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

the DNA of asymptotic safety Daniel F Litim based on work with Andrew Bond and Tugba Buyukbese ICTP 22 Sep 2016

standard model 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 local QFT for fundamental interactions strong nuclear force weak force electromagnetic force open challenges many... what comes beyond the SM? how does gravity fit in?

asymptotic freedom

asymptotic freedom triumph of QFT

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

asymptotic asymptotic freedom safety

asymptotic safety idea: some or all couplings achieve Wilson ’71 interacting UV fixed point Weinberg ’79 if so, new opportunities for BSM physics & quantum gravity proof of existence: 4D gauge-Yukawa theory with exact asymptotic safety Litim, Sannino, 1406.2337 (see talk by F Sannino)

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

asymptotic safety Bond, Litim 1608.00519 results *) *) *) provided certain auxiliary conditions hold true

fixed points gauge theory g 2 β = − B α 2 + C α 3 + O ( α 4 ) α = (4 π ) 2 0 < α ∗ = B/C ⌧ 1 weakly coupled fixed point competition between matter and gauge fields ✓ ◆ B = 2 2 − 1 11 C G 2 − 2 S F 2 S S 2 3 ✓ 10 ◆ ✓ 1 ◆ � 2 − 34 3 C G 2 + 2 C F S F 3 C G 2 + 2 C S S S 3 ( C G 2 ) 2 C = 2 2 + 2 2

fixed points gauge theory g 2 β = − B α 2 + C α 3 + O ( α 4 ) α = (4 π ) 2 0 < α ∗ = B/C ⌧ 1 weakly coupled fixed point competition between matter and gauge fields asymptotic freedom B, C > 0 : Caswell-Banks-Zaks IR FP asymptotic safety UV FP B, C < 0 : no examples

fixed points gauge theory g 2 β = − B α 2 + C α 3 + O ( α 4 ) α = (4 π ) 2 UV fixed point C = 2 h i 2 S F 11 C F 2 + 7 C G + 2 S S 11 C S 2 − C G − 17 B C G � � � � 2 2 2 2 2 11 fermions scalars 1-loop no go theorem, case a) Caswell ’74

fixed points gauge theory g 2 β = − B α 2 + C α 3 + O ( α 4 ) α = (4 π ) 2 UV fixed point C = 2 h i 2 S F 11 C F 2 + 7 C G + 2 S S 11 C S 2 − C G − 17 B C G � � � � 2 2 2 2 2 11 fermions scalars 1-loop 2 < 1 C S 11 C G must have 2

quadratic Casimirs C 2 ( Λ ) = 1 highest weights, 2( Λ , Λ + 2 δ ) Racah formula δ = (1 , 1 , . . . , 1) G ij u i v j X ( u, v ) ≡ weight metrics G ij ( Λ k ) i = δ i fundamental weights k

quadratic Casimirs result

quadratic Casimirs 1.0 1 E 8 χ = min C 2 ( R ) C 2 (adj) E 7 19 0.8 24 E 6 13 18 F 4 Χ 2 3 SO H N L 0.6 7 12 G 2 SU H N L 1 2 Sp H N L 0.4 3 8 5 10 15 20 N

no go theorems I 2 < 1 C S 11 C G must have 2 2 ≥ 3 C S 8 C G instead, we find 2 B ≤ 0 C > 0 ⇒ implication: no go theorem, case b)

no go theorems II β a = α 2 a ( − B a + C ab α b ) + O ( α 4 ) more gauge factors B a = C ab α ∗ interacting fixed points b ⇣ ⌘ C F b + C S b 2 S F a 2 S S a C ab = 4 ( a 6 = b ) non-trivial mixing 2 2 C ab ≥ 0 positive definite implication: B a ≤ 0 C ab ≥ 0 for all b ⇒ no go theorem, case c)

asymptotic safety result *) *) *) provided certain auxiliary conditions hold true

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) ∼ 1 β = α 2 ( − B + C α − 2 Y 4 ) 2( Y A ) JL φ A ψ J ζ ψ L 2 Y A ( Y A ) † ] /d ( G ) ≥ 0 Y 4 = Tr[ C F Yukawa’s slow down the running of the gauge B → B 0 = B + 2 Y ⇤ induced shift 4 > B α ⇤ = B 0 fixed point C

Yukawa couplings β A = E A ( Y ) − α F A ( Y ) Yukawa couplings Y A Gauss Yukawa nullclines β A = 0 ∗ = 0 ∗ = g Y A 4 π C A int. FP

Yukawa couplings β A = E A ( Y ) − α F A ( Y ) Yukawa couplings Y A Gauss Yukawa nullclines β A = 0 ∗ = 0 ∗ = g Y A 4 π C A int. FP ⇒ Y ∗ 4 = D · α 2 C A ( C A ) † ] /d ( G ) ≥ 0 D = Tr[ C F Yukawa contributions modify two-loop gauge term

Yukawa couplings β A = E A ( Y ) − α F A ( Y ) Yukawa couplings β = α 2 ( − B + C α − 2 Y 4 ) Y ∗ 4 = D · α ! = α 2 ( − B + C 0 α ) C → C 0 = C − 2 D < C induced shift

Yukawa couplings β A = E A ( Y ) − α F A ( Y ) Yukawa couplings β = α 2 ( − B + C α − 2 Y 4 ) Y ∗ 4 = D · α ! = α 2 ( − B + C 0 α ) C → C 0 = C − 2 D < C induced shift α ⇤ = B/C 0 reliable fixed point even if B<0 and C>0 impossible without Yukawa’s C 0 < 0 necessary condition for asymptotic safety case d)

more gauge couplings β a = α 2 a ( − B a + C ab α b − 2 Y 4 ,a ) B a → B 0 a = B a + 2 Y ⇤ Yukawa-induced shift 4 ,a B 0 a = C ab α ⇤ fixed points b novel solutions including UV FPs if B<0 and B’>0 Yukawas may compensate gauge contributions necessary condition for B 0 a > 0 asymptotic safety case e)

asymptotic safety result *) *) *) provided certain auxiliary conditions hold true

interacting FPs

phase diagrams phase diagrams of simple gauge theories B, C matter content parameters C 0 Yukawa structure

phase diagrams Y 4 G ( B > 0) �

phase diagrams Y 4 Y 4 G ( B, C > 0) G BZ � �

phase diagrams Y 4 Y 4 GY G G ( B, C, C 0 > 0) BZ � �

phase diagrams GY Y 4 Y 4 G G ( B < 0 , C > 0 > C 0 ) � �

extensions I interacting UV FPs with exact asymptotic safety exist for simple gauge theories Litim, Sannino, 1406.2337 but: do interacting UV FPs with exact asymptotic safety exist for semi-simple gauge theories? Yes! (talk by Andrew Bond) space of UV FP solutions is non-empty

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

extensions II Lagrangean further scalar invariants gauge Nc colours Nf flavours Yukawa Higgs Nf times Nf Litim, Sannino, 1406.2337 2 ( ) Buyukbese, Litim (in prep.)

extensions II results: exact eigenvalue spectrum ✓ n = D n + O ( ✏ ) n

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 Yukawa interactions pivotal for asymptotic safety asymptotic safety persist beyond canonically marginal invariants window of opportunities for BSM