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Timestamp: 2019-04-19 19:05:14+00:00

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The light-front holographic mapping of classical gravity in anti-de Sitter space, modified by a positive-sign dilaton background, leads to a nonperturbative effective coupling αAdSs(Q2). It agrees with hadron physics data extracted from different observables, such as the effective charge defined by the Bjorken sum rule, as well as with the predictions of models with built-in confinement and lattice simulations. It also displays a transition from perturbative to nonperturbative conformal regimes at a momentum scale ∼1 GeV. The resulting β function appears to capture the essential characteristics of the full β function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD. Commensurate scale relations relate observables to each other without scheme or scale ambiguity. In this paper we extrapolate these relations to the nonperturbative domain, thus extending the range of predictions based on αAdSs(Q2).
The concept of a running coupling αs(Q2) in QCD is usually restricted to the perturbative domain. However, as in QED, it is useful to define the coupling as an analytic function valid over the full spacelike and timelike domains. The study of the non-Abelian QCD coupling at small momentum transfer is a complex problem because of gluonic self-coupling and color confinement. Its behavior in the nonperturbative infrared (IR) regime has been the subject of intensive study using Dyson-Schwinger equations and Euclidean numerical lattice computation, (1) since it is a quantity of fundamental importance. We will show that the light-front (LF) holographic mapping of classical gravity in anti-de Sitter (AdS) space, modified by a positive-sign dilaton background exp(+κ2z2), leads to a nonperturbative effective coupling αAdSs(Q2) which is in agreement with hadron physics data extracted from different observables, as well as with the predictions of models with built-in confinement and lattice simulations.
The AdS/CFT correspondence (2) between a gravity or string theory on a higher dimensional AdS space-time and conformal gauge field theories in physical space-time has brought a new set of tools for studying the dynamics of strongly coupled quantum field theories, and it has led to new analytical insights into the confining dynamics of QCD. The AdS/CFT duality provides a gravity description in a (d+1)-dimensional AdS spacetime in terms of a flat d-dimensional conformally-invariant quantum field theory defined at the AdS asymptotic boundary. (3) Thus, in principle, one can compute physical observables in a strongly coupled gauge theory in terms of a classical gravity theory.
Since the quantum field theory dual to AdS5 space in the original correspondence (2) is conformal, the strong coupling of the dual gauge theory is constant, and its β function is zero. Thus, one must consider a deformed AdS space in order to have a running coupling αAdSs(Q2) for the gauge theory side of the correspondence. We assume a positive-sign confining dilaton background to modify AdS space, a model that gives a very good account of meson and baryon spectroscopy and form factors. We use LF holography (4); (5); (6); (7); (8) to map the amplitudes corresponding to hadrons propagating in AdS space to the frame-independent light-front wave functions (LFWFs) of hadrons in physical 3+1 space. This analysis utilizes recent developments in LF QCD, which have been inspired by the AdS/CFT correspondence. (2) The resulting LFWFs provide a fundamental description of the structure and internal dynamics of hadronic states in terms of their constituent quarks and gluons.
The definition of the running coupling in perturbative quantum field theory is scheme-dependent. As discussed by Grunberg, (9) an effective coupling or charge can be defined directly from physical observables. Effective charges defined from different observables can be related to each other in the leading-twist domain using commensurate scale relations (CSR). (10) A more challenging problem is to relate such observables and schemes over the full domain of momenta. An important part of this paper will be the application and test of commensurate scale relations and their tentative extension to the nonperturbative domain. Another important application is related to the potential between infinitely heavy quarks, which can be defined analytically in momentum transfer space as the product of the running coupling times the Born gluon propagator: V(q)=−4πCFαV(q)/q2. This effective charge defines a renormalization scheme – the αV scheme of Appelquist, Dine, and Muzinich. (11) In fact, the holographic coupling αAdSs(Q2) can be considered to be the nonperturbative extension of the αV effective charge defined in Ref. (11) .
