Commentary on rainbow-ladder truncation for excited states and exotics
Si-xue Qin, Lei Chang, Yu-xin Liu, Craig D. Roberts, David J. Wilson
aa r X i v : . [ nu c l - t h ] S e p Commentary on rainbow-ladder truncation for excited states and exotics
Si-xue Qin, Lei Chang, Yu-xin Liu,
1, 3
Craig D. Roberts,
1, 2, 4 and David J. Wilson Department of Physics, Center for High Energy Physics and State Key Laboratoryof Nuclear Physics and Technology, Peking University, Beijing 100871, China Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616-3793, USA
Ground-state, radially-excited and exotic scalar-, vector- and flavoured-pseudoscalar-mesons arestudied in rainbow-ladder truncation using an interaction kernel that is consonant with modernDSE- and lattice-QCD results. The inability of this truncation to provide realistic predictions forthe masses of excited- and exotic-states is confirmed and explained. On the other hand, its appli-cation does provide information that is potentially useful in proceeding beyond this leading-ordertruncation, e.g.: assisting with development of projection techniques that ease the computation ofexcited state properties; placing qualitative constraints on the long-range behaviour of the inter-action kernel; and highlighting and illustrating some features of hadron observables that do notdepend on details of the dynamics.
PACS numbers: 12.38.Aw, 14.40.Be, 14.40.Rt, 24.85.+p
I. INTRODUCTION
Meson spectroscopy is a keystone of extant and forth-coming programmes at numerous facilities worldwide,e.g.: the Beijing Spectrometer; the COMPASS detec-tor at CERN; Hall-D at Jefferson Laboratory; the Japanproton accelerator research complex (J-PARC); and thePANDA detector at GSI. Each identifies an essentiallyidentical primary motivation; namely, seeking answers totwo fundamental questions within the Standard Model:What matter is possible; and How is it constituted?The subtext is quantum chromodynamics (QCD), thestrongly-interacting part of the Standard Model, and theunique nature of the forces it seems to produce. WithQCD, Nature has prepared the sole known example of astrongly-interacting quantum field theory that is definedby degrees-of-freedom which cannot directly be detected;i.e., they are confined . One of the greatest challenges inmodern physics is to comprehend and explain the phe-nomenon of confinement.Following Ref. [1], confinement in mesons has typicallybeen associated with a linearly rising potential betweenthe quark-antiquark pair [2]. There are sound reasons forusing such potential model phenomenology in the studyof heavy quarkonia [3]. However, that is not true forlight-quark systems. The static potential measured insimulations of lattice-QCD is not related in any knownway to the question of light-quark confinement. Light-quark creation and annihilation effects are fundamentallynonperturbative. Hence it is impossible in principle tocompute a potential between two light quarks [4, 5]. Onthe other hand, confinement can be related to the ana-lytic properties of QCD’s Schwinger functions [6–14], sothe question of light-quark confinement may be trans-lated into the challenge of charting the infrared behaviorof QCD’s β -function.To a large degree, this is also true of explaining dynam-ical chiral symmetry breaking (DCSB), a phenomenon which has an enormous impact on the measurable prop-erties of mesons and baryons [12, 13]. It is known thatDCSB; namely, the generation of mass from nothing , doesoccur in QCD [15–17]. It arises primarily because adense cloud of gluons comes to clothe a low-momentumquark [11, 18]. This is readily seen by solving the Dyson-Schwinger equation (DSE) for the dressed-quark propa-gator; i.e., the gap equation. However, the origin of theinteraction strength at infrared momenta, which guaran-tees DCSB through the gap equation, is currently un-known. This relationship ties confinement to DCSB.The crucial role of DCSB means that reliable informa-tion about the β -function can only