GGI Florence, August 2012 WALKING VS. CONFORMAL — RESULTS FROM THE SCHR ¨ ODINGER FUNCTIONAL METHOD B. Svetitsky Tel Aviv University with Y. Shamir and T. DeGrand SU(2,3,4) gauge theories with N f = 2 fermions in the SYM 2 rep 1. Confining or conformal? And what lies in between 2. The running coupling at m = 0 : Schr¨ odinger Functional ( = background field method) 3. Phase diagrams on a finite lattice ( m, “ T ” � = 0 ) 4. Mass anomalous dimension γ ( g 2 )
POSSIBILITIES for IR PHYSICS • Confinement & χ SB = ⇒ RUNNING [QCD] – or WALKING [ETC — extended technicolor ] • IRFP — conformal theory = ⇒ STANDING STILL [unparticles?] WALKING and IRFP [ the conformal window ] are HARD CASES: • Running is slow — so strong coupling in IR is also strong coupling in UV (i.e., at lattice cutoff) i.e., we require L >>>>>> a for a weak-coupling continuum limit OTHERWISE you are looking at a narrow range of scales! • Scale invariance (approximate for WALKING) means all particle masses ∼ m 1 /y m with the same y m . Hard to q tell the two apart. • Gauge coupling is irrelevant; m q and 1 /L are relevant couplings. m q → 0 : really, really BAD finite-size effects. Schr¨ odinger functional turns finite volume from a hindrance to a method .
GAUGE GROUPS, REPs, and N f ( Dietrich & Sannino, PRD 2007 ) F AS SYM ADJ Our work: N = 2 , 3 , 4 ; REP = SYM = 3 , 6 , 10 ; N f = 2 Is there an IRFP? Ladder approx says NO
THE β FUNCTION in the MASSLESS THEORY: the Schr¨ odinger Functional Continuum SF definition of g ( L ) : ( L¨ uscher et al. , ALPHA collaboration ) • Hypercubical Euclidean box, volume L 4 , massless limit • Fix the gauge field on the two time boundaries ⇒ background field — unique classical minimum of S cl � d 4 x F 2 Y M = µν . Make sure L is the only scale. • Calculate (if you can) Γ ≡ − log Z = tree-level + one-loop + · · · � 1 b 1 � S cl = g 2 (1 /µ ) + 32 π 2 log( µL ) + · · · Y M 1 g 2 ( L ) S cl ≡ nonperturbatively! Y M
THE β FUNCTION in the MASSLESS THEORY: the Schr¨ odinger Functional Continuum SF definition of g ( L ) : ( L¨ uscher et al. , ALPHA collaboration ) • Hypercubical Euclidean box, volume L 4 , massless limit • Fix the gauge field on the two time boundaries ⇒ background field — unique classical minimum of S cl � d 4 x F 2 Y M = µν . Make sure L is the only scale. • Calculate (if you can) Γ ≡ − log Z = tree-level + one-loop + · · · � � 1 b 1 S cl = g 2 (1 /µ ) + 32 π 2 log( µL ) + · · · Y M 1 g 2 ( L ) S cl ≡ nonperturbatively! Y M LATTICE THEORY: • Wilson fermions + clover term + fat links ( nHYP = normalized HYPercubic ) • SF: fix spatial links U i on time boundaries t = 0 , L + give fermions a spatial twist
A PROPOS CHIRAL SYMMETRY: • Define m q via AWI 4 ( t ) O b ( t ′ = 0 , � A b � � ∂ 4 p = 0) � m q ≡ 1 ∂ µ A aµ = 2 m q P a � = ⇒ � P b ( t ) O b ( t ′ = 0 , � � 2 p = 0) � � t = L/ 2 • Find κ c ( β ) by setting m q = 0 . Work directly at κ c : stabilized by SF BC’s! EXTRACTING PHYSICS 0 , κ ≡ (8 + 2 m 0 a ) − 1 = κ c ( β ) 1. Fix lattice size L , bare couplings β = 6 /g 2 2. Calculate 1 /g 2 ( L ) and 1 /g 2 (2 L ) . Use common lattice spacing ( = UV cutoff) a . 3. Result: Discrete Beta Function 1 1 B ( u, 2) = g 2 (2 L ) − g 2 ( L ) , a function of u ≡ 1 /g 2 ( L ) .