This paper is organized as follows: after briefly reviewing in Sec. II the light-front quantization approach to the gauge/gravity correspondence, we identify a nonperturbative running coupling in Sec. III from the fifth-dimensional action of gauge fields propagating in AdS5 space modified by a positive-sign dilaton background exp(+κ2z2). In Sec. IV, we compare the results for the coupling αAdSs obtained in Sec. III with the effective QCD couplings extracted from different observables and lattice results. The nonperturbative results are extended to large Q2 by matching the holographic results to the perturbative results in the transition region. In Sec. V, we discuss the holographic results for the β function in the nonperturbative domain and compare the predictions with lattice and experimental results. In Sec. VI, we discuss the use of CSR to relate different effective charges. A discussion of experimental results, schemes and data normalization is given in Sec. VII. The CSR discussion is extended in Sec. VIII to configuration space. Some final remarks are given in the conclusions in Sec. IX. A check on the validity of CSR is carried out in the Appendix where the results for the g1, V, and ¯¯¯¯¯¯¯¯¯MS schemes are confronted in the perturbative domain.
invariant (R the AdS radius). Since the metric (1) is invariant under a dilatation of all coordinates xμ→λxμ, z→λz, the variable z acts like a scaling variable in Minkowski space: different values of z correspond to different energy scales at which the hadron is examined.
In order to describe a confining theory, the conformal invariance of AdS5 must be broken. A simple way to impose confinement and discrete normalizable modes is to truncate the regime where the string modes can propagate by introducing an IR cutoff at a finite value z0∼1/ΛQCD. Thus, the “hard-wall” at z0 breaks conformal invariance and allows the introduction of the QCD scale and a spectrum of particle states. (15) In this simplified approach the propagation of hadronic modes in a fixed effective gravitational background encodes the salient properties of the QCD dual theory, such as the ultraviolet (UV) conformal limit at the AdS boundary at z→0, as well as modifications of the background geometry in the large z infrared region which are dual to confining gauge theories. As first shown by Polchinski and Strassler, (15) the AdS/CFT duality, modified to incorporate a mass scale, provides a derivation of dimensional counting rules (16) for the leading power-law falloff of hard scattering beyond the perturbative regime. The modified theory generates the hard behavior expected from QCD, instead of the soft behavior characteristic of strings.
A dilaton profile exp(±κ2z2) of either sign also leads to a two-dimensional oscillator potential U(ζ)∼κ4ζ2 in the relativistic LF eigenvalue equation of Ref. (4) , which in turn reproduces the observed linear Regge trajectories in a Chew-Frautschi plot. Glazek and Schaden (19) have shown that in QCD a harmonic oscillator confining potential naturally arises as an effective potential between heavy quark states when higher gluonic Fock states are stochastically eliminated.
The soft-wall model of Ref. (17) also uses the AdS/QCD framework (22); (23) , where bulk fields are introduced to match the SU(2)L×SU(2)R chiral symmetry of QCD and its spontaneous breaking, but without an explicit connection to the internal constituent structure of hadrons. (24) Instead, axial and vector currents become the primary entities as in an effective chiral theory. In this “bottom-up” model only a limited number of operators are introduced, and consequently, only a limited number of fields are required to construct phenomenologically viable five-dimensional gravity duals.
Light-front holography provides a remarkable connection between the equations of motion in AdS space and the Hamiltonian formulation of QCD in physical spacetime quantized on the light front at fixed LF time τ=x+=x0+x3, the time marked by the front of a light wave. (25) This correspondence provides a direct connection between the hadronic amplitudes Φ(z) in AdS space with LFWFs ϕ(ζ) describing the quark and gluon constituent structure of hadrons in physical space-time. The mapping between the LF invariant variable ζ and the fifth-dimension AdS coordinate z was originally obtained by matching the expression for electromagnetic (EM) current matrix elements in AdS space with the corresponding expression for the current matrix element, using LF theory in physical spacetime. (5) It has also been shown that one obtains the identical holographic mapping using the matrix elements of the energy-momentum tensor, (6) thus verifying the consistency of the holographic mapping from AdS to physical observables defined on the light front. LF holography thus provides a direct correspondence between an effective gravity theory defined in a fifth-dimensional warped space and a physical description of hadrons in 3+1 spacetime.
