Symmetries of the Strong Interaction

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Symmetries of the strong interaction

H. Leutwyler

University of Bern

DFG-Graduiertenkolleg ”Quanten- und Gravitationsfelder” Jena, 19. Mai 2009

H. Leutwyler – Bern Symmetries of the strong interaction – p. 1/33 QCD with 3 massless quarks

As far as the strong interaction goes, the only difference between the quark ﬂavours u,d,...,t is the mass

mu, md, ms happen to be small For massless fermions, the right- and left-handed components lead a life of their own

Fictitious world with mu =md =ms =0: QCD acquires an exact chiral symmetry

No distinction between uL,dL,sL, nor between uR,dR,sR Hamiltonian is invariant under SU(3) SU(3) ⇒ L× R

H. Leutwyler – Bern Symmetries of the strong interaction – p. 2/33 Chiral symmetry of QCD is hidden

Chiral symmetry is spontaneously broken: Ground state is not symmetric under SU(3) SU(3) L× R symmetric only under the subgroup SU(3)V = SU(3)L+R

Mesons and baryons form degenerate SU(3)V multiplets ⇒ and the lowest multiplet is massless:

Mπ± =Mπ0 =MK± =MK0 =MK¯ 0 =Mη =0 Goldstone bosons of the hidden symmetry

H. Leutwyler – Bern Symmetries of the strong interaction – p. 3/33 Spontane Magnetisierung

Heisenbergmodell eines Ferromagneten Gitter von Teilchen, Wechselwirkung zwischen Spins Hamiltonoperator ist drehinvariant Im Zustand mit der tiefsten Energie zeigen alle Spins in dieselbe Richtung: Spontane Magnetisierung Grundzustand nicht drehinvariant ⇒ Symmetrie gegenüber Drehungen gar nicht sichtbar Symmetrie ist versteckt, spontan gebrochen Goldstonebosonen in diesem Fall: "Magnonen", "Spinwellen" Haben keine Energielücke: ω 0 für λ → → ∞ Nambu realisierte, dass auch in der Teilchenphysik Symmetrien spontan zusammenbrechen können

H. Leutwyler – Bern Symmetries of the strong interaction – p. 4/33 Chiral symmetry

H. Leutwyler – Bern Symmetries of the strong interaction – p. 5/33 Chiral symmetry

QCD with three massless quarks is invariant under G = SU(3) SU(3) L× R SU(3) has 8 parameters Symmetry under Lie group with 16 parameters ⇒ 16 conserved “charges” ⇒ V V Q1,... ,Q8 (vector currents, R+L) A A Q1,... ,Q8 (axial currents, R-L) Charges commute with the Hamiltonian: V A [Qi ,H0]=0 [Qi ,H0] = 0 Representation of charges in terms of conserved µ µ currents, such as uγ d and uγ γ5d: V 3 0 A 3 0 Q1 = d x uγ d, Q1 = d x uγ γ5d H. Leutwyler – Bern Z Z Symmetries of the strong interaction – p. 5/33 Symmetry properties of the ground state

Vafa and Witten 1984: state of lowest energy is invariant under the vector charges QV 0 = 0 i | i Axial charges ? QA 0 =? i | i

H. Leutwyler – Bern Symmetries of the strong interaction – p. 6/33 Two alternatives for axial charges

A 0 = 0 Qi | i Wigner-Weyl realization of G ground state is symmetric 0 0 = 0 h | qR qL | i ordinary symmetry spectrum contains parity partners degenerate multiplets of G

