Lattice: Field theories beyond QCD

Transcription

1 Yale University 4 th Electron-Ion Collider Workshop Hampton University, Hampton, VA 22 May 2008

2 Outline Dynamical Electroweak Symmetry Breaking (DEWSB) Wasn t technicolor ruled out more than a decade ago? The Conformal Window of SU(N) Yang-Mills Theories

3 Brief History of Electroweak Symmetry 1967: S. Weinberg publishes the electroweak (EW) sector of the standard model (SM). He speculated that it might be renormalizable. The paper was completely ignored for four years. It is now the top-cited paper in SPIRES. 1971: G. t Hooft proves renormalizability of EW theory. Mid 1970 s: Growing appreciation of the hierarchy problem for the standard model Higgs boson: New physics must appear at the TeV scale to explain why Higgs is so much lighter than the Planck/GUT scale: SUSY, : S. Weinberg describes dynamical electroweak symmetry breaking (DEWSB), dubbed technicolor (TC) by L. Susskind, obviated the need for a scalar Higgs field.

4 General features of DEWSB Dynamical Electroweak Symmetry Breaking ( DEWSB) signals new strong interactions. New strong sector has several general features at TeV scale: Spontaneously broken global chiral symmetry producing at least three Nambu-Goldstone Bosons ( NGB s) eaten to become longitudinal W, Z bosons. Extra pseudo-ngb s ( PNGB s) are massive (like Kaons). Additional resonances expected (e.g. vector mesons). Many possible gauge groups, colors, flavors, representations. Additional extended technicolor ( ETC) fields and interactions are needed to describe how fermions get their masses. Effectively generated by (QQ)(qq)/Λ 2 ETC. A scaled-up copy of QCD has all the ingredients, but...

5 Wasn t technicolor ruled out more than a decade ago? Aren t flavor-changing neutral currents ( FCNC s) too large? In ETC, FCNC s effectively generated by (qq)(qq)/λ 2 ETC For current limits on FCNC s: Λ ETC 1000 TeV. For top mass: m t QQ /Λ 2 ETC. In QCD-like TC, QQ (1 TeV) 3 implying Λ ETC 10 TeV. TC dynamics must have enhanced condensates. Aren t precision electroweak constraints violated? NLO ChiPT LEC l 5 related to π + π 0 mass-splitting leads to 3 σ discrepancy between EW and QCD-like TC. l5 yet to be accurately computed in lattice QCD, let alone other confining theories. Bottom line: So much is known about QCD, it s easy to argue against QCD-like TC. For other TC theories, so little is known that model-building is a baroque exercise based on conjecture and too many free parameters.

6 Flavor dependence of SU(3) Yang-Mills The running coupling g of QCD is characterized by two important features: asymptotic freedom ( g 0 in the UV) and confinement ( g in the IR.) These properties are strongly dependent on the number of fermion flavors, N f : Short-distance (UV) Long-distance (IR) 0 < N f < Nf c Free (g 0) Confined (g ) Nf c < N f < 16.5 Free (g 0) Fixed point (g g ) N f > 16.5 Divergent (g ) Trivial (g 0) Theories in the first row of the table are useful for building models of DEWSB. Theories in the second row of the table lie in the conformal window.

7 / :. - )*+, "#$%&' (! }~ klmnopqrstuvwxyz{j ^_àbcdefghi ]\ Dynamical Electroweak Symmetry Breaking Introduction (DEWSB) The conformal window of Yang-Mills Background Theories Lattice Strong Simulation Dynamics for and the Results LHC The beta function The conformal window Scale dependence Fixing A cartoon a reference of dynamical scale scales in SU(N) Yang-Mills g(l) g c L c The QCDpoint is a one-scale at which theory theory where confines L c is confinement provides us with scale. a scale, L c. L

8 / :. - )*+, "#$%&' (! }~ klmnopqrstuvwxyz{j ^_àbcdefghi ]\ Dynamical Electroweak Symmetry Breaking Introduction (DEWSB) The conformal window of Yang-Mills Background Theories Lattice Strong Simulation Dynamics for and the Results LHC The beta function The conformal window Scale dependence Fixing A cartoon a reference of dynamical scale scales in SU(N) Yang-Mills g(l) g c L Increasing N ff slightly pushes doesn t confinement change scale much. to longer distances.