be obtained via asymmetry-preserving treatment of the bound-state prob-lem that is capable of veraciously expressing DCSB. TheDSEs provide such a framework [7–13] and will be em-ployed herein.A considerable body of recent work (e.g., Refs. [11,12, 19–30]) has shown that in order to gain sensitiv-ity to the long-range part of the interaction, one shouldminimally study the properties of mesons with signifi-cant rest-frame quark orbital angular momentum, suchas scalar- and pseudovector-mesons, the radial excita-tions of pseudoscalar- and vector-mesons, and tensormesons. A challenging aspect of this problem is that theleading-order (rainbow-ladder) in the most widely usedsymmetry-preserving DSE truncation scheme [31, 32]fails to adequately express the full power of DCSB inthe kernels of the bound-state Bethe-Salpeter equations(BSEs) [25, 28, 33]. Consequently, the results pro-duced for systems other than ground-state flavoured-pseudoscalar- and vector-mesons have most often beenqualitatively and quantitatively incorrect.Is there any reason then to revisit the problem of thespectrum of excited and exotic mesons using the rainbow-ladder truncation? The answer is “no,” if the goal isto extract quantitatively reliable information about theinfrared behaviour of QCD’s β -function. On the otherhand, the answer is “yes,” if one can exploit the trunca-tion’s simplicity in order to: identify features of excitedand exotic states that are plausibly independent of thetruncation; or techniques that can be useful in connec-tion with more sophisticated truncations. Such is ouraim herein.In Sec. II we present the gap- and Bethe-Salpeter-equations in the symmetry-preserving rainbow-laddertruncation, explain the structure of their solutions anddefine their kernels. Section III reports and interprets ournumerical results, which include: masses and decay con-stants; an investigation of the relative importance of var-ious Dirac structures within meson Bethe-Salpeter am-plitudes; and an exploration of the pointwise behaviourand sign of the leading invariant amplitudes. Section IVis an epilogue. II. GAP AND BETHE-SALPETER EQUATIONS
The renormalised rainbow-gap- and ladder-Bethe-Salpeter-equations are, respectively: S ( p ) − = Z ( iγ · p + m bm )+ Z Z Λ ℓ G ( ℓ ) ℓ D free µν ( ℓ ) λ a γ µ S ( p − ℓ ) λ a γ ν , (1)Γ M ( k ; P ) = − Z Z Λ q G (( k − q ) ) ( k − q ) D free µν ( k − q ) × λ a γ µ S ( q + )Γ M ( q ; P ) S ( q − ) λ a γ ν , (2)where: we use a Euclidean metric [12]; R Λ ℓ := R Λ d ℓ (2 π ) represents a Poincar´e-invariant regularization of the in-tegral, with Λ the ultraviolet regularization mass-scale; Z ( ζ, Λ) is the quark wavefunction renormalisation con-stant, whose location and strength in these equationsmay be understood from Refs. [32, 34]; D free µν ( ℓ ) is theLandau-gauge free-gauge-boson propagator; one canchoose q ± = q ± P/ ℓ G ( ℓ ) = ℓ G IR ( ℓ ) + 4 π ˜ α pQCD ( ℓ ) (3)specifies the interaction, with ˜ α pQCD ( k ) a bounded,monotonically-decreasing regular continuation of theperturbative-QCD running coupling to all values ofspacelike- ℓ , and G IR ( ℓ ) an Ansatz for the interactionat infrared momenta, such that G IR ( ℓ ) ≪ ˜ α pQCD ( ℓ ) ∀ ℓ & . The form of G IR ( ℓ ) determines whether Landau gauge is used for many reasons [35, 36], for example, it is:a fixed point of the renormalisation group; that gauge for whichsensitivity to model-dependent differences between
