1. Strong CP problem in particle physics and its possible solutions*
Tatiana Seletskaia
Physics Department
California State University of East Bay
*This is pedagogical introduction into the problem with three possible solutions

2. Outline CPT symmetry and its importance
CP symmetry violation in electroweak decays
CP symmetry - a good symmetry of QCD
Possible solutions of strong CP problem:
Peccei-Quinn mechanism
String theory solution
Space-time with two time axis

3. References:
[1] R. D. Peccei, Helen Quinn, “CP Conservation in the Presence of Pseudoparticles”, Phys. Rev. Lett. 38 (25), , 1977
[2] R. D. Peccei, “The Strong CP Problem and Axions”,
[3] B. L. Ioffe, “Axial Anomaly: the current status”,
[4] David J. Gross, Robert D. Pisarski, Laurence G. Yaffe, “QCD and instantons at finite temperature”, Rev. Mod. Phys. 53 (1), 43-79, 1981
[5] Marc Kamionkowski, John March-Russel, “ Plank-Scale Physics and the Peccei-Quinn Mechanism”, Phys. Lett. B 282 (1-2), , 1992
[6] Juan M. Maldacena, “The Large N Limit of Superconformal Field Theories and Supergravity”, Int. Journal Of Theoretical Physics 38 (4), , 1999;
[7] D. Cremades, L. E. Ibanez, F. Marchesano, “Toward understanding quark masses, mixings and CP-violation”,
[8] I. Bars, C. Deliduman, O. Andreev, “Gauged Duality, Conformal Symmetry and Spacetime with Two Times”, Phys. Rev. D 58 (6), , 1998

4. What space-time symmetries tell us?
They define dynamics of the system
They define conservation laws
They tell us which systems are realizable and which are not

5. Internal symmetries of the particles
based on the space-time properties
Particles are fields
Fields are continuous and occupy the entire space-time
Nevertheless, particles are points and a free particle can have its position anywhere in space
Symmetry of the wave-function with respect to the coordinate transformations carried at an arbitrary point is internal symmetry of the particle
Examples: spin, parity, time reversal
Energy and momentum are due to translational symmetries of the space-time. They are not called internal because they have well defined classical limits. So far, spin, parity and time-reversal do not have classical analogs.

6. Internal symmetries of the particles
based on the properties of operators
In quantum field theory fields of the particles become operators
There are creation and annihilation operators corresponding to each particle state
Charge conjugation is a symmetry described by interchanging particles and antiparticles operators
It leaves the space-time coordinates unchanged and inverts the current
Charge conjugation does not have an analog in non-relativistic quantum mechanics.

7. Internal symmetries of fermions
based on the properties of interaction fields –
gauge symmetries
Fermions interact with each other
Being scattered at relativistic speed, a fermion can turn into another particle depending on its internal symmetry
Reactions between fermions are mediated by bosons
There are three fundamental forces discovered in particle reactions:
U(1) – electromagnetic (well studied at the macroscopic level)
SU(2) – weak ( radioactive decay, nuclear fission)
SU(3) – strong (stability of a proton, nuclear fusion)

8. Internal symmetries of fermions
based on the properties of interaction fields –
gauge symmetries
U(1), SU(2), SU(3) are gauge symmetries
From classical prospective:
Electromagnetic field is a gauge field because its potential is defined up to the gradient of some function. Gauge invariance gives us one extra condition.
From quantum field theory prospective:
If wave-function of the fermions is invariant under certain rotation, there must be a boson field that makes derivative covariant under this transformation.
Potentials and their gauge invariance play different role in classical and quantum theories!

9. Internal symmetries of fermions
based on the properties of interaction fields –
gauge symmetries
Fermion multiplets:
Leptons
Baryons
Bosons:

10. CPT symmetry
C charge conjugation P parity T time reversal

11. CPT theorem
All particle scattering processes are invariant under CPT transformation:
The theorem is deeply connected with the Lorentz invariance of the space-time and with the fact that masses of particles are equal to the masses of their anti-particles.

