aa r X i v : . [ nu c l - t h ] O c t Flavourful hadronic physics
B. El-Bennich a , M. A. Ivanov b and C. D. Roberts aca Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA b Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia c Department of Physics, Peking University, Beijing 100871, China
We review theoretical approaches to form factors that arise in heavy-meson decays and are hadronic expressionsof non-perturbative QCD. After motivating their origin in QCD factorisation, we retrace their evolution fromquark-model calculations to non-perturbative QCD techniques with an emphasis on formulations of truncatedheavy-light amplitudes based upon Dyson-Schwinger equations. We compare model predictions exemplarily forthe F B → π ( q ) transition form factor and discuss new results for the g D ∗ Dπ coupling in the hadronic D ∗ decay.
1. Flavour physics and strong phases
The last two decades have witnessed impor-tant advances in flavour physics and in partic-ular heavy-meson decays. From the first obser-vation of a B meson by the CLEO Collaborationin 1981 at the Cornell Electron Storage Ring [1](and their ongoing D -meson research program)to the dedicated B -physics facilities at SLAC inCalifornia and KEK in Japan, much progress hasbeen made. Of course, while B physics is themain focus of the Collaborations Belle at KEKand BaBar at SLAC, and of the CDF experimentat Fermilab, considerable efforts have also beendevoted to studies of D -meson decays, charmo-nium and τ physics.Naturally, the driving force is to confirm theelectroweak sector of the Standard Model whichhas established itself as the foremost paradigm todescribe CP violation; in other words, the maintask experimentalists strive for is the precise areaof the Cabibbo-Kobayashi-Maskawa (CKM) tri-angle and the weak CP violating phase codifiedwithin its angles. In order to measure the size andexact form of this triangle, the angles are derivedfrom branching fractions in a nowadays large va-riety of decay channels. We focus on non-leptonicdecays, B → M M , but our discussion also ap-plies to the cleaner semi-leptonic B → M ℓ ¯ ν ℓ case.From a theoretical point of view, heavy mesonscan be used to test simultaneously all the mani- festations of the Standard Model, namely the in-terplay between electroweak and strong interac-tions. They also provide an excellent playgroundto examine non-perturbative QCD effects alreadymuch studied in hadronic physics. It is notewor-thy to remind that no CP -violating amplitudecan be generated without strong phases. Sup-pose a heavy particle H decays into a mesonicfinal state M = M M ... , H → M , and that theStandard Model lagrangian contributes two terms(two Feynman diagrams) to this process. Then,the decay amplitude and its corresponding CP conjugate, written most generally, are A ( H → M ) = λ A e iϕ + λ A e iϕ (1)¯ A ( ¯ H → ¯ M ) = λ ∗ A e iϕ + λ ∗ A e iϕ . (2)The weak coupling λ i is a combination of possiblycomplex CKM matrix elements and Ae iϕ denotesthe strong (hadronic) parts of the transition am-plitude, where we emphasise that they too canhave both a real part, or magnitude, and a phase,or absorptive part, due to multiple rescatteringof the final-state quarks and mesons. These CP -related intermediate states must contribute thesame absorptive part to the two decays, thereforethe strong phases ϕ i are the same in Eqs. (1) and(2). Taking the difference of the absolute squares,known as direct CP violation, |A| − | ¯ A| = 2 A A Im( λ λ ∗ ) sin( ϕ − ϕ ) , (3)one sees that no such violation, | ¯ A / A| 6 = 1,1can occur if the weak couplings contain only realphases or the strong phases are the same. Hence,in order to extract the weak CKM phases withprecision from the decay amplitudes, it is crucialto evaluate the QCD contributions reliably .