Ans¨atze forthe fermion–gauge-boson vertex are least noticeable; and a co-variant gauge, which is readily implemented in numerical simu-lations of lattice regularised QCD [37]. confinement and/or DCSB are realised in solutions of thegap equation.The solution of the gap equation is a dressed-quarkpropagator S ( p ) = 1 iγ · p A ( p , ζ ) + B ( p , ζ ) = Z ( p , ζ ) iγ · p + M ( p ) , (4)which is obtained from Eq. (1) augmented by a renormal-isation condition. A mass-independent scheme is a usefulchoice and can be implemented by fixing all renormalisa-tion constants in the chiral limit. Notably, the mass func-tion, M ( p ) = B ( p , ζ ) /A ( p , ζ ), is independent of therenormalisation point, ζ ; and the renormalised current-quark mass is given by m ζ = Z m ( ζ, Λ) m bm (Λ) = Z − Z m bm , (5)wherein Z is the renormalisation constant associatedwith the Lagrangian’s mass-term. Like the running cou-pling constant, this “running mass” is a familiar con-cept. However, it is not commonly appreciated that m ζ is simply the dressed-quark mass function evaluated atone particular deep spacelike point; viz, m ζ = M ( ζ ) . (6)The renormalisation-group invariant current-quarkmass may be inferred viaˆ m f = lim p →∞ "
12 ln p Λ γ m M f ( p ) , (7)where f specifies the quark’s flavour, γ m = 12 / (33 − N f α ): N f α is the number of quark flavours em-ployed in computing the running coupling; and Λ QCD is QCD’s dynamically-generated renormalisation-group-invariant mass-scale. The chiral limit is expressed byˆ m f = 0 . (8)Moreover, ∀ ζ ≫ Λ , M f ( p = ζ ) M f ( p = ζ ) = m ζf m ζf = ˆ m f ˆ m f . (9)We would like to emphasise, however, that in the pres-ence of DCSB the ratio M f ( p ) /M f ( p ) is not indepen-dent of p : in the infrared; i.e., ∀ p . Λ , it thenexpresses a ratio of constituent-like quark masses, which,for light quarks, are two orders-of-magnitude larger thantheir current-masses and nonlinearly related to them[38, 39]. (See, e.g., the discussion following Eq. (15).)The BSE is an eigenvalue problem for the mesonmasses-squared; i.e., in a given channel Eq. (2) has so-lutions only at particular, isolated values of P = − m M .At these values, solving the equation produces the as-sociated meson’s Bethe-Salpeter amplitude, which canthen be used in the computation of observable proper-ties. Herein we consider flavoured-pseudoscalar-, scalar-and vector-meson ground-, radially-excited- and exotic-states, so that the following amplitudes arise:Γ J P =0 − ( k ; P ) = X i =1 γ τ i − ( k, P ) F i − ( k ; P ) , (10)Γ + ( k ; P ) = X i =1 τ i + ( k, P ) F i + ( k ; P ) , (11)Γ − ( k ; P ) = X i =1 τ i − ( k, P ) F i − ( k ; P ) , (12)with ( a Tµ := a µ − P µ a · P/P ) τ − = iτ + = i I D , (13a) τ − = γ · P, τ + = k · P τ − , (13b) τ − = k · P τ + , τ + = P γ · k − k · P γ · P, (13c) τ − = τ + = σ µν P µ k ν , (13d) τ − = iγ Tµ , (13e) τ − = i [3 k Tµ γ · k T − γ Tµ k T · k T ] , (13f) τ − = ik Tµ k · P γ · P, (13g) τ − = i [ γ Tµ γ · P γ · k T + k Tµ γ · P ] , (13h) τ − = k Tµ , (13i) τ − = k · P [ γ Tµ γ T · k − γ · k T γ Tµ ] , (13j) τ − = ( k T ) ( γ Tµ γ · P − γ · P γ Tµ ) − k Tµ γ · k T γ · P, (13k) τ − = k Tµ γ · k T γ · P. (13l)The canonical normalisation condition (see, e.g., Eq. (27)in Ref. [19] or, more generally, Ref. [41]) constrains thebound-state to produce a pole with unit residue in thequark-antiquark scattering matrix.It remains only to specify the interaction in order toproceed. We use that explained in Ref. [30]; viz., G ( s ) = 8 π ω D e − s/ω + 8 π γ m F ( s )ln[ τ + (1 + s/ Λ ) ] , (14)where: γ m = 12 /
25, Λ
QCD = 0 .
234 GeV; τ = e − F ( s ) = { − exp( − s/ [4 m t ]) } /s , m t = 0 . Masses and other properties of charge-neutral pseudoscalarmesons are affected by the non-Abelian anomaly. In the BSEcontext, this is discussed in Ref. [40]. Since the non-Abeliananomaly is a correction to rainbow-ladder truncation that is qual-itatively different to the focus of our study, herein we specialiseto flavoured pseudoscalars. confinement and DCSB. Moreover, it is consistent withmodern DSE and lattice studies, which indicate that thegluon propagator is a bounded, regular function of space-like momenta that achieves its maximum value on thisdomain at s = 0 [42–44], and the dressed-quark-gluonvertex does not possess any structure which can qualita-tively alter this behaviour [45, 46]. Notably, as illustratedin Ref. [30], the parameters D and ω are not independent:with Dω = constant, one can expect computed observ-ables to be practically insensitive to ω on the domain ω ∈ [0 . , .
