is responsiblefor the very accurate prediction of the spin-dependent quark-antiquark interactions.
Keywords:
Covariant quark model, meson spectrum, covariant spectator theory
PACS: q ¯ q states, sparking particular interest fromtheorists.The purpose of this work is twofold: First, we presentresults of relativistic calculations of q ¯ q bound states forsystems with at least one heavy ( b or c ) quark using themanifestly covariant framework of the Covariant Specta-tor Theory (CST) [2, 3, 4]. Second, we show that our ∗ Corresponding author
Email addresses: [email protected] (SofiaLeit˜ao), [email protected] (Alfred Stadler), [email protected] (M. T. Pe˜na), [email protected] (Elmar P. Biernat) covariant kernel correctly predicts the spin-dependent in-teractions when it is fitted to data that do not containany independent information about them. More precisely,when the kernel is fitted exclusively to pseudoscalar mesonstates, which are S -waves and thus insensitive to spin-orbitand tensor forces (and which do not allow to isolate thespin-spin interaction because here it always acts on sin-glets), the vector, scalar and axial-vector states which do depend on them are correctly described. We believe thatthis is an important test, performed here for the first time,which confirms the predictive power of covariant kernels.Most quark models are variations of the nonrelativis-tic Cornell potential [5] which consists of a short-rangecolor-Coulomb and a linear confining potential and wassurprisingly successful in describing heavy quarkonia. Be-cause light quarks require a relativistic description, in or-der to be applicable to all q ¯ q states these Cornell-typepotentials were “relativized” [6] by including a numberof relativistic corrections. For a more rigorous treatmentof relativity, a number of relativistic equations related tothe Bethe-Salpeter equation (BSE) were applied to calcu-late the meson spectrum [7, 8], and, more recently, alsocovariant two-body Dirac equations in the framework ofconstraint Hamiltonian dynamics [9, 10] gave very goodresults. The Lorentz structure of the confining interactionin these approaches is not quite settled, although in mostcases a scalar structure dominates.The influential Dyson-Schwinger-Bethe-Salpeter (DS-BS) approach [11, 12, 13, 14] is also covariant, but imple-ments confinement not through a confining interaction but Preprint submitted to Elsevier November 11, 2018 a r X i v : . [ h e p - ph ] A ug hrough the requirement that there be no real mass polein the dressed quark propagator. Formulated in Euclideanspace, the dynamics is driven by a pure Lorentz-vectorkernel, essentially a dressed gluon propagator.The CST belongs to the approaches related to the BSE,but is similar in spirit to the DS-BS framework in that itaims to incorporate the dynamical origin of the constituentquark masses by dressing the bare quark propagators withthe interquark kernel in a consistent fashion. However,the CST is formulated and solved directly in Minkowskimomentum space. This is advantageous over Euclideanformulations (although a number of singularities have tobe handled numerically) because no analytic continuationsare needed to calculate, e.g., form factors [15, 16], even inthe timelike region. The chosen interaction kernel is amanifestly covariant generalization of the Cornell poten-tial, and the full Dirac structure of the quarks is taken intoaccount.The Covariant Spectator Equation (CSE) is obtainedfrom the BSE [Fig. 1(a)] by carrying out the energy loopintegration such that only quark-propagator pole contri-butions are kept [Figs. 1(b) and 1(c)]. This prescriptionis motivated by partial cancellations between higher-orderladder and crossed-ladder kernels, implying that a CSTladder series effectively contains crossed-ladder contribu-tions which are necessary for the two-body equation toreach the correct one-body limit [3].In this work we are focussing on systems where onequark is typically much heavier than the other, so we areclose to the one-body limit. The BS ladder approxima-tion does not possess this limit, and it would not be agood choice to describe these mesons. On the other hand,heavy-light systems are ideal to apply a simplified versionof the CSE, the so-called one-channel spectator equation(1CSE): the positive-energy pole of the heavier quark dom-inates, such that the other three CST vertex functions canbe neglected. The 1CSE is shown in Fig. 1(c), inside thesolid rectangle.This equation retains most important properties of thecomplete CSE, namely manifest covariance, cluster separa-bility, and the correct one-body limit. It is also a good ap-proximation for equal-mass particles, as long as the bound-state mass is not too small (this excludes the pion from itsrange of applicability). In fact, in a properly symmetrizedform to account for the Pauli principle, it has been ap-plied very successfully to the description of the two- andthree-nucleon systems [17, 18, 19].A property the 1CSE does not maintain is charge-conjugation symmetry. Therefore, heavy quarkonium statescalculated with the 1CSE have no definite C-parity. Inprinciple, this problem is easily remedied by using insteadthe two-channel extension inside the dashed rectangle ofFig. 1(c). However, we decided that the considerable in-crease in computational effort would not be justified forthe purpose of this work: of the quarkonia with J P = 0 ± and 1 ± , only the axial-vector mesons ( J P = 1 + ) come inboth C-parities, and these pairs are separated by only a (a)(b)(c)Figure 1: Graphic representations of (a) the BSE for the q ¯ q boundstate vertex function Γ, where V represents the kernel of two-bodyirreducible Feynman diagrams; (b) the BS vertex function approxi-mated as a sum of CST vertex functions (crosses on quark lines indi-cate that a positive-energy pole of the propagator is calculated, lightcrosses in a dark square refer to a negative-energy pole); (c) the com-plete CST equation. The solid rectangle indicates the one-channelequation used in this work, the dashed rectangle a two-channel ex-tension with charge-conjugation symmetry. few MeV (5 to 6 MeV in bottomonium, 14 MeV in char-monium). Thus, as long as we do not seek an accuracybetter than about 10-20 MeV, the use of the 1CSE alsofor heavy quarkonia is perfectly justified. Consistent withthis level of accuracy, we also set m u = m d throughoutthis work.We use a kernel of the general form V = (cid:2) (1 − y ) (cid:0) ⊗ + γ ⊗ γ (cid:1) − y γ µ ⊗ γ µ (cid:3) V L − γ µ ⊗ γ µ [ V OGE + V C ] ≡ (cid:88) K V K Θ K ( µ )1 ⊗ Θ K µ ) , (1)where V L , V OGE , and V C are relativistic generalizationsof a linear confining potential, a short-range one-gluon-exchange (in Feynman gauge in this work), and a con-stant interaction, respectively. The confining interactionhas a mixed Lorentz structure, namely equally weightedscalar and pseudoscalar structures, and a vector struc-ture. The parameter y dials continuously between the twoextremes, y = 1 being pure vector coupling, and y = 0pure scalar+pseudoscalar coupling. The OGE and con-stant potentials are Lorentz-vector interactions. The signsare chosen such that—for any value of y —in the staticnonrelativistic limit always the same Cornell-type poten-tial V ( r ) = σr − α s /r − C is recovered.The reason for the presence of a pseudoscalar compo-nent is chiral symmetry. Although in general scalar inter-2ctions break chiral symmetry, it was shown in [20] thatthe CSE with our relativistic linear confining kernel sat-isfies the axial-vector Ward-Takahashi identity when it isaccompanied by an equal-weight pseudoscalar interaction.It has also been shown [21, 22] that, in the chiral limitof vanishing bare quark mass, a massless pion solution ofthe CSE emerges, while a finite dressed quark mass is dy-namically generated by the interaction kernel through aNJL-type mechanism.For simplicity, and to establish a reference calculation,we use fixed instead of dynamical, momentum-dependentconsituent quark masses in this work. For the same reason,we postpone the inclusion of a running coupling in V OGE and use a fixed value of α s instead.The 1CSE with quark 1 on its positive-energy massshell can be written in manifestly covariant formΓ(ˆ p , p ) = − (cid:90) d k (2 π ) m E k (cid:88) K V K (ˆ p , ˆ k )Θ K ( µ )1 × m + ˆ /k m Γ(ˆ k , k ) m + /k m − k − i(cid:15) Θ K µ ) , (2)where Θ K ( µ ) i = i , γ i , or γ µi , V K (ˆ p , ˆ k ) describes the mo-mentum dependence of the kernel K , m i is the mass ofquark i , and E ik ≡ (cid:112) m i + k . A “ˆ” over a momentumindicates that it is on its positive-energy mass shell.The kernel functions V K (ˆ p , ˆ k ) in (2) are V L (ˆ p , ˆ k ) = − σπ (cid:34) p − ˆ k ) − E p m (2 π ) δ ( p − k ) × (cid:90) d k (cid:48) (2 π ) m E k (cid:48) p − ˆ k (cid:48) ) (cid:35) , (3) V OGE (ˆ p , ˆ k ) = − πα s (ˆ p − ˆ k ) , (4) V C (ˆ p , ˆ k ) = (2 π ) E k m Cδ ( p − k ) . (5)Instead of solving (2) directly for the vertex functions,we introduce relativistic “wave functions” with definite or-bital angular momentum, defined as rather complicatedcombinations of spinor matrix elements of the vertex func-tion multiplied by the off-shell quark propagator [23]. Theyenable us to determine the spectroscopic identity of oursolutions, which is indispensable when comparing to themeasured states. In the nonrelativistic limit, they re-duce to the familiar Schr¨odinger wave functions. How-ever, our relativistic wave functions contain componentsnot present in nonrelativistic solutions. For example, the S -waves of our pseudoscalar states couple to small P -waves(with opposite intrinsic parity) that vanish in the nonrel-ativistic limit, whereas, for vector mesons, coupled S - and D -waves are accompanied by relativistic singlet and triplet P -waves. Table 1: Kernel parameters of models P1 and PSV1. Both modelsuse the quark masses m b = 4 .
892 GeV, m c = 1 .
600 GeV, m s = 0 . m u = m d = 0 .
346 GeV.
Model σ [GeV ] α s C [GeV]P1 0.2493 0.3643 0.3491PSV1 0.2247 0.3614 0.3377The 1CSE for the relativistic wave functions can bewritten as a generalized linear eigenvalue problem for thetotal bound-state mass. We solve this system by expand-ing the wave functions in a basis of B -splines, as describedin [23, 24]. Special attention is needed to treat the sin-gularities in the kernel at (ˆ p − ˆ k ) = 0. We apply tech-niques similar to the ones described in [25] to obtain stableresults. A standard Pauli-Villars regularization is appliedto divergent loop integrations, at the expense of a momen-tum cut-off parameter Λ. Our results are quite insensitiveto the exact value of Λ, and we simply fix it at twice theheavier quark mass.We calculated the pseudoscalar, scalar, vector, andaxial-vector meson states that contain at least one heavy(bottom or charm) quark, and whose mass falls below thecorresponding open-flavor threshold. As exceptions, a fewstates slightly above threshold but with very small widthsare considered as well.The model parameters are the four constituent quarkmasses m u = m d , m s , m c , and m b , the two couplingstrengths σ and α s , the constant C , and the mixing param-eter y . Early results clearly favored pure scalar+pseudoscalarconfinement, so throughout this work we set y = 0.Figure 2 shows the results of two different model calcu-lations with the 1CSE in comparison to the observed me-son masses. Model P1 was fitted to 9 pseudoscalar statesonly, whereas model PSV1 was fitted to the masses of 25pseudoscalar, scalar, and vector mesons. A solid circle(square) in Fig. 2 indicates a mass calculated with modelP1 (PSV1) that was used in the fit to the measured masses(solid lines), whereas the open symbols show predictions ofthe respective models. The parameters of the models arelisted in Tab. 1. Fitting the quark masses is much moretime-consuming than fitting the other parameters. There-fore, we first determined them in preliminary calculationsand then held them fixed in the final fits of σ , α s and C .This procedure is certainly good enough for the purposeof this work.Figure 2 clearly shows that both models give results invery good agreement with the experimental meson spec-trum. It is remarkable that a simple unified model withglobal parameters σ , α s , and C can describe heavy-lightand heavy mesons over such a large range of masses (calcu-lations in the literature often vary model parameters fromsector to sector).The rms differences to the measured masses are 0 . .
