1. Top meets Topology
C.T. Hill, Fermilab

2. The Standard Model Violates B+L
‘t Hooft instantons x u e+ u d
years

3. The puzzle: Is B+L Violation Out of Control???
(e.g., Topflavor: SU (2) SU (2) x SU (2) L L L
all fermions
3rd gen fermions
Instantonic Baryon Number Violation:
Is B+L Violation
Out of Control??? if
then
years

4. Topflavor breaking: SU(2) x SU(2) SU(2)
Bifundamental Higgs
The bifundamental Higgs is coupled to
the instanton. What role does it play in
the overall tunneling amplitude?

5. Important observation:
Topflavor (and Topcolor) can be viewed as
the first two modes of a pure Yang-Mills theory
in compactified D=5

6. Latticization (Deconstruction) of extra dimensions
Jacques Derride ( )
Hill,Pokorski,Wang;
Arkani-Hamed,Cohen,Georgi
“brane”
“link” U U U U U … …
SU(N)
SU(N)
SU(N)
SU(N)
SU(N)

7. Topflavor can be viewed as a deconstructed SU(2) theory in D=5:
3rd gen fermions
1st and 2nd gen fermions
SU(2)1
SU(2)2 h1 h2
Orbifold
Topflavor is equivalent to an SU(2)xSU(2)
gauged Chiral Lagrangian.

8. For Yang-Mills Theories Topology is maximally interesting in D=5 !!!
The reason: The instanton is the most important
topological object in Yang-Mills.
3(SU(2)): instanton naturally lives on a
D=4 Euclidean space, mapping SU(2) onto S3:
tunneling and Skyrmion
In D=5, the instanton becomes a
stable, static, soliton

9. Instantonic Soliton Skyrmion Gauge currents Chiral currents
The topology of the D=5 pure Yang Mills controls the topology of the D=4
Yang-Mills theory or chiral lagrangian
(via deconstruction):
CTH, CTH & Zachos
Instantonic Soliton Skyrmion
Gauge currents Chiral currents
Chern-Simons term WWZ term
Boundary Sources Anomalies
Insights into dynamical processes?

10. D=5 Yang-Mills instantonic soliton
Deser’s Theorem
Ramond and CTH t
Mass:
D=4 space
Associated Topological Currents:
Singlet:
Adjoint:

11. These currents come from a “completion”
of the Lagrangian (ala Noether):
Adjoint current - 2nd Chern character:
e.g. D=3, photon mass term
Deser, Jackiw, Templeton, Schonfeld, Siegel;
Niemi, Semenoff, Y.S. Wu
Quantization :
Singlet current - auxiliary character:
c’ = 3c/2

12. gauged chiral lagrangian
A Heuristic “derivation” of WWZ term from the D=5 Chern-Simons term on an orbifold:
left brane
right brane
Wilson line
gauged chiral lagrangian

13. = magnetic superconductng capacitor
Orbifold compactification
= magnetic superconductng capacitor
left brane
right brane
“London currents”

14. Chern-Simons term on the Left-hand side:
+ rhs+gauge
field terms
+ gauge field terms
+ gauge field terms
Witten’s coefficient !!!

15. Thus, D=4 Chiral Lagrangian topology is controlled by D=5 Yang Mills !!!
Kinetic term
Skyrme term

16. Topological currents in one-to-one Correspondence with D=5 Y-M:
Singlet: (global form of Goldstone and Wilczek)
(local form)
Adjoint: (global)
(local form)

17. Topological currents describe the Skyrmion
which coresponds to instantonic soliton:
Global Theory:
= 1
Local Theory:
GW = Bardeen’s current:

18. anomalies are bosonic in origin
Orbifolding produces nonzero current divergences
that arise as boundary terms; these are “anomalies”
anomalies are bosonic in origin
current flow
instanton L
instanton R

19. Summary of topological matching:
D=4 instantons are creation/annihilation vertices for the
D=5 instantonic soliton
D=5 Instantonic Soliton becomes the
a D=4 (gauged) Skyrmion under compactification
(which is built of massive KK-modes)
Instanton in D=4 SB theories
always create a Skyrmion either as a
vacuum twist or a physical object

20. Instantons Outgoing current Incoming visible current, e.g.
Always involves
overall conserved current,
e.g.:
Incoming visible
current, e.g.
J5 or JB+L
unbroken: Bardeen current
invisible
broken: Goldstone-Wilczek
current carried by a Skyrmion
visible

21. Action = The D=4 instanton tunneling process in unbroken
Yang-Mills theory is a vacuum to vacuum transition;
It can be viewed as creating a Skyrmionic
gauge configuration as the twisted vacuum: t
U=1
Action =

22. charge (twist) emerges
The anomaly in the process takes the form of an
exact differential:
Unobservable vacuum
charge (twist) emerges
Unobservable vacuum charge
changes by a unit reflecting
the twist:
Observable current
falls into instanton

23. This is now NOT vacuum tunneling (nonlinear sigma model limit)
If the theory is spontaneously broken then the
Higgs becomes twisted a physical Skyrmion
This is now NOT vacuum tunneling
(nonlinear sigma model limit)
‘t Hooft process is further suppressed by
the requirement that the Higgs VEV tunnel
to zero near origin so f(r) can unwind.
Consistently describes decay of Skyrmion
(e.g. proton into a positron) via anomaly
D’Hoker and Farhi

24. Skyrmion of the bifundamental Higgs in Topflavor is a boson (N=3c+1)
built of massive longitudinal gauge fields
Skyrmions bind fermionic zero modes, forming a
wave-function consistent with ‘t Hooft
determinant (B+L).
Instanton annihilates incoming
‘t Hooft determinant configuration
producing this state

25. Topflavor Topological selection rules:
Vacuum tunneling if
instanton x anti-instanton
D=4 Skyrmion production/decay
D=5 Instantonic Soliton producton/decay

26. Enhanced processes are the creation
Topflavor is safe
Enhanced processes are the creation
and annihilation of Skyrmions built of
the massive vector bosons via anomalies

27. Deconstructed fermions:
Fermion doubling

28. Wilson Term:

29. Chiral fermion = dislocation
orbifold
orbifold

30. QCD chiral lattice Gauge theory
R Chiral
dislocation
L Chiral
dislocation

31. Skyrmion motion in D=5 is “anomaly hopping”
outgoing L
incoming R
QCD U(1)
problem
‘t Hooft process
is a t-channel
exchange of IS. = =

32. Conclusions: Topflavor B+L violation is safe
Corresponds to production of a physical object, Skyrmion in bifundamental Higgs
=> D=5 Instantonic Soliton, mass > 100 TeV
Normal ‘t Hooft process is t-channel exchange of
the Skyrmion (IS) between branes (gauge groups)
IS decays into baryons+leptons.
New expt’l signatures?

33. The End

34. Deconstruction on a two slice lattice:
non-matrix
Representation:
Matrix
Representation:

35. Matrix
Representation: