Yue-Liang Wu

State Key Laboratory of Theoretical Physics (SKLTP)Kavli Institute for Theoretical Physics China （ KITPC）Institute of Theoretical Physics, Chinese Academy of

Sciences2012.04.12　　　　　

Quantum Structure of Field Theory

Symmetry & Quantum Field Theory

Symmetry has played an important role in elementary particle physics

All known basic forces of nature: electromagnetic, weak, strong & gravitational forces, are governed by

U(1)_Y x SU(2)_L x SU(3)_c x SO(1,3) Which has been found to be successfully described

by quantum field theories (QFTs)

Why Quantum Field Theory So Successful

Folk’s theorem by Weinberg: Any quantum theory that at sufficiently low

energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory.

Indication: existence in any case a characterizing energy scale (CES) Mc

So that at sufficiently low energy gets meaning: E << Mc QFTs

Why Quantum Field Theory So Successful

Renormalization group by Wilson/Gell-Mann & Low Allow to deal with physical phenomena at any

interesting energy scale by integrating out the physics at higher energy scales.

Allow to define the renormalized theory at any interesting renormalization scale .

Implication: Existence of sliding energy scale(SES) μs which is not related to masses of particles.

Physical effects above the SES μs are integrated in the renormalized couplings and fields.

How to Avoid Divergence

QFTs cannot be defined by a straightforward perturbative expansion due to the presence of ultraviolet divergences.

Regularization: Modifying the behavior of field theory at very large momentum so Feynman diagrams become well-defined quantities

String/superstring: Underlying theory might not be a quantum theory of fields, it could be something else.

Regularization Schemes Cut-off regularization Keeping divergent behavior, spoiling gauge symmetry &

translational/rotational symmetries Pauli-Villars regularization Modifying propagators, destroying non-abelian gauge symmetry Dimensional regularization: analytic continuation in dimension Gauge invariance, widely used for practical calculations

Gamma_5 problem: questionable to chiral theoryDimension problem: unsuitable for super-symmetric theoryDivergent behavior: losing quadratic behavior (incorrect gap eq.)

All the regularizations have their advantages and shortcomings

Criteria of Consistent Regularization

(i) The regularization is rigorous: It can maintain the basic symmetry principles

in the original theory, such as: gauge invariance, Lorentz invariance and translational invariance

(ii) The regularization is general: It can be applied to both underlying

renormalizable QFTs (such as QCD) and effective QFTs (like the gauged Nambu-Jona-Lasinio model and chiral perturbation theory).

Criteria of Consistent Regularization

(iii) The regularization is also essential: It can lead to the well-defined Feynman

diagrams with maintaining the initial divergent behavior of integrals, so that the regularized theory only needs to make an infinity-free renormalization.

(iv) The regularization must be simple: It can provide practical calculations.

Loop Regularization (LORE) Method with String Mode Regulators

Yue-Liang Wu, SYMMETRY PRINCIPLE PRESERVING AND INFINITY FREE REGULARIZATION AND RENORMALIZATION OF QUANTUM FIELD THEORIES AND THE MASS GAP. Int.J.Mod.Phys.A18:2003, 5363-5420.

Yue-Liang Wu, SYMMETRY PRESERVING LOOP REGULARIZATION AND RENORMALIZATION OF QFTS. Mod.Phys.Lett.A19:2004, 2191-2204.

J.~W.~Cui and Y.~L.~Wu, Int. J. Mod. Phys. A 23, 2861 (2008) J.~W.~Cui, Y.~Tang and Y.~L.~Wu, Phys. Rev. D 79, 125008 (2009) Y.~L.~Ma and Y.~L.~Wu, Int. J. Mod. Phys. A21, 6383 (2006) Y.~L.~Ma and Y.~L.~Wu, Phys. Lett. B 647, 427 (2007) J.W. Cui, Y.L. Ma and Y.L. Wu, Phys.Rev. D 84, 025020 (2011) Y.~B.~Dai and Y.~L.~Wu, Eur. Phys. J. C 39 (2004) S1 Y.~Tang and Y.~L.~Wu, Commun. Theor. Phys. 54, 1040 (2010) Y.~Tang and Y.~L.~Wu, JHEP 1111, 073 (2011), arXiv:1109.4001 [hep-

ph]. Y.~Tang and Y.~L.~Wu, arXiv:1012.0626 [hep-ph]. D. Huang and Y.L. Wu, arXiv:1108.3603

Irreducible Loop Integrals (ILIs)

Loop Regularization (LORE) MethodSimple Prescription: in ILIs, make the following replacement

With the conditions

So that

Gauge Invariant Consistency Conditions