The hadron four-momentum generator is P=(P+,P−,P⊥), P±=P0±P3, and the hadronic state |ψ⟩ is an expansion in multiparticle Fock eigenstates |n⟩ of the free light-front Hamiltonian: |ψ⟩=∑nψn|n⟩. The internal partonic coordinates of the hadron are the momentum fractions xi=k+i/P+ and the transverse momenta k⊥i, i=1,2,…,n, where n is the number of partons in a given Fock state. Momentum conservation requires ∑ni=1xi=1 and ∑ni=1k⊥i=0. It is useful to employ a mixed representation (27) in terms of n−1 independent momentum fraction variables xj and position coordinates b⊥j, j=1,2,…,n−1, so that ∑ni=1b⊥i=0. The relative transverse variables b⊥i are Fourier conjugates of the momentum variables k⊥i.
where U(ζ) is the effective potential, and L is the relative orbital angular momentum as defined in the LF formalism. The set of eigenvalues M2 gives the hadronic spectrum of the color-singlet states, and the corresponding eigenmodes ϕ(ζ) represent the LFWFs, which describe the dynamics of the constituents of the hadron. This first approximation to relativistic QCD bound-state systems is equivalent to the equations of motion, which describe the propagation of spin-J modes in a fixed gravitational background asymptotic to AdS space. (4) By using the correspondence between ζ in the LF theory and z in AdS space, one can identify the terms in the dual gravity AdS equations, which correspond to the kinetic energy terms of the partons inside a hadron and the interaction terms that build confinement. (4) The identification of orbital angular momentum of the constituents in the light-front description is also a key element in our description of the internal structure of hadrons using holographic principles.
As we will discuss, the conformal AdS5 metric (1) can be deformed by a warp factor exp(+κ2z2). In the case of a two-parton relativistic bound state, the resulting effective potential in the LF equation of motion is U(ζ)=κ4ζ2+2κ2(L+S−1). (7) There is only one parameter, the mass scale κ∼1/2 GeV, which enters the effective confining harmonic oscillator potential. Here S=0,1 is the spin of the quark-antiquark system, L is their relative orbital angular momentum, and ζ is the Lorentz-invariant coordinate defined above, which measures the distance between the quark and antiquark; it is analogous to the radial coordinate r in the Schrödinger equation. The resulting mesonic spectrum has the phenomenologically successful Regge form M2=4κ2(n+L+S/2), with equal slopes in the orbital angular momentum and the radial quantum number n. The pion with n=L=S=0 is massless for zero quark mass, consistent with chiral symmetry.
We will show in this section how the LF holographic mapping of effective classical gravity in AdS space, modified by a positive-sign dilaton background, can be used to identify an analytically simple color-confining nonperturbative effective coupling αAdSs(Q2) as a function of the spacelike momentum transfer Q2=−q2. As we shall show, this coupling incorporates confinement and agrees well with effective charge observables and lattice simulations. It also exhibits an infrared fixed point at small Q2 and asymptotic freedom at large Q2. However, the falloff of αAdSs(Q2) at large Q2 is exponential: αAdSs(Q2)∼e−Q2/κ2, rather than the perturbative QCD (pQCD) logarithmic falloff. We shall show in later sections that a phenomenological extended coupling can be defined which implements the pQCD behavior.
In contrast, the negative dilaton solution φ=−κ2z2 leads to an integral that diverges at large ζ. The essential assumption of this paper is the identification of αAdSs(Q2) with the physical QCD running coupling in its nonperturbative domain.
The flow Eq. (6) from the scale-dependent measure for the gauge fields can be understood as a consequence of field-strength renormalization. In physical QCD we can rescale the non-Abelian gluon field Aμ→λAμ and field strength Gμν→λGμν in the QCD Lagrangian density LQCD by a compensating rescaling of the coupling strength g→λ−1g. The renormalization of the coupling gphys=Z1/23g0, where g0 is the bare coupling in the Lagrangian in the UV-regulated theory, is thus equivalent to the renormalization of the vector potential and field strength: Aμren=Z−1/23Aμ0, Gμνren=Z−1/23Gμν0 with a rescaled Lagrangian density LrenQCD=Z−13L0QCD=(gphys/g0)−2L0. In lattice gauge theory, the lattice spacing a serves as the UV regulator, and the renormalized QCD coupling is determined from the normalization of the gluon field strength as it appears in the gluon propagator. The inverse of the lattice size L sets the mass scale of the resulting running coupling. As in lattice gauge theory, color confinement in AdS/QCD reflects nonpertubative dynamics at large distances. The QCD couplings defined from lattice gauge theory and the soft-wall holographic model are thus similar in concept, and both schemes are expected to have similar properties in the nonperturbative domain, up to a rescaling of their respective momentum scales.
