Anomalies and discrete chiral symmetries Michael Creutz BNL & - PowerPoint PPT Presentation

Anomalies and discrete chiral symmetries Michael Creutz BNL & U. Mainz Three sources of chiral symmetry breaking in QCD spontaneous breaking � ψψ � � = 0 • explains lightness of pions • implicit breaking of U (1) by the anomaly • explains why η ′ is not so light • explicit breaking from quark masses • pions are not exactly massless • Rich physics from the interplay of these three effects Michael Creutz BNL & U. Mainz 1

Talk based on very old ideas Dashen, 1971: possible spontaneous strong CP violation • before QCD! • ’t Hooft, 1976: ties between anomaly and gauge field topology • Fujikawa, 1979: fermion measure and the anomaly • Witten, 1980: connections with effective Lagrangians • MC 1995: Why is chiral symmetry so hard on the lattice • Why rehash old ideas? consequences have recently raised bitter controversies • Michael Creutz BNL & U. Mainz 2

Axial anomaly in N f flavor massless QCD leaves behind a residual Z N f flavor-singlet chiral symmetry • tied to gauge field topology and the QCD theta parameter • Consequences degenerate m � = 0 quarks: first-order transition at Θ = π • sign of mass relevant for odd N f : perturbation theory incomplete • • N f = 1 : no symmetry for mass protection  • m u = 0 cannot solve the strong CP problem     controversial nontrivial N f dependence: •   invalidates rooting  •  Michael Creutz BNL & U. Mainz 3

Consider QCD with N f light quarks and assume the field theory exists and confines • spontaneous chiral symmetry breaking � ψψ � � = 0 • • SU ( N f ) × SU ( N f ) chiral perturbation theory makes sense anomaly gives η ′ a mass • • N f small enough to avoid any conformal phase Use continuum language imagine some non-perturbative regulator in place (lattice?) • momentum space cutoff much larger than Λ QCD • lattice spacing a much smaller than 1 / Λ QCD • Michael Creutz BNL & U. Mainz 4

Construct effective potential V for meson fields • V represents vacuum energy density for a given field expectation formally via a Legendre transformation • assume regulator allows defining composite fields • For simplicity initially consider degenerate quarks with small mass m • • N f even interesting subtleties for odd N f • Michael Creutz BNL & U. Mainz 5

Work with composite fields • σ ∼ ψψ λ α : Gell-Mann matrices for SU ( N f ) • π α ∼ iψλ α γ 5 ψ • η ′ ∼ iψγ 5 ψ σ V( ) Spontaneous symmetry breaking at m = 0 • V ( σ ) has a double well structure vacuum has � σ � = v � = 0 • minimum of V ( σ ) = ± v • σ Ignore convexity issues phase separation occurs in a concave regions • Michael Creutz BNL & U. Mainz 6

Nonsinglet pseudoscalars are Goldstone bosons symmetry under flavored rotations • σ → cos( φ ) σ + sin( φ ) π α • ( N f = 2) π α → cos( φ ) π α − sin( φ ) σ • ψ → e iφγ 5 λ α ψ potential has N 2 f − 1 ‘‘flat’’ directions • one for each generator of SU ( N f ) • V π σ Michael Creutz BNL & U. Mainz 7

Small mass selects vacuum • V → V − mσ • � σ � ∼ + v � π � = 0 Goldstones acquire mass ∼ √ m • V π σ Michael Creutz BNL & U. Mainz 8

Anomaly gives the η ′ a mass even if m q = 0 • m η ′ = O (Λ QCD ) • V ( σ, η ′ ) not symmetric under • ψ → e iφγ 5 ψ • σ → σ cos( φ ) + η ′ sin( φ ) • η ′ → − σ sin( φ ) + η ′ cos( φ ) Expand the effective potential near the vacuum state σ ∼ v and η ′ ∼ 0 η ′ η ′ 2 + O (( σ − v ) 3 , η ′ 4 ) σ ( σ − v ) 2 + m 2 • V ( σ, η ′ ) = m 2 both masses of order Λ QCD • Michael Creutz BNL & U. Mainz 9

In quark language Classical symmetry • ψ → e iφγ 5 / 2 ψ • ψ → ψe iφγ 5 / 2 mixes σ and η ′ • • σ → σ cos( φ ) + η ′ sin( φ ) • η ′ → − σ sin( φ ) + η ′ cos( φ ) This symmetry is ‘‘anomalous’’ any valid regulator must break chiral symmetry • remnant of the breaking survives in the continuum • Michael Creutz BNL & U. Mainz 10

Variable change alters fermion measure • dψ → | e − iφγ 5 / 2 | dψ = e − iφ Tr γ 5 / 2 dψ But doesn’t Tr γ 5 = 0 ??? Fujikawa: Not in the regulated theory!!! � γ 5 e D 2 / Λ 2 � i.e. • lim Λ →∞ Tr � = 0 Dirac action ψ ( D + m ) ψ • D † = − D = γ 5 Dγ 5 Use eigenstates of D to define Tr γ 5 • D | ψ i � = λ i | ψ i � • Tr γ 5 = � i � ψ i | γ 5 | ψ i � Michael Creutz BNL & U. Mainz 11

