Process-independent strong running coupling
Daniele Binosi, Cedric Mezrag, Joannis Papavassiliou, Craig D. Roberts, Jose Rodriguez-Quintero
aa r X i v : . [ nu c l - t h ] D ec Process-independent strong running coupling
Daniele Binosi, C´edric Mezrag, Joannis Papavassiliou, Craig D. Roberts, and Jose Rodr´ıguez-Quintero European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT ∗ ) and Fondazione Bruno KesslerVilla Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy Physics Division, Argonne National Laboratory, Argonne IL 60439, USA Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100, Valencia, Spain Department of Integrated Sciences; University of Huelva, E-21071 Huelva, Spain (Dated: 13 December 2016)We unify two widely different approaches to understanding the infrared behaviour of quantumchromodynamics (QCD), one essentially phenomenological, based on data, and the other computa-tional, realised via quantum field equations in the continuum theory. Using the latter, we explainand calculate a process-independent running-coupling for QCD, a new type of effective charge thatis an analogue of the Gell-Mann–Low effective coupling in quantum electrodynamics. The result isalmost identical to the process-dependent effective charge defined via the Bjorken sum rule, whichprovides one of the most basic constraints on our knowledge of nucleon spin structure. This re-veals the Bjorken sum to be a near direct means by which to gain empirical insight into QCD’sGell-Mann–Low effective charge.
1: Introduction . — In quantum gauge field theories de-fined in four spacetime dimensions, the Lagrangian cou-plings and masses do not remain constant. Instead, ow-ing to the need for ultraviolet (UV) renormalisation, theycome to depend on a mass scale, which can often be re-lated to the energy or momentum at which a given pro-cess occurs. The archetype is quantum electrodynamics(QED), for which a sensible perturbation theory can bedefined [1]. Within this framework, owing to the Wardidentity [2], there is a single running coupling, measur-ing the strength of the photon–charged-fermion vertex,which can be obtained by summing the collection of vir-tual processes that change the bare photon into a dressedobject, viz . by computing the photon vacuum polarisa-tion. QED’s running coupling is known to great accuracy[3] and the running has been observed directly [4, 5].A new coupling appears when electromagnetism iscombined with weak interactions to produce the Stan-dard Electroweak Model [6]. It may be characterised bysin θ W , where θ W is a scale-dependent angle which spec-ifies the particular mixing between the model’s definingneutral gauge bosons that produces the observed photonand Z -boson. A perturbation theory can also be de-fined for the electroweak theory [7] so that sin θ W canbe computed and compared with precise experiments [3].At first sight, the addition of quantum chromodynam-ics (QCD) [8] to the Standard Model does not quali-tatively change anything, despite the presence of fourpossibly distinct strong-interaction vertices (gluon-ghost,three-gluon, four-gluon and gluon-quark) in the renor-malised theory. An array of Slavnov-Taylor identities(STIs) [9, 10], implementing BRST symmetry [11, 12](a generalisation of non-Abelian gauge invariance for thequantised theory) ensures that a single running couplingcharacterises all four interactions on the domain withinwhich perturbation theory is valid. The difference hereis that whilst QCD is asymptotically free and extant ev- idence suggests that perturbation theory is valid at largemomentum scales, all dynamics is nonperturbative atthose scales typical of everyday strong-interaction phe-nomena, e.g . ζ . m p , where m p is the proton’s mass.The questions that arise are how many distinct run-ning couplings exist in nonperturbative QCD, and howcan they be computed? Given that there are four individ-ual, apparently UV-divergent interaction vertices in theperturbative treatment of QCD, there could be as manyas four distinct couplings at infrared (IR) momenta. (Ofcourse, if nonperturbatively there are two or more cou-plings, they must all become equivalent on the perturba-tive domain.) In our view, nonperturbatively, too, QCDpossesses a unique running coupling. The alternative ad-mits the possibility of a different renormalisation-group-invariant (RGI) intrinsic mass-scale for each coupling andno guarantee of a connection between them. In such cir-cumstances, BRST symmetry would likely be irreparablybroken by nonperturbative dynamics and one would bepressed to conclude that QCD was non-renormalisableowing to IR dynamics. There is no empirical evidenceto support such a conclusion: QCD does seem to be awell-defined theory at all momentum scales, owing to thedynamical generation of gluon [13–18] and quark masses[19–21], which are large at IR momenta.
