A Consistency Check of Commensurate Scale Relations

Nonperturbative QCD Coupling and its $β$ function from Light-Front Holography

Abstract

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 . 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 GeV. The resulting function appears to capture the essential characteristics of the full 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 .

pacs:
11.15.Tk, 11.25Tq, 12.38Aw, 12.40Yx
1

I Introduction

The concept of a running coupling 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 , leads to a nonperturbative effective coupling 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 ()-dimensional AdS spacetime in terms of a flat -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 AdS space in the original correspondence (2) is conformal, the strong coupling of the dual gauge theory is constant, and its function is zero. Thus, one must consider a deformed AdS space in order to have a running coupling 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 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: . This effective charge defines a renormalization scheme – the scheme of Appelquist, Dine, and Muzinich. (11) In fact, the holographic coupling can be considered to be the nonperturbative extension of the effective charge defined in Ref. (11).

We shall also make extensive use of the scheme, where the strong coupling is determined from the Bjorken sum rule. (12) The coupling has the advantage that it is the best-measured effective charge, and it can be used to extrapolate the definition of the effective coupling to large distances. (13) It has been measured at intermediate energies, and it is therefore particularly useful to study the transition from short distances, where partons are the relevant degrees of freedom, to large distances, where the hadronic degrees of freedom are present. (14)

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 AdS space modified by a positive-sign dilaton background . In Sec. IV, we compare the results for the coupling obtained in Sec. III with the effective QCD couplings extracted from different observables and lattice results. The nonperturbative results are extended to large by matching the holographic results to the perturbative results in the transition region. In Sec. V, we discuss the holographic results for the 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 , , and schemes are confronted in the perturbative domain.

Ii Light-Front Holography and QCD

The basic principle underlying the AdS/CFT approach to conformal gauge theories is the isomorphism of the group of Poincaré and conformal transformations to the group of isometries of AdS space, the group of transformations that leave the AdS metric

(1)

invariant ( the AdS radius). Since the metric (1) is invariant under a dilatation of all coordinates , , the variable acts like a scaling variable in Minkowski space: different values of correspond to different energy scales at which the hadron is examined.

In order to describe a confining theory, the conformal invariance of AdS 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 . Thus, the “hard-wall” at 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 , as well as modifications of the background geometry in the large 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.

The conformal metric of AdS space can be modified within the AdS/QCD framework to simulate confinement forces. (17) The introduction of a dilaton profile in the AdS action can be considered equivalent to modifying the AdS metric (1) by introducing an additional warp factor  (18)

(2)

A dilaton profile of either sign also leads to a two-dimensional oscillator potential 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 modified metric induced by the dilaton can be interpreted in AdS space as a gravitational potential for an object of mass in the fifth dimension: . In the case of the negative solution the potential decreases monotonically, and thus an object in AdS will fall to infinitely large values of . For the positive solution, the potential is nonmonotonic and has an absolute minimum at . Furthermore, for large values of the gravitational potential increases exponentially, thus confining any object to distances (7) We thus use the positive-sign dilaton solution opposite to that of Ref. (17). This additional warp factor leads to a well-defined scale-dependent effective coupling. Introducing a positive-sign dilaton background is also relevant for describing chiral symmetry breaking in the soft-wall model, (20) since the expectation value of the scalar field associated with the quark mass and condensate does not blow up in the far infrared region of AdS, (21) in contrast with the original model. (17)

The soft-wall model of Ref. (17) also uses the AdS/QCD framework (22); (23), where bulk fields are introduced to match the 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 , the time marked by the front of a light wave. (25) This correspondence provides a direct connection between the hadronic amplitudes in AdS space with LFWFs describing the quark and gluon constituent structure of hadrons in physical space-time. The mapping between the LF invariant variable and the fifth-dimension AdS coordinate 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 spacetime.