The DISCRETE BETA FUNCTION — SU(2)/triplet 6->12 0.03 6->12, shifted κ 2 loops 6 4 − → 12 4 0.02 0.01 B ( u ,2) S L O W running . . . 0 B ( u, 2) crosses zero near the BZ -0.01 coupling -0.02 = ⇒ IRFP -0.03 0 0.1 0.2 0.3 0.4 0.5 0.6 2 (6 4 or 8 4 ) u = 1/ g
The DISCRETE BETA FUNCTION — SU(2)/triplet 6->12 0.03 6->12, shifted κ 8->16 8->16, shifted κ 6 4 − → 12 4 0.02 2 loops 8 4 − → 16 4 0.01 B ( u ,2) S L O W running . . . 0 B ( u, 2) crosses zero near the BZ -0.01 coupling -0.02 = ⇒ IRFP -0.03 0 0.1 0.2 0.3 0.4 0.5 0.6 2 (6 4 or 8 4 ) u = 1/ g
SLOW RUNNING IS ALMOST NO RUNNING Let u ( s ) ≡ 1 /g 2 ( s ) , and ˜ β ( u ) ≡ du/d log s = 2 β ( g 2 ) /g 4 . [We have been plotting B ( u, 2) = u (2) − u (1) .] Slow running: ˜ β ( u ( s )) ≃ ˜ β ( u (1)) — quasi-conformal! Then u ( s ) − u (1) ≃ ˜ β ( u (1)) log s
SLOW RUNNING IS ALMOST NO RUNNING Let u ( s ) ≡ 1 /g 2 ( s ) , and ˜ β ( u ) ≡ du/d log s = 2 β ( g 2 ) /g 4 . [We have been plotting B ( u, 2) = u (2) − u (1) .] Slow running: ˜ β ( u ( s )) ≃ ˜ β ( u (1)) — quasi-conformal! Then u ( s ) − u (1) ≃ ˜ β ( u (1)) log s 0.5 0.4 ⇒ linear fit to 1 /g 2 (log L ) = 2 1/ g 0.3 (improved action is crucial) 0.2 0.1 0 12 8 16 6 L / a
SLOW RUNNING IS ALMOST NO RUNNING Let u ( s ) ≡ 1 /g 2 ( s ) , and ˜ β ( u ) ≡ du/d log s = 2 β ( g 2 ) /g 4 . [We have been plotting B ( u, 2) = u (2) − u (1) .] Slow running: ˜ β ( u ( s )) ≃ ˜ β ( u (1)) — quasi-conformal! Then u ( s ) − u (1) ≃ ˜ β ( u (1)) log s = ⇒ collapse data for different s . ⇒ Reduced DBF R ( g 2 ) ≃ ˜ β ( g 2 )
NOW FOR SU(3)/sextet 0.02 plaquette action 0 Fits from L = 6 , 8 , 12 , 16 S L O W running . . . 2 ) -0.02 R(g but does it cross zero? -0.04 Why did we stop? -0.06 0 0.1 0.2 0.3 0.4 0.5 2 1/g
PHASE DIAGRAM: (SU(3)/sextet) THE WALL m q < 0 in strong coupling: m q = 0 m q discontinuous in κ , never zero 1st κ confining κ c cf. SU(3) with large N f fund rep ( Iwasaki, Kanaya, Kaya, Sakai, and non−conf. Yoshie 1992, 2003 ) 1st No critical point [ cf. SU(2)/triplet: critical point at intersection ] β
PHASE DIAGRAM: (SU(3)/sextet) Cf. QCD m q < 0 m q < 0 m q = 0 m π = 0 1st m = 0 q 2nd κ κ confining κ c confining κ c non−conf. non−conf. 1st 1st with xover No critical point N t β β
PHASE DIAGRAM: (SU(3)/sextet) m q < 0 m q = 0 MOVING THE WALL: 1st κ Change the gauge action — confining κ c β Tr U p + β f � � S g = Tr V p non−conf. 2 N c 2 d f 1st where V p is made of fat links in the No critical point fermion rep (e.g. β f = +0 . 5 ) β
= ⇒ pushes the wall to stronger coupling: 0.02 An IRFP in the SU(3)/sextet theory* mixed action plaquette action 0 2 ) -0.02 R(g -0.04 -0.06 *at low significance 0 0.1 0.2 0.3 0.4 0.5 2 1/g
MASS ANOMALOUS DIMENSION Expected: γ ( g 2 ∗ ) → 1 at sill of conformal window ( Cohen & Georgi 1988; Kaplan, Lee, Son, Stephanov 2010 ) Work with correlation functions on lattice: P b ( t ) O b ( t ′ = 0) t = L/ 2 = Z P Z O e − m π L/ 2 � �� � O b ( t = L ) O b ( t ′ = 0) O e − m π L = Z 2 � � Take ratio, extract Z P ( L ) , whence � L � − γ Z P ( L ) Z P ( L 0 ) = L 0 assuming γ ≃ const as L 0 → L , since the running is S L O W
MASS ANOMALOUS DIMENSION — SU(2)/triplet = ⇒ Cf. one loop: γ = 6 C 2 ( R ) slope = − γ m ( g 2 ) g 2 16 π 2
MASS ANOMALOUS DIMENSION — SU(3)/sextet Mass renormalization 0.3 β=5.8 β=5.4 β=5.0 β=4.8 0.2 β=4.6 β=4.4 β=4.3 Z = ⇒ 0.1 6 8 8 12 16 16 L slope = − γ m ( g 2 ) Cf. one loop: γ = 6 C 2 ( R ) g 2 16 π 2
FINALLY, SU(4)/decuplet — compare all 3 theories 1 2 N = 2 N = 2 N = 3 N = 3 N = 4 N = 4 0.8 N = ∞ N = ∞ 1 0.6 ~( u ) γ m 0 2 b 8 π 0.4 -1 0.2 -2 0 0 10 20 30 40 0 0.1 0.2 0.3 2 N 2 N ) u = 1/( g g d � 1 � beta function ˜ γ − → ∼ 0 . 45 — new universality? b ≡ d log s g 2 N
SUMMARY 1. SU(2) gauge theory with N f = 2 fermions in the SYM 2 rep has an IRFP . SU(3), SU(4) might — at least, they run very slowly. 2. In each case, the mass anomalous dimension γ flattens out well short of 1. THEORETICAL POINTS Schwinger–Dyson eqns say these theories have no IRFP . • Our fixed point(s) contradict the Schwinger–Dyson analysis. SDEs also predict γ ≃ 1 near the sill of the conformal window ( walking technicolor ). • For each N = 2 , 3 , 4 — γ � 0 . 5 means: 1. We are deep in the conformal phase, or 2. S–D eqns, model calculations are inapplicable here, too.
SUMMARY 1. SU(2) gauge theory with N f = 2 fermions in the SYM 2 rep has an IRFP . SU(3), SU(4) might — at least, they run very slowly. 2. In each case, the mass anomalous dimension γ flattens out well short of 1. THEORETICAL POINTS Schwinger–Dyson eqns say these theories have no IRFP . • Our fixed point(s) contradict the Schwinger–Dyson analysis. SDEs also predict γ ≃ 1 near the sill of the conformal window ( walking technicolor ). • For each N = 2 , 3 , 4 — γ � 0 . 5 means: 1. We are deep in the conformal phase, or 2. S–D eqns, model calculations are inapplicable here, too. FOR THE FUTURE γ is much easier to calculate than β . More anomalous dimensions are waiting . . . ( = ⇒ “spectrum” of conformal theories) . . . and also more gauge theories.
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