The gauge/gravity correspondence has also been used to study the running coupling of the dual field theory. One can modify the dynamics of the dilaton in the AdS space to simulate the QCD β function in the UV domain. (30); (31); (32); (33); (34); (35); (36) For example, a β-function ansatz of the boundary field theory is used as input in Refs. (32); (33); (34); (35); (36) to modify the AdS metrics assuming the correspondence between the AdS variable z and the energy scale E of the conformal field theory, E∼1/z, as discussed in Ref. (37) . In our paper, the effective QCD coupling is identified by using the precise mapping from z in AdS space to the transverse impact variable ζ in LF QCD.
A similar value for the normalization constant is derived in Ref. (22) from the AdS/CFT prediction for the current-current correlator. The value of the five-dimensional coupling found in (22) for a SU(2) flavor gauge theory is (g25)SU(2)=12π2R/NC, and thus (g254π)SU(2)=π for NC=3 in units R=1.
Figure 1: The effective coupling from LF holographic mapping for κ=0.54 GeV is compared with effective QCD couplings extracted from different observables and lattice results. Details on the comparison with other effective charges are given in Ref. (39) .
The couplings in Fig. 1 agree well in the strong coupling regime up to Q∼1 GeV. The value κ=0.54 GeV has been determined from the vector meson principal Regge trajectory. (7) The lattice results shown in Fig. 1 from Ref. (38) have been scaled to match the perturbative UV domain. The effective charge αg1 has been determined in Ref (39) from several experiments. Figure 1 also displays other couplings from different observables as well as αg1, which is computed from the Bjorken sum rule (12) over a large range of momentum transfer (continuous band). At Q2=0 one has the constraint on the slope of αg1 from the Gerasimov-Drell-Hearn (GDH) sum rule (40) , which is also shown in the figure. The results show no sign of a phase transition, cusp, or other nonanalytical behavior, a fact which allows us to extend the functional dependence of the coupling to large distances. The smooth behavior of the holographic strong coupling also allows us to extrapolate its form to the perturbative domain. This is discussed further in Sec. VI.
The hadronic model obtained from the dilaton-modified AdS space provides a semiclassical first approximation to QCD. Color confinement is introduced by the harmonic oscillator potential, but effects from gluon creation and absorption are not included in this effective theory. The nonperturbative confining effects vanish exponentially at large momentum transfer [Eq. (9)], and thus the logarithmic falloff from pQCD quantum loops will dominate in this regime.
Here αAdSg1 is given by Eq. (9) with the normalization (10) [continuous line in Fig. 1] and αfitg1 is the analytical fit to the measured coupling αg1. (39) These couplings have the same normalization at Q2=0, given by Eq. (10). We use the fit from (39) rather than using pQCD directly since the perturbative results are meaningless near or below the transition region and thus would not allow us to obtain a smooth transition and analytical expression of αg1. In order to smoothly connect the two contributions (dot-dashed line in Fig. 1), we employ smeared step functions. For convenience we have chosen g±(Q2)=1/(1+e±(Q2−Q20)/τ2) with the parameters Q20=0.8 GeV2 and τ2=0.3 GeV2.
Figure 2: Holographic model prediction for the β function compared to JLab and CCFR data, lattice simulations and results from the Bjorken sum rule.
Also shown on Fig. 2 are the β functions obtained from phenomenology and lattice calculations. For clarity, we present on Fig. 2 only the LF holographic predictions, the lattice results from, (38) and the experimental data supplemented by the relevant sum rules. The width of the continuous band is computed from the uncertainty of αg1 in the perturbative regime. The dot-dashed curve corresponds to the extrapolated approximation given by Eq. (11). Only the point-to-point uncorrelated uncertainties of the JLab data are used to estimate the uncertainties, since a systematic shift cancels in the derivative in (12). The data have been recombined in fewer points to improve the statistical uncertainty; nevertheless, the uncertainties are still large. Upcoming JLab Hall A and Hall B data (43) should reduce further this uncertainty. The β function extracted from LF holography, as well as the forms obtained from the works of Cornwall, Bloch, Fisher et al., (44) Burkert and Ioffe (45) and Furui and Nakajima, (38) are seen to have a similar shape and magnitude.
are satisfied by our model β function obtained from LF holography.