2: Process-independent running coupling . — Poincar´e co-variance is of enormous importance in modern physics, e.g . it places severe limitations on the nature and numberof those independent amplitudes that are required to fullyspecify any one of a gauge theory’s n -point Schwingerfunctions (Euclidean Green functions). Analyses andquantisation procedures that violate Poincar´e covariancelead to a rapid proliferation in the number of such func-tions. For example, the gluon 2-point function (propaga-tor, D µν ) is completely specified by one scalar functionin the class of linear covariant gauges; but, in the class ofaxial gauges, two unconnected functions are required andunphysical, kinematic singularities are present in the as-sociated tensors [22]. Consequently, covariant gauges aretypically preferred for concrete calculations in both con-tinuum and lattice-regularised studies of QCD. In fact,Landau gauge is the most common choice because, in-ter alia , it is a fixed point of the renormalisation groupand readily implemented in lattice-QCD [23]. Herein,therefore, we use Landau gauge; and, moreover, employ aphysical momentum-subtraction renormalisation scheme,detailed elsewhere [24].As noted in Sec. 1, there is a particular simplicityto QED, viz . the unique running coupling, a process-independent effective charge, can be obtained simplyby computing the photon vacuum polarisation. This isbecause ghost-fields decouple in Abelian theories; and,consequently, one has the Ward identity, which guaran-tees that the electric-charge renormalisation constant isequivalent to that of the photon field. Stated physically,the impact of dressing the interaction vertices is absorbedinto the vacuum polarisation. This is not generally truein QCD because ghost-fields do not decouple.There is one approach to analysing QCD’s Schwingerfunctions, however, that preserves some of QED’s sim-plicity; namely, the combination of pinch technique (PT)[25–30] and background field method (BFM) [31, 32].This framework can be seen as a means by which QCDcan be made to “look” Abelian: one systematically re-arranges classes of diagrams and their sums in order toobtain modified Schwinger functions that satisfy linearSTIs. In the gauge sector, in Landau gauge, this pro-duces a modified gluon dressing function from which onecan compute the QCD running coupling, i.e . the polarisa-tion captures all required features of the renormalisationgroup. Furthermore, the coupling is process independent:one obtains precisely the same result, independent of thescattering process considered, whether gluon+gluon → gluon+gluon, quark+quark → quark+quark, etc. Thisclean connection between the coupling and the gluon vac-uum polarisation relies on another particular feature ofQCD, viz . in Landau gauge the renormalisation constantof the gluon-ghost vertex is not only finite but unity [9], inconsequence of which the effective charge obtained fromthe PT-BFM gluon vacuum polarisation is directly con-nected with that deduced from the gluon-ghost vertex[24], sometimes called the “Taylor coupling,” α T [33–35].Writing these statements explicitly, with T µν ( k ) = δ µν − k µ k ν /k , one has [36, 37] α ( ζ ) D PB µν ( k ; ζ ) = b d ( k ) T µν ( k ) , (1a) I ( k ) := k b d ( k ) = α T ( k )[1 − L ( k ; ζ ) F ( k ; ζ )] , (1b)where: α ( ζ ) = g ( ζ ) / [4 π ], ζ is the renormalisa-tion scale; D PB µν is the PT-BFM gluon two-point func-tion; b d ( k ) is the RGI running-interaction discussed inRef. [24]; F is the dressing function for the ghost prop- agator; and L is a longitudinal piece of the gluon-ghostvacuum polarisation that vanishes at k = 0. In terms ofthese quantities, QCD’s matter-sector gap equation canbe written ( k = p − q ) S − ( p ) = Z ( iγ · p + m bm ) + Σ( p ) , (2a)Σ( p ) = Z Z Λ dq π b d ( k ) T µν ( k ) γ µ S ( q )ˆΓ aν ( q, p ) , (2b)where the usual Z Γ aν has become Z ˆΓ aν , with the lat-ter being a PT-BFM gluon-quark vertex that satisfiesan Abelian-like Ward-Green-Takahashi identity [30] and Z , are, respectively, the gluon-quark vertex and quarkwave function renormalisation constants.The RGI interaction, b d ( k ), in Eqs. (1) has beencomputed. The most up-to-date result is discussed inRefs. [36, 37]. These analyses make explicit a remark-able feature of QCD; namely, the interaction saturatesat infrared momenta: b d ( k = 0) = α ( ζ ) /m g ( ζ ) = α /m , (3)where α := α (0) ≈ . π , m := m g (0) ≈ m p / i.e . the gluon sector of QCD is characterised by anonperturbatively-generated infrared mass-scale [13–18].With this in mind, we define a RGI function D ( k ) = ∆ F ( k ; ζ ) m