Confining and conformal models Lecture 4 Hasenfratz University of Colorado, Boulder 2011 Schladming Winter School
Technicolor models are candidates for dynamical electroweak symmetry breaking gauge coupling must be walking need large anomalous mass dimension 4-fermion interactions are natural in extended TC Simplest models are just gauge + many fermions
Fermion-gauge system The RG β function could be: QCD-like IRFP(conformal) walking What is the dependence on N f?
Roadmap: Many-fermion models Schwinger-Dyson approach
Many-fermion models Functional RG Braun, Gies SU(3) with fundamental fermions conformal window is at N f =10.0 +1.6/ -0.7
The lattice phase diagram confining m conformal =6/g 2 Critical line (arrows: UV to IR)
Spectrum Can t we use the hadron (bound state) spectrum to distinguish confining and conformal? M(bound state) vs m fermion : (β fixed, m varies) M β fixed in QCD M β >β fixed m N m rho m N m rho m pion m pion m m
Spectrum Can t we use the hadron (bound state) spectrum to distinguish confining and conformal? M(bound state) vs m fermion : Conformal M β fixed M β >β fixed m N m N m rho m rho m pion m pion m m In the conformal system β is irrelevant; all masses 0 as m 0
Spectrum Can t we use the hadron (bound state) spectrum to distinguish confining and conformal? M(bound state) vs m fermion : **Except** in a finite volume QCD deconfines M QCD confined M QCD deconfined m N m rho m N m rho m pion m pion m m In the deconfined phase of QCD the mesons are degenerate, all hadrons 0 as m 0
Spectrum The spectrum of QCD in small volume is similar to a conformal one. They scale differently, but distinguishing them is hard. Same applies to quark potential, string tension as well. (DelDebbio et al) RG techniques work in small volumes (relative to the correlation length) since we are looking at short distance continuum behavior
RG methods on the lattice II We need different methods with different systematical issues to control & understand the systems at the conformal/confining boundary 1. 2-lattice matching MCRG 2. Schroedinger functional method
The step scaling function around a UVFP The bare differential step scaling function s b ( ) s b ( ) = - where»( ) =»( )/s ( =2N c /g 0 2 ) s b ( ) versus the RG β function: s b ( ) <0 where β(g)>0 s b ( ) =0 where β(g)=0 s b ( ) const at small g 2
»=1 2-lattice matching: Do simulations at β and β (m=0) RG block and compare the blocked actions if S( β (n) )= S( β (n-1) )--> ξ(β)=2ξ(β ) the step scaling function is s b (β )=lim nb 1 (β - β ) The location of the FP on the critical surface depends on the RG transformation Tuning free parameters in the RG transformation can pull the FP and its RT close, reducing systematical errors Along a relevant direction s b (β) is universal (up to lattice artifacts)
Matching the actions == matching the configurations Lattice volume Configurations at β Configurations at β Lattice volume 16 d {S} block block 8 d {S (1) } {S } 8 d block block block block 4 d {S (2) } {S (1) } 4 d block block block block 2 d {S (3) } {S (2) } 2 d Blocking eats up the volume and give new results only in factor 2 quanta
Schroedinger functional method Define a renormalized coupling as a response to some boundary field: L t Fixed boundary conditions η(t=0), η(t=l) The scale (energy) is given by the lattice size; g 2 (L) runs as the lattice size increases, g 0 fixed; Continuum limit: tune g 0 0! g 2 (L) = K d ln(z) $ # & " d! % g 2 (L) can be connected to some continuum renormalization scheme (MOM) perturbatively '1!=0
Schroedinger functional step scaling function g 2 (L) runs with L; at some point L will be too large to simulate Change g 0 and L together: β β ξ =ξ/b a =ba Calculate g 2 (L) ξ a Calculate g 2 (L) } In the continuum limit g 2 (L/b) = g 2 (L) Step scaling function s(g 2 (L))=g 2 (L/b) As β, L the lattice artifacts are removed and we get an infinite cutoff running coupling.
