Continuing our previous QCD Hamiltonian studies in the gluonic and quark sectors, we describe a new renormalization procedure which generates an effective Hamiltonian. The formulation, which is in the Coulomb gauge, provides an improved framework for investigating hadron structure

Robertson, David G.

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Otterbein University

Otterbein University Digital Commons @ Otterbein Physics Faculty Scholarship Physics 6-7-1998 Renormalized Effective Hamiltonian Approach to QCD David G. Robertson Otterbein University Follow this and additional works at: https://digitalcommons.otterbein.edu/phys_fac  Part of the Physics Commons Repository Citation Robertson, David G., "Renormalized Effective Hamiltonian Approach to QCD" (1998). Physics Faculty Scholarship. 3. https://digitalcommons.otterbein.edu/phys_fac/3 This Conference Proceeding is brought to you for free and open access by the Physics at Digital Commons @ Otterbein. It has been accepted for inclusion in Physics Faculty Scholarship by an authorized administrator of Digital Commons @ Otterbein. For more information, please contact digitalcommons07@otterbein.edu. RENORMALIZED EFFECTIVE HAMILTONIAN APPROACH TO QCDD. G. Robertson,1 E. S. Swanson,1 A. P. Szczepaniak,2 C.-R. Ji1 and S. R. Cotanch11Department of Physics, North Carolina State University, Raleigh, NC 27695, USA2Department of Physics and Nuclear Theory Center, Indiana University,Bloomington, IN 47405, USAABSTRACTContinuing our previous QCD Hamiltonian studies in the gluonic and quarksectors, we describe a new renormalization procedure which generates an effec-tive Hamiltonian. The formulation, which is in the Coulomb gauge, provides animproved framework for investigating hadron structure.1. IntroductionFor the past few years the NCSU group has focused upon the development ofa comprehensive relativistic, many-body description of hadron structure1,2. This ap-proach has many virtues, particularly the explicit incorporation of gluonic degrees offreedom and a rigorous connection to QCD at short distances. Calculations2 of theglueball spectrum have been quite encouraging, reproducing the available lattice data.This talk details recent results from a new renormalization procedure for theCoulomb gauge QCD Hamiltonian. There are several reasons for selecting the Coulombgauge. First, it is completely physical; there are no spurious gauge degrees of freedom orghosts. Second, both qq and gluon-gluon interactions appear explicitly in the canonicalHamiltonian which facilitates contact with quark model phenomenology. Finally, thereappears to be a natural scenario for confinement in the Coulomb gauge, as originallysuggested by Gribov3 and recently formalized by Zwanziger4.This Hamiltonian must be supplemented with a regulator, which is applicablenonperturbatively and is consistent with the many-body techniques we eventually wishto employ. Next the renormalized Hamiltonian is constructed, which includes the cor-rect cut-off dependent counterterms such that physical quantities (e.g., eigenvalues)are cut-off independent. It is inevitable that the regulators at our disposal will vio-late subsets of Lorentz covariance and gauge invariance, so that the structure of thecounterterm Hamiltonian may be expected to be quite complicated. Due to asymptoticfreedom, however, it should be reasonable to construct these operators using pertur-bation theory, provided the energy cut-off is not too low.The result is a Hamiltonian formulation of QCD which is fully covariant and gaugeinvariant through some order in perturbation theory. To effectively apply this Hamil-tonian, further approximations and phenomenological considerations are necessary. Anapproach we have initiated involves making a variational Ansatz for the physical vac-uum state, based on the BCS “pairing” form. Quasi-particle degrees of freedom emergenaturally with constituent-scale masses and may be identified with constituent quarksand gluons. One then can construct approximate bound states of these constituentsutilizing standard many-body techniques such as Tamm-Dancoff and RPA.2. Regularization and RenormalizationOur starting point is the canonical QCD Hamiltonian in the Coulomb gauge∇ ·A = 0. The dynamical degrees of freedom are the transverse gauge fields A, theirconjugate momenta Π, and the quark field ψ. The canonical Hamiltonian may bewritten asHcan = H0 +Hgg +Hqg +HC , (1)where H0 is the free Hamiltonian for quarks and transverse gluons, Hgg contains theself-interactions between transverse gluons, and Hqg contains quark-gluon interactions.HC is the Coulomb term,HC =12g2∫d3xd3y J −1ρa(x)Kab(x,y)J ρb(y) , (2)where J = det [∇ ·D] is the Fadeev-Popov determinant, and the kernel K is given byKab(x,y) = 〈x, a|(∇ ·D)−1(−∇2)(∇ ·D)−1|y, b〉 . (3)The color charge density ρa includes both matter and gluonic contributions, ρa(x) =ψ†(x)T aψ(x)+ fabcAb(x) ·Πc(x), so that HC represents a complicated set of instanta-neous quark-quark, quark-gluon, and gluon-gluon interactions.Perturbative schemes for renormalizing Hamiltonians typically suffer from tech-nical problems related to the occurrence of vanishing energy denominators. One way toavoiding this difficulty is the cut-off method of G lazek and Wilson5. In this approachone considers the basis formed by eigenstates of the free Hamiltonian H0, with eigen-values En. The theory is then regulated by suppressing matrix elements of H betweenstates for which the absolute value of the free energy difference Enm ≡ En − Em islarge. Specifically, we define matrix elements of the regulated Hamiltonian H(Λ) by〈n|H(Λ)|m〉 ≡ Enδnm + fmn(Λ)〈n|HI |m〉 , (4)where HI = H −H0 is the interaction. Here fnm(Λ) is some convenient function whichis unity for |Enm| ≪ Λ and vanishes for |Enm| ≫ Λ. In Ref.1 we chosefnm(Λ) = e−E2nm/Λ2. (5)Note that, since the cut-off is defined in terms of free energy differences, it is notfully Lorentz covariant (though it is of course rotationally invariant). It also violatesgauge invariance, though this issue is difficult to address directly in the present contextsince we work in a fixed gauge. The counterterm Hamiltonian will therefore also containLorentz- and gauge-noninvariant terms, which are necessary to correct for the violationsinduced by the regulator. These complications are in practice unavoidable if one wishesto work with Hamiltonians in a relativistic field theory.At this point we have a fully regulated formulation of QCD in the Coulombgauge. Our next task is to remove the dependence on the cut-off parameter Λ byadding counterterms to the Hamiltonian. We thus obtain the renormalized effectiveHamiltonian, Heff(Λ), which may then be analyzed nonperturbatively using many-bodytechniques.An elegant way of determining the counterterms involves designing a similaritytransformation which changes the cut-off in all matrix elements of H . Because thetransformed Hamiltonian is equivalent to the original one, physical predictions (e.g.,eigenvalues) are unchanged. One then searches for Hamiltonians which are form in-variant, or “coherent,” under this transformation. Such self-similar Hamiltonians willautomatically yield cut-off independent predictions and thus represent renormalizedHamiltonians. Specifically, we writeH(Λ′) = U(Λ′; Λ)H(Λ)U †(Λ′; Λ) , (6)where U is a unitary matrix constructed so that the matrix elements 〈m|H(Λ′)|n〉 are allproportional to exp(−E2nm/Λ′2). Construction of a suitable U is may be accomplishedfollowing the formulation of Wegner6. We define the operator T viadU(Λ; Λ′)d(Λ−2)≡ T (Λ)U(Λ; Λ′) . (7)The evolution of H under a change in Λ is then given bydH(Λ)dΛ−2= [T (Λ), H(Λ)] , (8)from which T is seen to be a generator of infinitesimal scale transformations. It isstraightforward to show that the choiceT (Λ) = [H0, H(Λ)] (9)results