2. QCD factorisation
As simple as these mesons appear to be—a bound colourless heavy-light ¯ qq pair—the dif-ference in quark masses and the energetic lightmesons produced in their decays lead to a dis-parate array of energy scales. A central as-pect of heavy-meson phenomenology are factori-sation theorems which allow for a disentangle-ment of short-distance or hard physics, which en-compasses electroweak interactions and perturba-tive QCD, from long-distance or soft physics, gov-erned by non-perturbative hadronic effects. Inthe following, we illustrate the factorisation withnon-leptonic decays of a heavy meson H .In the hamiltonian formulation of heavy-quarkeffective theory (HQET) [2], in which amplitudesare expanded in powers of Λ QCD /m h and theheavy quark is a static particle in the limit m h →∞ , the H → M M decay amplitude is given by A = G F √ X p λ p X i C i ( ζ ) h M M | O i | H i ( ζ ) , (4)where λ p = V pb V ∗ pk ( p = u, c ; k = d, s ) are prod-ucts of CKM matrix elements and G F is the Fermicoupling constant. The dimension-six four-quarkoperators O i result from integrating out the weakgauge bosons W ± in the operator product expan-sion and the Wilson coefficients C i ( ζ ) encode per-turbative QCD effects above the renormalisationpoint ζ .In what is called “naive” factorisation, thehadronic matrix element h M M | O i | H i is ap-proximated by the product of two bilinear cur-rents, h M | ¯ qγ µ (1 − γ ) b | B i ⊗ h M | ¯ q ′ γ µ (1 − γ ) q | i + ( M ↔ M ), where colour indices have beenomitted. This factorisation simply expresses thematrix element of a local four-quark operator asa product of a decay constant and a transitionform factor. However, as has long been known,the saturation by vacuum insertion fails in thecase of most D decay modes and is largely in- sufficient to reproduce the experimentally moreprecise data on B → M M branching fractions.In fact, any hard final-state gluon interaction hasbeen neglected and soft-gluon exchange is at bestincorporated into an effective colour parameter orform factors. Moreover, the renormalisation scaleand scheme dependence of C i ( ζ ) are not cancelledby those of the matrix element h M M | O i | H i ( ζ ).A major improvement over this simple fac-torisation Ansatz comes from the systematic re-organisation of weak and QCD interactions inthe HQET. Three distinctive approaches haveemerged in recent years: QCD factorisation(QCDF) [3], perturbative QCD (pQCD) [4] andsoft-collinear effective theory (SCET) [5]. Wehere focus on the form factors that emerge inthese factorisation approaches and solely remarkthat QCD corrections beyond naive factorisationentail, in the limit m h ≫ Λ QCD , the B → M M decay amplitude can be schematically written as h M M | O i | B i = h M | j | B ih M | j | i× " X n r n α ns + O (Λ QCD /m b ) , (5)where j and j are the bilinear currents. Thishas been shown explicitly to leading order in α s [3,6] and including the one-loop correction( α s ) to the tree-diagram scattering between theemitted meson and the one containing the spec-tator quark [7].The factorisation theorem derived using SCETagrees with QCDF if perturbation theory is ap-plied at the hard m b and hard-collinear m b Λscales, with Λ typically of the order of 100 MeV.It is evident from Eq. (5) that higher orders in α s break the factorisation but these corrections canbe systematically supplemented; the analogy withperturbative factorisation for exclusive processesin QCD at large-momentum transfer is not acci-dental [8]. Further contributions that break thefactorisation, formally suppressed in Λ QCD /m b yet not irrelevant, are weak annihilation decayamplitudes and final-state interactions betweendaughter hadrons [9]. Neglecting power correc-tions in α s and taking the limit m b → ∞ , thenaive factorisation is recovered.
