Snmek 1

1
Dynamical symmetry breaking and fermion mass
generation
Tomá Brauner Nuclear Physics Institute AS CR, Re
based on T.B., J. Hoek, hep-ph/0407339
2
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

3
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

4
Spontaneous symmetry breaking
Occurs when the ground state does not share the
symmetry of the Lagrangian.

• Symmetry not implemented by unitary operators.
• Degenerate, unitarily inequivalent ground states.
• Massless modes as long-wavelength fluctuations
• of the symmetry-breaking order parameter.

5
Fermion mass generation

• Dirac fermion kinetic term
  
• invariant under independent rotations of left and
  
• right handed fields chiral symmetry.
• Mass term
  couples left
• and right handed fields and locks their allowed
• symmetry transformations.
• Non-zero masses may appear in a
  chirally-invariant
• theory only after the chiral symmetry has been
• spontaneously broken.

6
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

7
Goldstone theorem
For any spontaneously broken continuous symmetry
there is a massless mode in the spectrum of the
theory.
For approximate symmetries one expects nearly
massless modes to appear, but no theorem
analogous to Goldstones exists.
8
Technical assumptions

• Translation invariance is not entirely broken.
• Commutativity condition for any two local
  operators

• Automatically satisfied for Lorentz-invariant
  theories
• where Goldstone modes broken generators.
• For non-relativistic theories the Goldstone mode
• counting is more complex, see
• H. B. Nielsen, S. Chadha, Nucl. Phys. B105 (1976)
  445.

9
Examples

• Ferromagnet 2 broken generators (spin
  rotations),
• 1 Goldstone mode (magnon)
• Antiferromagnet 2 broken generators (spin
  rotations),
• 2 Goldstone modes (magnons)
• Crystal lattice 3 broken generators (space
  translations),
• 3 Goldstone modes (acoustic phonons)

For internal symmetries, anomalous number of
Goldstone modes is connected with non-zero
density of some of the symmetry generators and
broken T-invariance.
10
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

11
Our model

• Theory of a massless Dirac fermion and a heavy
  complex
• scalar interacting via the Yukawa interaction
• Motivated by the symmetry-breaking sector of the
• Standard model.
• Attempt at dynamical symmetry breaking in
• electroweak interactions.

12
Symmetries of the model
Global U(1)VU(1)A symmetry upon suitable
definition of the transformation law of the
scalar.
vector U(1)V
axial U(1)A
Noether currents
13
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

14
Ward identities
To large extent are independent on the detailed
form of the interaction.
Must hold whether the symmetry is broken
spontaneously or not.
Analogous identity exists for the vertex of the
axial current with two scalar fields.
15
cSB by the Higgs mechanism
Phenomenological approach analogous to
Ginzburg-Landau. Design the scalar potential so
that it has a non-trivial minimum and expand
around it.
f2 is now the Goldstone boson, f1 is the massive
Higgs, and the Yukawa interaction contains the
mass term for the fermion.
16
Ward identities for the Higgs mechanism
Additional term in the vertex function coming
from the coupling of the GB to the broken
symmetry current.
This term is right enough to saturate the Ward
identity for the broken symmetry.
When symmetry is spontaneously broken, the vertex
function has a pole due to the massless
Goldstone boson.
17
Ward identities for our model
The low-energy interactions of the Goldstone
boson are fixed by the broken symmetry.
Use the Ward identities to express the GB
interactions in terms of the fermion and scalar
self-energies. These are in turn found by
solving the Schwinger-Dyson equations.
18
Effective vertices of the Goldstone boson
Assume no symmetry preserving quantum
corrections. The GB effective vertices are now
proportional to the symmetry-breaking
self-energies.
Consistency check when symmetry is restored, the
Goldstone mode decouples and vanishes from the
spectrum.
19
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

20
Schwinger-Dyson equations
Neglect vertex corrections ! a set of coupled
non-linear integral equations for the
self-energies of the fermion and the scalar.
To be solved approximately either analytically
(approximating the SD kernel), or numerically.
21
Numerical solution
The form of SDEs suggests iterative solution.
Physical mass given by the pole solve
We calculate S(p2) iteratively with an initial
ansatz for P(p2) and then evaluate P(p2) from
the second SDE.
22
Outline

• Spontaneous symmetry breaking.
• Goldstone theorem and Goldstone boson.
• Model of dynamical symmetry breaking.
• Green functions and Ward identities.
• Schwinger-Dyson equations.
• Conclusions.

23
Conclusions

• Chiral symmetry may be broken by Yukawa
  interaction.
• Dynamical fermion mass generation without the
• Higgs mechanism.
• Possible application to electroweak symmetry
  breaking.
• Phenomenological predictions still far ahead
  our
• approximation are too crude.
• Applications to other systems?