Weyl Symmetry, Planck Scale, & Inflation. Christopher T. Hill Fermilab Eclipse Conference Celebration of Thom Curtright August, 2017

Inflation and Planck Scale Generation as a Unified Phenomenon No fifth force in a scale invariant universe By Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross. arXiv:1612.03157 [gr-qc], to appearTPhys. Rev .D. Weyl Current, Scale-Invariant Inflation and Planck Scale Generation By Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross. arXiv:1610.09243 [hep-th]. Phys.Rev. D95 (2017) no.4, 043507. Scale-Independent Inflation and Hierarchy Generation By Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross. arXiv:1603.05983 [hep-th]. Phys.Lett. B763 (2016) 174-178.

Related earlier works …..

Weyl Invariance in a nutshell: Coordinates are scale free numbers. Length is defined by the covariant metric. Fields have canonical mass (length)-1 dimension Local Weyl transformation: Global Weyl transformation:

Some Local Weyl Invariants:

Exercise: Construct a locally Weyl invariant action Integrate by parts:

A globally Weyl = Scale invariant action in D=4:

A globally Weyl = Scale invariant action in D=4: Define:

Canonical normalization of s: A globally Weyl = Scale invariant action in D=4: Define: Total divergence Canonical normalization of s: f

Result: Weyl symmetry in Jordan frame is hidden ``Einstein frame” Weyl symmetry in Jordan frame is hidden in the Einstein frame. Can show that s decouples from Fermions and gauge fields. (No Brans-Dicke constraints.)

Note singularity: a - 1 = 0 (local Weyl invariance) s terms cancel Note wrong sign when a = 1

A conventional scalar-gravity action: (notation g = (1,-1,-1,-1))

(notation g = (1,-1,-1,-1)) A conventional scalar-gravity action: A scale invariant scalar-gravity action:

Classically Weyl Equivalent A conventional scalar-gravity action: (notation g = (1,-1,-1,-1)) A scale invariant scalar-gravity action: These theories are Classically Weyl Equivalent

How do we understand the dynamics of the first theory (Einstein frame) A conventional scalar-gravity action: (notation g = (1,-1,-1,-1)) A scale invariant scalar-gravity action: How do we understand the dynamics of the first theory (Einstein frame) in the second theory (Jordan frame)?

Compute in the Jordan Frame (!)!!! Einstein Equations Compare:

Trace of Einstein Equations

Klein-Gordon Equation

Trace and KG Equations: Combine to eliminate R:

Trace and KG Equations Combine to eliminate R:

If then conserved Conserved Current : Combined Trace of Einstein and KG Equations yields a current conservation law: If then conserved Conserved Current :

The Kernel of the Current can be written as where is the “kernel”

The Kernel of the Current can be written as where is the “kernel” FRW: becomes

“Inertial” spontaneous scale generation The Kernel of the Current constant constant “Inertial” spontaneous scale generation

The Kernel of the current Spatially constant Einstein equations Theory leads to eternal inflation

The Kernel of the current constant

Our action is globally Weyl invariant: Noether: Perform a local infinitesimal Weyl Transformation:

Our action is globally Weyl invariant: Noether Current: properties: constant

Inertial spontaneous breaking of global Weyl symmetry leads to dilaton: Define:

follows from Weyl current. A conventional scalar-gravity action: A scale free action Summary: These theories are classically equivalent; dynamical equivalence follows from Weyl current.

Can our world have a hidden Weyl invariance? How can we inflate and exit eternal inflation?

Two Scalars, globally Weyl symmetric Effective Planck Mass

Two Scalar K-current Two Scalar Potential Current divergence Kernel

Potential with a flat direction:

Fixed K defines an ellipse

Fixed K defines an ellipse

More detailed studies in progress (Ferreira) below. More detailed studies in progress (Ferreira) Scale-Independent Inflation and Hierarchy Generation By Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross. arXiv:1603.05983 [hep-th]. Phys.Lett. B763 (2016) 174-178. Other phenomena? Dilaton stars and clumps? (maybe seed primordial black holes) Dilatonic hair? (hard) Dilatonic radiation from BH annihilation? (final burst)

Weyl symmetry must be exact: For this to work Weyl symmetry must be exact: (Failure is not an option) Does this conflict with Quantum Mechanics?

Scale = Weyl Symmetry appears ab initio to be problematic due to loop divergences. However, loop divergences are subtle and are often confused with physics.

An Operating Principle (W. Bardeen) Quantum loop divergences are unphysical; they are artifacts of the method of calculation. The Allowed Symmetries of a Renormalized Quantum Field Theory are determined by anomalies, or absence thereof. There are no anomalies associated with quadratic or quartic divergences; Scale Symmetry is permitted modulo Trace Anomalies; trace anomalies are D=4 ops.