g ( ζ ) /m , (4)where ∆ F is a parametrisation of continuum- and/orlattice-QCD calculations of the canonical gluon two-pointfunction, built such that the IR behaviour is preservedand 1 / ∆ F ( k ; ζ ) = k + O(1) on k ≫ m . Using Eq. (4),Σ( p ) = Z Z Λ dq π b α PI ( k ) D µν ( k ) γ µ S ( q )ˆΓ aν ( q, p ) , (5)where D µν = D T µν and the dimensionless product b α PI ( k ) = b d ( k ) / D ( k ) (6)is a RGI running-coupling (effective charge): by construc-tion, b α PI ( k ) = I ( k ) on k ≫ m .The product in Eq. (6) has many important qualities.For instance, it is process independent: as noted above,the same function appears irrespective of the initial andfinal parton systems. Moreover, it unifies a diverse andextensive array of hadron observables [36]; a propertythat is evident in the fact that the dressed-quark self-energy serves as a generating functional for the Bethe-Salpeter kernel in all meson channels and the product b α PI ( k ) is untouched by the generating procedure in allflavoured systems [38–41]. Finally, although b α PI ( k ) isRGI and process-independent in any gauge, it is sufficientto know b α PI ( k ) in Landau gauge (the choice for easi-est computation) because b α PI ( k ) is form-invariant undergauge transformations [42] and, crucially, gauge covari-ance ensures that such transformations produce nothing (cid:1)(cid:2) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1)(cid:2) ( (cid:1)(cid:2) = (cid:3) ) (cid:1) (cid:1)(cid:2)(cid:1)(cid:3) (cid:1)(cid:2)(cid:4) (cid:4) (cid:4)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:8)(cid:4)(cid:2)(cid:1) (cid:1) [ (cid:1)(cid:2)(cid:3) ] (cid:1) ( (cid:1) ) / (cid:2) FIG. 1. Solid (blue) curve, complete effective charge inEq. (6); and dot-dashed (black) curve, Taylor-scheme effec-tive charge, i.e . computed in the absence of crucial pieces ofthe gluon-ghost vacuum polarisation [ LF ≡ but an overall “phase” in the gap equation’s solution,which may be absorbed into the dressed-quark two-pointfunction.
3: Computing the running coupling . — The effectivecharge defined in Eq. (6) is a product of known quantities:both b d ( k ) and the canonical gluon two-point functionhave been extensively studied and tightly constrained us-ing continuum and lattice methods [36, 37, 43]. Indeed,the known forms of these functions provide a unified,quantitatively reliable explanation of numerous hadronphysics observables [36, 43]. It is therefore straightfor-ward to combine existing results and compute b d ( k ),a procedure [37] which yields the function depicted inFig. 1. For this purpose we used a [ n, n + 1], n = 1,Pad´e approximant to simultaneously interpolate the IRbehaviour of contemporary lattice results for D µν ( k ) [37]and express the UV constraint on ∆ F ( k ; ζ ). (Using n ≥ n = 0 isincapable of representing modern lattice data.)It is worth highlighting some important features of theeffective charge in Fig. 1. First, it is a parameter-freeprediction: the curve is completely determined by resultsobtained for the gluon and ghost two-point functions us-ing continuum and lattice-regularised QCD. Second, itis physical, in the sense that there is no Landau pole,and it saturates in the IR: b α PI ( k = 0) = α ≈ . π , i.e . the coupling possesses an infrared fixed point [44].Third, the prediction is equally concrete and sound at allspacelike momenta, connecting the IR and UV domains,and precisely reproducing the known behaviour of theTaylor coupling at large k [33–35], with no need for an ad hoc “matching procedure,” such as that employed in models [45]. Finally, our result is essentially nonpertur-bative, obtained by combining self-consistent solutions ofgauge-sector gap equations with lattice simulations, aug-mented only by a physical procedure for setting a singlemass-scale [37]. There are indications [46–48] that theeffective charge in Fig. 1 could prove useful in developinga modern dynamical perturbation theory [49].It is evident in Fig. 1 that ghost-gluon interactions arecritical. The RGI product LF in Eq. (1b) expresses ef-fects of gluon-ghost scattering that are essential to ensur-ing b α PI is process-independent. It is also quantitativelyimportant, introducing a roughly 60% enhancement of b α PI ( k ) for k ≃ m . It must also, therefore, be physi-cally significant because the strength of the running cou-pling at IR momenta determines the magnitude of dy-namical chiral symmetry breaking (DCSB) [36, 37, 43];and DCSB is a crucial emergent phenomenon in QCD,possibly inseparable from confinement in the unquenchedtheory [50], i.e . when dynamical light quarks are active.
