Cours gabriele veneziano

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Particules Élémentaires, Gravitation et CosmologieAnnée 2004-2005Interactions fortes et chromodynamique quantique I:Aspects perturbatifsCours VIII: 19 avril 2005Symmetries & Anomalies1. Global vs. local symmetries2. Classical global symmetries, conserved currents3. Explicit breaking, accidental symmetries,anomalies4. Explicit vs. spontaneous symmetry breaking5. CP & U(1) problems in QCD19 avril 2005 1G. Veneziano, Cours no. 81. Global vs. local symmetries In QFT we have to make the important distinctionbetween global (rigid) and local symmetries Example of former is Lorentz(Poincaré) invariance Example of latter are the gauge symmetries(NB. General Relativity can be seen as a gaugetheory with G = local Poincaré-group: x -->x’(x)) Recall: local symmetries mean that some d.o.f. areunphysical => they are sacred (have to be broken«carefully» ) Global symmetries are not sacred: we are allowedto break them explicitly19 avril 2005 2G. Veneziano, Cours no. 8    Occasionally symmetries of S are broken at thecl quantum level (i.e. in S ). One talks about aneff«anomaly» We have seen an example all along: Classical-CDcontains no dimensionful parameter, but QCDcontains a scale, : one talks indeed about a scale-invariance (or trace) anomaly, anomalousdimensions etc. Anomalies are acceptable in global symmetries Anomalies unacceptable in local symmetries (animportant constraint on the standard model andits possible extensions)19 avril ...

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• Publié par: Jicel
• Nombre de lectures: 16
• Langue: English

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osmologie

ParticulesÉlémentaires, Gravitatio   n et   C Année 2004-2005 Interactions fortes et chromodynamique quantiqu Aspects perturbatifs

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CoursVIII: 19 avril 2005

Symmetries& Anomalies

Global vs. local symmetries

Classical global symmetries, conserved curren

Explicit breaking, accidental symmetries, anomalies

Explicit vs. spontaneous symmetry breaking

CP & U(1) problems in QCD

19 avril 2005

G. Veneziano, Cours no. 8

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1. Global vs. locasly mmetries In QFT we have to make the important distincti between global (rigid) and local symmetries

Example of former is Lorentz(Poincaré) invaria

Example of latter are the gauge symmetries (NB. General Relativity can be seen as a gauge theory with G   = local Poincaré-group: x -- ( > x) x)’

Recall: local symmetries mean that some d.o.f. unphysical => they are sacred (have to be bro   carefully   ») «

Global symmetries are not sacred: we are allo to break them explicitly

19 avril 2005

G. Veneziano, Cours no. 8

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Occasionally symmetries o S f c   l are broken at the quantum level (i.e. i S n e   ff ). One talks about an «anomaly»

We have seen an example all along: Classical- contains no dimensionful parameter, but QCD contains a scale Λ , : one talks indeed about a sca invariance (or trace) anomaly, anomalous dimensions etc. Anomalies are a   cceptable in glob   a s l ymmetries Anomalies u   nacceptable in loc   a s l ymmetries (an important constraint on the standard model an its possible extensions)

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2.1Classicalglobalsymmetries

Recall (again from lecture 1) the general definition of a gauge theory (without scalar fields for simplicity) Besides specifying the gauge gro G upwe had to assign all our l.h. fermionsto somereps. of G (the r.h. antiparticles will be automatically in the c.c. reps.) Suppose that there are 1 Nl.h. fermionsin the rep. r 1 , N 2 in r 2 etc. and, for the moment, let’s not give them mas The only fermionic terms i S n cl will be of the type

(actually   ψ∗  in the notationof lect. 1)

19 avril 2005

G. Veneziano, Cours no. 8

G. Veneziano, Cours no. 8

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A globa  l phase rotation of any fermion leaves the lagrangia invariant. Even better, any transformation of the form:

leaves L class unchanged. => We have identified a large global-symmetry group:   G global = U(N 1 )xU(N 2 )xU(N 3 )x… What is G global for QCD (w/ gauge group SU(N)) with f N massless quarks? It has f Nl.h. quarks in the F-rep. and f   iNn its c.c. rep. F*. Thus the classical symmetry is G= U(N f ) F xU(N f ) F* (more traditionally called U( f )N L xU(N f ) R )

