Francesco Sannino

Exact results in (super) safe QFT &

(The naturally safe Higgs)

In collaboration with: Litim, 1406.2337 Intriligator 1508.07411Bajc 1610.09681Pelaggi, Strumia, Vigiani 1701.01453

Plan• Controllable asymp. safe theory in 4D

• a-theorem for asymptotic safety

• Asymptotically safe thermodynamics and thermal d.o.f. count

• QCD conformal window 2.0 (adding the asymp. safe window)

• Exact nonperturbative results for N=1 supersymmetric safety

• Super GUTs with R-parity

• Naturally safe gauged Higgs

Gauge: SU(3) x SU(2) x U(1) at EW scale

Standard Model

Interactions:

Gauge fields + fermions + scalars

Yukawa: Fermion masses/Flavour

Scalar self-interaction

Fields:

Culprit: Higgs

Gauge - Yukawa theoriesL = !1

2F 2 + iQ!µD

µQ+ y(QLHQR + h.c.)

Tr!DH†DH

"! !uTr

!(H†H)2

"! !vTr

!(H†H)

"2

4D: standard model, dark matter, …

3D: condensed matter, phase transitions

2D: graphene, …

4plusD: extra dimensions, string theory, …

Gauge

Yukawa

Scalar selfinteractions

Universal description of physical phenomena

Standard Model (blind spots)L = !1

2F 2 + iQ!µD

µQ+ y(QLHQR + h.c.)

Tr!DH†DH

"! !uTr

!(H†H)2

"! !vTr

!(H†H)

"2Gauge

Yukawa

Scalar selfinteractions

Gauge structure is established

Yukawa structure partially constrained

Higgs self-coupling is not directly constrained

Unsafe field theory

But it does work well, so far!

Wilson: A fundamental theory has an UV fixed point

Trivial fixed point Interacting fixed point

!!"# !#"$ #"##"#

#"!

#"%

#"&

#"'

()*"!#!!$

""!$

Asymptotic freedom

!!"# !#"$ #"# #"$#"#

#"!

#"%

#"&

#"'

()*"!#!!$

!!""

Asymptotic safety

Non-interacting in the UV

Logarithmic scale depend.

Integrating in the UV

Power law

Fundamental theory

(Safe) Standard Model?

Pelaggi, Sannino, Strumia, Vigiani 1701.01453

Do theory like these exist?

Exact 4D Interacting UV Fixed Point

Litim and Sannino, 1406.2337, JHEP

Tr!!H†!H

"! uTr

!(H†H)2

"! vTr

!(H†H)

"2L = !F 2 + iQ! ·DQ+ y(QLHQR + h.c.)+

Antipin, Gillioz, Mølgaard, Sannino 1303.1525 PRD

Pelaggi, Sannino, Strumia, Vigiani, 1701.01453

Veneziano Limit

Normalised couplings

At large N NF

NC! "+

v

u=

!v

!hNF

Small parameters

Landau Pole ?

B < 0 ! > 0

!"! !"# $"!!"!

!"$

!"%

!"&

!"'

()*!!"!!#

" !!!#

! =NF

NC! 11

2

0 ! ! " 1

B = !4

3!

Can NL help?

!

!!

" !

!g = !B"2g + C"3

g B = !4

3!

!!g =

B

C! "

0 ! !!g " 1 i! C < 0

Impossible in Gauge Theories with Fermions alone Caswell, PRL 1974

Phase Diagram

Relevant

Irrelev

ant

Separatrix = Line of PhysicsGlobally defined line connecting two FPs

Separatrix

Complete asymptotic safety

Scalars are needed to make the theory fundamental

Gauge + fermion + scalars theories can be fund. at any energy scale

Litim and Sannino, 1406.2337, JHEP

!

a-theorem

gi = gi(x)

!µ! ! e2"(x)!µ!

gi(µ) ! gi(e!!(x)µ)

W = log

!"D!ei

!d4xL

#

L = LCFT + giOi Quantum correct., marginal oper.

Tool: Curved backgrounds

Conformal transformation

Variation of the generating functional

Weyl (anomaly) relations!!W !

!d4x!(x)

"2"µ"

#W

#"µ"" $i

#W

#gi

#= !

$aE(") + %ij&µgi&"gjG

µ"%+ &µ!w

i &"giGµ" + . . .