6] GeV.
III. NUMERICAL RESULTS FORBOUND-STATES PROPERTIESA. Ground states
Using the method of Ref. [47], we solved the gap equa-tion for light u = d quarks and the s -quark, withtheir current-quark masses fixed by requiring that thepion and kaon BSEs produce m π ≈ .
138 GeV and m K ≈ .
496 GeV. This is straightforward in rainbow-ladder truncation because there is no coupling betweenthe separate gap equations and no feedback from theBSEs [48]; and yields m ζu = d = 3 . , m ζs = 82 MeV (15)quoted at our renormalisation point ζ = 19 GeV, a valuechosen to match the bulk of extant studies. These valuescorrespond to renormalisation-group-invariant masses ofˆ m u,d = 6 MeV, ˆ m s = 146 MeV, one-loop-evolved massesof m u = d = 5 MeV, m s = 129 MeV; and give m s /m u = 24. They are consequently comparable withcontemporary estimates by other means [49]. NB. With ω = 0 . M Es /M Eu = 1 . ≪ ˆ m s / ˆ m u , where theconstituent-quark mass M Ef := { s | s > , s = M f ( s ) } .In Table I we report selected results related to ground-state pseudoscalar-, scalar- and vector-mesons. The me-son masses are obtained in solving the BSEs. Regardingthe other meson quantities, in terms of the canonicallynormalised Bethe-Salpeter amplitudes and with χ J P ( k ; P ) = S f ( k + )Γ J P ( k ; K ) S f ( k − ) , (16)where f , f are the meson’s valence-quark and ω A (0) 2.07 1.70 1.38 1.16 M (0) 0.62 0.52 0.42 0.29 m π f π ρ / π m K f K ρ / K m σ ρ / σ m κ f κ + ρ / κ m ρ f ρ m φ f φ Dω = (0 . . The current-quark masses at ζ =19 GeV are given in Eq. (15). Dimensioned quantities are re-ported in GeV. For comparison, some experimental valuesare [49]: f π = 0 .
092 GeV, m π = 0 .
138 GeV; f K = 0 .
113 GeV, m K = 0 .
496 GeV; f ρ = 0 .
153 GeV, m ρ = 0 .
777 GeV; and f φ = 0 .
168 GeV, m φ = 1 .
02 GeV. NB. The scalar mesonslisted here are not directly comparable with the lightestscalars in the hadron spectrum because the rainbow-laddertruncation is a priori known to be a poor approximation inthis channel: nonresonant corrections [25, 28] and resonantfinal-state interactions are both important [39]. -antiquark, respectively, one has [19, 50, 51] f − P µ = Z tr CD Z Λ k iγ γ µ χ − ( k ; P ) , (17) iρ ζ − = Z tr CD Z Λ k γ χ − ( k ; P ) , (18) f +12 P µ = Z tr CD Z Λ k iγ µ χ +12 ( k ; P ) , (19) ρ ζ +12 = − Z tr CD Z Λ k χ +12 ( k ; P ) , (20) f − m − = Z tr CD Z Λ k γ µ χ − ( k ; P ) . (21)The Table confirms that, with Dω = constant, observableproperties of ground-state scalar-, vector- and flavoured-pseudoscalar-mesons computed with Eq. (14) are practi-cally insensitive to variations of ω ∈ [0 . , .