19 avril 2005

G. Veneziano, Cours no. 8

G. Veneziano, Cours no. 8

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2.2 ConservedCurrents

A general theorem by (Mme)Noether tells us that we ca associate to any symmetry a conserved curre (i n ) µ t   :J

In our casethese currents are easilyidentified. In QCD:

R) = and similarly for J ( µ J (F*) µ  

whereT a ff’  is hermitian,  

In Dirac notation they are:

Let us split the N f 2  matrices T a ff’ into the (N f 2 -1) traceless ones and the unit matrix, corresponding to the decomposi

G= U(N f ) L xU(N f ) R = [SU(N f ) L x U(1) L ]x[SU(N f ) R x U(1) R ]  

19 avril 2005

G. Veneziano, Cours no. 8

G. Veneziano, Cours no. 8

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3.1Explicit classicalsymmetrybreakingand accidentalsymmetries

What happens when we add (quark) masses? Recallthat massterms involvetwo fermionsof the same handness (and cc reps.). In QCD the most general mas term (using reality of the action) iss (um over f, f’ !)

In general this terms breaks badly the global symmetr leaving just a small unbroken subgroup (see below) If, however,m ff’ = m δ ff ’ it is quite clear that a full U( f N) V subgroup of G= U( f  N) F xU(N f ) F* is preserved (V = + )U This is nothing but the isospin symmetry f   (=N2) or the = Gell-Mann-Ne’emaSn U(3) flavour symmetry for N f  3.

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These symmetries looked very fundamenitna tl he sixties. At first sight their presencein QCD seemsvery unlikely. Why should f m f’ = m δ ff ? One can show, however, that even i f f/ m f ’   ≠ 1, the U(N f ) V symmetry becomes very good a f s - -m> 0 ( f m/ Λ   -->0) Thus the existence of an approximat U e (   N f ) flavour symmetry is a mer c e o   nsequence of having f   lNight quarks on the scale of QCD, Λ   . One talks about accidental (i.e. not at all fundamental) symmetries of the strong interactions, i.e. of symmetri which are there because of the terms we can possibly write down in the lagrangian, but do not require any precise fine-tuning of parameters.

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3 . 2  Explicit quantums ymmetryb reaking: anomalies In the late 60’sAdler Bell andJackiw found a very puzzling result, knowntodayas the ABJ anomaly: Even in the absence of masses some global classical symmetries are broken at the quantum (loop) . l   e T v h e e l effect is dueto a «triangle»graphwith fermionscirculatingin the loop.The onerelevantto QCD is as shownin the figure (the original one was in QED and it’s crucial to account π f 0 o-r> 2 γ )

fermion (a) J µ

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the gluonis blind to flavour this diagram gluon i S s i   n p c ro e  portionalt o Tr(T a ff’ ) anddoesnot contribute to SU(N) currents. It also cancelsif we take the right combinationof U(1) L xU(1) R gluon ( c i. al e l . e  b d a  U r ( y 1 o ) V n   ) ,    n c u or m re b s e p r. o B  n u d t  i ( n q gt +  o q  ( * L q )- - ( q q * + ) L q + * ( R )  q i - s  q n * R ) o longerconserved(e.g.creationof (q+q* L )or   (q+q* R )   from the vacuum) G. Veneziano, Cours no. 8  9

The anomaly is best defined as a symmetry c o la f s   s Swhich is not a symmetry of the effective action e   f S f (the one that includes quantum corrections). Under a A U   (tr1a)nsformation:

the effective actionchangesby:

At the levelof the currents it canbe expressedas a quantum-non-conservation of the classicall conserved A U( current: The symmetry and the conservation of the corresponding current are apparently lost…but the story is not yet over. 19 avril 2005  G. Veneziano, Cours no. 8  10

4.1 Explicit vs. spontaneou s  ymmetryb reaking

• Another very basic distinctionis that of explicit breaking  (what we have discussed so far both at the classical and a the quantum level) and spontaneous breaking

•In the former casethe (effective) actionis not invariantand the current is not conserved.

•In the latter case the opposite is true. One should not tal about symmetry breaking but of a hidden (or secret) symmetry.The «   breaking   »is due tothe non-invariance of the ground state under the spontaneouslyb roken symmetry transformations.

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