Light-front quantization is the ideal framework for describing the structure of hadrons in terms of their quark and gluon degrees of freedom. LFWFs play the same role in hadron physics that Schrödinger wave functions play in atomic physics. (26) The simple structure of the LF vacuum provides an unambiguous definition of the partonic content of a hadron in QCD. A physical hadron in four-dimensional Minkowski space has four-momentum and invariant hadronic mass states, , determined by the Lorentz-invariant Hamiltonian equation for the relativistic bound-state system

(3)

The hadron four-momentum generator is , , and the hadronic state is an expansion in multiparticle Fock eigenstates of the free light-front Hamiltonian: . The internal partonic coordinates of the hadron are the momentum fractions and the transverse momenta , , where is the number of partons in a given Fock state. Momentum conservation requires and . It is useful to employ a mixed representation (27) in terms of independent momentum fraction variables and position coordinates , , so that . The relative transverse variables are Fourier conjugates of the momentum variables .

In AdS space the physical states are represented by normalizable modes , with plane waves along the Poincaré coordinates and a profile function along the holographic coordinate . Each LF hadronic state is dual to a normalizable string mode . The hadronic mass is found by solving the eigenvalue problem for the corresponding wave equation in AdS space, which, as we discuss below, is equivalent to the semiclassical approximation to the light-front bound-state Hamiltonian equation of motion in QCD. One can indeed systematically reduce the LF Hamiltonian eigenvalue Eq. (3) to an effective relativistic wave equation (4) by observing that each -particle Fock state has an essential dependence on the invariant mass of the system and thus, to a first approximation, LF dynamics depend only on . In impact space the relevant variable is the boost invariant transverse variable which measures the separation of the quark and gluonic constituents within the hadron at the same LF time and which also allows one to separate the dynamics of quark and gluon binding from the kinematics of the constituent internal angular momentum. In the case of two constituents, where is the LF fraction. The result is the single-variable light-front relativistic Schrödinger equation (4)

(4)

where is the effective potential, and is the relative orbital angular momentum as defined in the LF formalism. The set of eigenvalues gives the hadronic spectrum of the color-singlet states, and the corresponding eigenmodes 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- modes in a fixed gravitational background asymptotic to AdS space. (4) By using the correspondence between in the LF theory and 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 AdS metric (1) can be deformed by a warp factor . In the case of a two-parton relativistic bound state, the resulting effective potential in the LF equation of motion is (7) There is only one parameter, the mass scale GeV, which enters the effective confining harmonic oscillator potential. Here is the spin of the quark-antiquark system, is their relative orbital angular momentum, and is the Lorentz-invariant coordinate defined above, which measures the distance between the quark and antiquark; it is analogous to the radial coordinate in the Schrödinger equation. The resulting mesonic spectrum has the phenomenologically successful Regge form with equal slopes in the orbital angular momentum and the radial quantum number . The pion with is massless for zero quark mass, consistent with chiral symmetry.

Iii Nonperturbative QCD Coupling from Light-Front Holography

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 as a function of the spacelike momentum transfer . 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 and asymptotic freedom at large . However, the falloff of at large is exponential: , 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.

As will be explained in Sec. V, the function derived from light-front holography becomes significantly negative in the nonperturbative regime , where it reaches a minimum, signaling the transition region from the IR conformal region, characterized by hadronic degrees of freedom, to a pQCD conformal UV regime where the relevant degrees of freedom are the quark and gluon constituents. The function is always negative; it vanishes at large consistent with asymptotic freedom, and it vanishes at small consistent with an infrared fixed point. (28); (29)

Let us consider a five-dimensional gauge field propagating in AdS space in the presence of a dilaton background which introduces the energy scale in the five-dimensional action. At quadratic order in the field strength the action is

(5)

where the metric determinant of AdS is , , and the square of the coupling has dimensions of length. On general grounds we would expect that the value of the five-dimensional coupling in units is determined by a geometrical factor scaled by . We can identify the prefactor

(6)

in the AdS action (5) as the effective coupling of the theory at the length scale . The coupling then incorporates the nonconformal dynamics of confinement. The five-dimensional coupling is mapped, modulo a constant, into the Yang-Mills (YM) coupling of the confining theory in physical space-time using light-front holography. One identifies with the invariant impact separation variable which appears in the LF Hamiltonian: . Thus

(7)

In contrast with the three-dimensional radial coordinates of the nonrelativistic Schrödinger theory, the natural light-front variables are the two-dimensional cylindrical coordinates and the light-cone fraction . The physical coupling measured at the scale is the two-dimensional Fourier transform of the LF transverse coupling (7). Integration over the azimuthal angle gives the Bessel transform

