Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining...

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Quark masses and the strong CP problem Michael Creutz Brookhaven Lab Michael Creutz BNL 1

Transcript of Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining...

Page 1: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Quark masses and the strong CP problem

Michael Creutz

Brookhaven Lab

Michael Creutz BNL 1

Page 2: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Conventional picture

• quark mass matrix can have a phase |M | = ||M ||eiΘ

• U(1) anomaly prevents rotating this phase away

• gives rise to CP violation

• if one quark is massless |M | = 0

• Θ is irrelevant

Michael Creutz BNL 2

Page 3: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Confinement: quarks are not free

• defining quark masses is subtle

• related to ambiguities with gauge field topology

If mu = 0 is to solve the strong CP problem

• need a non-perturbative definition of mu

Entwined with chiral symmetry and anomalies

Michael Creutz BNL 3

Page 4: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

MC (1995):

• naive variable change ψ −→ eiγ5θψ

• mass term ψψ −→ cos(θ) ψψ + i sin(θ) ψγ5ψ

Study QCD dependence on m1 and m5

• m ψψ → m1 ψψ + im5 ψγ5ψ

Does physics depend only on√

m21 +m2

5? X

Michael Creutz BNL 4

Page 5: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Tool: effective chiral Lagrangean

Consider 2 flavors with effective potential

• V = (σ2 + ~π2 − v2)2

• mass term m1ψψ −→ m1σ tilts the sombrero

V

π

σ

=⇒π

σ

V

• pion becomes massive M2π ∝ m1

Michael Creutz BNL 5

Page 6: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

What does m5 do?

• im5ψγ5ψ −→ m5η not in above effective potential

• m5 will give η an expectation value

• 〈η〉 ∝ m5/M2η

Add an effective coupling (~π · ~a0 + ση)2 → 〈η〉2σ2

• form dictated by chiral symmetry and parity

Michael Creutz BNL 6

Page 7: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

V → V − αm25σ

2(sign related to pi eta mixing)

V

π

σ

=⇒

V

π

σ

m5 also gives pions a mass

• M2π ∝ m2

5 not linear in m5

Michael Creutz BNL 7

Page 8: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

m5 induces a barrier between σ > 0 and σ < 0V

π

σ

Transition at m1 = 0 becomes first orderm

m

5

1

σ < 0 σ > 0

Michael Creutz BNL 8

Page 9: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Transition occurs at conventional Θ = π

• m5

m1

= tan(Θ/2)

• physics not only dependent on√

m21 +m2

5

• variable change is anomalous

Physics depends non-trivially on Θ

• m1 and m5 are physically independent parameters

Michael Creutz BNL 9

Page 10: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

The strong CP problem:

• m5 is CP violating

• why is this parameter so small?

• would mu = 0 fix this?

Mπ 6= 0 means mu and md cannot both be zero

• must include up-down mass difference

Michael Creutz BNL 10

Page 11: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Introduce an up-down mass difference

• m ψψ → m1 ψψ +m2 ψτ3ψ

• ψ~τψ ∼ ~a0 isovector scalar

• also not in starting effective potential

• m2 will give a03 an expectation value

• 〈a03〉 ∝ m2/M2a0

Michael Creutz BNL 11

Page 12: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Effective coupling (~π · ~a0 + ση)2 (form dictated by chiral symmetry)

• V → V − βm22π

23

V

π

σ

=⇒

V

π

σ

3

Without tilt from m1,• π3 gains an expectation value!• the CP violating “Dashen phase”

Michael Creutz BNL 12

Page 13: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Conventional QCD parameters αs mu md Θ

• αs tied to overall scale: mp

• mq determines pseudoscalar spectrum: mπ mK . . .

• Θ determines neutron electric dipole moment

Michael Creutz BNL 13

Page 14: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

In chiral Lagrangean language:

αs mu md Θ map onto effective potential• overall scale• tilt• warp• angle between tilt and warp

V

π

σ

Michael Creutz BNL 14

Page 15: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Full 2 flavor phase diagram

m

m

m

1

m = 0

5

2

m = 0u

d

M.C (2011)

Michael Creutz BNL 15

Page 16: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Concentrate on m5 = 0 plane

CP

CP

m

m

d

u

σ > 0

σ < 0

Second order transition at mumd < 0; i.e. Θ = π

• order parameter 〈π0〉• massless neutral pion along transition line

M.C. (2004), S. Aoki, Sharpe, Horkel

Michael Creutz BNL 16

Page 17: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Symmetries

CP

CP

m

m

d

u

σ > 0

σ < 0

mu ↔ md

• if mu = md isospin is exact

• m2 = 0 protected from additive renormalization

Michael Creutz BNL 17

Page 18: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Symmetries

CP

CP

m

m

d

u

σ > 0

σ < 0

mu ↔ −md

• mu = −md isospin symmetry at Θ = π

• m1 = 0 also protected: mu +md

Michael Creutz BNL 18

Page 19: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Symmetries

CP

CP

m

m

d

u

σ > 0

σ < 0

mu,md ↔ −mu,−md

• ψ → eiπτ3γ5ψ

• flavored chiral symmetry not anomalous

• Trτ3 = 0 ⇒∣

∣eiπτ3γ5

∣ = 1

Michael Creutz BNL 19

Page 20: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

CP

CP

m

m

d

u

σ > 0

σ < 0

NO symmetry under mu ↔ −mu

• mu = 0 not protected by any symmetry

• Ward identities mix in FF̃

Michael Creutz BNL 20

Page 21: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Perturbatively ψψ → ψ eiπγ5 ψ flips sign of the mass

