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Quark masses and the strong CP problem
Michael Creutz
Brookhaven Lab
Michael Creutz BNL 1
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Full 2 flavor phase diagram
m
m
m
1
m = 0
5
2
m = 0u
d
M.C (2011)
Michael Creutz BNL 15
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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