(8)

in the light-front frame where is the square of the spacelike four-momentum transferred to the hadronic bound state. Using this ansatz we then have from Eq. (8)

(9)

In contrast, the negative dilaton solution leads to an integral that diverges at large . The essential assumption of this paper is the identification of 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 and field strength in the QCD Lagrangian density by a compensating rescaling of the coupling strength The renormalization of the coupling where 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: , with a rescaled Lagrangian density . In lattice gauge theory, the lattice spacing 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 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 function in the UV domain. (30); (31); (32); (33); (34); (35); (36) For example, a -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 and the energy scale of the conformal field theory, , as discussed in Ref. (37). In our paper, the effective QCD coupling is identified by using the precise mapping from in AdS space to the transverse impact variable in LF QCD.

Iv Comparison of the Holographic Coupling with Other Effective Charges

The effective coupling (solid line) is compared in Fig. 1 with experimental and lattice data. For this comparison to be meaningful, we have to impose the same normalization on the AdS coupling as the coupling. This defines normalized to the scheme

(10)

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 flavor gauge theory is , and thus for in units .

Figure 1: The effective coupling from LF holographic mapping for 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 GeV. The value 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 has been determined in Ref (39) from several experiments. Figure 1 also displays other couplings from different observables as well as , which is computed from the Bjorken sum rule (12) over a large range of momentum transfer (continuous band). At one has the constraint on the slope of 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.

It is interesting to illustrate what one expects in an augmented model which contains the standard pQCD contributions. We can use the similarity of the AdS coupling to the effective charge at small scales as guide on how to join the perturbative and nonperturbative regimes. The fit to the data from Ref. (39) agrees with pQCD at high momentum. Thus, the coupling provides a guide for the analytic form of the coupling over all . We write

(11)

Here is given by Eq. (9) with the normalization (10) [continuous line in Fig. 1] and is the analytical fit to the measured coupling (39) These couplings have the same normalization at , 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 . In order to smoothly connect the two contributions (dot-dashed line in Fig. 1), we employ smeared step functions. For convenience we have chosen with the parameters GeV and GeV.

V Holographic function

The function for the nonperturbative effective coupling obtained from the LF holographic mapping in a positive dilaton-modified AdS background is

(12)

The solid line in Fig. 2 corresponds to the light-front holographic result Eq. (12). Near GeV, we can interpret the results as a transition from the nonperturbative IR domain to the quark and gluon degrees of freedom in the perturbative UV regime. The transition momentum scale is compatible with the momentum transfer for the onset of scaling behavior in exclusive reactions where quark counting rules are observed. (16) For example, in deuteron photo-disintegration the onset of scaling corresponds to momentum transfer of 1.0 GeV to the nucleon involved. (41) Dimensional counting is built into the AdS/QCD soft and hard-wall models, since the AdS amplitudes are governed by their twist scaling behavior at short distances, (15) A similar scale for parton-hadron transition region has been observed in inclusive reactions. (42)

Figure 2: Holographic model prediction for the function compared to JLab and CCFR data, lattice simulations and results from the Bjorken sum rule.

Also shown on Fig. 2 are the 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 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 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.

Judging from these results, we infer that the actual function of QCD will interpolate between the nonperturbative results for GeV and the pQCD results for GeV. We also observe that the general conditions

(13)
(14)
(15)
(16)

are satisfied by our model 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 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 because of asymptotic freedom and near 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 (29) Condition (14) for large , expresses the basic antiscreening behavior of QCD where the strong coupling vanishes. The 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 where the 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 function for an effective QCD coupling whose scheme/definition is similar to that of the scheme.

Vi Effective Charges and Commensurate Scale Relations

As noted by Grunberg, one can use observables such as heavy quark scattering or the Bjorken sum rule to define effective charges each with its own physical scale. (9) This generalizes the convention in QED where the Gell Mann-Low coupling (47) 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: to and to relates to , independent of the choice of the intermediate scheme .

Although the perturbative function for every effective charge (9) is universal up to two loops at high , each effective charge has specific characteristics, which influence its behavior at small . For example, the value and derivative of the coupling at are both constrained since the Bjorken sum vanishes at and its derivative is given by the GDH sum rule. (39); (40)

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 function of the two couplings is synchronized. This factor can be determined by the Brodsky-Lepage-Mackenzie procedure, (49) where all and -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 and the Adler function effective charge which is defined from 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 could be seen as the nonperturbative extension of the 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 given by Eqs. (9) and (11) to well-measured effective charges such as the coupling even in the nonperturbative domain.