A 0 = 0 Qi | i 6 Nambu-Goldstone realization of G ground state is asymmetric 0 0 = 0 h | qR qL | i 6 “order parameter” spontaneously broken symmetry spectrum contains Goldstone bosons

degenerate multiplets of SU(3)V ⊂ G = G SU(3)R × SU(3)L

H. Leutwyler – Bern Symmetries of the strong interaction – p. 7/33 Goldstone bosons

QCD chooses the Nambu-Goldstone mode: QA 0 = 0 i | i 6 Immediate consequence: H QA 0 = QA H 0 = 0 for i = 1,..., 8 0 i | i i 0 | i Spectrum must contain 8 states QA 0 ,...,QA 0 ⇒ 1 | i 8 | i with E = 0, spin 0, negative parity, octet of SU(3)V "Goldstone bosons" Indeed, the 8 lightest mesons do have these quantum numbers: + 0 + 0 0 π , π , π−,K ,K , K¯ ,K−,η All other one-particle states must form degenerate

multiplets of SU(3)V

H. Leutwyler – Bern Symmetries of the strong interaction – p. 8/33 Side remark: mathematics used is slippery

Argument given for the occurrence of Goldstone bosons is not quite water tight: 0 QA QA 0 = d3xd3y 0 A0(x) A0(y) 0 h | i k | i Z h | i k | i 0 A0(x) A0(y) 0 only depends on ~x ~y h | i k | i − 0 QA QA 0 is proportional to the volume of the ⇒ h | i k | i universe, QA 0 = | i | i| ∞ Rigorous proof of Goldstone theorem given later

H. Leutwyler – Bern Symmetries of the strong interaction – p. 9/33 Quark condensate

uR uL dR uL sR uL

qR qL =  uR dL dR dL sR dL 

 uR sL dR sL sR sL    Transforms like (3, 3)¯ under SU(3) SU(3) L × R If the ground state were symmetric, the matrix 0 qR qL 0 would have to vanish, because it singles out ha direction| | i in ﬂavour space

0 qR qL 0 is referred to as the “quark condensate”, quantitativeh | | i measure of the strength of spontaneous symmetry breaking, “order parameter” 0 q q 0 is the analog of magnetization h | R L | i

H. Leutwyler – Bern Symmetries of the strong interaction – p. 10/33 Quark condensate

Ground state is invariant under SU(3)V 0 q q 0 is proportional to unit matrix ⇒ h | R L | i 0 u u 0 = 0 d d 0 = 0 s s 0 h | R L | i h | R L | i h | R L | i 0 u d 0 = . . . = 0 h | R L | i

H. Leutwyler – Bern Symmetries of the strong interaction – p. 11/33 Quark masses

Real world = paradise 6 In reality, the multiplets are split and the lightest mesons are not massless m , m , m = 0 u d s 6 Quark masses break chiral symmetry allow the left to talk to the right Chiral symmetry broken in two ways: ⇒ spontaneously 0 q q 0 = 0 h | R L | i 6 explicitly m , m , m = 0 u d s 6

H. Leutwyler – Bern Symmetries of the strong interaction – p. 12/33 Quark masses

Only the diagonal vector currents are strictly conserved in QCD: Nu, Nd, Ns, Nc, Nb, Nt baryon number, electric charge, strangeness, charm,→ ...

It so happens that mu, md, ms are small H has an approximate SU(3) SU(3) symmetry ⇒ QCD L× R Masses of the light quarks enter the Hamiltonian via

HQCD = H0 + H1 3 H = d x mu uu + md dd + ms ss 1 Z { }

H0 describes u,d,s as massless, c,b,t as massive H is invariant under SU(3) SU(3) 0 L× R

H. Leutwyler – Bern Symmetries of the strong interaction – p. 13/33 Pattern of light quark masses

With the discovery of QCD, the mass of the quarks became an unambiguous concept: quark masses occur in the Hamiltonian of the theory. First crude estimate within QCD relied on a model for the wave functions of π, K, ρ, based on SU(6) (spin-ﬂavour-symmetry) 2 1 FπMπ 2(mu + md) = 5 MeV, ms 135 MeV 3FρMρ ≃ ≃ “Is the quark mass as small as 5 MeV ?” 1974 Not very different from the pattern found within the Nambu-Jona-Lasinio model (1961) or the one obtained from sum rules by Okubo (1969)