• this rotation is anomalous

• topology gives chiral zero modes

• Trγ5 = ν

• −1ν measure factor at negative m

Michael Creutz BNL 21

Page 22: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

The strong CP question

• the m5 term violates CP: 〈η〉 ∝ m5

• no strong CP violation observed

• m5 << Λqcd

Claim: mu = 0 eliminates the problem

• mu ≡ m1+m2

2+ im5

Michael Creutz BNL 22

Page 23: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Symmetries protect m1, m2, m5 separately

• renormalizations not in general equal

• no symmetry to protect m1 +m2

m1, m2, m5 physically distinct parameters

• independent renormalizations

• m1+m2

2+ im5 is an artificial construct

Michael Creutz BNL 23

Page 24: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Question: Can any experiment tell if mu = 0?

• is mu = 0 well-defined?

• MS is perturbative, cannot answer this

Non-perturbative issues require the lattice

• adjust lattice parameters for hadron spectrum

• read off quark masses and see if mu = 0

Michael Creutz BNL 24

Page 25: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Complication: md can induce an effective mu

������������������������������

u

u d

d

R

L

R

L

π, η=⇒

u

u d

d

R

L

R

L

π, ηmd

���������������������������������������������

���������������������������������������������

Mass ratios not renormalization group invariant

mu

md

→mu + ǫmd

md + ǫmu

Effect is non-perturbative, from topology

Michael Creutz BNL 25

Page 26: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Mass renormalization is not flavor blind !

• Mass independent renormalization?

• allowed but not natural

• hides non-perturbative mixing from topology

• physical particle mass ratios not constant

Michael Creutz BNL 26

Page 27: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

But there are many lattice formulations

• how do we define the quark mass

Can we use the operator product expansion

• no symmetry to rule out terms

• lattice currents involve neighboring sites

• details depend on specific formulation

Michael Creutz BNL 27

Page 28: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Can we use topology?

• mu = 0 equivalent to vanishing susceptibility

How to define lattice topology?

Space of lattice fields simply connected

Topology lost at the outset

• small instantons can fall through the lattice

Michael Creutz BNL 28

Page 29: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Combine loops around a hypercube• to give FF̃ in continuum limit• resulting charge not generally an integer

0

5

10

15

20

25

30

35

-3 -2 -1 0 1 2 3

Winding number distribution for 500 12^4 lattices at beta 2.3

MC (2011)

Michael Creutz BNL 29

Page 30: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Cooling (Wilson flow, . . . ) to remove UV fluctuations• Action settles to multiple instantons

0

50

100

150

200

0 100 200 300 400 500

total action versus cooling steps, 5 lattices at beta=2.3

actio

n

cooling steps

Many studies over the years: M. Teper (1985); de Forcrand, Garcia-Perez, Stamatescu;

Del Debbio, Giusti, Pica; Bruckmann, Gruber, Jansen, Marinkovic, Urbach, Wagner;

Ilgenfritz, Martemyanov, Muller-Preussker, Veselov, ...

Michael Creutz BNL 30

Page 31: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Often stable but ambiguities appear• winding can depend on cooling algorithm• with which action should we cool? How long?

-7

-6

-5

-4

-3

-2

-1

0 20 40 60 80 100 120 140 160

winding evolution of one lattice under three cooling algorithms

under relaxover relax

Michael Creutz BNL 31

Page 32: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Can we use the index theorem?

• count small eigenvalues of the Wilson operator

• at finite cutoff not exact zeros

• how to define “small”?

• depends on eigenvalues in first Wilson “circle”

Count zero modes of the overlap operator

• operator not unique: “domain wall height”

• reverts to Wilson eigenvalue distribution

Michael Creutz BNL 32

Page 33: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Should we care if topology is ambiguous?

• not measured in laboratory experiments

• concentrate on Mη′ , which is physical

• Witten-Veneziano formula a large Nc result

• tied to the mu = 0 issue

Michael Creutz BNL 33

Page 34: Quark masses and the strong CP problem - Miami · Confinement: quarks are not free • defining quark masses is subtle • related to ambiguities with gauge field topology If mu

Summary

mu = 0 ambiguous between lattice schemes

• not an appropriate solution to strong CP

• entangled with ambiguities in defining topology

Intuition from perturbation theory fails

• mass renormalization is not flavor blind

• matching lattice and perturbative quantities tricky

Review: Acta Physica Slovaca 61, 1 (2011), arXiv:1103.3304;

Michael Creutz BNL 34