Vii Experimental Results, Schemes and Data Normalization

The effective charges and shown in Figs. 1 and 2 are extracted in Ref. (39) following the prescription of Grunberg. (9) Data on the spin structure function , from JLab (52) are used to form . CCFR data on the structure function  (53) are used to form , which is then related to using a CSR. The GDH and Bjorken sum rules constrain, respectively, the small (40) and large (12) behavior of the integral of and provide a description of over a large domain.

We note that the works of (9) and (10) pertain to the UV domain, whereas Ref. (39) extends them to the IR region based on the analytical behavior of the coupling. The effective charge is found to be approximately scale invariant in the IR domain, in agreement with an IR fixed point behavior. (29) The shape of the coupling agrees with other predictions of the running coupling at small , including lattice QCD, (38) the Schwinger-Dyson formalism, (28); (44) and the coupling of a constituent quark model which is consistent with hadron spectroscopy. (54) We point out that the essential difference between these running couplings is their value at : if normalized to the same point at , their dependences agree within their relative uncertainties. (39)

The continuous band in Fig. 1 for is computed with the Bjorken sum rule using the relation between and (39); (10) The pQCD leading-twist expression of the Bjorken sum up to third order in is used to estimate the Bjorken sum. The sum rule is then used to extract at large Q. In the pQCD expression of the Bjorken sum rule, is retained up to second order in (i.e. up to ). The uncertainty in the band comes from the uncertainty on and the truncation of the series. (55)

Although the effective coupling has specific features of deep inelastic lepton-proton scattering, it nevertheless appears to closely mimic the shape and magnitude of the AdS/QCD coupling near the transition region . In particular, it illustrates how one can have a coupling which flows analytically from the IR strong coupling domain with an IR fixed point to the UV domain controlled by pQCD. (57)

The value of at 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 and schemes are numerically close.

Despite the different physics underlying the light-front holographic coupling and the effective charge 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 and is understandable if we recall that is a natural nonperturbative extension of . The scale shift in the CSR between and is small, making them numerically very close. Furthermore every effective charge satisfies the same pQCD function to two loops. Thus, the extended and are also very close at high scales. The AdS and couplings share other common features: their 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 and to consistently extend the AdS coupling to the UV domain, consistent with pQCD.

Viii Holographic Coupling in Configuration Space

In order to obtain modifications to the instantaneous Coulomb potential in configuration space from the running coupling, one must transform the coupling defined by the static quark potential in the nonrelativistic limit and extract the coefficient of to define the coupling in the scheme. The couplings are related by the Fourier transform (60)

(17)

From (9) we find the expression

(18)

where since . We have written explicitly the normalization at in the scheme since it is not expected to be equal to the normalization in the scheme for the reasons discussed in Sec. VII.

The couplings in the and schemes are related at leading twist by the CSR: (10)

(19)

with , , and we set . We have verified that this relation numerically holds at least down to GeV, as shown in the figure in the Appendix (Fig. 7). In order to transform into over the full range, we extrapolate the CSR to the nonperturbative domain. For guidance, we use the fact that QCD is near conformal at very small ; thus, the ratio is independent. A model for the ratio is shown in Fig. 4. We apply this ratio to , Eq. (11), and then Fourier transform the result using Eq. (17) to obtain . We find .

Figure 4: Ratio . 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.

viii.1 Comparison of and Results

The right panel of Fig. 5 displays (dashed line) and 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 scheme: 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 is exemplified in this figure in which in the scheme and in the scheme freeze to the IR fixed point values of and respectively.

Figure 5: Holographic model predictions for in configuration space in the scheme (left panel) and 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 and at . 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 coming from: a) the uncertainty in the value , b) the truncation of the pQCD -series used to calculate in the perturbative region, c) the truncation of the pQCD CSR at which, has been estimated by using the difference between the and orders and d) the experimental uncertainties on the JLab data for . The uncertainty coming from the truncation of the pQCD series for the Bjorken sum rule is negligible.