6] GeV.It is noteworthy, and readily verified using entriesin the Table, that the pseudoscalar- and scalar- mesonmasses satisfy the following identities, exact in QCD ω σ m π m −− m π m σ m + − m σ m ρ m − + m ρ Dω = (1 . .The subscript “1” indicates first radial excitation. The lastcolumn measures sensitivity to variations in r ω := 1 /ω : σ ≪ σ ≈
1, immaterialsensitivity. Dimensioned quantities reported in GeV. [19, 50]: f − m − = ( m ζf + m ζf ) ρ ζ − , (22) f +12 m +12 = − ( m ζf − m ζf ) ρ ζ +12 . (23)Furthermore, the products f ± ρ ± describe in-mesoncondensates [19, 50, 54]. B. Radial excitations and exotics
In addition to properties of the ground-states, we havecomputed selected quantities associated with J = 0 , m ( ω ) = constant and m ( ω ) = ω ( c + c ω ) , (24)then computed the standard-deviation of the relative er-ror in each fit, σ for the constant and σ for Eq. (24),and finally formed the ratio: σ = σ /σ .In preparing the table we used Dω = (1 . . Thishas the effect of inflating the π - and ρ -meson ground-state masses to a point wherefrom corrections to rainbow-ladder truncation can plausibly return them to the ob-served values [56, 57]. It is therefore notable that, incontrast to Table I, the value reported for m σ in Table II Notwithstanding complexities associated with the structure oflight-quark scalars [39, 52, 53], the identity written here appliesto any scalar meson that can be produced via e + e − annihilation.It is not of experimental significance, however, if the pole is deepin the complex plane. τ i M a ss s h i ft ( % ) −20−15−10−505 Remove π π e π FIG. 1. Pseudoscalar mesons. Relative difference betweenthe mass computed with all the amplitudes in Eq. (10) andthat obtained when the identified i ≥ circles – ground-state pion; squares – J PC = 0 −− exotic;and diamonds – first pseudoscalar radial excitation. In allcases, ω = 0 . Dω = (1 . . There is only minorquantitative variation with ω ∈ [0 . , .
6] GeV. NB. The i = 1amplitude is never omitted, it specifies the reference value. matches estimates for the mass of the dressed-quark-corecomponent of the σ -meson obtained using unitarised chi-ral perturbation theory [52, 53].A comparison between the ω -dependence of ground-state properties and those of excited- and exotic-stateswas drawn in Ref. [30] and we only summarise it here.Ground-state masses of light-quark pseudoscalar- andvector-mesons are quite insensitive to ω ∈ [0 . , .
6] GeV.Any minor variation is described by a decreasing func-tion. In the case of exotics and radial excitations, thevariation with ω is described by an increasing functionand the variation is usually significant. This is readilyunderstood. The quantity r ω := 1 /ω is a length-scalethat measures the range over which the infrared part ofEq. (3), G IR , is active. For ω = 0 this range is infinite, butit decreases with increasing ω . One expects exotic- andexcited-states to be more sensitive to long-range featuresof the interaction than ground-states and, additionally,that their masses should increase if the magnitude andrange of the strong piece of the interaction is reducedbecause there is less binding energy.Table II confirms a known fault with the rainbow-ladder truncation; viz., whilst it binds in exotic channels,it produces masses that are too light, just as it does foraxial-vector mesons. It is similarly noticeable that m π is far more sensitive to variations in ω than is m ρ ; andalthough m π < m ρ for ω = 0 . m π > m ρ in rainbow-ladder truncation.This, too, is a property of the truncation, which is insen-sitive to the details of G ( k ); e.g., the same ordering isobtained with a momentum-independent interaction [57]. τ i M a ss s h i ft ( % ) Remove σ σ e σ FIG. 2. Scalar mesons. Relative difference between the masscomputed with all the amplitudes in Eq. (11) and that ob-tained when the identified i ≥ circles – ground-state u = d scalar; squares – J PC = 0 + − exotic;and diamonds – first pseudoscalar radial excitation. In allcases, ω = 0 . Dω = (1 . . There is only minorquantitative variation with ω ∈ [0 . , .