H. Leutwyler – Bern Symmetries of the strong interaction – p. 14/33 Notion of quark mass

Quarks do not occur in isolation Mass not directly observable like mass of electron Quark mass is a theoretical notion ⇒ Bare masses in the Hamiltonian cannot serve to deﬁne mu, md, ms, need to be renormalized Quark masses depend on renormalization convention Generally accepted convention: compare values in MS scheme (based on dimensional regularization) In this scheme, the quark masses depend on the running scale µ, much like the coupling constant gs Estimates quoted in the 2008 edition of the PDG tables: 1(m + m ) = 2.5 5 MeV, m = 70 130 MeV 2 u d ÷ s ÷ refer to µ=2 GeV

H. Leutwyler – Bern Symmetries of the strong interaction – p. 15/33 Pattern of light quark masses

Difference between mu and md ? e.m. self energy: proton > neutron M > M cannot be due to the e.m. interaction ⇒ n p Mn > Mp must be due to md > mu Isospin is not a symmetry of the strong interaction ! ⇒ Gasser & L. 1975 In fact a very strong breaking appears to be needed: m 2.5 MeV, m 5 MeV PDG 2008 u ≃ d ≃

H. Leutwyler – Bern Symmetries of the strong interaction – p. 16/33 Crude picture for mu,md,ms

m 2.5 MeV, m 5 MeV, m 100 MeV u ≃ d ≃ s ≃ mu and md are very different mu and md are small compared to ms “constituent masses” / Lagrangian of QCD ∈

H. Leutwyler – Bern Symmetries of the strong interaction – p. 17/33 Approximate symmetries are natural in QCD

Why is isospin such a good quantum number ? (a) Dimensional transmutation, logarithmic divergences of perturbation theory physics ∈ QCD has an intrinsic scale ⇒ (b) m m scale of QCD, not m + m d − u ≪ ≪ u d Why is the eightfold way a decent approximate symmetry ? m m scale of QCD s − u ≪ Isospin is an even better symmetry because m m m m d − u ≪ s − u m m m , m , m scale of QCD u ≪ s ⇒ u d s ≪ Masses of the light quarks represent perturbations ⇒ Can neglect these in a ﬁrst approximation

H. Leutwyler – Bern Symmetries of the strong interaction – p. 18/33 Quark masses as perturbations

HQCD = H0 + H1 3 H = d x mu uu + md dd + ms ss 1 Z { } Expansion in Perturbation series

powers of mu, md, ms ⇐⇒ in powers of H1

H0 treats π,K,η as massless, H1 gives them a mass

H. Leutwyler – Bern Symmetries of the strong interaction – p. 19/33 Magnitude of the perturbations due to mu,md,ms

0 dγµγ u π+ = i pµ√2 F h | 5 | i π 0 sγµγ u K+ = i pµ√2 F h | 5 | i K Value of F , F known from π+ µ+ν, K+ µ+ν π K → → (and CKM matrix elements Vud, Vus) + + Difference between π and K comes from ms > md FK Observed ratio: = 1.19 0.01 Fπ ± Branching fraction of K → πeν changed by > 3 σ in 2004 ! 1.22 → 1.19 m m generates correction of order 20% ⇒ s − d m , m m correction mainly comes from m u d ≪ s ⇒ s effects from mu, md are tiny

H. Leutwyler – Bern Symmetries of the strong interaction – p. 20/33 Gell-Mann-Oakes-Renner formula

First order perturbation theory yields: 1 M 2 = (m + m ) 0 uu 0 π u d 2 × |h | | i| × Fπ explicit⇑ spontaneous ⇑ Gell-Mann, Oakes & Renner 1968. At that time, the quarks very still considered with suspicion ⇒ formula does not appear like this in the paper Postpone derivation (involves Goldstone theorem)

H. Leutwyler – Bern Symmetries of the strong interaction – p. 21/33 Consequences of GMOR formula

2 2 Mπ = (mu + md) B + O(m ) The energy gap of QCD is small because mu, md ⇒ happen to be small

H. Leutwyler – Bern Symmetries of the strong interaction – p. 22/33 Consequences of GMOR formula

2 2 Mπ = (mu + md) B + O(m ) The energy gap of QCD is small because mu, md ⇒ happen to be small 2 2 MK+ = (mu + ms) B + O(m ) 2 2 MK0 = (md + ms) B + O(m ) M 2 is much larger than M 2, because m happens to ⇒ K π s be large compared to mu, md