E(!) = Rµ!"#Rµ!"# ! 4Rµ!Rµ! +R2

Gµ! = Rµ! ! 1

2!µ!R

Euler density

Einstein tensor

a, !ij , "i

!i

Functions of couplings

Beta functions

Weyl relations from abelian nature of Weyl anomaly

!!!"W = !"!!W

Perturbative a-theorem

a-tilde is RG monotonically decreasing if chi is positive definite

!a

!gi=

!!"ij +

!wi

!gj! !wj

!gi

"#ja ! a" wi!i

d

dµa = !!ij"i"j

Cardy 88, conjecture

True in lowest order PT Osborn 89 & 91, Jack & Osborn 90

Analyticity: a-tilde bigger in UV Komargodski & Schwimmer 11, Komargodski 12

Safe variation for the a-theorem function

Positive and growing with epsilon

Antipin, Gillioz, Mølgaard, Sannino 13

To leading order

!a

!gg=

104

171"2

!gg =N2

C ! 1

128"2aUVaIR

Bootstrap and composite operators Antipin, Mølgaard, Sannino 14

Sannino, in preparation

Asymptotically Safe Thermodynamics

Pressure and Entropy to NNLORischke & Sannino 1505.07828, PRD

! = 0.08! = 0.07

! = 0.05! = 0.05

! = 0.03

! = 0.07

NLOIdeal gas NNLO

Violation of the thermal d.o.f. count

Rischke & Sannino 1505.07828, PRD

Thermal d.o.f. is violated

Thermal d.o.f. conjecture Appelquist, Cohen, Schmaltz, th/9901109 PRD

Although the thermal d.o.f. count is violated the a-theorem works!

Corrected SU(2) GB count in Sannino 0902.3494 PRD

QCD Conformal Window vs 2.0

‘If’ large Nf QCD is safeNf

Nc

Critical Asymp. Safe Nf must exist

Unsafe region in Nf-Nc

Continuous (Walking) transition?NAF

f

N IRf

NSafef

On Large Nf safety of QCD Pica and Sannino 1011.5917, PRD Litim and Sannino, 1406.2337, JHEP

Supersymmetric (un)safety

Intriligator and Sannino, 1508.07413, JHEP

Beyond perturbation theory

Bajc and Sannino, 1610.09681, JHEP

Unitarity constraintsOperators belong to unitary representations of the superconf. group

Dimensions have different lower bounds

Gauge invariant spin zero operators

Chiral primary operators have dim. D and U(1)R charge R

Central chargesPositivity of coefficients related to the stress-energy trace anomaly

‘a(R)’ Conformal anomaly of SCFT = U(1)R ’t Hooft anomalies [proportional to the square of the dual of the Rieman Curvature]

‘c(R)’ [proportional to the square of the Weyl tensor]

‘b(R)’ [proportional to the square of the flavor symmetry field strength]

a-theoremFor any super CFT besides positivity we also have, following Cardy

ri = dim. of matter rep.

+(-) for asymptotic safety (freedom)

Stronger constraint for asymp. safety, since at least one large R > 5/3

Beta functionsGauge coupling beta function proportional to ABJ anomaly

Beta function of the holomorphic Wy coupling y

SQCD with H

W = yTrQH!Q

Nf > 3Nc

AF is lost

No perturbative UV fixed point

SQCD with HAssume a nonperturbative fixed point, however

D(H) =3

2R(H) = 3

Nc

Nf< 1 for Nf > 3Nc

Violates the unitarity bound

D(O) ! 1

Potential loophole: H is free and decouples at the fixed point

Check if SQCD without H has an UV fixed point

SQCDUnitarity bound is not sufficient

Non-abelian SQED with(out) H cannot be asymptotically safe

Can be ruled out via a-theorem

aUV!safe ! aIR!safe < 0

Generalisation to several susy theories using a-maximisation*

Super safe GUTs

Exact results

Bajc and Sannino, 1610.09681, JHEP

Gaining R parity… butR-symmetry from SO(10) Cartan subalgebra generator B-L

M = matter parity

Elegant breaking of SO(10) preserving R-parity:

Introduce 126 + 126* Higgs in SO(10)

126(126*) SM and SU(5) singlet has B-L=-2(2) preserving R-parity

asymp. freedom is lost

To fully break SO(10) to SM add 210 of SO(10)

In summary: 3 x 16 + 126 + 126* + 10 + 210 contributes

!1!loop = !109

Asymptotic freedom is badly lost!

a, b run over generations

Exact resultsMinimal SO(10) without super potential

3 x 16 + 126 + 126* + 10 + 210 is unsafe.

Minimal SO(10) with general 3-linear super potential

• All trilinear present R=2/3 for all fields and no UV, NSVZ zero •Eliminate one 16 from super potential passes the constraints

Exotic examples exist requiring thousands of generations!

