WALKING VS. CONFORMAL RESULTS FROM THE SCHR ODINGER FUNCTIONAL - - PowerPoint PPT Presentation

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 Nf = 2 fermions in the SYM2 rep

• 1. Confining or conformal? And what lies in between
• 2. The running coupling at m = 0: Schr¨
• dinger Functional (= background field method)
• 3. Phase diagrams on a finite lattice (m, “T” = 0)
• 4. Mass anomalous dimension γ(g2)

POSSIBILITIES for IR PHYSICS

• Confinement & χSB =

⇒ 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 ∼ m1/ym

q

with the same ym. Hard to tell the two apart.

• Gauge coupling is irrelevant; mq and 1/L are relevant couplings.

mq → 0: really, really BAD finite-size effects. Schr¨

• dinger functional turns finite volume from a hindrance to a method.

GAUGE GROUPS, REPs, and Nf (Dietrich & Sannino, PRD 2007)

ADJ F AS SYM

Our work: N = 2, 3, 4; REP=SYM=3, 6, 10; Nf = 2 Is there an IRFP? Ladder approx says NO

THE β FUNCTION in the MASSLESS THEORY: the Schr¨

• dinger Functional

Continuum SF definition of g(L): (L¨

uscher et al., ALPHA collaboration)

• Hypercubical Euclidean box, volume L4, massless limit
• Fix the gauge field on the two time boundaries

⇒ background field — unique classical minimum of Scl

Y M =

• d4x F 2

µν. Make sure L is the only scale.

• Calculate (if you can)

Γ ≡ − log Z = tree-level + one-loop + · · · =

• 1

g2(1/µ) + b1 32π2 log(µL) + · · ·

• Scl

Y M

≡ 1 g2(L)Scl

Y M

nonperturbatively!

THE β FUNCTION in the MASSLESS THEORY: the Schr¨

• dinger Functional

Continuum SF definition of g(L): (L¨

uscher et al., ALPHA collaboration)

• Hypercubical Euclidean box, volume L4, massless limit
• Fix the gauge field on the two time boundaries

⇒ background field — unique classical minimum of Scl

Y M =

• d4x F 2

µν. Make sure L is the only scale.

• Calculate (if you can)

Γ ≡ − log Z = tree-level + one-loop + · · · =

• 1

g2(1/µ) + b1 32π2 log(µL) + · · ·

• Scl

Y M

≡ 1 g2(L)Scl

Y M

nonperturbatively! LATTICE THEORY:

• Wilson fermions

+ clover term + fat links (nHYP = normalized HYPercubic)

• SF: fix spatial links Ui on time boundaries t = 0, L

+ give fermions a spatial twist

A PROPOS CHIRAL SYMMETRY:

• Define mq via AWI

∂µAaµ = 2mqP a = ⇒ mq ≡ 1 2 ∂4

• Ab

4(t) Ob(t′ = 0,

p = 0)

• P b(t) Ob(t′ = 0,

p = 0)

• t=L/2
• Find κc(β) by setting mq = 0. Work directly at κc: stabilized by SF BC’s!

EXTRACTING PHYSICS

• 1. Fix lattice size L, bare couplings β = 6/g2

0, κ ≡ (8 + 2m0a)−1 = κc(β)

• 2. Calculate 1/g2(L) and 1/g2(2L). Use common lattice spacing (= UV cutoff) a.
• 3. Result: Discrete Beta Function

B(u, 2) = 1 g2(2L) − 1 g2(L) , a function of u ≡ 1/g2(L).

The DISCRETE BETA FUNCTION — SU(2)/triplet

0.1 0.2 0.3 0.4 0.5 0.6

u = 1/g

2 (6 4 or 8 4)

• 0.03
• 0.02
• 0.01

0.01 0.02 0.03

B(u,2)

6->12 6->12, shifted κ 2 loops

64 − → 124 S L O W running . . . B(u, 2) crosses zero near the BZ coupling = ⇒ IRFP

The DISCRETE BETA FUNCTION — SU(2)/triplet

0.1 0.2 0.3 0.4 0.5 0.6

u = 1/g

2 (6 4 or 8 4)

• 0.03
• 0.02
• 0.01

0.01 0.02 0.03

B(u,2)

6->12 6->12, shifted κ 8->16 8->16, shifted κ 2 loops

64 − → 124 84 − → 164 S L O W running . . . B(u, 2) crosses zero near the BZ coupling = ⇒ IRFP