Equation (13) expresses the fact that QCD approaches a conformal theory in both the far ultraviolet and deep infrared regions. In the semiclassical approximation to QCD without particle creation or absorption, the β function is zero, and the approximate theory is scale invariant in the limit of massless quarks. (46) When quantum corrections are included, the conformal behavior is preserved at very large Q because of asymptotic freedom and near Q→0 because the theory develops a fixed point. An infrared fixed point is in fact a natural consequence of color confinement: (28) since the propagators of the colored fields have a maximum wavelength, all loop integrals in the computation of the gluon self-energy decouple at Q2→0. (29) Condition (14) for large Q2, expresses the basic antiscreening behavior of QCD where the strong coupling vanishes. The β function in QCD is essentially negative, thus the coupling increases monotonically from the UV to the IR where it reaches its maximum value: it has a finite value for a theory with a mass gap. Equation (15) defines the transition region at Q0 where the β function has a minimum. Since there is only one hadronic-partonic transition, the minimum is an absolute minimum; thus the additional conditions expressed in Eq. (16) follow immediately from Eqs. (13-15). The conditions given by Eqs. (13-16) describe the essential behavior of the full β function for an effective QCD coupling whose scheme/definition is similar to that of the V scheme.
As noted by Grunberg, one can use observables such as heavy quark scattering or the Bjorken sum rule to define effective charges αO(Q2) each with its own physical scale. (9) This generalizes the convention in QED where the Gell Mann-Low coupling (47) αQED(Q2) is defined at all scales from the scattering of infinitively heavy charged particles. Since physical quantities are involved, the relation between effective charges cannot depend on theoretical conventions such as the of the choice of an intermediate scheme. (48) This is formally the transitivity property of the renormalization group: A to B and B to C relates A to C, independent of the choice of the intermediate scheme B.
The relations between effective charges in pQCD are given by commensurate scale relations (10) . The relative factor between the scales of the two effective charges in the CSR is set to ensure that the onset of a new quark pair in the β function of the two couplings is synchronized. This factor can be determined by the Brodsky-Lepage-Mackenzie procedure, (49) where all nF and β-dependent nonconformal terms in the perturbative expansion are absorbed by the choice of the renormalization scale of the effective coupling.
This procedure also eliminates the factorial renormalon growth of perturbation theory. The commensurate scale relation between αg1(Q2) and the Adler function effective charge αD(Q2) which is defined from Re+e− data is now known to four loops in pQCD (50) . The relation between observables given by the CSR is independent of the choice of the intermediate renormalization scheme. CSR are thus precise predictions of QCD without scale or scheme ambiguity; they thus provide essential tests of the validity of QCD.
The holographic coupling αAdSs(Q2) could be seen as the nonperturbative extension of the αV effective charge defined by Appelquist et al., (11) and it thus can be compared to phenomenological models for the heavy quark potential such as the Cornell potential (51) and lattice computations. Thus, an important question is how to extend the relations between observables and their effective charges to the nonperturbative domain. We can also use the CSR concept to understand the relation of αAdS(Q2) given by Eqs. (9) and (11) to well-measured effective charges such as the αg1 coupling even in the nonperturbative domain.
The effective charges αg1 and αF3 shown in Figs. 1 and 2 are extracted in Ref. (39) following the prescription of Grunberg. (9) Data on the spin structure function g1, from JLab (52) are used to form αg1. CCFR data on the structure function F3 (53) are used to form αF3, which is then related to αg1 using a CSR. The GDH and Bjorken sum rules constrain, respectively, the small (40) and large (12) Q2 behavior of the integral of g1 and provide a description of αg1 over a large domain.
The value of αAdSs(Q) at Q=0 was not determined by our holographic approach. (58) It is also well known that even in the pQCD domain the value of running coupling is significantly scheme dependent when the momentum transfer becomes small. It is thus reasonable to assume that such differences propagate in the IR domain and consequently the IR value of different effective charges can differ. Such differences between schemes can naturally explain the smaller IR fixed point values obtained in other computations of the strong coupling, e.g., in Ref. (28) , as qualitatively illustrated on Fig. 3.
Figure 3: How different schemes can lead to different values for the IR fixed point. The couplings are computed in the UV region. They freeze in the IR region. The interpolation between UV and IR is drawn freely and is meant to be illustrative, as are the various IR fixed point values. We note that the V and g1 schemes are numerically close.