Super GUTs with R-charge are challenging

Higgs as shoelace

OutlookExtend to other (chiral) gauge theories/space-time dim [Ebensen, Ryttov, Sannino,1512.04402 PRD, Codello, Langaeble, Litim, Sannino, JHEP 1603.03462, Mølgaard and Sannino 1610.03130]

N=1 Susy GUTs safety

Wilson loops, critical exponents, MHV

Similarities and differences w.r.t. N=4

Go beyond P.T. [Lattice, dualities, holography, truncations]

New ways to unify flavour?

Models of DM and/or Inflation

Hope for asymptotic safe quantum gravity*?* Weinberg

Backup slides

Phenomenological Applications

Safe QCD

QCDQCD is not IR conformal because

Asymptotic freedom verified < TeV

Hadronic spectrum/dyn. mass

Pions <-> Spont. ChSB

If above TeV asymptotic freedom is lost, then what?

Asymptotic safety

!!"# !#"$ #"##"#

#"!

#"%

#"&

#"'

()*"!#!!$

""!$

!!"# !#"$ #"# #"$#"#

#"!

#"%

#"&

#"'

()*"!#!!$

!!""

ChSB/Confinement

1 GeV ~ TeV Before Planck

!s

µ

New coloured states

Higgs mechanismLight quarks

Top

Top partners

Colorons

Gluino-like

Unexpected

Safe QCD scenario

Sannino, 1511.09022

CosmologyCosmic raysLHC

Asymptotic safety

!!"# !#"$ #"##"#

#"!

#"%

#"&

#"'

()*"!#!!$

""!$

!!"# !#"$ #"# #"$#"#

#"!

#"%

#"&

#"'

()*"!#!!$

!!""

ChSB/Confinement

1 GeV~ TeV

Before Planck

!s

µ

Is the safe QCD scenario testable?Sannino, 1511.09022

Asymptotic freedom is not a must for UV complete theories

Large Nf, QCD, Holdom 1006.2119 PLB & Pica & Sannino,1011.5917 PRD

Safe Dark Matter

Safe DM

! ! "q"X

m4V

µ2

X X

SM SM

V

!!annv" #"q"X

m4V

m2X

X

XSM

SM

V

Offset direct detection

Sannino & Shoemaker, 1412.8034, PRD

Anomalous dimensions

HB = Z12HH

!H = 1 + !H

!H = !1

2

d lnZH

d lnµ

!H =4"

19+

14567! 2376"23

6859"2 +O("3)

Mass dimensions

!F = 3! !F !F =d lnM

d lnµ

!F =4

19"+

4048!23" 59711

6859"2 +O("3)

MQQ

Fermion

Mass dimensions

Small perturb., hence m2= 0 at the UV-FP

Scalar

m2Tr!H†H

"

!(1)m = 2"y + 4"h + 2"v

!m =1

2

d lnm2

d lnµ

UV critical surface(Ir)relevant directions implies UV lower dim. critical

`

Near the fixed point

Double - trace and stability

Is the potential stable at FP?

Which FP survives?

ModuliClassical moduli space

Use U(Nf)xU(Nf) symmetry

If V vanishes on Hc it will vanish for any multiple of it

Litim, Mojaza, Sannino 1501.03061 JHEP

Ground state conditions at any Nf

Hc ! !ij

Hc ! !i1

!!h + !!

v2 < 0 < !!h + !!

v1

Stability for !!v1

Quantum Potential

The QP obeys an exact RG equation

Hc = !c"ij ! = !1

2d lnZ/d lnµ

Litim, Mojaza, Sannino 1501.03061, JHEP

Resumming logs

Dimensional analysis

The Potential

Lambert Function

Effective gauge coupling

Visualisation

!"! !"# !"$ !"% !"& '"!'"!!

'"!(

'"'!

'"'(

!!!"!

!"## "!!#!$% "!!#

NLONNLO

!"! !"# !"$ !"% !"& '"!!"!

!"#

!"$

!"%

!"&

'"!

!!!"!

!"## "!!#!$% ""!#

QFT is controllably defined to arbitrary short scales

Gauge - Yukawa theories/Gradient Flow

Relations among the modified ! of different couplings

Precise prescription for expanding beta functions in perturb. theory

!a

!gi=

!!"ij +

!wi

!gj! !wj

!gi

"#j " !a

!gi= !#i , #i # "ij#j

!"j

!gi=

!"i

!gj,Gradient flow fundamental relation

Antipin, Gillioz, Mølgaard, Sannino 13

omega is an exact form Osborn 89 & 91, Jack & Osborn 90

Jack and Poole 15