Co-authors: T. Grandou, Y. Gabellini, Y-M Sheu A New, Analytic, Non-Perturbative, Gauge-Invariant Formulation of Realistic QCD Presented by H. M. Fried Co-authors: T. Grandou, Y. Gabellini, Y-M Sheu

In the time allowed, this can only be a brief – and rapid – description of each of the five adjectives of the Title. 1) New: < 3 years old. But it could have been done decades ago, for the input information existed … but was overlooked. 2) Analytic: Physically-reasonable approximations can be estimated with paper and pencil; exact amplitudes can be calculated as Meijer G-functions, of various orders. 3) Non-Perturbative: Sums over all possible gluon exchanges between any pair of quarks and/or antiquarks, including cubic and quartic gluon interactions, are performed. These multiple gluon exchanges combine into “Gluon Bundles” (GBs); Feynman graphs are replaced by “Bundle graphs”. In effect, gluons disappear from the formalism, GBs remain, as the effective carrier of all interactions between quark lines.

4) Gauge-invariance: A simple rearrangement of the Schwinger/Symanzik formal, functional solution for the Generating Functional of QCD – possible in QCD but not in QED – produces a formal statement of gauge invariance, even though the formulation contains gauge-dependent gluon propagators. After the non-perturbative sums produce GBs, one sees explicit cancelation of all gluon gauge dependent propagators; gauge-invariance is achieved as gauge-independence.

5) Realistic: A new insight into non-perturbative QCD 5) Realistic: A new insight into non-perturbative QCD. Conventional Lagrangian generates absurdities in the non-perturbative domain, because quarks do not have individual asymptotic states; they are always asymptotically bound, and their transverse coordinates cannot, in principle, be measured exactly. In QED, Yes; in QCD, No. Introduce “transverse imprecision” into the basic Lagrangian, in terms of an integral over an unknown transverse probability amplitude, later determined by quark binding parameters. All the above absurdities then disappear, and the quark-binding potentials for the construction of mesons and nucleons can be defined and evaluated. And One More Surprise: 6) A new, non-perturbative property appears: EFFECTIVE LOCALITY, with the result that all Functional Integrals reduce to (a few) sets of ordinary integrals, easy to estimate approximately, or calculate on a desktop computer. A FANTASTIC SIMPLIFICATION!

METHODOLOGY: How can all this be done? In actual sequence: a) Begin with Schwinger/Symanzik formal functional solution for the Generating Functional (GF) of QCD. b) Rearrange S/S solution so that gauge-invariance is (formally) obtained. A small trick, overlooked for decades - it is possible in QCD but not in QED. GF still contains gauge-dependent gluon propagators, but they will all cancel – explicitly – after sums over all exchanged gluons, in all relevant Feynman graphs, are performed. c) How to sum over ALL relevant Feynman graphs? By using Functional Methods! The gauge-invariant rearrangement of the S/S functional solution for the GF requires a Gaussian-weighted functional operation upon the A-dependence of two functionals: G_c[A] and L[A]; these are Potential Theory functionals which appear in the S/S GF after the quark fields are integrated out. One also requires a Halpern functional representation of the familiar factor: exp{-i(/4)Int(F^2)}=>Norm. Int{d[xi]Exp[(i/4)int{xi^2} +(i/2)Int{F*xi}]}, where the A-dependence inside the RHS term is Gaussian.

But there have existed – for a good 50 years – Fradkin functional representations for both G_c[A] and L[A] which are also Gaussian in A. THEREFORE the needed functional operation over the A dependence can be carried through EXACTLY, and this : = the sum of all Feynman graphs in a specific process defined by the functional operation over groups of G_c[A] factors – one for each quark or antiquark – and as many loop factors of L[A] as desired (or, using the functional cluster expansion, all L[A]s). And then one sees - explicitly - cancellation of all the gauge-dependent gluon propagators, and the emergence of Effective Locality, with Functional Integrals reduced to Ordinary Integrals. NB: The results are expressed in terms of the functional representations of G_c and L. Is this progress? YES, because these can be approximated in certain physical situations, e.g., at high energies, G_c -> B-N Eikonal approximation; and there are some tricks for L. How do we calculate quark binding potentials? And nucleon-nucleon binding?