Despite the different physics underlying the light-front holographic coupling αAdSs(Q2) and the effective charge αg1(Q2) determined empirically from measurements of the Bjorken sum, the shapes of the two running couplings are remarkably close in the infrared regime. The resemblance of αAdSs and αg1 is understandable if we recall that αAdSs is a natural nonperturbative extension of αV. The scale shift in the CSR between αV and αg1 is small, making them numerically very close. Furthermore every effective charge satisfies the same pQCD β function to two loops. Thus, the extended αAdSs and αg1 are also very close at high scales. The AdS and g1 couplings share other common features: their β functions have similar structures: zero in the IR, strongly negative in the GeV domain, and zero in the far UV. We can exploit all of these similarities to fix the normalization αAdSs(Q=0)=π and to consistently extend the AdS coupling to the UV domain, consistent with pQCD.
where C=αV(Q=0)=αV(r→∞) since erf(x→+∞)=1. We have written explicitly the normalization at Q=0 in the V scheme since it is not expected to be equal to the normalization in the g1 scheme for the reasons discussed in Sec. VII.
with Q∗=1.18Q, Q∗∗=2.73Q, and we set Q∗∗∗=Q∗∗. We have verified that this relation numerically holds at least down to Q2=0.6 GeV2, as shown in the figure in the Appendix (Fig. 7). In order to transform αg1(Q2) into αV(Q2) over the full Q2 range, we extrapolate the CSR to the nonperturbative domain. For guidance, we use the fact that QCD is near conformal at very small Q; thus, the ratio αV/αg1 is Q independent. A model for the ratio αV(Q)/αg1(Q) is shown in Fig. 4. We apply this ratio to αAdSModified,g1(Q), Eq. (11), and then Fourier transform the result using Eq. (17) to obtain αAdSModified,V(r). We find C≃2.2.
Figure 4: Ratio αV(Q)/αg1(Q). The continuous line represents the domain where the CSR are computed at leading twist [Eq. (19)]. The dashed line is the extrapolation to the nonperturbative domain using the fixed point IR conformal behavior of QCD.
The right panel of Fig. 5 displays αAdSV(r) (dashed line) and αV(r) obtained with the same procedure but applied to the JLab data (lower cross-hatched band). Also shown for comparison are, on the left panel, the results in the g1 scheme: αg1(r) from JLab data (lower cross-hatched band), the light-front holographic result from Eqs. 11 and 18 (continuous line) and lattice results from (38) (upper cross-hatched band). The same scales are used on both panels. The fact that different schemes imply different values for the IR fixed point of αs is exemplified in this figure in which αs(r) in the V scheme and in the g1 scheme freeze to the IR fixed point values of αV(Q=0)=2.2 and αg1(Q=0)=π respectively.
Figure 5: Holographic model predictions for αs(r) in configuration space in the g1 scheme (left panel) and V scheme (right panel). The dashed lines are the holographic AdS results, the continuous lines correspond to the modified holographic results from Eq. (11) normalized, respectively, to αg1 and αV at Q=0. The lower bands (cross-hatched pattern) correspond to the JLab data and the higher (sparser pattern) to lattice results.
The width of the lower band on the right hand panel is the combined uncertainty on αV coming from: a) the uncertainty in the value ΛQCD, b) the truncation of the pQCD β-series used to calculate α¯¯¯¯¯¯¯¯MS in the perturbative region, c) the truncation of the pQCD CSR at Q∗∗∗ which, has been estimated by using the difference between the Q∗∗ and Q∗∗∗ orders and d) the experimental uncertainties on the JLab data for αg1. The uncertainty coming from the truncation of the pQCD series for the Bjorken sum rule is negligible.
The experimental results for αg1(r) follow from the integrated JLab data according to Eq. (17). The contributions to the integral from the unmeasured low Q (Q<0.23 GeV) and high Q (Q>1.71 GeV) regions are computed using the sum rules (40) and (12) respectively. The total experimental uncertainties, as well as the uncertainty on the large Q region, are added in quadrature. This underestimates somewhat the final uncertainty. Since αg1(r) can be computed for any r, the experimental data and lattice results now appear as bands on Fig. 5 rather than a set of data points.
The quark-antiquark Coulomb potential V(r)=−4αV(r)/3r is shown in Fig. 6 for the running coupling computed from light-front holography and the JLab g1 measurement. The results can be compared at large distances to the phenomenological Cornell potential (51) and, in the deep UV region, to the two-loop calculation of Peter (61) as well as with the three-loop calculation of Anzai et al. (62) . Other recent three-loop calculations (63) are consistent with the results from Ref. (62) ; the central values of the three-loop parameter a3 agree within 3 %. The uncertainty in Peter’s result is mainly due to the uncertainty in Λ¯¯¯¯¯¯¯¯MS, with negligible contributions from the truncations of the pQCD β¯¯¯¯¯¯¯¯MS series and the CSR series. The truncation uncertainties are estimated as the values of the last known order of the series. All contributions to the uncertainty are added in quadrature.
Figure 6: Contribution of the running coupling to the quark-antiquark Coulomb potential. The continuous line represents the result from light-front holography modified for pQCD effects using Eq. (11) and transformed to the V scheme using the extrapolated CSR results shown in Fig. 4. The continuous band is the JLab results transformed to the V scheme using the same CSR. The dashed line is the Cornell potential. The inserted figure zooms into the deep UV domain. PQCD results at two loops (Peter, Ref. (61) ) and three loops (Anzai et al., Ref. (62) ) are shown, respectively, by the cross-hatched band and the dot-dashed line. An arbitrary offset is applied to the Anzai et al. results.
where Vconf for a soft-wall dilaton background is the potential for a three-dimensional harmonic oscillator, Vconf≃12mredω2r2. Here mred is the reduced mass of the heavy ¯¯¯¯Q−Q system, mred=mQm¯¯¯¯Q/(mQ+m¯¯¯¯Q), and ω=κ2/(mQ+m¯¯¯¯Q). Remarkably, the explicit holographic confining potential Vconf, which is the dominant interaction for light quarks, vanishes as the inverse of the quark mass for heavy quark masses.
For finite quark masses both contributions will appear. This will bring the effective potential closer to the phenomenological Cornell potential. Thus, the comparison of the Coulomb results in Fig. 6 with the Cornell potential only holds in the limit of infinite quark masses. A detailed discussion of the confining interaction, its implication for the study of the heavy meson mass spectrum, and other aspects of the instantaneous quark-antiquark potential will be discussed elsewhere.
We have shown that the light-front holographic mapping of effective classical gravity in AdS space, modified by a positive-sign dilaton background exp(+κ2z2), can be used to identify a nonperturbative effective coupling αAdSs(Q) and its β function. The same theory provides a very good description of the spectrum and form factors of light hadrons. Our analytical results for the effective holographic coupling provide new insights into the infrared dynamics and the form of the full β function of QCD.
We also observe that the effective charge obtained from light-front holography is in very good agreement with the effective coupling αg1 extracted from the Bjorken sum rule. Surprisingly, the Furui and Nakajima lattice results (38) also agree better overall with the g1 scheme rather than the V scheme as seen in Fig. 5. Our analysis indicates that light-front holography captures the essential dynamics of confinement, showing that it belongs to a universality class of models with built-in confinement. The holographic β function shows the transition from nonperturbative to perturbative regimes at a momentum scale Q∼1 GeV and captures some of the essential characteristics of the full β function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD.
We have made extensive use of commensurate scale relations, which allows us to relate observables in different schemes and regimes. In particular, we have extrapolated the CSR to extend the relation between observables to the nonperturbative domain. In the pQCD domain, we checked that the CSR are valid. This validity provides a fundamental check of QCD since the CSR are a central pQCD prediction independent of theoretical conventions.
The normalization of the QCD coupling αAdSs at Q2=0 appears to be considerably higher than that suggested in Ref. (1) , a difference probably stemming from the different scheme choices. However, αg1(Q2) has the advantage that it is the most precisely measured effective charge. As we have noted, there is a remarkable similarity of αg1(Q2) to the nonperturbative strong coupling αAdSs(Q2) obtained here except at large Q2 where the contribution from quantum loops is dominant. To extend its utility, we have provided an analytical expression encompassing the holographic result at low Q2 and pQCD contributions from gluon exchange at large Q2. The value of the confining scale of the model κ is determined from the vector meson Regge trajectory, so our small Q2-dependence prediction is parameter free.
