The strong CP problem
a seminar in EPII course
28/2/01
Outline:
1. The q vaccua.
2. Should it be taken seriously?
The U(1)A problem.
3. The strong CP problem.
4. Some solution attempts:
a. QCD properties (Vacuum dynamics).
b. Additional chiral symmetry (Axions).
c. Spontaneous CP breaking.
5. Conclusions.
1. The q vacuum
QCD SU(3) gauge transformation of a gluon:
A "semiclassical" approach:
Vacuum Û Pure gauge states
Many, Many possible field configurations…
W(x) Contains an mapping:
x ® ¥ / ® / WÎSU(2)
/ ® /
an mapping can be classified into HOMOTOPY CLASSES:
All mappings within the same homotopy class can be continuously distorted to one another.
A simple example:
n=0 n=1 n=2
A characteristic property for each class:
The WINDING NUMBER.
In :
Using W as our f:
Taking at x ® ¥, G0=0 (this will not change the class):
Now we can classify W and Gm by their class:
Both Wn and Gn m represent a class!
We can use W1 to build all classes:
Which leads to:
i.e.:
The main issue:
We want to associate a vacuum state with each gauge class:
We will see if this is justifiable later…
But since:
We have:
So for the vacuum to be invariant we must take:
And then:
Actually, we're allowed to have a phase:
Denoting the vacuum by q we get:
q is a property of the QCD vacuum.
Each q value labels a separate theory:
The green's function for an operator O:
But from gauge invariance of O:
ß
So is some function of n=m-n only and:
q cannot be changed by any gauge invariance operator!
The effective lagrangian term:
with Ssm being the Standard Model action.
As it turn out, taking:
One can show (Bardeen's identity 1974):
Where:
For pure gauge and in G0=0, the only non-vanishing term:
This looks familiar…
At the edges ±¥ we have pure gauge. So we get:
So:
And we got the famous expression:
· The Lq term represents a contribution due to "vacuum tunneling".
· Configuration with such finite Sq are called "INSTANTONS".
· Since Lq is a total derivative it will not enter any perturbation order.
2. Should it be taken seriously?
The U(1)A problem.
We have built a lot on:
Is there a way to justify it?
There is! - The U(1)A problem…
1 generation QCD with mu = md = 0:
a global symmetry:
Since the actual bare masses are small, mLQCD, we can treat the mass terms as perturbations. The global symmetry is therefor an approximate one.
Let's check:
The U(1)A (chiral) problem:
U(1)A
We need something else…
Indeed, taking:
There is an anomaly (Adler-Bell-Jackiw 1969):
So
So is everything OK?
No! Because we saw:
So we have a modified symmetry that is conserved, denoted by :
is a "wider" transformation. It includes chiral transformation of quarks + more…
It is conserved at the quantum level.
The problem lives on…
The SM is really sick…
The solution:
As can be guess -
- Has to do with q ('t Hooft 1976).
is indeed conserved in terms of fields.
But it does get contribution from vaccua states.
Since:
We get:
is not gauge invariant! - for every gauge there is a different conserved .
But even more:
So the conclusion: induce change in q.
Even though the charge is conserved for fields, the vacuum is changed by it…
The symmetry is broken with no Goldstone boson produced.
(To really show it: demonstrate that there are no poles in the appropriate green function…)
Anomaly q vacuum
Insights:
· We have diagonalized the mass terms using general transformation:
Part of this is chiral: .
So after diagonalization we get:
We can also do the opposite, take no q term but with:
As mass terms.
· If is an exact symmetry - all terms are equivalent. We can choose q = 0.
i.e.: No mass terms = No q term.
· 3. The strong CP problem.
is P and CP violating.
Experimental constraints:
Neutron electric dipole moment
From:
We get:
4. Some solution attempts.
a. QCD properties (Vacuum dynamics).
· No q vacuum - but what about the u(1)A problem?
· Un-confinement in in CPN model (Schierholz 1994) - not clear how to implement in QCD.
· VEV of quarks redefine (Lee 2000) - ???.
· mu = 0 and therefor by chiral transformation (Banks, Nir, Seiberg 1994) - problems with hadronic spectrum.
b. Additional chiral symmetry (Axions).
Chiral symmetry Þ no q term.
But we don't have chiral symmetry.
The solution:
Assume some chiral symmetry U(1)PQ, which is spontaneously broken (Peccei, Quinn 1977).
A Goldstone boson, the axion, with the symmetry transformation:
The Lagrangian contains a term due to Adler-Bell-Jackiw anomaly:
Due to the anomaly the axion get a VEV:
Then by:
The q term is gone:
The anomaly also gives the physical axion a mass:
The larger vPQ is, the lighter and weakly coupled the axion is Þ an invisible axion…
Experimentally:
Theoretical constraints from astrophysics and cosmology:
c. Spontaneous CP breaking:
We can take by imposing CP invariance on the SM.
But we know CP is violated…
Impose CP and break it carefully -
such that both and CKM matrix are obtained.
( enters only at loop level).
Introduce more complicated Higgs sector with complexes VEVs:
4. Conclusions.
· q vaccua is both the cure to the problem, and the cause to the strong CP problem.
· Those problems are good indication that we have incomplete understanding of the full theory.
· Especially, they can be related to low-energy CPV.
· With no new experimental evidence, this full theory is still elusive.
26 The strong CP problem