The experimental results for follow from the integrated JLab data according to Eq. (17). The contributions to the integral from the unmeasured low ( GeV) and high ( GeV) regions are computed using the sum rules (40) and (12) respectively. The total experimental uncertainties, as well as the uncertainty on the large region, are added in quadrature. This underestimates somewhat the final uncertainty. Since can be computed for any , the experimental data and lattice results now appear as bands on Fig. 5  rather than a set of data points.

viii.2 Contribution to the Instantaneous Quark-Antiquark Potential

The quark-antiquark Coulomb potential is shown in Fig. 6 for the running coupling computed from light-front holography and the JLab 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 agree within 3 %. The uncertainty in Peter’s result is mainly due to the uncertainty in , with negligible contributions from the truncations of the pQCD 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 scheme using the extrapolated CSR results shown in Fig. 4. The continuous band is the JLab results transformed to the 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.

In the case of heavy quarks the light-front holographic equations reduce to a nonrelativistic Schrödinger equation in configuration space with potential

(20)

where for a soft-wall dilaton background is the potential for a three-dimensional harmonic oscillator, . Here is the reduced mass of the heavy system, , and . Remarkably, the explicit holographic confining potential , 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.

Ix Conclusions

We have shown that the light-front holographic mapping of effective classical gravity in AdS space, modified by a positive-sign dilaton background , can be used to identify a nonperturbative effective coupling and its 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 function of QCD.

We also observe that the effective charge obtained from light-front holography is in very good agreement with the effective coupling extracted from the Bjorken sum rule. Surprisingly, the Furui and Nakajima lattice results (38) also agree better overall with the scheme rather than the 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 function shows the transition from nonperturbative to perturbative regimes at a momentum scale GeV and captures some of the essential characteristics of the full 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 at appears to be considerably higher than that suggested in Ref. (1), a difference probably stemming from the different scheme choices. However, has the advantage that it is the most precisely measured effective charge. As we have noted, there is a remarkable similarity of to the nonperturbative strong coupling obtained here except at large where the contribution from quantum loops is dominant. To extend its utility, we have provided an analytical expression encompassing the holographic result at low and pQCD contributions from gluon exchange at large . The value of the confining scale of the model is determined from the vector meson Regge trajectory, so our small -dependence prediction is parameter free.

There are many phenomenological applications where detailed knowledge of the QCD coupling and the renormalized gluon propagator at relatively soft momentum transfer are essential. This includes the rescattering (final-state and initial-state interactions), which create the leading-twist Sivers single-spin correlations in semi-inclusive deep inelastic scattering, (64); (65) the Boer-Mulders functions which lead to anomalous contributions to the lepton pair angular distribution in the unpolarized Drell-Yan reaction, (66) and the Sommerfeld-Sakharov-Schwinger correction to heavy quark production at threshold. (67) The confining AdS/QCD coupling from light-front holography can lead to a quantitative understanding of this factorization-breaking physics. (68)

Acknowledgements.
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 CP-Origins Institute for their support at Southern Denmark University.

*

Appendix A Consistency Check of Commensurate Scale Relations

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 from the two-loop pQCD calculation of Ref. (61) and obtained using the CSR with as input. Bottom panel: Comparison of 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 from Ref. (61) with the coupling resulting from applying the CSR to down to . The width of the bands gives the uncertainties. For the sparse cross-hatched band (two-loop pQCD calculation), the uncertainty stems from , the truncation of the series to , and the truncation of the series to two loops ( coefficient) in (61). All these contributions are added in quadrature. For the dense cross-hatched band (CSR), the uncertainties come from , the truncation of the series to , and the truncation of the CSR series to order . 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 so that no IR terms appear.

A similar test of CSR is also shown on the bottom panel of Fig. 7. It shows computed using four different methods. 1) The continuous band corresponds to the results using the Bjorken sum rule and . 2) cross-hatched band (70): using the CSR to obtain as a function of . 3) Dashed line: using the CSR to obtain as a function of . This latter is computed from the two-loop computation of Ref. (61). 4) Continuous line: using from pQCD (61) as an input to the appropriate CSR to form . This later is used as input in another CSR to form . 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.

Footnotes

  1. preprint: SLAC–PUB–13840     JLAB–PHY–10–1128

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