We thank A. Radyushkin for helpful, critical remarks. We also thank V. Burkert, J. Cornwall, H.G. Dosch, J. Erlich, P. Hägler, W. Korsch, J. Kühn, G. P. Lepage, T. Okui, and J. Papavassiliou for helpful comments. We thank S. Furui for sending us his recent lattice results. This research was supported by the Department of Energy under Contract No. DE–AC02–76SF00515. A.D.’s work is supported by the U.S. Department of Energy (DOE). The Jefferson Science Associates (JSA) operates the Thomas Jefferson National Accelerator Facility for the DOE under Contract No. DE–AC05–84ER40150. S.J.B. thanks the Hans Christian Andersen Academy and the CP3-Origins Institute for their support at Southern Denmark University.
In this appendix we verify, within the uncertainties discussed in the text, the validity of the CSR predictions in the pQCD domain.
Figure 7: Check of the CSR validity. Top panel: Comparison of αV(Q2) from the two-loop pQCD calculation of Ref. (61) and αV(Q2) obtained using the CSR with αg1 as input. Bottom panel: Comparison of αg1(Q2) computed using four different methods. The good agreement on top and bottom panels is a fundamental check of QCD.
The verification of CSR for different schemes is illustrated on Fig. 7. On the top panel of Fig. 7 we compare the full two-loop computation of αV(Q2) from Ref. (61) with the coupling αV(Q2) resulting from applying the CSR to αg1 down to Q2=0.6 GeV2. The width of the bands gives the uncertainties. For the sparse cross-hatched band (two-loop pQCD calculation), the uncertainty stems from ΛQCD, the truncation of the α¯¯¯¯¯¯¯¯MS series to β2, and the truncation of the αV series to two loops (a2 coefficient) in (61) . All these contributions are added in quadrature. For the dense cross-hatched band (CSR), the uncertainties come from ΛQCD, the truncation of the α¯¯¯¯¯¯¯¯MS series to β2, and the truncation of the CSR series to order Q∗∗∗. All these contributions are again added in quadrature. The various truncation uncertainties are estimated by taking the value of the last known term of the series. The very good agreement of the results (69) allows us to check the consistency and the applicability of CSR, even into the IR-UV transition region, albeit with large uncertainties. Throughout the paper, we limit the order of our calculation to α3s so that no IR terms appear.
A similar test of CSR is also shown on the bottom panel of Fig. 7. It shows αg1 computed using four different methods. 1) The continuous band corresponds to the results using the Bjorken sum rule and α¯¯¯¯¯¯¯¯MS. 2) cross-hatched band (70) : using the CSR to obtain αg1 as a function of α¯¯¯¯¯¯¯¯MS. 3) Dashed line: using the CSR to obtain αg1 as a function of αV. This latter is computed from the two-loop computation of Ref. (61) . 4) Continuous line: using αV from pQCD (61) as an input to the appropriate CSR to form α¯¯¯¯¯¯¯¯MS. This later is used as input in another CSR to form αg1. There is again excellent agreement. In addition, that the dashed and continuous lines are on top of each other verifies the transitivity property of the CSR. These agreements are nontrivial consistency checks of QCD since the CSR are central predictions of pQCD.
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Upcoming JLab data: JLab experiments E97-110, J-P Chen, A. Deur, F. Garibaldi spokespersons; EG1b, V. Burkert, D. Crabb, M. Tauiti, G. Dodge and S. Khun spokespersons; EG4, M. Battaglieri, A. Deur, R. DeVita, G. Dodge, M. Ripani and K. Slifer spokespersons.
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The uncertainties of the dense and sparse cross-hatched bands are partially correlated because they share the same contribution from ΛQCD and α¯¯¯¯¯¯¯¯MS. The correlated contributions should be removed when checking the consistency of the two bands. The agreement is still excellent when only the uncertainties from the αV series truncation at two loops for the sparse cross-hatched band and the uncertainties on the CSR series truncation for the dense cross-hatched band are used to estimate the band widths.
The cross-hatched band could be computed only for Q2≳3 GeV2 because, as already noticed, the perturbative series in the ¯¯¯¯¯¯¯¯¯MS scheme is less suitable for low Q2 studies: the leading-order scale shift in the present case is Q∗=0.368Q.

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