the DNA of asymptotic safety Daniel F Litim based on work with - - PowerPoint PPT Presentation

Daniel F Litim

the DNA of asymptotic safety

based on work with Andrew Bond and Tugba Buyukbese ICTP 22 Sep 2016

local QFT for fundamental interactions strong nuclear force weak force electromagnetic force degrees of freedom spin 0 (the Higgs has finally arrived) spin 1/2 (quite a few) spin 1 perturbatively renormalisable & predictive

standard model

standard model

local QFT for fundamental interactions strong nuclear force weak force electromagnetic force

• pen challenges

what comes beyond the SM? how does gravity fit in? many...

asymptotic freedom

asymptotic freedom

triumph

• f QFT

asymptotic freedom

complete asymptotic freedom in 4D scalars scalars with fermions non-Abelian gauge fields non-Abelian fields with fermions non-Abelian fields, fermions, scalars all couplings achieve non-interacting UV fixed point no no yes yes* yes* fields cAF

*) provided certain conditions hold true

asymptotic freedom asymptotic safety

asymptotic safety

some or all couplings achieve interacting UV fixed point

Wilson ’71 Weinberg ’79

if so, new opportunities for BSM physics & quantum gravity 4D gauge-Yukawa theory with exact asymptotic safety

Litim, Sannino, 1406.2337 (see talk by F Sannino)

idea: proof of existence:

asymptotic safety

weakly interacting fixed points of general 4d gauge theories conditions for asymptotic safety today:

asymptotic safety

*) provided certain auxiliary conditions hold true *) *)

results

Bond, Litim 1608.00519

gauge theory competition between matter and gauge fields β = −B α2 + C α3 + O(α4) α = g2 (4π)2 weakly coupled fixed point B = 2 3 ✓ 11CG

2 − 2SF 2 − 1

2SS

2

◆ C = 2 ✓10 3 CG

2 + 2CF 2

◆ SF

2 +

✓1 3CG

2 + 2CS 2

◆ SS

2 − 34

3 (CG

2 )2

• 0 < α∗ = B/C ⌧ 1

fixed points

competition between matter and gauge fields β = −B α2 + C α3 + O(α4) α = g2 (4π)2 B, C > 0 : B, C < 0 : asymptotic freedom Caswell-Banks-Zaks IR FP asymptotic safety UV FP no examples gauge theory weakly coupled fixed point 0 < α∗ = B/C ⌧ 1

fixed points

β = −B α2 + C α3 + O(α4) α = g2 (4π)2 UV fixed point gauge theory C = 2 11 h 2SF

2

• 11CF

2 + 7CG 2

• + 2SS

2

• 11CS

2 − CG 2

• − 17B CG

2

i fermions scalars 1-loop no go theorem, case a)

fixed points

Caswell ’74

β = −B α2 + C α3 + O(α4) α = g2 (4π)2 UV fixed point gauge theory C = 2 11 h 2SF

2

• 11CF

2 + 7CG 2

• + 2SS

2

• 11CS

2 − CG 2

• − 17B CG

2

i CS

2 < 1

11CG

2

fermions scalars 1-loop must have

fixed points

fundamental weights highest weights, Racah formula

quadratic Casimirs

C2(Λ) = 1 2(Λ, Λ + 2δ)

(u, v) ≡ X

ij

Gij ui vj

(Λk)i = δi

k

weight metrics G

δ = (1, 1, . . . , 1)

quadratic Casimirs

result

quadratic Casimirs

E8 E7 E6 F4 G2 SOHNL SpHNL SUHNL

5 10 15 20 0.4 0.6 0.8 1.0

3 8 1 2 2 3 7 12 13 18 19 24

1 N Χ

χ = min C2(R) C2(adj)

no go theorems I

CS

2 < 1

11CG

2

must have CS

2 ≥ 3

8CG

2

instead, we find

B ≤ 0 ⇒ C > 0

implication: no go theorem, case b)

no go theorems II

more gauge factors non-trivial mixing implication: no go theorem, case c) βa = α2

a (−Ba + Cab αb) + O(α4)