A Common Complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry !!! What are these people talking about? “

A Common Complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry !!! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “

These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry ! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ It is False to conclude that the theory does not exist because the regulator doesn’t respect the defining symmetry.

These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry ! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ The regulator one uses may break symmetries. For gauge theories consistent regulators exist that moderate quadratic divergences, eg, Pauli-Villars, Dim. reg.

These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry ! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ No regulator unambiguously deals with chiral anomalies !!! These always require human intervention (ie, counterterms to define loops)

These are the same complaint: “I computed a Feynman loop in high school and discovered it had quadratic and quartic divergences! So there is no scale symmetry! What are these people talking about? “ “I computed the photon vacuum polarization in high school and discovered it had a quadratic divergence that caused the photon to acquire a mass! So there is no Local Gauge Invariance! What are these people talking about? “ For scale symmetry we do not have a consistent regulator. The physics cannot depend upon the choice of regulator! Scale symmetry can be maintained at loop level if trace anomaly cancels.

“The Higgs Boson Mass receives a contribution coming from the cutoff used to compute loops”

“The Higgs Boson Mass receives a contribution coming from the cutoff used to compute loops” FALSE. However the Higgs can mix with GUT or Gravity sectors, and this is a real physics problem that must be controlled.

The trace of the stress tensor is the divergence of the scale current and is the true measure of scale breaking in a quantum theory.

The trace anomaly underlies Coleman-Weinberg Symmetry Breaking Coleman & Weinberg PR D7 (1973) 1888

Coleman Weinberg Symmetry Breaking

Coleman Weinberg Symmetry Breaking

Coleman Weinberg Symmetry Breaking

External mass entering from Renormalization Coleman & Weinberg PR D7 (1973) pg. 1892: M is external to the action

Injection of external renormalization scale leads to Scale Anomaly: Coleman-Weinberg Potential (one loop): Divergence of the K-current: Scale Anomaly due to M

Weyl Invariant Renormalization Coleman & Weinberg PR D7 (1973) pg. 1892: c c VEV M is replaced by the VEV of a field in the action: M => c c c c

Schematically, a Weyl invariant Coleman-Weinberg Potential requires two or more scalars or phi/R. Invariant renormalization: Beta-function governs the potential, but there is No trace anomaly! Divergence of the K-current

The Renormalization Group remains intact as usual, but: All dimensionless parameters run in Weyl invariant ratios: Usual one-loop solutions with external mass: One-loop solutions with Weyl invariant ratio:

Reinterpreted renormalization group: F’s are Weyl invariants. They are arbitrary, but define the physical coupling constants.

Compute conventional CW Potential:

Compute a Weyl Invariant CW Potential:

Compute a Weyl Invariant CW Potential: Now perform the Weyl transformation:

Weyl Invariant CW Potential: Current divergence Quantum implementation of the idea that there are no fundamental mass scales in nature

New Physics? Consider approximate running in a1 with a2 constant Classical ellipse Vacuum minimum: Quantum deformed ellipse Flat direction

A possible new way to generate intermediate scales? “Quantum deformation of the ellipse” can lead to hierarchical intermediate scales in nature! Solution for running of a with Weyl invariant ratio gives relationship between VEVs and MP :

Conclusions The Planck Mass and all other mass scales are dynamical if there is a hidden exact Weyl invariance. Only ratios of masses are physical. Multi-scalar models can dynamically generate the Planck Mass at the same time as inflation occurs. The two phenomena are unified. The dilaton decouples with exact Weyl symmetry; there is no 5th force. Through renormalization without external mass scales Weyl symmetry of the overall quantum theory is maintained. This is the quantum implementation of the idea that there are no fundamental mass scales. There remains RG running, but running only in Weyl invariant quantities like f/c ; hence no scale anomalies occur. The Weyl current is then exactly conserved.

Conclusions Nontrivial new mechanisms can arise that may sculpt intermediate mass scales, eg, by quantum deformation of the ellipse. Consistent with inflation; may yield new phenomena associated with dilaton. E.g., can dilaton lumps seed primordial BH formation? Is dark matter black holes? (Kamionkowski) A new UV completion may be required: Is Weyl R2 gravity a good Candidate? Asymptotically Safe Gravity (Weinberg)? Or, Can we make String Theory Weyl Invariant in D=4 ? Can string tension be dynamically generated by inertial SSB?

CERN Theory Picnic, circa Fall 1987. That old devil, Time…. CERN Theory Picnic, circa Fall 1987.