12. CPT symmetry C = PT ct absolute future x spatially spatially separated
interval between two causally
connected events, e. g. dynamics of macroscopic body
absolute future
In different Lorentz frames the vector looks
as if it was rotated in the upper cone x
spatially
separated
events
spatially
separated -
no causal connection
inversed interval
Relativistic invariance =>
invariance with respect to 4-D inversion
absolute past
C = PT

13. CP symmetry in electroweak decay
Particles taking part in weak interactions: C symmetry is maximally violated: There are no right-handed neutrinos! There are no left-handed anti-neutrinos! P symmetry is maximally violated: Interaction term is a sum of vector and axial vector

14. CP symmetry in electroweak decay
CP symmetry should be a GOOD symmetry in electroweak reactions: p
W e Beta-decay of a neutron n
CP symmetry turns left-handed neutrinos into right-handed anti-neutrinos.

15. CP violation in electroweak decay
CP violation is observed for kaons Kaon is a meson consisting of one strange quark and up or down anti-quark. Kaons:
1 “Modern Elementary Particle Physics”, Gordon Kane, 1993

16. CP violation is a result of Adler-Bell-Jackiw anomaly
Anomalous symmetries arise when the symmetry of the classical field is broken as a result of quantization
Product of the field and its conjugate at the same space-time point is necessarily singular. Finite values can be recovered by renormalization procedure after which symmetry is broken in the presence of the gauge field
CP violation in QED is due to radiation corrections
It comes from the third order-term describing interaction of fermion axial current with external electromagnetic field
For this triangular diagram either vector current or axial current can be conserved, but not both
Axial current of fermions
Photons

17. Why CP violation is important?
It could help us to explain dominance of matter over anti-matter. CP violation is one of Saharov’s conditions for baryogenesis.
It could help us to understand the reasons of time-reversal symmetry violation and its nature. It might be relevant to describing non-linear thermodynamic processes.

18. CP symmetry in quantum chromodynamics
CP symmetry should be violated in chromodynamics: Axial current:
Axial currents – quarks
External field – colored vector gluons
Axial current
Vector
gluons
Product of the gluon field and its dual is a divergence of the axial current of the quark. It is different from zero due to U(1) anomalous symmetry of the quark. Since vector current is conserved in all particle reactions, axial current of quarks should be violated.

19. CP symmetry in quantum chromodynamics
But CP violation is not observed in quantum chromodynamics! Gluons self-interactions keep quarks together inside protons and neutrons. CP violation would produce electric dipole moment in the neutron ~10-16ecm Measured upper bound of the electric dipole moment of the neutron ~2.9x10-26ecm

20. Possible solutions of strong CP problem
Peccei-Quinn mechanism #1
String theory #2
Space-time with two time axis #3

21. Peccei-Quinn mechanism for preserving CP symmetry#1
The major assumptions are that at least one quark is massless and the Lagrangian has global U(1) symmetry described by parameter Peccei-Quinn symmetry
The symmetry is spontaneously broken by dynamical CP conserving field - axion
Axion can be a massless Goldstone boson or in the case when global Peccei-Quinn symmetry becomes local gauge symmetry, it can become longitudinal component of massive gauge boson
Mass of the quarks is a result of their interaction with the axion

22. Peccei-Quinn mechanism for preserving CP symmetry#1
Under U(1) Peccei-Quinn symmetry transformation,
axion translates by some constant times the order parameter:
CP violating Lagrangian term and the terms arising due to axion
field are:
Expectation value of axion vacuum:
It cancels CP-violating term and makes masses of the quarks real!

23. Instantons in QCD #1
Instantons are pseudo-particles that appear as a result of transition between two states in topologically non-trivial vacuum. Instantons explicitly break Peccei-Quinn symmetry by adding axion some very small mass.

