Sextet QCD: slow running and the mass anomalous dimension

Transcription

1 Lattice 2010, Sardinia Sextet QCD: slow running and the mass anomalous dimension B. Svetitsky Tel Aviv University with Y. Shamir and T. DeGrand SU(3) gauge theory with N f = 2 fermions in the 6 rep Wilson clover fermions, nhyp fat links 1. In the conformal window? 2. Phase diagram on a finite lattice (m,t 0) 3. The running coupling at m = 0: Schrödinger Functional = runs slowly; = IRFP? Inconclusive 4. Mass anomalous dimension γ(g 2 ): Too small for walking technicolor! (arxiv: )

2 MAPPING THE CONFORMAL WINDOW (Dietrich & Sannino, PRD 2007) F AS SYM ADJ Our work: N = 3, REP=SYM=6, N f = 2 Is there an IRFP? Ladder approx says NO: g 2 10.

3 PHASE DIAGRAM: m q < 0 m q = 0 Bulk / finite temperature transition κ confining κ c FIRST ORDER No critical point non conf. cf. SU(3) with large N f fund rep (Damgaard, Heller, Krasnitz, Olesen 1997) (Iwasaki et al. 2002) β

4 PHASE DIAGRAM: Cf. QCD m q < 0 m q < 0 κ No critical point confining m q = 0 non conf. κ c κ 2nd m = 0 π mq = 0 confining xover non conf. with N t κ c β β

5 AS N t INCREASES: Cf. QCD doesn t move? m q < 0 m q < 0 κ confining m q = 0 non conf. with N t κ c κ 2nd m = 0 π mq = 0 confining xover non conf. with N t κ c β β

6 AS N t INCREASES: doesn t move? m q < 0 m q = 0 Intersection point moves slowly (or not at all) κ confining non conf. with N t κ c = slow running (or conformal phase on κ c line!) β

7 The DISCRETE BETA FUNCTION (2008) 0.04 thin links 4->8 2 loops B(u,2) Thin links B(u,2) crosses zero at g a weak coupling = IRFP u = 1/g 2 (4 4 )

8 The DISCRETE BETA FUNCTION ( ) thin links 4->8 fat links 6->12 2 loops FAT links B(u,2) Rules out old IRFP 2. Go to stronger coupling? Stopped by order phase transition! 3. No clear IRFP 4. SLOW running u = 1/g 2 (4 4 or 6 4 )

9 The DISCRETE BETA FUNCTION ( ) 0.04 thin links 4->8 fat links 6->12 fat links 8->16 2 loops B(u,2) FAT links Not much help u = 1/g 2 (4 4, 6 4, or 8 4 )

10 How SLOW is SLOW? g 2 (L) one loop β=8.0 β=5.8 β=5.4 β=5.0 β=4.8 β=4.6 β=4.4 β=4.3 < 15% variation as L = 6 16 Remember this a/l

11 MASS ANOMALOUS DIMENSION A technicolor theory needs to enhance the techniquark condensate from Λ TC Λ ETC according to ΨΨ ETC = ΨΨ [ ] ΛET C exp dµ TC Λ T C µ γ(g2 (µ)) [enhance m q while suppressing FCNC] Walking technicolor: g 2 g 2 nearly constant during running, so ΨΨ ETC ΨΨ ( )γ(g 2 TC ΛETC ) Λ TC WANTED: γ(g 2 ) = 1 (or very close to 1) (Chivukula & Simmons, arxiv: ) What does the lattice say about this?

12 MASS ANOMALOUS DIMENSION the calculation (Bursa et al. arxiv: ) Correlation functions on lattice: P b (t) O b (t = 0) t=l/2 = Z P Z O e m πl/2 O b (t = L) O b (t = 0) = Z 2 O e m πl Take ratio, extract Z P (L), whence ( ) γ Z P (L) L Z P (L 0 ) = L 0 assuming γ const as L 0 L, since the coupling (almost) doesn t run: g 2 (L) one loop β=8.0 β=5.8 β=5.4 β=5.0 β=4.8 β=4.6 β=4.4 β= a/l

13 MASS ANOMALOUS DIMENSION result Mass renormalization 1 Mass anomalous dimension Z β=5.8 β=5.4 β=5.0 β=4.8 β=4.6 β=4.4 β=4.3 = γ m 0.4 one loop from SF from FSS L g 2 SF slope = γ m (g 2 ) Cf. one loop: γ = 6C 2(R) 16π 2 g 2 (FSS: DeGrand, arxiv: )

14 WHAT HAVE WE LEARNED? Two possibilities: 1. We have NO IR fixed point: order transition will slide towards β = as N t Theory has confinement/χsb β function doesn t approach zero; it s just smaller than two loops, and much smaller than in QCD. The theory doesn t walk (but it runs slowly). γ m 0.6: no good for extended technicolor [cf. SU(2)/adj (Bursa et al. arxiv: )] 2. We have an IR fixed point: order transition (m q = 0) is stuck: can t penetrate conformal phase The zero of the β function is around the corner (change lattice action?) γ m 0.6 at IRFP: The theory is well inside the conformal window.* * since γ m should = 1 at the edge of the window (Cohen & Georgi 1989; Kaplan, Lee, Son, Stephanov arxiv: )