3. Separating scales: the softer the harder
While factorisation theorems elaborated withSCET provide the means to systematically inte-grate out energy scales in the perturbative do-main, yielding approximations valid in the heavy-quark limit for a given decay in terms of productsof hard and soft matrix elements, a reliable eval-uation of the latter is notoriously difficult. Infact, it is the soft physics of the bound statesthat renders the task hard, as it implies non-perturbative QCD contributions. Full ab initio calculations are currently out of reach and for thetime being one is left with modelling the heavy-to-light amplitudes with as much input from non-perturbative QCD as possible. Just how muchsoft physics is included depends on the theoreti-cal
Ansatz and techniques employed.A variety of theoretical approaches have beenapplied to this problem, recent amongst which areanalyses using light-front and relativistic quarkmodels, light-cone sum rules (LCSR) and lattice-QCD simulations. In Section 4 we briefly sum-marise these approaches while Section 5 dealsin more detail with studies of heavy-to-lightform factors within the framework of the Dyson-Schwinger equation (DSE). We refer to a recentreview [10] for a summary of transition form fac-tor data from lattice-regularised QCD and justnote that current results are obtained at largesquared momentum transfer, i.e. , q ≃
16 GeV for B → π transitions. Hence, values at low q must necessarily be extrapolated by means of a(pole-dominance) parametrisation [11,23].
4. Hadronic transition form factors
Quark models : relativistic quark models [12,13,14,15,16,17,18] have in common that their onlydegrees of freedom are constituent quarks whosemasses are parameters of the hamiltonian. Thehadronisation of the two valence quarks is de-scribed by vertex wave functions or phenomeno-logical Bethe-Salpeter amplitudes (BSA). Theapproaches in [13,14,15] represent heavy-to-lighttransition amplitudes by triangle diagrams, a 3-point function between two meson BSA and theweak coupling, which yields the transition ampli-tude h M ( p ) | ¯ q Γ I h | H ( p ) i and reads generally, A ( p , p ) = tr CD Z d k (2 π ) ¯Γ ( µ ) M ( k ; − p ) S q ( k + p ) × Γ I ( p , p ) S Q ( k + p )Γ H ( k ; p ) S q ′ ( k ) , (6)where S ( k ) are quark propagators, Q = c, b ; q = q ′ = u, d, s ; M = S, P, V, A and the index µ indicates a possible vector structure in the final-state BSA. Γ I is the interaction vertex whoseLorentz structure depends on the operator O i inthe HQET and Γ H is the heavy meson BSA. Thetrace is over Dirac and colour indices.An analogous approach represents the ampli-tude in Eq. (6) by relativistic double-dispersionintegrals over the initial- and final-mass vari-ables p and p , where the integration kernelarises from the double discontinuity of the tri-angle diagram (putting internal quark propaga-tors on-shell via the Landau-Cutkosky rule). Themeson-vertex functions are given by one-covariantBSA [16,17]. Other quark models [12] represent B → M form factors by overlap integrals of me-son wave functions, obtained from confining po-tential models, and appropriate weak interactionvertices. Similar quark model calculations wereperformed on the light cone [18].All these approaches model soft contributionswith vertex functions, while the propagation ofthe constituent quark, S ( k ) = ( /k − m q ) − , isscale independent and does not describe confine-ment and dynamical chiral symmetry breaking(DCSB). As noted in Refs. [17,19,20], this canlead to considerable model dependance at largermomentum transfer. Light-cone sum rules : In a LCSR the operator-product expansion of a given correlation functionis combined with hadronic dispersion relations.The quark-hadron duality is invoked: the corre-lator function is calculated twice, as a hadronicobject and with subhadronic degrees of freedom.After separation in HQET of the heavy meson’sstatic part, P H = p + q = m h v h + k , where v h is the four-velocity and k is the residual momen-tum, and likewise redefinition of the momentumtransfer q = m h v h + ˜ q ⇒ p + ˜ q = k , one obtainsthe heavy-limit correlation function,Π H ( p, q ) = ˜Π H v ( p, ˜ q ) + O (1 /m h ); (7) q [GeV ] Ref. [12] Ref. [16] Ref. [18] Ref. [20] Ref. [22] Ref. [23] Ref. [21]0 0.217 0.29 0.247 0.24 0 . ± .
05 0 . ± .
