edms from the qcd θ term · •the qcd lagrangian contains in principle the following term: 4 the...
TRANSCRIPT
![Page 1: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/1.jpg)
1
EDMs from the QCD θ term
Vincenzo CiriglianoLos Alamos National Laboratory
ACFI EDM SchoolNovember 2016
![Page 2: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/2.jpg)
2
Lecture II outline
• The QCD θ term
• Toolbox: chiral symmetries and their breaking
• Estimate of the neutron EDMs from θ term
• The “Strong CP” problem: understanding the smallness of θ• Peccei-Quinn mechanism and axions
• Induced θ term
![Page 3: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/3.jpg)
3
The QCD θ term
![Page 4: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/4.jpg)
• The QCD Lagrangian contains in principle the following term:
4
The θ term
gs = strong coupling constant
εμναβ = 4-dim Levi-Civita
symbol
![Page 5: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/5.jpg)
• The QCD Lagrangian contains in principle the following term:
4
The θ term
• Multiple reasons for the presence of θ term:
• EFT perspective: at dimension=4, include all terms built out of quarks and gluons that respect SU(3)C gauge invariance
• Diagonalization of quark mass matrix mq induces Δθ = arg det mq (will discuss this later)
• Structure of QCD vacuum (won’t discuss this)
gs = strong coupling constant
εμναβ = 4-dim Levi-Civita
symbol
![Page 6: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/6.jpg)
• The QCD Lagrangian contains in principle the following term:
5
The θ term
• Transformation properties under discrete symmetries: analogy with Electrodynamics
gs = strong coupling constant
εμναβ = 4-dim Levi-Civita
symbol
P-even, T-even
P-odd, T-odd
E is P-odd, T-even
B is P-even, T-odd
![Page 7: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/7.jpg)
• The QCD Lagrangian contains in principle the following term:
6
The θ term
gs = strong coupling constant
εμναβ = 4-dim Levi-Civita
symbol
• θ term is P-odd and T-odd, and hence CP-odd (CPT theorem)
• How do hadronic CP-violating observables depend on θ? (After all, no breaking of P and T observed in strong interactions)
![Page 8: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/8.jpg)
• Relevant to understand
1. How to compute the neutron EDM from the θ term
2. How the Peccei-Quinn mechanism works
7
Toolbox: chiral symmetries and their breaking
Technical subject: I will present the main concepts and implications
![Page 9: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/9.jpg)
8
Chiral symmetry
![Page 10: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/10.jpg)
8
Chiral symmetry
• For mq = 0, action invariant under independent U(3) transformations of left- and right-handed quarks:
• Conserved vector and axial currents (Ta: SU(3) generators and identity)
L,R ∈ U(3)
![Page 11: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/11.jpg)
9
Chiral symmetry
• For mq = 0, action invariant under independent U(3) transformations of left- and right-handed quarks:
• Symmetry is broken by mq ≠0 and by more subtle effects
L,R ∈ U(3)
![Page 12: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/12.jpg)
10
Symmetry breaking
• In general, three known mechanisms for symmetry breaking
• Explicit symmetry breaking
• Symmetry is approximate; still very useful
• Spontaneous symmetry breaking
• Equations of motion invariant, but ground state is not
• Anomalous (quantum mechanical) symmetry breaking
• Classical invariance but no symmetry at QM level
![Page 13: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/13.jpg)
10
Symmetry breaking
• In general, three known mechanisms for symmetry breaking
• Explicit symmetry breaking
• Symmetry is approximate; still very useful
• Spontaneous symmetry breaking
• Equations of motion invariant, but ground state is not
• Anomalous (quantum mechanical) symmetry breaking
• Classical invariance but no symmetry at QM level
All relevant to the discussion of chiral symmetry in QCD and Peccei-Quinn symmetry
![Page 14: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/14.jpg)
11
Spontaneous symmetry breaking
• Action is invariant, but ground state is not!
• Continuous symmetry: degenerate physically equivalent minima
• Excitations along the valley of minima → massless states in the spectrum (Goldstone Bosons)
• Many examples of Goldstone bosons in physics: phonons in solids (translations); spin waves in magnets (rotations); …
![Page 15: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/15.jpg)
12
Spontaneous symmetry breaking
• Pions, kaons, mesons: Goldstone bosons associated with SSB of chiral symmetry
• In case of SSB currents are still conserved. Massless states appear in the spectrum. What about the U(1)A symmetry?
• Axial subgroup is broken. Vector subgroup SU(3)V stays unbroken (symmetry approximately manifest in the QCD spectrum)
Figure from M. Creutz, 1103.3304
![Page 16: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/16.jpg)
13
Anomalous symmetry breaking• Action is invariant, but path-integral measure is not!
![Page 17: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/17.jpg)
13
Anomalous symmetry breaking• Action is invariant, but path-integral measure is not!
• Chiral anomaly [U(1)A]: in mq=0 limit axial current not conserved
Axial transformation induces a shift in the θ term
![Page 18: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/18.jpg)
13
Anomalous symmetry breaking• Action is invariant, but path-integral measure is not!
• Chiral anomaly [U(1)A]: in mq=0 limit axial current not conserved
Axial transformation induces a shift in the θ term
![Page 19: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/19.jpg)
14
Implications for θ term
• Diagonal mq matrix has complex eigenvalues
• To make them real, additional axial rotation is needed
• This induces shift in θ proportional to
• Diagonalization of quark mass matrix mq induces Δθ = arg det mq
![Page 20: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/20.jpg)
14
Implications for θ term
• Diagonal mq matrix has complex eigenvalues
• To make them real, additional axial rotation is needed
• This induces shift in θ proportional to
• Diagonalization of quark mass matrix mq induces Δθ = arg det mq
• Can put it in the gluonic θ term or in a complex quark mass!
• Physics depends only on the combination
![Page 21: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/21.jpg)
15
Estimate of the neutron EDM from θ term
Crewther, Di Vecchia, Veneziano, Witten Phys. Lett. 88B, 123 (1979)
![Page 22: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/22.jpg)
16
Rotating CPV to quark mass
• In order to analyze pion-nucleon couplings, it is more convenient to put the strong CPV in the form of pseudoscalar quark densities
![Page 23: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/23.jpg)
16
Rotating CPV to quark mass
• In order to analyze pion-nucleon couplings, it is more convenient to put the strong CPV in the form of pseudoscalar quark densities
• Use freedom in SU(3)A transformation to ensure that perturbation introduces no mixing of the vacuum to Goldstone Bosons (“Vacuum alignment”)
![Page 24: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/24.jpg)
17
Rotating CPV to quark mass
• This requires A to be proportional to the identity, with
Effect disappears if one of the quark masses vanishes
![Page 25: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/25.jpg)
• Use chiral symmetry (soft pion theorem) to relate CPV pion-nucleon coupling to baryon mass splittings
18
CPV pion-nucleon coupling
Crewther-DiVecchia-Veneziano-Witten 1979
![Page 26: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/26.jpg)
• Use chiral symmetry (soft pion theorem) to relate CPV pion-nucleon coupling to baryon mass splittings
18
CPV pion-nucleon coupling
• Equivalent way to see this: θ and mass splitting are chiral partners. Low-energy couplings controlling the two are related
Crewther-DiVecchia-Veneziano-Witten 1979
![Page 27: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/27.jpg)
• Use chiral symmetry (soft pion theorem) to relate CPV pion-nucleon coupling to baryon mass splittings
19
CPV pion-nucleon coupling
⇓
Mereghetti, van Kolck 1505.06272
and refs therein
Crewther-DiVecchia-Veneziano-Witten 1979
![Page 28: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/28.jpg)
20
Chiral loop and estimate of dn
• Leading contribution (for mq →0) to neutron EDM via chiral loop
Counter-term (of same order) and sub-leading contributions
Crewther-DiVecchia-Veneziano-Witten 1979
E. Mereghetti et al Phys. Lett. B 696 (2011) 97
![Page 29: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/29.jpg)
20
Chiral loop and estimate of dn
• Leading contribution (for mq →0) to neutron EDM via chiral loop
Counter-term (of same order) and sub-leading contributions
Crewther-DiVecchia-Veneziano-Witten 1979
E. Mereghetti et al Phys. Lett. B 696 (2011) 97
![Page 30: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/30.jpg)
21
• Leading contribution (for mq →0) to neutron EDM via chiral loop
Counter-term (of same order) and sub-leading contributions
Crewther-DiVecchia-Veneziano-Witten 1979
E. Mereghetti et al Phys. Lett. B 696 (2011) 97
Recent lattice QCD results** do not change qualitative picture
Guo et al., 1502.02295 Akan et al., 1406.2882 Alexandrou et al., 151005823
Chiral loop and estimate of dn
![Page 31: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/31.jpg)
21
• Leading contribution (for mq →0) to neutron EDM via chiral loop
Counter-term (of same order) and sub-leading contributions
Crewther-DiVecchia-Veneziano-Witten 1979
E. Mereghetti et al Phys. Lett. B 696 (2011) 97
Recent lattice QCD results** do not change qualitative picture
Chiral loop and estimate of dn
![Page 32: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/32.jpg)
22
The “strong CP” problem: understanding the smallness of θ
_
![Page 33: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/33.jpg)
• The small value of begs for an explanation
23
Understanding the smallness of θ
• Possible ways out:
• One of the quark masses vanishes (so can “rotate away” θ): this is strongly disfavored by phenomenology of light quark masses**
• Invoke some symmetry principle
• P or CP exact at high scale, broken spontaneously at lower scale. Difficulty: keep θ<10-10 while allowing large CKM phase
• Peccei-Quinn scenarios
** See Wilczek-Moore 1[601.02937] for a reincarnation of this idea through “cryptoquarks": massless quarks confined in super-heavy bound states
![Page 34: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/34.jpg)
• Basic idea: promote θ to a field and make sure that it dynamically relaxes to zero
24
Peccei-Quinn mechanism_
• How to get there: extend the SM with additional fields so that the model has an axial U(1)PQ global symmetry with these features:
• U(1)PQ is broken spontaneously at some high scale → axion is the resulting Goldstone mode
• U(1)PQ is broken by the axial anomaly → the axion acquires interactions with gluons, which generate an axion potential
• Potential induces axion expectation value such that θ=0_
• Salient features can be captured by effective theory analysis
![Page 35: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/35.jpg)
25
Axion effective theory
• At energies below the U(1)PQ breaking scale fa, axion effective Lagrangian is given by
We can ignore derivative terms
irrelevant for strong CP problem, such as
The presence of this term is required by the
axial anomaly
Goldstone nature of the axion requires the effective Lagrangian to be invariant under
a(x) → a(x) + constant ** (up to the anomaly term)
** In simplest models, the axion is the phase of a complex scalar
charge under U(1)PQ
Hence the transformation property
![Page 36: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/36.jpg)
26
Axion effective theory
• At energies below the U(1)PQ breaking scale fa, axion effective Lagrangian is given by
We can ignore derivative terms
irrelevant for strong CP problem, such as
The presence of this term is required by the
axial anomaly
Goldstone nature of the axion requires the effective Lagrangian to be invariant under
a(x) → a(x) + constant (up to the anomaly term)
• Key point: in LQCD +La , a(x) leads to a field-dependent shift of θ
Through interactions with gluons this quantity acquires a potential
![Page 37: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/37.jpg)
27
Axion effective theory
• In absence of other sources of CP violation, the potential is an even function of
![Page 38: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/38.jpg)
28
Axion effective theory
• Minimum of the potential when vanishes. This solves the strong CP problem, independently of the initial value of θ
• In absence of other sources of CP violation, the potential is an even function of
![Page 39: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/39.jpg)
28
Axion effective theory
• Minimum of the potential when vanishes. This solves the strong CP problem, independently of the initial value of θ
• In absence of other sources of CP violation, the potential is an even function of
• Axion mass given by with
![Page 40: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/40.jpg)
29
Induced θ term
• In presence of other sources of CP violation beyond the θ term, the potential is not an even function:
⊗
![Page 41: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/41.jpg)
29
Induced θ term
• In presence of other sources of CP violation beyond the θ term, the potential is not an even function:
⊗ This needs to be taken into account when computing the impact of BSM
operators on EDMs
![Page 42: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/42.jpg)
30
Status of axion searches
ArXiv: 1311.0029
Disfavored by astrophysics / cosmological observations
(grey) or argument (blue)
Axion as cold dark matter lives here
Sensitivity of planned experiments
![Page 43: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/43.jpg)
31
Backup slides
![Page 44: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/44.jpg)
32
Abelian gauge theory• Recall U(1) (abelian) example
• Form of the interaction:
conserved current associated with global U(1)
![Page 45: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/45.jpg)
33
Non-abelian gauge theory• Generalize to non-abelian group G (e.g. SU(2), SU(3), …).
• Invariant dynamics if introduce new vector fields transforming as
![Page 46: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/46.jpg)
34
Anomalous breaking of B and L• Action is invariant, but path-integral measure is not!
![Page 47: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/47.jpg)
34
Anomalous breaking of B and L• Action is invariant, but path-integral measure is not!
• Baryon (B) and Lepton (L) number are anomalous in the SM
• Only B-L is conserved; B+L is violated; negligible at zero temperature
![Page 48: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/48.jpg)
35
θ term and topology
• θ term is total derivative (surface term) but can’t ignore it due to non-trivial topological effects
Difference in winding number of gauge fields t = ±∞
![Page 49: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/49.jpg)
36
CP and chiral symmetry
• Degenerate vacua. Each spontaneously breaks all but one CPχ = χ-1CPχ
• Choice of fermion phases: CP0 (standard CP) is preserved ( | iΨγ5Ψ|Ω) = 0 ) This defines a “reference vacuum” |Ω
• Chiral symmetry (ΨL,R→e±χ ΨL,R) is spontaneously broken
Figure from M. Creutz, 1103.3304
![Page 50: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/50.jpg)
36
CP and chiral symmetry
• Degenerate vacua. Each spontaneously breaks all but one CPχ = χ-1CPχ
• Choice of fermion phases: CP0 (standard CP) is preserved ( | iΨγ5Ψ|Ω) = 0 ) This defines a “reference vacuum” |Ω
• Chiral symmetry (ΨL,R→e±χ ΨL,R) is spontaneously broken
• Explicit chiral symmetry breaking δL lifts degeneracy, i.e. selects “true” vacuum and the associated unbroken CP
• If we want true vacuum to be |Ω then δL cannot be arbitrary. It satisfies
“Vacuum alignment”
• Chiral symmetry is explicitly broken by quark masses and BSM operators
Figure from M. Creutz, 1103.3304
![Page 51: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/51.jpg)
37
Chiral symmetry relations
• Prototype: theta term and mass splitting are chiral partners
• Nucleon matrix elements are related. At LO (soft pion theorem)Crewther-DiVecchia-
Veneziano-Witten 1979
• Corrections appear at NNLO, not log enhanced
⇓
(with LQCD input)
Mereghetti, van Kolck 1505.06272
and refs therein
![Page 52: EDMs from the QCD θ term · •The QCD Lagrangian contains in principle the following term: 4 The θ term • Multiple reasons for the presence of θ term: • EFT perspective: at](https://reader031.vdocuments.pub/reader031/viewer/2022013003/5f6c4de8f72802687232ab9a/html5/thumbnails/52.jpg)
38
Toy model of invisible axionShifman-Vainshtein-Zakharov Nucl. Phys. B 166 (1980) 493
Field content: new quark (only strong interactions) + New complex scalar
Yukawa interactions invariant under axial U(1)PQ
φ acquires VEV
Quark and “radial” scalar excitations super-heavy. Axion is identified the phase of the scalar field:
Super-heavy quarks mediates axion-gluon interaction via triangle diagram:
From this point on, the analysis proceeds as in the EFT description