H. Leutwyler – Bern Symmetries of the strong interaction – p. 22/33 Consequences of GMOR formula

π and K belong to an octet of SU(3)V Goldstone boson masses strongly violate SU(3) ⇒ V

H. Leutwyler – Bern Symmetries of the strong interaction – p. 22/33 Check of SU(3)V

Goldstone boson masses strongly break SU(3)V

Nevertheless, SU(3)V is a decent approximate symmetry Check: ﬁrst order perturbation theory also yields 2 1 2 Mη = 3 (mu + md + 4ms)B + O(m ) M 2 4M 2 + 3M 2 = O(m2) ⇒ π − K η Gell-Mann-Okubo formula for M 2 √

H. Leutwyler – Bern Symmetries of the strong interaction – p. 23/33 Lattice

Simulations of QCD on a lattice now reach sufﬁciently small lattice spacings, sufﬁciently small quark masses to make contact with physics GMOR formula can now be checked on the lattice:

determine Mπ as a function of mu =md =m

0.08 2 ﬁt to 5 points (amPS) ﬁt to 4 points 0.08 ∼676 2 mπ MeV 0.06 (amπ ) 0.06

0.04 0.04 484 381 0.02 0.02 294

(aµ) 0 0 0 0.01 0.02 0.03 am 0 0.004 0.008 0.012 0.016 Lüscher, Lattice conference 2005 ETM collaboration, hep-lat/0701012

H. Leutwyler – Bern Symmetries of the strong interaction – p. 24/33 Lattice

Quality of data is impressive No quenching, quark masses are sufﬁciently light Legitimate to use χPT for the extrapolation to the ⇒ physical values of mu, md 2 Proportionality of Mπ to the quark mass appears to hold out to values of mu, md that are an order of magnitude larger than in nature Main limitation: systematic uncertainties in particular: N = 2 N = 3 f → f

H. Leutwyler – Bern Symmetries of the strong interaction – p. 25/33 Summary

H has an approximate symmetry: G=SU(3) SU(3) QCD L× R 0 approximately symmetric only under SU(3) G | i V ⊂ World we live in is close to the paradise Light quark masses amount to a small perturbation

H. Leutwyler – Bern Symmetries of the strong interaction – p. 26/33 Summary

Only the subgroup SU(3)V G is an approximate ⇒ symmetry of spectrum and⊂ matrix elements “Eightfold way”, u d s ↔ ↔

H. Leutwyler – Bern Symmetries of the strong interaction – p. 26/33 Summary

Only the subgroup SU(3)V G is an approximate ⇒ symmetry of spectrum and⊂ matrix elements “Eightfold way”, u d s ↔ ↔ mu, md are particularly small SU(2) SU(2) is a nearly exact symmetry of H ⇒ L× R QCD Expansion in powers of m , m converges very rapidly ⇒ u d H. Leutwyler – Bern Symmetries of the strong interaction – p. 26/33 Appendix: Proof of Goldstone theorem

0 q q 0 = 0 massless particles h | R L | i 6 ⇒ ∃ 3 0 Q = d xuγ γ5d Z [Q, dγ u] = uu dd 5 − − First determine the form of the 2-point-function F µ(x y) 0 u(x)γµγ d(x) d(y)γ u(y) 0 − ≡ h | 5 5 | i Lorentz invariance F µ(z) = zµf(z2) ⇒ Chiral symmetry ∂ F µ(z) = 0 ⇒ µ zµ F µ(z) = constant (for z2 = 0) ⇒ z4 × 6 Symmetry ﬁxes the 2-point-function up to a constant ⇒

H. Leutwyler – Bern Symmetries of the strong interaction – p. 27/33 Proof of Goldstone theorem ctd.

Compare Källen–Lehmann representation: 0 u(x)γµγ d(x) d(y)γ u(y) 0 h | 5 5 | i 3 4 µ 2 ip(x y) = (2π)− d p p ρ(p )e− − Z ∞ = ds ρ(s)∂µ∆+(x y,s) Z0 − ∆+(z,s): positive frequency part of propagator