03 0 . ± . ± . >
10 1.75 1.83 1.58 –Table 1
Numerical comparison for the transition form factor, F B → π + ( q ); the q values of Ref. [20] are calculated, whereasfor Refs. [12,16,18,22,23] the value F B → π + (0) and the corresponding extrapolation in these references are employed. ˜Π H v ( p, ˜ q ) = i Z d x e ipx h | T [ J M ( x ) J h v (0)] | H v i , whereas a hadronic correlator can be written,˜Π had . ( p, q ) = h | J M | M ( p ) ih M ( p ) | J h | H ( P H ) i m M − p , (8)where J h (0) and J h v (0) are heavy-light currentsand J M ( x ) the interpolating current for a pseu-doscalar or vector meson.In Eq. (8) only the light-meson contributionis represented but higher and continuum statescan also be taken into account. In Eqs. (7) and(8), the usual role of the correlation functions hasbeen reversed [21,22]: the correlation function istaken between the vacuum and the on-shell B -state vector using its light-cone distribution am-plitude (DA) expansion and the pion is interpo-lated with the light-quark (axialvector) current J M ( x ). The B -meson DAs are universal non-perturbative objects introduced within HQET.In Ref. [23], however, the correlation functionis taken between the vacuum and the light-mesonstate, whereas the B meson is interpolated bythe heavy-light quark current J h (0). As a re-sult, the long-distance dynamics in the correla-tion function is described by a set of light-meson( π, K, ρ, K ∗ ) DA. In the last step, a Borel trans-formations is applied to both, Eqs. (7) and (8),from which one derives a transition form factorexpressed as a sum rule. The transformation in-troduces a scale via the Borel parameter which,in turn, is fixed with sum rules for light-mesondecay constants.Besides a systematic uncertainty owing to theduality assumption, the main incertitude lieswithin the DAs which encode the relevant non-perturbative effects. Only their asymptotic formis known exactly from perturbative QCD. As of yet, the first two Gegenbauer moments of the DAfor various light pseudoscalar and vector mesonshave been obtained from QCD sum rules withvery large errors, though the first moment is con-sistent with lattice calculations [24]. Moreover,the transition form factors must be extrapolatedto space-like momenta.For purpose of comparison, we list B → π tran-sitions form factors, F + ( q ), for various models inTable 1. As observed therein, there is a 30% vari-ation within the quark models [12,16,18] at q = 0which increases at larger q values. The LCSRpredictions [22,23] agree at q = 0 but their re-spective slopes for q >
5. Flavourful Dyson-Schwinger equations
The elements entering the amplitude in Eq. (6)can be motivated by the solutions of DSEs ap-plied to QCD. A general review of the DSEs canbe found in Refs. [25,26] and their applicationsto heavy-light transition form factors have beeninvestigated in [20,27,28].