in the transformation properly changing the value of the cut-off parameter inthe regulating functions used for all matrix elements of the Hamiltonian1.Eqn. (8) can then be analyzed order-by-order in perturbation theory by expandingthe interaction HI =∑p gpHp. At second order, for example, we find〈n|δV2|m〉 =12∑l(1Enl+1Eml) (e−2EnlEml/Λ2− e−2EnlEml/Λ′2)〈n|V1(Λ)|l〉〈l|V1(Λ)|m〉 ,(10)where 〈n|Hp(Λ)|m〉 ≡ exp (−E2nm/Λ2) 〈n|Vp(Λ)|m〉 defines the “reduced” interactionVp, and 〈n|δV2|m〉 ≡ 〈n|V2(Λ′)|m〉−〈n|V2(Λ)|m〉. This shift in the interaction explicitlyaccounts for the physics removed by changing the cut-off from Λ to Λ′.As discussed above, requiring form-invariance of the Hamiltonian under the sim-ilarity transformation determines the counterterm structure uniquely. We have imple-mented this constraint and determined the complete effective Hamiltonian throughsecond order. The operators that appear in the counterterm Hamiltonian include, asexpected, all possible one-body operators which are rotationally invariant (and respect,e.g., time-reversal invariance, charge conjugation, etc.). These are a gluon “mass” term,A2, the operators Π2 and (∇iAj)2, and, for fermions, ψψ. Each has a calculable coeffi-cient, except for the particular combination of Π2 and (∇iAj)2 which has the structureof a gluon wavefunction renormalization. For this combined operator, the variationof the coefficient under changes in the cut-off is calculable, and gives the gauge fieldanomalous dimension: γA = −(g2/16pi2) (4CA/3), where CA is the Casimir invariant ofthe adjoint representation. See Ref.1 for full details.3. Work in Progress and Future DirectionsHaving obtained the renormalized Hamiltonian, we can now proceed to constructa model for the ground state and analyze excitations such as glueballs, mesons andhybrids. In Ref.1, we focused attention on the gluonic sector and made a variationalBCS vacuum Ansatz. This leads to a gap equation which may be solved numericallyto give the spectral function of the quasi-particle constituent gluons.Work in progress includes extending the similarity analysis to higher orders, ap-plying this smooth cut-off treatment to the quark sector and designing improved modelsfor the vacuum. At third order, one should see the running of the coupling, as well asthe appearance of many new operators in the counterterm Hamiltonian. Going beyondBCS is expected to be particularly important in a combined treatment of the quarkand gluon sectors. In addition, other composite operators, such as currents, must beproperly renormalized which is also possible using the formalism presented in Ref.1.Finally, there are a host of phenomenological applications to be explored. Notonly meson spectra but also form factors, electro-weak and hadronic decays, etc. canbe predicted and confronted with available data. This comprehensive analysis shouldprovide significant insight into hadron structure as well as a key linkage between fun-damental QCD and the successful phenomenological quark model.AcknowledgmentsFinancial support is provided by U. S. Department of Energy.References1. D. G. Robertson, E. S. Swanson, A. P. Szczepaniak, C.-R. Ji, and S. R.Cotanch, submitted to Phys. Rev. D.2. A. Szczepaniak, E. S. Swanson, C.-R. Ji, and S. R. Cotanch, Phys. Rev.Lett. 76, 2011 (1996); A. Szczepaniak and E. S. Swanson, Phys. Rev. D55, 1578 (1997); E. S. Swanson and A. Szczepaniak, hep-ph/9804219.3. V. N. Gribov, Nucl. Phys. B 139, 1 (1978).4. D. Zwanziger, Nucl. Phys. B 485, 185 (1997); A. Cucchieri and D.Zwanziger, Phys. Rev. Lett. 78, 3814 (1997).5. St. D. G lazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993); ibid. 49,4214 (1994).6. F. Wegner, Ann. Physik 3, 77 (1994).

Renormalized Effective Hamiltonian Approach to QCD

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Renormalized Effective Hamiltonian Approach to QCD

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