6] GeV. NB. The i = 1amplitude is never omitted, it specifies the reference value. C. Structure of bound states
In order to develop insight, both into the structure ofexcited- and exotic-states, and for progressing beyondrainbow-ladder truncation, it is useful to know whichof the invariant amplitudes in Eqs. (10)-(12) are domi-nant. One useful measure of an amplitude’s importanceis the contribution it makes to a given meson’s mass.Figure 1 displays the result for pseudoscalar mesons: inall cases a good approximation is obtained by retaining F − and F − . This outcome is in agreement with ex-tant ground-state computations [19] but extends thoserainbow-ladder conclusions to excited- and exotic-states.Evidently, there is little here to distinguish between theexotic and the radial excitation. Curiously, F − playsa role of similar magnitude in each state and the am-plitudes F − and F − are always largely unimportant.These last two, in this instance small, amplitudes arethose most directly associated with nonzero quark orbitalangular momentum in the meson’s rest-frame.For scalar mesons, on the other hand, one reads fromFig. 2 that F + , F + and F + should be included if a re-liable approximation is to be obtained. The latter twoamplitudes are directly associated with significant rest-frame quark orbital angular momentum. Notably, inquantum mechanical models, scalar mesons are identifiedas P states, in contrast to S for pseudoscalar mesons.The vector meson ( S ) situation is displayed in Fig. 3.In agreement with Ref. [58], a good approximation for thevector-meson ground-state is obtained by retaining F − , F − , F − . The last two amplitudes are associated with P -wave components in the rest-frame. However, for thefirst radial excitation, F − is also important: this am-plitude is directly associated with a D -wave component τ i M a ss s h i ft ( % ) Remove ρ ρ e ρ FIG. 3. Vector mesons. Relative difference between the masscomputed with all the amplitudes in Eq. (12) and that ob-tained when the identified i ≥ circles – ground-state u = d vector; squares – J PC = 1 − + exotic;and diamonds – first vector radial excitation. In all cases, ω = 0 . Dω = (1 . . Whilst there are quantita-tive changes with ω , the pattern of amplitude importance isunchanged. NB. The i = 1 amplitude is never omitted, itspecifies the reference value. in the radially-excited vector-meson’s rest frame. Theseobservations suggest that a BSE might be built whichprojects selectively onto the first radially excited state.The additional information contained in these figuresindicates that the shortcomings identified above, of therainbow-ladder truncation for states other than ground-state vector- and flavoured-pseudoscalar-mesons, can beattributed to this truncation’s inadequate expression inthe Bethe-Salpeter kernels of effects which in quantummechanics would be described as spin-orbit interactions.Namely, treating the quark-gluon vertex as effectivelybare in both the gap- and Bethe-Salpeter-equations leadsto omission of critically important helicity-flipping inter-actions that are dramatically enhanced by DCSB, as dis-cussed in Refs. [25, 28, 33].One may readily expand on this. For example, vec-tor meson bound states possess nonzero magnetic- andquadrupole-moments [59]. This fact, Fig. 3 and the as-sociated discussion together indicate that there is ap-preciably more dressed-quark orbital angular momen-tum within these states than within pseudoscalar mesons.Hence, spin-orbit repulsion could significantly boost m ρ and thereby produce the correct level ordering; viz., m ρ > m π . Moreover, since exotic states appear aspoles in vertices generated by interpolating fields with“unnatural time-parity,” the importance of orbital an-gular momentum within these states is magnified. Thesecomments apply with equal force to tensor mesons, whichcannot be formed without rest-frame quark orbital angu-lar momentum.At present the best hope for a realistic descriptionof the meson spectrum within a Poincar´e covariant ap- E π E π E π F ( k ) k [GeV ] E π E π E π F ( k ) k [GeV ] E π E π E π F ( k ) k [GeV ] E π E π E π F ( k ) k [GeV ] E π E π e E π e F ( k ) k [GeV ] E π E π e E π e F ( k ) k [GeV ] E π E π e E π e F ( k ) k [GeV ] E π E π e E π e F ( k ) k [GeV ] FIG. 4. Pseudoscalar mesons. ω -dependence of low-orderChebyshev-projections of leading invariant amplitude forground-, radially-excited- and exotic-states: upper four pan-els , ground and radial; lower four panels , ground and exotic.In all panels, solid – zeroth moment, ground-state; dashed – leading moment, comparison state; dash-dot – sublead-ing moment, comparison state. Row-1, left , ω = 0 . Row-1, right , ω = 0 . Row-2, left , ω = 0 . Row 2, right , ω = 0 . E π ( p = 0) = 1; and Dω = (1 . . proach is provided by the essentially nonperturbativeDSE truncation scheme whose use is illustrated mostfully in Ref. [28]. That symmetry-preserving schemedeeply embeds effects associated with DCSB into theBethe-Salpeter kernel. A lattice-QCD perspective on the meson spectrum may be drawnfrom Ref. [60].