+ i 4 0 2 ipz ∆ (z,s) = d p θ(p ) δ(p s) e− (2π)3 Z − Massless propagator:

+ 1 + zµ ∆ (z, 0) = ∂µ∆ (z, 0) = constant 4πiz2 ⇒ z4 ×

H. Leutwyler – Bern Symmetries of the strong interaction – p. 28/33 Proof of Goldstone theorem ctd.

Result: 0 u(x)γµγ d(x) d(y)γ u(y) 0 = C∂µ∆+(z, 0) h | 5 5 | i Only massless intermedate states contribute: ⇒ ρ(s) = Cδ(s) Why only massless intermediate states ? 0 u(x)γµγ d(x) d(y)γ u(y) 0 h | 5 5 | i = 0 uγµγ d n n dγ u 0 e ipn(x y) nh | 5 | ih | 5 | i − − n Pdγ u 0 = 0 only if n has spin 0 h | 5 | i 6 h | If n has spin 0 0 u(x)γµγ d(x) n pµ e ipx | i ⇒ h | 5 | i ∝ − ∂ (uγµγ d) = 0 p2 = 0 µ 5 ⇒ Either massless particles or C = 0 ⇒ ∃

H. Leutwyler – Bern Symmetries of the strong interaction – p. 29/33 Proof of Goldstone theorem ctd.

Claim: 0 q q 0 = 0 C = 0 h | R L| i 6 ⇒ 6 Interchange the two operators: 0 d(y)γ u(y)u(x)γµγ d(x) 0 = C ∂µ∆ (z) h | 5 5 | i ′ − 0 [u(x)γµγ d(x), d(y)γ u(y)] 0 ⇒ h | 5 5 | i = C∂µ∆+(z, 0) C ∂µ∆ (z, 0) − ′ − Causality: if x y is spacelike, then − 0 [u(x)γµγ d(x), d(y)γ u(y)] 0 = 0 h | 5 5 | i C = C ⇒ ′ − 0 [u(x)γµγ d(x), d(y)γ u(y)] 0 = C∂µ∆(z, 0) ⇒ h | 5 5 | i 0 [ Q, d(y)γ u(y)] 0 = C ⇒ h | 5 | i 0 [ Q, d(y)γ u(y)] 0 = 0 uu + dd 0 h | 5 | i −h | | i Hence 0 uu + dd 0 = 0 implies C = 0 qed. h | | i 6 6

H. Leutwyler – Bern Symmetries of the strong interaction – p. 30/33 Appendix: Derivation of GMOR relation

Pion matrix elements in massless theory: µ µ 0 uγ γ d π− =i √2 F p h | 5 | i 0 uiγ d π− =√2 G h | 5 | i Only the one–pion intermediate state 0 u(x)γµγ d(x) d(y)γ u(y) 0 = C∂µ∆+(z, 0) h | 5 5 | i ⇑ π π | −ih −| contributes. Hence 2 F G = C Value of C ﬁxed by quark condensate C = 0 uu + dd 0 −h | | i Exact result in massless theory: F G = 0 uu 0 −h | | i

H. Leutwyler – Bern Symmetries of the strong interaction – p. 31/33 Derivation of GMOR relation ctd.

Turn quark masses on 0 uγµγ d π = i √2 F pµ h | 5 | −i π 0 uiγ d π = √2 G h | 5 | −i π Current conservation µ ∂µ(uγ γ5d) = (mu + md)uiγ5d F M 2 = (m + m ) G ⇒ π π u d π

2 Gπ Mπ = (mu + md) exact for m = 0 Fπ 6

H. Leutwyler – Bern Symmetries of the strong interaction – p. 32/33 Proof of GMOR relation ctd.

Collect the results F F , G G for m 0 π → π → → F G = 0 uu 0 −h | | i Gπ 0 uu 0 = h | | i + O(m) 2 ⇒ Fπ − Fπ

2 Gπ Mπ = (mu + md) Fπ 0 uu 0 M 2 = (m + m ) −h | | i + O(m2) √ π u d 2 ⇒ Fπ

H. Leutwyler – Bern Symmetries of the strong interaction – p. 33/33