SLOW RUNNING IS ALMOST NO RUNNING Let u(s) ≡ 1/g2(s), and ˜ β(u) ≡ du/d log s = 2β(g2)/g4. [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) log s ≃ ˜ β(u(1))

SLOW RUNNING IS ALMOST NO RUNNING Let u(s) ≡ 1/g2(s), and ˜ β(u) ≡ du/d log s = 2β(g2)/g4. [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) log s ≃ ˜ β(u(1)) = ⇒ linear fit to 1/g2(log L) (improved action is crucial)

6 8 12 16

L/a

0.1 0.2 0.3 0.4 0.5

1/g

2

SLOW RUNNING IS ALMOST NO RUNNING Let u(s) ≡ 1/g2(s), and ˜ β(u) ≡ du/d log s = 2β(g2)/g4. [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) log s ≃ ˜ β(u(1)) = ⇒ collapse data for different s. ⇒ Reduced DBF R(g2) ≃ ˜ β(g2)

NOW FOR SU(3)/sextet

0.1 0.2 0.3 0.4 0.5 1/g

2

• 0.06
• 0.04
• 0.02

0.02 R(g

2)

plaquette action

Fits from L = 6, 8, 12, 16 S L O W running . . . but does it cross zero? Why did we stop?

PHASE DIAGRAM: (SU(3)/sextet)

q = 0 m

κ β κc critical point

q < 0 m

No confining

1st

non−conf.

1st

THE WALL in strong coupling: mq discontinuous in κ, never zero

• cf. SU(3) with large Nf fund rep

(Iwasaki, Kanaya, Kaya, Sakai, and

Yoshie 1992, 2003)

[cf. SU(2)/triplet: critical point at intersection]

PHASE DIAGRAM: (SU(3)/sextet)

q = 0 m

κ β κc critical point

q < 0 m

No confining

1st

non−conf.

1st

• Cf. QCD

xover m π = 0 q

κ β κc

q < 0 m

confining non−conf.

t

with N

= 0 2nd m 1st

PHASE DIAGRAM: (SU(3)/sextet)

q = 0 m

κ β κc critical point

q < 0 m

No confining

1st

non−conf.

1st

MOVING THE WALL: Change the gauge action — Sg = β 2Nc

• Tr Up + βf

2df

• Tr Vp

where Vp is made of fat links in the fermion rep (e.g. βf = +0.5)

= ⇒ pushes the wall to stronger coupling:

0.1 0.2 0.3 0.4 0.5 1/g

2

• 0.06
• 0.04
• 0.02

0.02 R(g

2)

mixed action plaquette action

An IRFP in the SU(3)/sextet theory* *at low significance

MASS ANOMALOUS DIMENSION Expected: γ(g2

∗) → 1 at sill of conformal window

(Cohen & Georgi 1988; Kaplan, Lee, Son, Stephanov 2010) Work with correlation functions on lattice:

• P b(t) Ob(t′ = 0)
• t=L/2 = ZP ZO e−mπL/2
• Ob(t = L) Ob(t′ = 0)
• = Z2

O e−mπL

Take ratio, extract ZP (L), whence ZP (L) ZP (L0) = L L0 −γ assuming γ ≃ const as L0 → L, since the running is S L O W

MASS ANOMALOUS DIMENSION — SU(2)/triplet slope = −γm(g2) = ⇒

• Cf. one loop: γ = 6C2(R)

16π2 g2

MASS ANOMALOUS DIMENSION — SU(3)/sextet

6 8 12 16 8 16

L

0.1 0.2 0.3

Z

β=5.8 β=5.4 β=5.0 β=4.8 β=4.6 β=4.4 β=4.3

Mass renormalization

slope = −γm(g2) = ⇒

• Cf. one loop: γ = 6C2(R)

16π2 g2

FINALLY, SU(4)/decuplet — compare all 3 theories

0.1 0.2 0.3

u = 1/(g

2N)

• 2
• 1

1 2

8π

2 b

~(u)

N = 2 N = 3 N = 4 N = ∞

beta function ˜ b ≡ d d log s

• 1

g2N

• 10

20 30 40

g

2N

0.2 0.4 0.6 0.8 1

γm

N = 2 N = 3 N = 4 N = ∞

γ − → ∼ 0.45 — new universality?

SUMMARY

• 1. SU(2) gauge theory with Nf = 2 fermions in the SYM2 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 Nf = 2 fermions in the SYM2 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.