Top meets Topology C.T. Hill, Fermilab

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

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

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?

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

Latticization (Deconstruction) of extra dimensions Jacques Derride (1930-2004) Hill,Pokorski,Wang; Arkani-Hamed,Cohen,Georgi “brane” “link” U U U U U … … SU(N) SU(N) SU(N) SU(N) SU(N)

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.

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

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?

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

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

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

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

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

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

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)

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

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

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

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

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 =

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

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

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

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

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

Deconstructed fermions: Fermion doubling

Wilson Term:

Chiral fermion = dislocation orbifold orbifold

QCD chiral lattice Gauge theory R Chiral dislocation L Chiral dislocation

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. = =

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?

The End

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

Matrix Representation: