exact results in (super) safe qft (the naturally safe higgs)dietrich/slides3/sannino.pdf ·...
TRANSCRIPT
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