4: Comparison of effective charges . — Another approachto determining an “effective charge” in QCD was intro-duced in Ref. [51]. This is a process-dependent proce-dure; namely, an effective running coupling is definedto be completely fixed by the leading-order term in theperturbative expansion of a given observable in terms ofthe canonical running coupling. An obvious difficulty,or perhaps drawback, of such a scheme is the process-dependence itself. Naturally, effective charges from dif-ferent observables can in principle be algebraically con-nected to each other via an expansion of one coupling interms of the other. However, any such expansion containsinfinitely many terms [45]; and this connection does notimbue a given process-dependent charge with the abilityto predict any other observable, since the expansion isonly defined a posteriori , i.e . after both effective chargesare independently constructed.One such process-dependent effective charge is α g ( k ),which is defined via the Bjorken sum rule [52, 53]: Z dx (cid:2) g p ( x, k ) − g n ( x, k ) (cid:3) = g A (cid:2) − π α g ( k ) (cid:3) , (7)where g p,n are the spin-dependent proton and neutronstructure functions, whose extraction requires measure-ments using polarised targets, and g A is the nucleonflavour-singlet axial-charge [54]. The merits of this defini-tion are outlined in Ref. [45]. They include the existenceof data for a wide range of k [55–80]; tight sum-rulesconstraints on the behaviour of the integral at the IR andUV extremes of k ; and the isospin non-singlet feature ofthe difference, which suppresses contributions from nu-merous processes that are hard to compute and hencemight muddy interpretation of the integral in terms ofan effective charge.The world’s data on the process-dependent effective (cid:1)(cid:2)(cid:3)(cid:3) (cid:4) / (cid:5)(cid:6)(cid:4)(cid:7)(cid:8)(cid:6)(cid:2)(cid:9) (cid:5)(cid:6)(cid:4)(cid:7) ( (cid:10)(cid:11)(cid:11)(cid:12) ) (cid:8)(cid:6)(cid:2)(cid:9) (cid:5)(cid:6)(cid:4)(cid:7) ( (cid:10)(cid:11)(cid:13)(cid:14) ) (cid:15)(cid:16)(cid:7)(cid:17) (cid:1)(cid:16)(cid:18)(cid:19)(cid:16)(cid:7)(cid:5)(cid:16)(cid:18)(cid:20) (cid:5)(cid:21)(cid:19)(cid:22)(cid:4)(cid:7)(cid:7)(cid:5)(cid:16)(cid:18)(cid:20) (cid:7)(cid:19)(cid:5)(cid:5)(cid:16)(cid:18)(cid:20) (cid:21)(cid:22)(cid:4)(cid:6)(cid:7)(cid:6)(cid:4)(cid:5) (cid:16)(cid:13)(cid:14)(cid:10) / (cid:16)(cid:13)(cid:14)(cid:23)(cid:7)(cid:6)(cid:4)(cid:5) (cid:16)(cid:13)(cid:24)(cid:14) / (cid:16)(cid:13)(cid:24)(cid:24)(cid:8)(cid:6)(cid:2)(cid:9) (cid:18)(cid:7)(cid:7)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:3)(cid:2)(cid:9) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) (cid:3)(cid:4) (cid:1) (cid:1)(cid:2)(cid:1)(cid:3) (cid:1)(cid:2)(cid:4) (cid:4) (cid:4)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:8)(cid:4)(cid:2)(cid:1) (cid:1) [ (cid:1)(cid:2)(cid:3) ] (cid:1) ( (cid:1) ) / (cid:2) FIG. 2. Solid (blue) curve: predicted process-independentRGI running-coupling b α PI ( k ), Eq. (6). The shaded (blue)band bracketing this curve combines a 95% confidence-levelwindow based on existing lattice-QCD results for the gluontwo-point function with an error of 10% in the continuumextraction of the RGI product LF in Eqs. (1). World dataon α g [55–80]. The shaded (yellow) band on k > α g obtained from the Bjorken sum by using QCDevolution [81–83] to extrapolate high- k data into the depictedregion, following Refs. [55, 56]; and, for additional context, thedashed (red) curve is the light-front holographic model of α g canvassed in Ref. [45]. charge α g ( k ) are depicted in Fig. 2 and therein com-pared with our prediction for the process-independentRGI running-coupling b α PI ( k ). Owing to asymptoticfreedom, all reasonable definitions of a QCD effectivecharge must agree on k & and our approachguarantees this connection. To be specific, in terms ofthe widely-used MS running coupling [3]: α g ( k ) = α MS ( k )(1 + 1 . α MS ( k ) + . . . ) , (8a) b α PI ( k ) = α MS ( k )(1 + 1 . α MS ( k ) + . . . ) , (8b)where Eq. (8a) may be built from, e.g . Refs. [84, 85].Significantly, there is also near precise agreement withdata on the IR domain, k . m , and complete accordon k ≥ m . Fig. 1 makes plain that any agreement on k ∈ [0 . ,