Index theorem with gauge winding ν , D has ν zero modes D | ψ i � = 0 • modes are chiral: γ 5 | ψ i � = ±| ψ i � • • ν = n + − n − Non-zero eigenstates in chiral pairs • D | ψ � = λ | ψ � • Dγ 5 | ψ � = − λγ 5 | ψ � = λ ∗ γ 5 | ψ � Space spanned by | ψ � and | γ 5 ψ � gives no contribution to Tr γ 5 • � ψ | γ 5 | ψ � = 0 when λ � = 0 only the zero modes count! • Tr γ 5 = � i � ψ i | γ 5 | ψ i � = ν Michael Creutz BNL & U. Mainz 12

Where did the opposite chirality states go? continuum: lost at ‘‘infinity’’ ‘‘above the cutoff’’ • Wilson: real eigenvalues in doubler region • overlap: modes on opposite side of unitarity circle • • Dγ 5 = − ˆ γ 5 D Tr ˆ γ 5 = 2 ν This phenomenon involves both short and long distances zero modes compensated by modes lost at the cutoff • Cannot uniquely separate perturbative and non-perturbative effects small instantons can ‘‘fall through the lattice’’ • scheme and scale dependent • Michael Creutz BNL & U. Mainz 13

Back to effective field language At least two minima in the σ, η ′ plane ( σ, η ′ ) = (0 , ± v ) η ? σ −v v ? Question: do we know anything else about the potential in the σ, η ′ plane? • Yes! there are actually N f equivalent minima • Michael Creutz BNL & U. Mainz 14

Define ψ L = 1+ γ 5 ψ 2 Singlet rotation ψ L → e iφ ψ L not a good symmetry for generic φ • Flavored rotation ψ L → g L ψ L = e iφ α λ α ψ L is a symmetry for g L ∈ SU ( N f ) • For special discrete values of φ these rotations can cross • g = e 2 πi/N f ∈ Z N f ⊂ SU ( N f ) A valid discrete singlet symmetry: σ → + σ cos(2 π/N f ) + η ′ sin(2 π/N f ) η ′ → − σ sin(2 π/N f ) + η ′ cos(2 π/N f ) Michael Creutz BNL & U. Mainz 15

V ( σ, η ′ ) has a Z N f symmetry • N f equivalent minima in the ( σ, η ′ ) plane • N f = 4 : η V 1 σ V V 2 0 V 3 At the chiral lagrangian level • Z N is a subgroup of both SU ( N ) and U (1) At the quark level measure gets a contribution from each flavor (’t Hooft vertex) • • ψ L → e 2 πi/N f ψ L is a valid symmetry Michael Creutz BNL & U. Mainz 16

η V 1 σ V V 2 0 V 3 Mass term mψψ tilts effective potential picks one vacuum ( v 0 ) as the lowest • in n ’th minimum m 2 • π ∼ m cos(2 πn/N f ) highest minima are unstable in the π α direction • multiple meta-stable minima when N f > 4 • Michael Creutz BNL & U. Mainz 17

Anomalous rotation of the mass term • mψψ → m cos( φ ) ψψ + im sin( φ ) ψγ 5 ψ twists tilt away from the σ direction • An inequivalent theory! η m V 1 φ σ V V 0 2 V 3 as φ increases, vacuum jumps from one minimum to the next • Michael Creutz BNL & U. Mainz 18

Here each flavor has been given the same phase Conventional notation uses Θ = N f φ • • Z N f symmetry implies 2 π periodicity in Θ Degenerate light quarks ⇒ first order transition at Θ = π η Θ = π V 1 σ V V 0 2 V 3 Michael Creutz BNL & U. Mainz 19

� � Discrete symmetry in mass parameter space m → m exp 2 πiγ 5 N f for N f = 4 : • • mψψ and imψγ 5 ψ mass terms give equivalent theories true if and only if N f is a multiple of 4 • η V 1 σ V V 2 0 V 3 Michael Creutz BNL & U. Mainz 20

η Odd number of flavors, N f = 2 N + 1 • − 1 is not in SU (2 N + 1) N =3 f V 1 • m > 0 and m < 0 not equivalent! σ • m < 0 represents Θ = π V 0 an inequivalent theory • spontaneous CP violation: � η ′ � � = 0 • V 2 Inequivalent theories can have identical perturbative expansions! Theta dependence invisible to perturbation theory • Michael Creutz BNL & U. Mainz 21

Center of SU ( N f ) is a subgroup of U (1) 10,000 random SU (3) and SU (4) matrices: • 3 4 SU(3) SU(4) 2 3 2 1 1 Im Tr g Im Tr g 0 0 -1 -1 -2 -2 -3 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 Re Tr g Re Tr g region for SU (3) bounded by exp( iφλ 8 ) • all SU ( N ) points enclosed by the U (1) circle e iφ • boundary reached at center elements • Michael Creutz BNL & U. Mainz 22

η N f = 1 : No chiral symmetry at all! N =1 f unique vacuum • σ • � ψψ � ∼ � σ � � = 0 from ’t Hooft vertex V 0 not a spontaneous symmetry breaking • No singularity at m = 0 • m = 0 not protected: ‘‘renormalon’’ ambiguity For small mass no first order transition at Θ = π • larger masses? • N f = 0 : pure gauge theory • Θ = π behavior unknown Michael Creutz BNL & U. Mainz 23

Michael Creutz BNL & U. Mainz 24