D. Connecting amplitudes with observables
Whilst not directly observable, the momentum-dependence of meson Bethe-Salpeter amplitudes is a cru-cial determinative factor in the computation of measur-able quantities. In Figs. 4 and 5, therefore, we depict the ω -dependence of a few low-order Chebyshev moments ofthe leading invariant amplitude for the pseudoscalar andvector mesons: n F M ( p ) := 2 π Z − dx p − x U n ( x ) F M ( k , x ; P ) , (25)where k · P = x √ k P and U n ( x ) is a Chebyshev poly-nomial of the second kind. NB. For pseudoscalar andvector states with natural C -parity, only the even mo-ments are nonzero, whereas it is the odd moments whichare nonzero for the exotic partners of these states.The upper four panels in Fig. 4 compare the ampli-tudes of the ground-state and first-radially-excited pseu-doscalar mesons. The ground-state is clearly insensitiveto ω . However, as hoped for and anticipated, the ra-dial excitation reacts strongly to variations in ω . Mostnotable is the suppression of E π with decreasing ω , tobe replaced by an increasingly large E π . Indeed, at ω = 0 . E π is almost negligible and possesses twozeros, instead of the single zero expected in the amplitudeof a first radial excitation since the work of Ref. [20]. Insuch circumstances, the radial excitation may even pos-sess a smaller charge radius than the ground state [21].In our view these features signal that values of ω . . β -function cannot dramaticallysuppress the radial excitation’s leading amplitude norinduce it to have a second zero. This perspective issupported by the following considerations. Neither thehomogeneous BSE nor the canonical normalisation con-dition fix the sign of the Bethe-Salpeter amplitude at k = 0. As in quantum mechanics, this is arbitrary andcannot affect observables. Another parallel with quan-tum mechanics is also relevant. Namely, for a ground-state, the sign of the radial wave function at the originin configuration space is the same as that of its analogueat the origin in momentum space, whereas these signsare opposite for the first radial excitation. This patternrepeats for higher even- and odd-numbered radial excita-tions. Here, a direct solution of the inhomogeneous BSEis instructive because this equation does determine signs.For example, consider the pseudoscalar vertex: Fig. 6 ofRef. [24] illustrates a case in which the residue associatedwith the pseudoscalar meson ground-state is positive andthat connected with the first radial excitation is negative,which is the behaviour found herein for ω & . k = 0, Γ − (0; P ),and ρ − . The latter is the expression in quantum fieldtheory for the value of the Bethe-Salpeter wave func-tion at the origin in configuration space. Thus, the pat- tern exposed by the inhomogeneous BSE parallels thatin quantum mechanics.It is straightforward to see that this pattern is realisedin the second, third and fourth panels of Fig. 4, whichdepict results obtained with ω ≥ . k = 0 values of the leading amplitudes’ lowest Cheby-shev projections are positive; and whilst that for theground-state remains positive, that for the first radialexcitation changes sign, so that it is a negative-definitefunction for k & . In performing a Fourier trans-form, large- k maps onto small x and hence this be-haviour guarantees that the Bethe-Salpeter wave func-tion for the first radial excitation is negative at the originin configuration space.These observations reemphasise the peculiar characterof the ω = 0 . k , the first radial excitation is negative at large- k , and so on. With this convention, one necessarily finds ρ ζπ > ρ ζπ <
0, etc., and hence, from Eq. (22), f π > f π <
0. We depict the ω -dependence of the leptonicdecay constants in Fig. 6.The bottom four panels of Fig. 4 display low-order mo-ments of the exotic-pseudoscalar-meson’s leading invari-ant amplitude, contrasted with the ground-state’s zerothmoment. So long as ω & . E π and thethird moment is negative definite. This is the first timethese features have been exposed but we expect themto be characteristic of the rainbow-ladder truncation. Itwill be important to learn whether this pattern persistsbeyond rainbow-ladder truncation.The top four panels in Fig. 5 compare the ampli-tudes of the ground-state and first-radially-excited vec-tor mesons. The ground-state is insensitive to ω so longas ω & . ω . In this case, natural be-haviour for