1] GeV is non-trivial because ghost-gluon in-teractions produce as much as 40% of b α PI ( k ) on thisdomain: if these effects were omitted from the gluonvacuum polarisation, then α g and b α PI would differ byroughly a factor of two on the critical domain of transi-tion between strong and perturbative QCD.
5: Conclusions . — We have defined and calculated aprocess-independent running-coupling for QCD, b α PI ( k )[Eq. (6), Fig. 1]. This is a new type of effective charge,which is an analogue of the Gell-Mann–Low effective cou-pling in QED, being completely determined by the gauge-boson two-point function. Our prediction for b α PI ( k ) is parameter-free, being obtained by combining the self-consistent solution of a set of Dyson-Schwinger equa-tions with results from lattice-QCD; and it smoothly uni-fies the nonperturbative and perturbative domains of thestrong-interaction theory. This process-independent run-ning coupling is known to unify a vast array of observ-ables, e.g . the pion mass and decay constant, and thelight meson spectrum [86]; the parton distribution am-plitudes of light- and heavy-mesons [87–89], associatedelastic and transition form factors [90, 91], etc.Finally, and perhaps surprisingly at first sight, b α PI ( k )is almost pointwise identical at infrared momenta to theprocess-dependent effective charge, α g , defined via theBjorken sum rule, one of the most basic constraints onour knowledge of nucleon spin structure, and in com-plete agreement on the domain of perturbative momenta[Fig. 2]. Equivalence on the perturbative domain is guar-anteed for any two reasonable definitions of QCD’s ef-fective charge, but here the subleading terms differ byjust 4% [Eqs. (8)]. An excellent match at infrared mo-menta, i.e . below the scale at which perturbation theorywould locate the Landau pole, is non-trivial; and crucialto this agreement is the careful treatment and incorpo-ration of a special class of gluon-ghost scattering effects.One is naturally compelled to ask how these two appar-ently unrelated definitions of a QCD effective charge canbe so similar? We attribute this outcome to a physi-cally useful feature of the Bjorken sum rule, viz . it isan isospin non-singlet relation and hence contributionsfrom many hard-to-compute processes are suppressed,and these same processes are omitted in our computa-tion of b α PI ( k ).The analysis herein unifies two vastly different ap-proaches to understanding the infrared behaviour ofQCD, one essentially phenomenological and the other de-liberately computational, embedded within QCD. Thereis no Landau pole in our predicted running coupling.In fact, there is an inflection point at √ k = 0 . b α PI ( k = 0) ≈ . π [Fig. 2]. This unification identifiesthe Bjorken sum rule as a near direct means by which togain empirical insight into a QCD analogue of the Gell-Mann–Low effective charge. Acknowledgments . — We are grateful for comments fromS. J. Brodsky, L. Chang, A. Deur and S.-X. Qin. Thisstudy was conceived and initiated during the 3 rd Work-shop on Non-perturbative QCD, University of Seville,Spain, 17-21 October 2016. This research was supportedby: Spanish MEYC, under grants FPA2014-53631-C-1-P, FPA2014-53631-C-2-P and SEV-2014-0398; Generali-tat Valenciana under grant Prometeo II/2014/066; andU.S. Department of Energy, Office of Science, Office ofNuclear Physics, contract no. DE-AC02-06CH11357. [1] S. Lundqvist (Editor),
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