A new geometrical interpretation of chiral perturbation theory based on a topological QCD model is explained in pictures.Objašnjava se slikama novo geometrijsko tumačenje kiralne teorije smetnje zasnovano na topološkom modelu QCDa

Tang, Alfred

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ISSN1330–0016CODEN FIZBE7CHIRAL PERTURBATION ON THE LIGHTFRONT1ALFRED TANGMax Planck Institut für Kernphysik, Postfach 103980, D-69120 Heidelberg, GermanyReceived 25 August 2003; Accepted 26 April 2004Online 22 October 2004A new geometrical interpretation of chiral perturbation theory based on a topolog-ical QCD model is explained in pictures.PACS numbers: 12.38.Bx, 11.30.Rd, 12.39.Fe UDC 539.12Keywords: chiral perturbation theory, topological QCD model1. IntroductionChiral perturbation provides a framework to calculate corrections to nuclearobservables such as nucleon masses, charge radii and form factors. When chiralsymmetry is broken, intermediate mesons are created as Goldstone bosons whichcontribute to the tree-level values of relevant physical observables. Traditional chi-ral perturbation theory based on Feynman diagrams provides a means to estimatethe corrections that come from the Goldstone bosons. Unfortunately, the calcula-tions become intractable rapidly beyond the one-loop level. The current doctrineof particle physics is a bottom-up approach that tries to analyze the most fun-damental units of nature in terms of elementary particles with all the associatingdynmanics and symmetries. The bottom-up approach of the particle picture (par-ticularly in the framework of perturbative QCD) cannot solve the nucleons becausesuch systems are characterized by highly non-linear many-body problems. Recentdevelopments of non-critical dimension string theory in AdS5 × S5 with N = 1shows some promises in solving the nucleons by virtue of avoiding the the difficultperturbative QCD calculations by effectively summing the infinite perturbativeQCD series on the string world sheets. This approach is a paradigm shift fromtraditional wisdom in a sense that the bottom-up approach of particle physics thatbegins from elementary symmetries is replaced by the top-down approach of stringtheory that starts with global symmetries that hide the details on the particlelevel. Although supersymmetry is a beautiful symmetry, it may pose too many1This work is a written summary, with minor changes, of a talk presented at NAPP 2003,Dubrovnik, Croatia.FIZIKA B 13 (2004) 1, 265–270 265tang: chiral perturbation on the lightfrontconstraints when modeling the nucleons. The present work suggests an alternativetop-down approach by summing the infinite series on the string worldsheets withno assumption of any supersymmetry nor extra dimensions. This research is workin progress. This paper explains the logic of this new approach in picture format.The introductory remarks, such as a brief review of chiral perturbation theory andrecent works related to χpt on the lightfront, are skipped in this paper so that themain concept, i.e. topological QCD, is addressed right away.2. Topological QCDGeometrical interpretation of QCD is quite old. For instance, the gauge field ofa nucleon is usually represented as a torus T 2 in lattice QCD. In the same spirit, thedisconnected graphs of the tadpoles can be modeled as S2 spheres. A gauge field isrepresented by a worldsheet called the gauge surface from now on. In the instantform, the geometry of the nucleon is represented in Fig. 1. The plane of initialdata intersects the gaugeballs representing the vacuum. The Hamiltonian evolvesthe plane of initial data along the torus in the time-like direction. In this case,the plane sweeps through the gaugeballs so that the structure of the vacuum is notsimple. On the other hand, the initial data in the front form are parameterized alonga plane tangent to the the torus as shown in Fig. 2. The front form Hamiltonianevolves the plane of initial data around the torus in spiral directions with variouswinding modes. Since the gaugeballs are disconnected from the torus, the plane ofinitial data does not necessarily intersect the gaugeballs during the evolution. Thispicture recovers the typical result that lightfront QCD is uniquely endowed with asimpler vacuum.At the end, the worldsheets of the gauge fields can be summed using pathintegrals. The integrals are categorized according to geni. A genus 0 surface hasno hole, a genus 1 surface has 1 hole and so on. All topologically equivalent surfacesFig. 1. The geometry of a nucleon in the instant form. The torus represents thenucleon and the spheres (gaugeballs) symbolize the vacuum. The plane is the surfaceof initial data that intersects the torus. The instant form Hamiltonian evolves planeof initial data along the torus in the time-like direction and sweeps it through thegaugeballs so that the instant form Hamiltonian sees a complicated vacuum.266 FIZIKA B 13 (2004) 1, 265–270tang: chiral perturbation on the lightfrontFig. 