9 / :. - )*+, "#$%&' (! }~ klmnopqrstuvwxyz{j ^_àbcdefghi ]\ Dynamical Electroweak Symmetry Breaking Introduction (DEWSB) The conformal window of Yang-Mills Background Theories Lattice Strong Simulation Dynamics for and the Results LHC The beta function The conformal window Scale dependence Fixing A cartoon a reference of dynamical scale scales in SU(N) Yang-Mills g(l) g c L I This For large time, Nwe f canappropriate fix a scale scale L I using is the the inflection point point. L I. L

10 / :. - )*+, "#$%&' (! }~ klmnopqrstuvwxyz{j ^_àbcdefghi ]\ Dynamical Electroweak Symmetry Breaking Introduction (DEWSB) The conformal window of Yang-Mills Background Theories Simulation and Results The beta function The conformal window Scale dependence Fixing A cartoon a reference of dynamical scale scales in SU(N) Yang-Mills g(l) g c...and A walking we have theory L c! Two can have distinct both scalesarise L I and naturally. L c. L I L c L

11 U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { } ~ Dynamical Electroweak Symmetry Breaking (DEWSB) Setup and Methods Schrödinger Functional The Schrödinger Functional x 4 T = L Schrödinger Functional x i Schrödinger Functional E k simulations simulations introduce introduce Dirichlet Dirichlet boundaries boundaries in time; in boundary time; boundary gauge gauge fields fields are chosen are chosen to give to give a constant a constant chromoelectric chromoelectric T =0 background background field. field. Running coupling Running coupling The The SF running SF running coupling coupling g 2 (L) is g 2 defined (L) is to defined vary inversely to vary with inversely the with the response response of theofaction action to the to strength the strength η of the background η of the background field, field, ds dη = k ds g 2 (L). dη = k g 2 η=0 (L) η=0 Ethan Neil (Yale) Conformal Window in QCD-like Theories May 2, / 13

12 Results for N f = 8 SU(3) Yang-Mills Step scaling results, Nf 8 2-loop univ. 3-loop SF g 2 L Log L L 0 No evidence for an IR fixed point at N f = 8. Shaded area is systematic error of continuum extrapolation. 1 T. Appelquist, G. Fleming and E. Neil, Phys. Rev. Lett. 100, (2008)

13 Results for N f = 12 SU(3) Yang-Mills Step scaling results, Nf 12 2-loop univ. 3-loop SF g 2 L Log L L 0 The conformal window extends to N f = 12. Running consistent with 3-loop PT in SF scheme. 2 T. Appelquist, G. Fleming and E. Neil, Phys. Rev. Lett. 100, (2008)

14 Status of conformal window studies Lower edge of conformal window constrained: 8 < N c f < 12. No evidence yet for walking at N f = 8. N f = 10 studies getting underway. Other lattice groups working with fermions in different representations. But, model builders need to know more than just if a theory confines at long distances.

15 Lattice Strong Dynamics (LSD) Collaboration J. C. Osborn Argonne National Laboratory R. Babich, R. C. Brower, M. A. Clark, C. Rebbi, D. Schaich Boston University M. Cheng, T. Luu, R. Soltz, P. M. Vranas Lawrence Livermore National Laboratory T. Appelquist, G. T. Fleming, M. Kastoryano, E. T. Neil Yale University Albert Hofmann

16 Workshop on Lattice Gauge Theory for LHC Physics May 2 3, 2008, Wente Vineyards, Livermore, CA

17 Flavor dependence of confined SU(3) Yang-Mills A few years ago, the RIKEN-BNL-Columbia ( RBC) lattice collaboration began simulating QCD with full chiral symmetry. As chiral symmetry plays a crucial role in the walking conjecture, the LSD collaboration would like to follow RBC s lead and simulate Yang-Mills at larger values of N f. Cost of calculation (N f /2) 3/2. Moore s law: relative cost of N f = 4, 6 today are same as N f = 2 was 2.25 or 3.5 yrs ago, resp. Initial focus on light hadron spectrum, pion decay constant, static potential, and ChiPT LEC l 5.

18 Flavor dependence of confined SU(3) Yang-Mills SU(3) Yang-Mills with N f =2 flavors is quite familiar. SU(3) Yang-Mills with N f =4 flavors is not too different? The non-abelian Coulomb phase exists for N f > 8. What is the nature of the quantum phase transition 3 as N f Nf c? Is it first-order or continuous? lim M/Λ = 0. N f Nf c M represents the scale of confinement, e.g. nucleon mass M N, and Λ represents some UV scale, e.g. g 2 (Λ) = 1.0. If continuous, approximate scale invariance or walking may occur for 8 N f N c f. 3 S. Sachdev, Quantum Phase Transitions, Cambridge Univ. Press, 2000.

19 3-loop SF running coupling in Yang-Mills 3 loop SF running coupling with N f light fermions g 2 (T -1 c ) 10 N f = 0 N f = 2 N f = 4 N f = 6 N f = 8 N f = 10 N f = 12 g 2 L 8 6 g 2 (r 0 ) 4 2 L/a= Log L L 0 (Ethan Neil, Yale U.) M/Λ 0 as N f Nf c as a natural consequence of the N f dependence of running.

20 Final Thoughts: DEWSB, LHeC and triple-boson vertices. In DEWSB, the vector bosons are composite GB s, like pions. LHC can produce resonances but cannot say much about the structure of the new mesons. ILC, if built, can study timelike form factors of the vector bosons. LHeC a, if built, can study spacelike form factors. e + e - p + e - γ,z F W (Q 2 ) W γ,z W W + F W (Q 2 ) W - e - a ν