24. Peccei-Quinn Mechanism at the Plank-Scale#1
At the Plank-scale general relativity predicts formation of black holes and worm-holes. As a consequence of no-hair black hole theorem, the global charge is not defined and global symmetries should be violated
In the string theory, symmetry-breaking is due to the higher dimension operators suppressed by the appropriate power of the Plank mass
Since Peccei-Quinn is a global symmetry, contributions from higher dimension operators modify axion mass
Assuming that and coupling constants of the higher dimensional fields are of the order of the Standard Model couplings, one obtains that

25. Bounds on the axion mass from the experiments#1
Michael S. Turner, “Windows on the Axion”, Phys. Rep. 197 (2), 67-97, 1990
To satisfy these bounds, the coupling of the higher dimension symmetry-breaking terms should be extremely small that substitutes one problem by another.

26. CP violation in string theory#2
World-sheets of the string are invariant with respect to parameterization
Diffeomorphism makes strings invariant with respect to any continuous coordinate transformation
Thus, scattering reactions between strings necessarily preserve T and CP symmetry
Strong CP problem is resolved
Point-particle
String
CP violating term arises due to quantization rules and singularities of the wave-function products at the same space-time point. Symmetry of the classical field disappears after renormalization.
Making particles extended brings singularities from the physical space-time to parameters’ space.

27. CP violation in string theory#2
How to get CP violation term and explain that the angle is significantly small?
Fermions arise at the D-branes intersections
There are three baryonic branes, two left branes, one right brane and one leptonic brane
SU(2)R is a brane intersection responsible for mixing and CP-violation
CP symmetry violation can be due to the Wilson loops of U(1) symmetry in compactified dimensions
Bad news: This CP violation mechanism cannot to be tested in AMO experiments

28. CP violation in string theory#2
AdS/CFT duality:
Duality between the dynamics of a string moving in anti-de Sitter space and conformal field theory – quantum field theory of elementary particles. Conformal map is a function that preserves angles locally.
Conformal map is can be represented by complex function of complex variables and symmetry operations involve translations, rotations, and scaling.
AdS/CFT duality has a great success in solid state physics!
Structure of the Standard Model plays secondary role. Symmetries of the elementary particles and Yukawa couplings can be reproduced by fitting moduli space of the string vacuum to the observed data.

29. Comparison of #1 and #2
String theory within AdS/CFT duality gives the same solution to the strong CP problem as Peccei-Quinn model. To be consistent with the experiments, Peccei-Quin model needs to have Θ going to zero. String theory needs to have the size of the compactified dimensions to be extremely small. The difference is that Peccei-Quin model predicts existence of axion field associated with symmetry breaking. In the string theory axion fields belong to compactified dimensions.

30. Space-time with the two time axis#3
Preliminaries:
Quantization rules are symmetric under interchange of coordinate and momentum:
The electric and magnetic field are generalized coordinate and momenta
Physics laws are the same if we replace E -> -B, B -> E
This is called duality
Transformations of (x, p) doublet in the phase space form symplectic Sp(2) group
Local Sp(2) symmetry:

31. Space-time with the two time axis#3
Consequence:
There is a global SO(d, 2) symmetry
Generators of SO(d, 2) symmetry are:
These symmetry generators are gauge invariant under the local Sp(2) transformations
Minkowski space-time get enlarged by one space-like and one time-like dimensions
Duality between different physical systems such as free-particle, H atom, harmonic oscillator is due to the different choice of the time gauge

32. Space-time with the two time axis
Standard Model in 3+1 dimensions is a gauged fixed version of quantum field theory in 4+2 dimensions
CP violating term does not appear in such a theory –
No axion field required for explanation
In 4+2 dimensions Levi-Cevita symbol has 6 indexes
Yang-Mills symmetry of elementary particles is combined with Sp(2,R) gauge symmetry
As the result CP term is identical to zero
- CP violating term in 3+1 quantum field theory

33. SUMMARY
We presented three solutions to strong CP-problem based on QCD, string and conformal gravity research
Peccei-Quinn axion is likely to have mass different from zero due to instanton effect
In string theory axion is represented by Wislon loop in compactified dimensions
In space-times with two time axis the CP violating term is identical to zero