2. The geometry of a nucleon in the front form. The torus represents thenucleon and the spheres (gaugeballs) symbolize the vacuum. The plane is the surfaceof initial data that is tangential to the torus. The front form Hamiltonian evolvesthe plane of initial data around the torus in a spiral direction so that the front formHamiltonian sees a simpler vacuum.are grouped together. Topological equivalence greatly simplifies the calculations.For instance, the geometry of all hairpins and sea quarks are equivalent to a planeand are effectively cancelled as shown in Figs. 3a and c.(a)(b)(c)Fig. 3. Simplications that comes from topological equivalence. (a) A hairpin, (b) apion loop and (c) sea quarks.3. Non-Abelian gauge as torsionIn non-Abelian gauge theory, the partial derivatives are modified by the additionof an extra term that comes from minimal coupling as inDµ = ∂µ + i τc Acµ. (1)FIZIKA B 13 (2004) 1, 265–270 267tang: chiral perturbation on the lightfrontWith the extra term, the total derivatives no longer commute,DµDν − DνDµ = ∂µ∂ν − ∂ν∂µ︸ ︷︷ ︸0+i τc Fcµν . (2)The commutation of the total derivatives measures the shift of the gauge fieldaround a loop. A non-zero shift indicates an implict twist in the gauge configuration.In general relativity, the torsion-free condition is written as▽a ▽b f −▽b ▽a f = 0 , (3)where f is a scalar field. By analogy, we can interpret the right hand side of Eq. (2)geometrically as torsion. It is hoped that this torsion can be gauged away by anappropriate choice of transformation. Figure 4 shows heuristically how torsion canbe canceled with a carefully chosen topology. In general relativity, the mass of thefield is related to curvature as inm ∼1r, (4)so that the curvature of the edge of the folded surface is interpreted as the ef-fective gluon mass. This interpretation is slightly different from that given in theFig. 4. The torsion vector from the loop on the upper surface is in the oppositedirection of that on the lower surface so that torsion is effectively canceled in thistopology.Fig. 5. A pion loop (shown as a wormhole) on a surface of torsion-free topology.268 FIZIKA B 13 (2004) 1, 265–270tang: chiral perturbation on the lightfrontoriginal talk at NAPP 2003 where the curvature was interpreted as the mass ofthe constituent quark. Figure 3b suggests the possibility that the pion loop can beconstructed inside the gauge surface as shown in Fig. 5.4. Global torsion-free QCD topologyIt is reasonable to assume that the presence of constituent quarks must affectthe geometry of the gauge field. A good guess is that the global topology of ameson has two edges and that of a baryon has three. Although torsion (the non-Abelian part of the gauge theory) is gauged away by a suitable transform, it survivesas a twist on the transformed field configuration. Figure 6 shows a sketch of thetopological structures of the gauge fields of a meson and a baryon. The twist causes(a)(b)1221132132Fig. 6. The topological structures of the gauge fields of a meson and a baryon. Theidentifications of the boundaries are labeled.the topology of the gauge field to have an unusual boundary condition similar toa Möbius strip. The topology of the baryon has been called the 3-Möbius stripand the “triniton” in previous presentations. In this picture, chiral perturbation onthe lightfront is represented by a sum of these graphs of different geni as shown inFig. 7.genus 1 genus 2 genus 3+ +Fig. 7. Chiral perturbation on the lightfront is reduced to a sum of graphs of variousgeni.FIZIKA B 13 (2004) 1, 265–270 269tang: chiral perturbation on the lightfront5. ConclusionThe present work illustrates the concept of topological QCD in pictures. Thisresearch work is in progress. Many claims made in this presentation are based onintuition but are now being subjected to the rigor of mathematics. The contribu-tions of the constituent quarks have not yet been included in the pictures of thiswork. These deficiencies will be amended in a future publication. Some colleagueshave also pointed out that this research should aim at producing relevant predic-tions. It suffices to say that so far topological QCD is suggestive and is promisingas a candidate for the analysis of the structure of the hadrons. Currently, a searchis underway to find a suitable transform that maps the non-Abelian gauge fieldconfiguration to the torsion-free twisted n-Möbius surface. Then the mathematicsof the path integrals over the surfaces of the twisted tori needs to be formulated.And lastly, it is hoped that this model can explain traditionally difficult theoreticalproblems such as confinement and asymptotic freedom. It is also hoped that newtechnology of non-perturbative QCD calculations will be discovered in the process.References[1] A. Tang, hep-th/0309206.KIRALNA SMETNJA NA VALNOJ FRONTIObjašnjava se slikama novo geometrijsko tumačenje kiralne teorije smetnje zasno-vano na topološkom modelu QCDa.270 FIZIKA B 13 (2004) 1, 265–270

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