Gerard ’t Hooft Spinoza Institute Yukawa – Tomonaga Workshop, Kyoto, December 11, 2006 Utrecht University

What to do with the Infinities in Quantum Field theory?

The principle of renormalization (Kramers) 1.We do not know the “bare masses” or “bare coupling strengths” 2. We can only observe up to an energy limit: 3. This implies a limit in our resolution in space and time: 4. For our descriptions of what we can observe, a theory with a cut-off in space suffices, e.g. a lattice in space: 5.The bare masses and coupling strengths may well depend on. 6.The fact that these dependences diverge as has no physical consequences.

The principle of renormalization (Kramers) 1.We do not know the “bare masses” or “bare coupling strengths” 2. We can only observe up to an energy limit: 3. This implies a limit in our resolution in space and time: 4. For our descriptions of what we can observe, a theory with a cut-off in space suffices, e.g. a lattice in space: 5.The bare masses and coupling strengths may well depend on. 6.The fact that these dependences diverge as has no physical consequences.

One may suspect that this phenomenon is an artifact of the perturbation expansion, but In most cases where one can check, it persists beyond this expansion (or might get worse!) And for many theories (such as QED), perturbation theory is all that matters, in practice.

How do calculations ? 1.Cut-off in momentum space: or, after Wick rotation to Euclidean spacetime: ( This is Lorentz-covariant ! ) Question: are these two cut-offs equivalent ? 2. Easier for calculations: modify your propagators: 3. Bare masses and coupling strengths now depend on Λ : but diverge as

For theories such as, this is good enough. The following prescription works: 1. Compute the first quantum correction for a mass m : x Add a counter term:

Coupling constant: It goes the same way at higher orders

How does this go for Q uantum E lectro D ynamics ? One can change the photon propagator analogously: But, if you do this for the electron, it destroys gauge-invariance: This is allowed because the photon is electrically neutral !

?? But the following is gauge-invariant: _ Pauli - Villars

How does Renormalization work in the Yang-Mills theory ? The Yang-Mills theory : Is a mass term allowed ? It’s not gauge-invariant, but so what ?? the term goes away at high energy !! When is a theory renormalizable ??

You have to understand the rules ! The most restrictive demands are: - Unitarity - Causality Cutting rules dispersion relations Do the Book keeping ! Veltman:

S M. Veltman:

In “ordinary” theories such as this automatically gives unitarity, if: all propagators come with positive masses and positive overall signs !!

But the propagators of Yang-Mills theories are not of this type ! “Massive Yang-Mills theory” now MUST have the propagator: Because the numerator is local in x-space, and has 3 positive eigenvalues: the 3 helicities of a spin 1 particle. But this diverges MUCH worse at than in any renormalizable theory ! We want to replace it by But the second part does NOT obey the cutting rules

Define “propagators”: The Discovery of Ghosts (feynman 1963, veltman 1968)

After a couple of manipulations, they claim: Tree But a real disaster comes at higher orders:

To relate the two different propagators, ABSOLUTE GAUGE INVARIANCE is needed. “Massive Yang-Mills” is not completely gauge-invariant. The “gauge ghosts” would not couple in the UV limit, but they are UNPHYSICAL, and they do couple at low energies. Therefore, a Gauge-invariant UV limit is not enough !! But a completely gauge-invariant Yang-Mills theory where the vector particles have mass DOES exist: it is the Brout-Englert-Higgs theory. And ONLY THAT theory may be renormalizable !!

Massive case : Exact gauge transformations vector Higgs F.P. ghost Higgs ghost

The Feynman/Veltman ghost is a combination of two ! Miraculously, these two different ghosts obey exactly the same Feynman rules at the one-loop level, if the Higgs mass is sent to infinity ! So Veltman’s claim for the one-loop diagrams in “pure massive gauge Yang-Mills theory” is correct.

We saw that the Brout-Englert-Higgs theory is formally renormalizable, since unitary Feynman rules are formally equivalent to renormalizable ones. However, we have seen that introducing cut-offs might break gauge-invariance. To verify that the theory remains renormalizable at higher orders, it is imperative to use gauge-invariant cut-offs All cut-off procedures introduced so-far are not gauge-invariant. How to find a gauge-invariant cut-off ? (Numerous dead alleys... )

How to find a gauge-invariant cut-off ? 1.Pauli-Villars works fine for the fermions ! Advanced formulation: choose  ii i m e   How do we cut-off (“regularize”) the boson lines ? N.B.: Unless chiral symmetry is asked for !

Step 1: Observe that (non-chiral) YM theory is gauge-invariant in any number of dimensions: Consider a diagram contributing to an amplitude. The momenta of the external lines are in 4 dimensions. But the internal lines may have components in dimensions. The most natural thing to try first: take the one-loop diagrams. Add one extra dimension for the internal lines. Then: Give a FIXED VALUE á la Pauli-Villars !

That this is gauge-invariant is NOT obvious, but can be proven by checking that gauge-invariance does not require integration over all values of the internal momentum But how do we do the diagrams with more, overlapping loops ?? Try 6, 7, or more internal dimensions ? That is NOT gauge-invariant ! The last straw... Note that the general formula for any number of dimensions is

Poles at: This expression keeps everything finite as long as integer

not only are these expressions finite, All formal proofs of gauge-invariance, unitarity and causality work without a glitch for ALL values of n, except when or are needed ; these are special for n = 4. in the limit, counter terms of the form are needed, where L is the number of overlapping loops.

It had to be proven that this procedure yields finite, gauge-invariant, unitary and causal amplitudes at all orders of the perturbation expansion. It does, although special precautions are needed when chiral symmetry is required. Anomalous currents may not be coupled to gauge fields ! Thus, one has to demand anomaly cancellation in the Standard Model. All our proofs were initially formulated without the use of B ecchi – R ouet – S tora symmetry :

The formal proofs: first the hard way... Combinatorics of Feynman diagrams. An outline: == = = = = = = = This requires the Jacobi identity for Lie-groups:

New identities for gauge field diagrams = On mass shell, transverse or longitudinal On mass shell, longitudinal

Slavnov - Taylor: do it off mass shell... = Off mass shell, transverse or longitudinal off mass shell, longitudinal =

Unitarity in the renormalizable gauge follows:

Becchi-Rouet-Stora (1974): In every gauge theory, after gauge fixing, “There’s a supersymmetry” Fixes the gauge BRST Quantization !

Thank you THE END