General technique: In Potential Theory, given a V(r) acting between two particles, there exists a simple relation to calculate the Eikonal Function Xi(b) relevant in HE scattering: Xi(b) = g Int{dz V(b + z)}, where the integration is over all longitudinal z, and where g depends on the CM energy and the spin structure of the interaction. Here, we calculate Xi(b), and then just reverse the procedure, using analytic continuation to obtain V(r), both for the q-qbar model of a pion, and for a qqq model of a nucleon. (N-Body Eikonal formalism in Phys. Rev. A28, 738 (1983).) It’s easy: it took one year to understand what we were doing, to write the first paper of this series: EPJC (2010) 65, 395-411, in which non-perturbative Asymptotic Freedom is clear. Then it took one month more to calculate the pion’s quark-binding potential: V (r) = x m (rm)^(1+x), where m is approximately the pion mass, and x is on the order of 1/10. The parameter x is not a coupling constant; it appears in the transverse imprecision probability integral, and is needed to express the form of transverse fluctuations which are associated with quark binding.

How to construct the V(r) for nucleon-nucleon scattering and binding? Quark binding into mesons and nucleons is represented by GB exchange, important over distances on the order of an inverse pion mass. And for larger distances, the GB amplitude falls away sharply. But nucleon binding involves distance up to 5 times that length. How to arrange this? Between the two nucleons, each composed of three bound quarks, insert a closed loop, bound by a GB to each nucleon: As the distance between the nucleons increases, the loop stretches – intuitively resembles pion exchange between nucleons. The loop must be renormalized, and the net potential between the nucleons Is proportional to a_s. Using the simplest method of estimation, a “deuteron” bound state of 2.2 MeV requires a_s = 12.5, a “Proof of Principle” result, since this estimation neglects nucleon spin, assumes only one flavor of quark, and the x parameter has been suppressed. To our knowledge this is the first analytic example of nucleon binding directly from basic QCD; and it immediately suggests bases for the Nuclear Shell Model and the Independent Boson Model.

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References: 1) EPJC (2010) 65, 395-411 2) arXiv (het-ph): 1204.2038; to be published in Annals of Physics. 3) arXiv (het-ph): 1203.6137 #1 is a calculation of q-q scattering in a quenched, eikonal approximation, displaying the gauge-invariance, and the non-perturbative meaning of asymptotic freedom. #2 is a compilation of three other papers in the arXiv, describing Effective Locality, Realistic QCD, and – as a simple example – deriving the quark-binding potential. #3 removes quenching (retaining one quark loop) to obtain the simplest approximation to the nucleon-nucleon binding potential. (After calculating the quark binding potential, this n-n potential required 4 months.) Other problems under consideration: a) Inclusion of all possible closed quark loops, to make the analysis truly non-pert. b) Derivation of the Nuclear Shell Model and Independent Boson Model. c) Improvements of binding potentials, taking into account quark flavors and spins. d) Non-perturbative renormalization theory, with gluons replaced by GBs. e) Quark-gluon plasma Two intuitive comments: Cubic and Quartic gluon interactions are crucial for the construction of flux tubes… This approach to QCD appears to yield results far simpler than does QED. Why? Because of those cubic and quartic gluon interactions! EG, check out self-effects using GBs…

In the remaining time, I thought it might be most interesting to see how a simple rearrangement of the Schwinger/Symanzik GF can lead to MGI in QCD. The first step is this long-overlooked rearrangement; the second step is the use of Fradkin’s (Gaussian in A) representations to sum over ALL gluon exchanges between any pair of quarks and/or antiquarks, including cubic and quartic gluon interactions. And then one sees the explicit cancellation of all gauge-dependent gluon propagators, following the Gaussian functional operations which can be performed exactly, corresponding to the sum over ALL such Feynman graphs, now represented by a ‘Gluon Bundle’. And then there appears the simplifying property of Effective Locality, which permits simple, pencil-and-paper approximations to relevant amplitudes, which can also be given exact representations in terms of Meijer G-functions.