Cab = 4 ⇣ CFb

2 SFa 2

+ CSb

2 SSa 2

⌘ (a 6= b) positive definite

Cab ≥ 0

Ba ≤ 0 ⇒ Cab ≥ 0 for all b interacting fixed points Ba = Cab α∗

b

asymptotic safety

*) provided certain auxiliary conditions hold true *) *)

result

matter couplings

scalar self-couplings No (start at 3- or 4-loop)

Yukawa couplings

scalar self-couplings No (start at 3- or 4-loop) Yukawa couplings Yes (start at 2-loop)

Yukawa couplings

scalar self-couplings No (start at 3- or 4-loop) Yukawa couplings Yes (start at 2-loop) β = α2 (−B + C α − 2 Y4) Yukawa’s slow down the running of the gauge Y4 = Tr[CF

2 YA (YA)†]/d(G) ≥ 0

∼ 1 2(YA)JLφA ψJ ζ ψL

fixed point α⇤ = B0 C B → B0 = B + 2Y ⇤

4 > B

induced shift

Yukawa couplings

Yukawa couplings βA = EA(Y ) − α FA(Y ) Yukawa nullclines βA = 0 YA

∗ = g

4π CA YA

∗ = 0

Gauss

• int. FP

Yukawa couplings

Yukawa couplings Yukawa contributions modify two-loop gauge term βA = EA(Y ) − α FA(Y ) Yukawa nullclines βA = 0 YA

∗ = g

4π CA ⇒ Y ∗

4 = D · α

D = Tr[CF

2 CA (CA)†]/d(G) ≥ 0

YA

∗ = 0

Gauss

• int. FP

Yukawa couplings

Yukawa couplings induced shift βA = EA(Y ) − α FA(Y ) C → C0 = C − 2D < C β = α2 (−B + C α − 2 Y4) = α2(−B + C0α) !

Y ∗

4 = D · α

Yukawa couplings

Yukawa couplings reliable fixed point induced shift βA = EA(Y ) − α FA(Y ) C → C0 = C − 2D < C β = α2 (−B + C α − 2 Y4) α⇤ = B/C0 necessary condition for asymptotic safety

C0 < 0

= α2(−B + C0α) !

Y ∗

4 = D · α

even if B<0 and C>0 impossible without Yukawa’s case d)

more gauge couplings

Yukawas may compensate gauge contributions fixed points Yukawa-induced shift βa = α2

a (−Ba + Cab αb − 2 Y4,a)

Ba → B0

a = Ba + 2 Y ⇤ 4,a

B0

a = Cabα⇤ b

novel solutions including UV FPs if B<0 and B’>0 necessary condition for asymptotic safety

B0

a > 0

case e)

asymptotic safety

*) provided certain auxiliary conditions hold true *) *)

result

interacting FPs

phase diagrams

phase diagrams of simple gauge theories parameters B, C C0 matter content Yukawa structure

phase diagrams

• Y4

G (B > 0)

phase diagrams

• Y4

G

• Y4

G BZ (B, C > 0)

phase diagrams

• Y4

G

• Y4

GY G BZ (B, C, C0 > 0)

phase diagrams

• Y4

G

• Y4

GY G (B < 0, C > 0 > C0)

extensions I

but: do interacting UV FPs with exact asymptotic safety exist for semi-simple gauge theories? (talk by Andrew Bond) Yes! space of UV FP solutions is non-empty interacting UV FPs with exact asymptotic safety exist for simple gauge theories

Litim, Sannino, 1406.2337

extensions II

what is the impact of higher-dimensional invariants? tool: functional RG (see poster by Tugba Buyukbese) fixed point persists effective potential remains stable results:

Lagrangean gauge Yukawa Higgs Nc colours Nf flavours Nf times Nf

extensions II

further scalar invariants

Litim, Sannino, 1406.2337

2

( )

Buyukbese, Litim (in prep.)

extensions II

results: exact eigenvalue spectrum n

✓n = Dn + O(✏)

conclusions

identified all weakly interacting fixed points of general 4D gauge theories - rich spectrum strict no go theorems together with necessary and sufficient conditions for asymptotic safety for general 4D gauge theories asymptotic safety persist beyond canonically marginal invariants Yukawa interactions pivotal for asymptotic safety window of opportunities for BSM