the excited state’s amplitudes is only ob-tained for ω & . f ρ > f ρ < ω & . E ρ and the third moment is negative definite. The sim-ilarity to the lower panels of Fig. 4 encourages us in theexpectation that these features are characteristic of therainbow-ladder truncation. Moreover, they suggest againthat there is too much similarity between natural and ex-otic C -parity states in rainbow-ladder truncation.In Fig. 6 we depict the ω -dependence of pseudoscalar-and vector-meson leptonic decay constants. Those for theground-states are positive whilst those for the first radial E ρ E ρ E ρ F ( k ) −0.500.51.0 k [GeV ] E ρ E ρ E ρ F ( k ) k [GeV ] E ρ E ρ E ρ F ( k ) k [GeV ] E ρ E ρ E ρ F ( k ) k [GeV ] E ρ E ρ e E ρ e F ( k ) k [GeV ] E ρ E ρ e E ρ e F ( k ) k [GeV ] E ρ E ρ e E ρ e F ( k ) k [GeV ] E ρ E ρ e E ρ e F ( k ) k [GeV ] FIG. 5. Vector mesons. ω -dependence of low-orderChebyshev-projections of leading invariant amplitude forground-, radially-excited- and exotic-states: upper four pan-els , ground and radial; lower four panels , ground and exotic.In all panels, solid – zeroth moment, ground-state; dashed – leading moment, comparison state; dash-dot – subleadingmoment, comparison state. Row-1, left , ω = 0 . Row-1,right , ω = 0 . Row-2, left , ω = 0 . Row 2,right , ω = 0 . E ρ ( p = 0) = 1;and Dω = (1 . . excitations are negative. The origin of this outcome inan internally consistent treatment of bound-states wasexplained above. Notable, too, is the small magnitude ofthe decay constant for the pion’s first radial excitation: f π ≈ − τ → π (1300) ν τ [61] andnumerical simulations of lattice-regularised QCD [23]. f J P [ G e V ] −0.100.10.2 ω [GeV] f π f π f ρ f ρ FIG. 6. ω -dependence of leptonic decay constants for pseu-doscalar and vector mesons: ground-state pion, solid line;radially-excited pion, dashed line; ground-state rho-meson,dotted line; and radially-excited rho-meson, dash-dot line.( Dω = (1 . .) IV. CONCLUSION
Using an interaction kernel that is consonant withmodern DSE- and lattice-QCD results, we employed arainbow-ladder truncation of QCD’s Dyson-Schwingerequations in an analysis of ground-state, radially-excitedand exotic scalar-, vector- and flavoured-pseudoscalar-mesons. We confirmed that rainbow-ladder truncation isincapable of providing realistic predictions for the massesof excited- and exotic-states; e.g., the ordering betweenpseudoscalar and vector radially-excited states is incor-rect, and computed masses for exotic states are too lowin comparison with other estimates. Indeed, in rainbow-ladder truncation, it appears that exotic states are inmost respects too much like their C -parity partners.On the other hand, rainbow-ladder results do pro-vide information that is useful in proceeding beyondthis leading-order. For example, in each channel therainbow-ladder truncation indicates those invariant am-plitudes which are likely to dominate in any solution ofthe Bethe-Salpeter equation. This knowledge can be usedin developing integral projection techniques that sup-press ground-state contamination when searching for ex-cited states. Moreover, the response of observables, andthe Bethe-Salpeter amplitudes which produce them, tochanges in the infrared evolution of the interaction ker-nel can be used effectively to demarcate the domain ofphysically allowed possibilities for that evolution. This isvaluable in qualitatively constraining the long-range be-haviour of QCD’s β -function. In addition, the symmetry-preserving character of the rainbow-ladder truncationand the ready access it provides to Bethe-Salpeter ampli-tudes for bound-states enable one to highlight and illus-trate features of hadron observables that do not dependon details of the dynamics.There are many indications that dynamical chiralsymmetry breaking (DCSB), of which the momentum-dependence of the dressed-quark mass-function is a strik-ing signal, has an enormous impact on hadron proper-ties. This study is one of a growing body which indicatesthat the veracious expression of DCSB in the bound-stateproblem is essential if one is to reliably predict and under-stand the spectrum and properties of excited and exotichadrons. Achieving this will provide the power to use ex-tant and forthcoming data as a tool with which to chart the nonperturbative evolution of QCD’s β -function. ACKNOWLEDGMENTS
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