030 GeV for PSV1, which is compa-3 � η � ( �� ) η � ( �� ) η � ( �� ) Υ ( �� ) Υ ( �� ) Υ ( �� ) Υ ( �� ) χ �� ( �� ) χ �� ( �� ) � � ( �� ) χ �� ( �� ) � � ( �� ) χ �� ( �� ) χ � ( �� ) Υ ( �� ) ���� ������� π ��� πη � ( �� ) η � ( �� ) � / ψ ( �� ) ψ ( �� ) ψ ( ���� ) χ �� ( �� ) χ �� ( �� ) � � ( �� ) χ �� ( �� ) � � � � ( �� ) � �� � � * � �� ( ���� ) � � � * � � � � * � � * ( ���� ) � �� ( ���� ) � � * � - � - � + � + ������ � � �� ( � � � ) Figure 2: Masses of heavy-light and heavy mesons with J P = 0 ± and 1 ± . Circles are 1CSE results with model P1, squares of model PSV1.Solid symbols represent states used in the model fits, open symbols are predictions. Solid horizontal lines are the measured meson masses[26]. The two dashed levels are estimates taken from Ref. [27]. There is weak evidence (at 1 . σ ) that the Υ(1 D ) has been seen [28, 29].Both models predict a so far unobserved Υ(2 D ) between Υ(3 S ) and Υ(4 S ). Dashed horizontal lines across the figure indicate open flavorthresholds. The axial-vector quarkonia are shaded because the 1CSE does not define a specific C-parity for these states. χ c (1 P )and h c (1 P )) and in bottomonium ( χ b (1 P ) and h b (1 P ),as well as χ b (2 P ) and h b (2 P )).The fact that the fit exclusively to pseudoscalar mesonsof model P1 yields almost the same result as the moregeneral fit of model PSV1 allows us to draw a more funda-mental conclusion: the covariant form of the kernel (1) isresponsible for a correct prediction of the spin-dependentinteractions.It is of course very well known that the Lorentz struc-ture of a kernel determines the spin-dependent interactions(see, e.g., [30]), and it is certainly one of the many attrac-tive features of a covariant formalism that they are nottreated perturbatively but on an equal footing with thespin-independent interactions. But in a general fit to alltypes of mesons one cannot really test the predictive powerof the covariant kernels in this regard because all interac-tions are fitted simultaneously.However, pseudoscalar states are nearly pure S -waves(the tiny relativistic P -wave admixture has almost no ef-fect), such that spin-orbit and tensor forces are exactlyzero, whereas the spin-spin interaction is probed only insinglet states and cannot be separated from the centralforce. A fit in which the spin-dependent interactions arecompletely unconstraint is therefore the ideal case to testtheir prediction.To summarize, from our results one can conclude notonly that our covariant kernel is a very efficient way toderive the spectrum and wave functions of heavy-light andheavy mesons, but also that it correctly predicts the spin-dependent interactions solely based on their relation to thespin-independent interactions as dictated by covariance. Acknowledgements
We thank F. Gross and J. E. Ribeiro for helpful dis-cussions, and the Jefferson Lab Theory Group for theirhospitality during recent visits when part of this work wasperformed. This work was supported by the Portuguese
Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) under con-tracts SFRH/BD/92637/2013, SFRH/BPD/100578/2014,and UID/FIS/0777/2013.
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