Schrodinger functional was defined to calculate the running coupling at the g 0 =0 continuum limit. In conformal theories we want g 2 near the IRFP, not tuned around the Gaussian UVFP The step scaling function still can be used to find the running coupling, though backward running is tricky IRFP Gaussian Calculate g 2 (L) on the other side to find backward Running. But there is no UVFP to tune to! Tune to g 0 0 to find g 2 (L) up to the IRFP
MCRG vs Schrodinger functional I am biased.. Schrodinger functional Can be used with many lattice volumes Expensive but precise Has NO built-in checks Theoretically unclear on the back side of an IRFP MCRG Needs volumes increasing by factor of 2 Has a lot of built-in checks Optimization is essential, but there are many RG transformations! Works with bare quantities, understandable based on Wilson RG
The 3 Renormalization Group transformations A real space block transformation averages out the short distance modes Many possibilities - I tried 3 types: Original HYP optimize with HYP2 like HYP, but with twice blocked links optimize with 1 (play with 2, 3 )
Some illustrative results Most are still in progress
SU(3) pure gauge The bare step scaling function can be calculated in many ways - Schrodinger fn; Wilson loop ratios, - physical observables r 0, T c - RG matching: 32 4 16 4 and 16 4 8 4 Perturbative Excellent agreement between r 0, T c and MCRG Both SF and MCRG approach the perturbative value Since at =6 we can test confinement, we know there is no physical IRFP
Compare different RG transformations: When the flow is governed by a UVFP, s b ( ) is universal (up to lattice corrections). Compare 3 different RG transformations: Excellent agreement between the 3 RG blockings attractive region of a UVFP
N f =16 flavors Expected to be conformal 16 4 8 4 MCRG ORIG blocking shows s b ( )=0 around =7.0 HYP blocking has an IRFP around =8.0 Different block transformations predict different s b ( )=0 but they both show an IRFP in the function
N f =8 flavors Thought to be QCD-like (analytical & numerical results) Compare the different RG transformations (m 0) s b >0 everywhere - no IRFP Is it confining? Look at the anomalous mass dimension String tension
N f =8 flavors, anomalous mass 4 different couplings ( =4.8,5.0,5.8,6.0), optimal RG from m=0 data y m =2 m 2 =m 1 2-1/y m m = y m -1 All 3 couplings predict the same value y m = 1.02(5) close to free field exponent
N f =12 fundamental fermions, SU(3) gauge Very controversial Schrodinger functional method, Unimproved staggered fermions Backward flow found conformal Appelquiest et al Yale group
N f =12 flavors with MCRG Use the same techniques as before; s b (β) hovers around 0 IRFP? matching on 16 4-8 4, 32 4 16 4 show large lattice artifacts On the other hand spectral measurements suggest confining, chiral broken behavior (Kuti et al)
SU(2) gauge, adjoint fermions Schroedinger functional results: Unimproved Wilson action results are inconclusive HYP smeared improved Wilson action shows IRFP Discrete β function vs 1/g 2 Degrand, Shamir, Swetitsky
Summary & Outlook Gauge-fermion and gauge-fermion+4 fermion systems are important Can describe dynamical electroweak symmetry breaking Can show a non-trivial UVFP in 4 dimensions But near the conformal-confining boundary they are difficult. We need many different methods: analytical, numerical, RG or not These are exciting times
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Optimization of the RG transformations = - at =7.0 as the function of the RG parameter α Optimizing the RG transformation is essential opt =0.65 Optimized RG gives the same matching value at each level, for each operator a( =7.0) = a( =6.49)/2 s b = - =0.51
The plaquette* after 1-4 levels of blocking 32 4 16 4 8 4 4 4 2 4 (symbols) compare to 16 4 8 4 4 4 2 4 (lines) Repeat with many different operators. If they all give the same result, we found matching s b ( =7.0) = = - = 7.0-6.49 = 0.51 *plaquette: Tr(U )