Dressed quark propagator : The mesons arebound states of a quark and antiquark pair, wherefor a given quark flavour their dressing is de-scribed by the DSE (in Euclidean metric), S − ( p ) = Z ( iγ · p + m bm ) + Σ( p ) , (9)with the dressed quark self energy,Σ( p ) = Z g Z Λ k D µν ( p − k ) λ a γ µ S ( k )Γ aν ( k, p ) , (10)where R Λ k represents a Poincar´e invariant regu-larisation of the integral with the regularisationmass scale Λ. The current quark bare mass m bm receives corrections from the self energy Σ( p ) inwhich the integral is over the dressed gluon prop-agator, D µν ( k ), the dressed quark-gluon vertex,Γ aν ( k, p ), and λ a are the usual SU(3) colour ma-trices. The solution to the gap equation (9) reads S ( p ) = − iγ · p σ V ( p ) + σ S ( p )= (cid:2) iγ · p A ( p ) + B ( p ) (cid:3) − . (11)The renormalisation constants for the quark-gluon vertex, Z ( ζ, Λ ), and quark-wave function, Z ( ζ, Λ ), depend on the renormalisation point, ζ , the regularisation scale, Λ, and the gauge pa-rameter, whereas the mass function M ( p ) = B ( p ) /A ( p ) is independent of ζ . Since QCD isasymptotically free, it is useful to impose at largespacelike ζ the renormalisation condition, S − ( p ) | p = ζ = iγ · p + m ( ζ ) , (12)where m ( ζ ) is the renormalised running quarkmass, so that for ζ ≫ Λ quantitative match-ing with pQCD results can be made.Infrared dressing of light quarks has profound consequences for hadron phenomenology [29]: thequark-wave function renormalisation, Z ( p ) =1 /A ( p ), is suppressed whereas the dressed quark-mass function, M ( p ) = B ( p ) /A ( p ), is en-hanced in the infrared which expresses dynam-ical chiral symmetry breaking (DCSB) and iscrucial to the emergence of a constituent quarkmass scale. Both, numerical solutions of thequark DSE and simulations of lattice-regularisedQCD [30], predict this behaviour of M ( p ) andpointwise agreement between DSE and lattice re-sults has been explored in Refs. [31,32]. Studiesthat do not implement light-quark dressing runinto artefacts caused by rather large light-quarkmasses [17,19] to emulate confinement. This isbecause unphysical thresholds in transition am-plitudes can only be overcome with the prescrip-tion that m H < m q + m q , which poses problemsfor a description of light vector mesons ( ρ, K ∗ ),heavy flavoured vector mesons ( D ∗ , B ∗ ) and for P -wave and excited charmonium states.Whereas the impact of gluon dressing is strik-ing for light quarks, its effect on the heavy quarksis barely notable. This can be appreciated, for in-stance, in Fig. 1 of Ref. [28]: for light quarks, mass can be generated from nothing, i.e. , theHiggs mechanism is irrelevant to their acquiringof a constituent-like mass. Bethe-Salpeter amplitudes : The BSA can bedetermined reliably by solving the Bethe-Salpeterequation (BSE) in a truncation scheme consistentwith that employed in the gap equation (9). Con-sider the inhomogeneous BSE for the axialvectorvertex Γ fg µ in which pseudoscalar and axialvectormesons appear as poles:Γ fg µ ( k ; P ) = Z γ γ µ − g Z Λ q D αβ ( k − q ) λ a γ α × S f ( q + ) Γ fg µ ( q ; P ) S g ( q − ) λ a gβ ( q − , k − ) (13)+ g Z Λ q D αβ ( k − q ) λ a γ α S f ( q + ) λ a fg µβ ( k, q ; P ) . In Eq. (13), P is the total meson momentum, q ± = q ± P/ , k ± = k ± P/
2, Λ fg µβ is a 4-pointSchwinger function entirely defined via the quarkself energy [33] and f, g denote the flavour indicesof a light-light or heavy-light bound state. Thesolutions of the vertex Γ fg µ must satisfy the axial-vector Ward-Takahashi identity, P µ Γ fg µ ( k ; P ) = S − f ( k + ) iγ + iγ S − g ( k − ) − i [ m f ( ζ ) + m g ( ζ )] Γ fg ( k ; P ) , (14)where Γ fg solves the pseudoscalar analogue toEq. (13). A systematic, symmetry-preservingtruncation of the DSE and BSE is given by theRainbow ladder [34] which is their leading-orderterm with the dressed quark-gluon vertex, Γ fµ , re-placed by γ µ . It can be shown that Λ fg µβ ≡ ε ( k ; P ), of the light pseu-doscalar BSA with the scalar part, B ( p ), of thedressed-quark propagator (11) in the chiral limit.This motivates an effective parametrisation of thelight mesons ( M = π, K ),Γ M ( k ; P ) = iγ ε M ( k ) = iγ B M ( k ) / ˆ f M , (15)ˆ f M = f M / √
2, which has been capitalised on intransition form factor calculations [20,27,28].Simultaneous solutions of the quarks’sDSE and the heavy meson’s BSA withrenormalisation-group improved ladder trunca-tions, obtained for the kaon [35], prove to bedifficult. The truncations do not yield the Diracequation when one of the quark masses is large.A recent attempt to calculate BSA for D and B mesons [36] reproduces well the respective massesbut underestimates experimental leptonic decayconstants by 30 − H ( k ; P ) are currentlyemployed in Eq. (6), which reproduce leptonicdecay constants in a simultaneous calculation.
6. Hadronic decays
The decay D ∗ → Dπ can be used to extractthe strong coupling ˆ g between heavy vector andpseudoscalar mesons to a low-momentum pion inthe heavy meson chiral lagrangian [37]. One con-siders the matrix element, h D ( p ) π ( q ) | D ∗ ( p , λ ) i = g D ∗ Dπ ǫ λ · q , (16)where the coupling, g D ∗ Dπ = 17 . ± . ± .
9, isexperimentally known [38] and related to ˆ g . Sim-ilarly, one may also extract ˆ g from the unphysicaldecay B ∗ → Bπ in the chiral limit and where m b / Λ QCD corrections are better controlled.The coupling g D ∗ Dπ is related to a heavy-to-heavy transition form factor via the LSZ reduc-tion of the pion and the use of PCAC, π ( x ) = ∂ µ A µ ( x ) / ( f π m π ), which leads to: h D ( p ) π ( q ) | D ∗ ( p ) i = q µ ( m π − q ) f π m π × Z d x e iq · x h D ( p ) | A µ ( x ) | D ∗ ( p ) i . (17)Hence, the matrix element in Eq. (16) has beenreduced to the Fourier transform of a transitionmatrix element between the D ∗ and D mesonsin the chiral limit with the axial QCD current A µ ( x ) = ¯ q a γ µ γ q b . Results from this reductionprocedure have been obtained on the lattice [39,40] and most recently in a simulation with n f =2 [41] which yields g D ∗ Dπ = 20 ±
2. This form factor can also be calculatedstraightforwardly without reduction of the pionemploying Eq. (6) with the dressed quark prop-agators in Eq. (11) and substituting the pion’sBA (15) for Γ I . In a reassessment and improve-ment of a calculation of g D ∗ Dπ within a Dyson-Schwinger model [28], we obtain g D ∗ Dπ = 21 [42]in agreement with the lattice result [41] and about16% larger than the experimental value.
7. Conclusive remarks
We have stressed the importance of hadroniceffects in decays of heavy-flavoured mesons andportrayed the various theoretical
Ans¨atze for theheavy-to-light transition form factors. In short,the main obstacle to their precise calculation,which veraciously reproduces the infrared fea-tures of QCD, are the uncertainties of the light-cone DA in the case of LCSR and the lack ofmodel-independent wave functions in relativis-tic quark model calculations. We have arguedthat the running quark mass of the DSE quarkpropagators is crucial to include confinement andDCSB effects in the transition amplitudes; an un-finished task are consistent solutions of the BSEfor the D and B mesons within the DSE formal-ism, which will reduce model dependence. Acknowledgments
Based on the talk given at Light Cone 2009:Relativistic Hadronic and Particle Physics, 8–13 July 2009, S˜ao Jos´e dos Campos, S˜ao Paulo,Brazil. B. E. thanks the organisers at the Insti-tuto Tecnol´ogico de Aeron´autica for the welcom-ing atmosphere and in particular Tobias Fredericoand Jo˜ao Pacheco de Melo for their hospital-ity. Several stimulating discussions with ArleneAguilar, Adriano Natale, Fernando Navarra, Ma-rina Nielsen, Joannis Papavassiliou and LauroTomio were greatly appreciated. This work wassupported by the Department of Energy, Office ofNuclear Physics, No. DE-AC02-06CH11357.
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