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

2. 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: [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: [hep-th]. Phys.Rev. D95 (2017) no.4, Scale-Independent Inflation and Hierarchy Generation By Pedro G. Ferreira, Christopher T. Hill, Graham G. Ross. arXiv: [hep-th]. Phys.Lett. B763 (2016)

3. Related earlier works
…..

4. 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:

5. Some Local Weyl Invariants:

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

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

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

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

10. 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.)

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

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

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

14. 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

15. 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)?

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

17. Trace of Einstein Equations

18. Klein-Gordon Equation

19. Trace and KG Equations:
Combine to eliminate R:

20. Trace and KG Equations
Combine to eliminate R:

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

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

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

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

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

26. The Kernel of the current
constant

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

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

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

30. 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.

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

32. Two Scalars, globally Weyl symmetric
Effective Planck Mass

33. Two Scalar K-current
Two Scalar Potential
Current divergence
Kernel

34. Potential with a flat direction:

35. Fixed K defines an ellipse

36. Fixed K defines an ellipse

39. 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: [hep-th]. Phys.Lett. B763 (2016)
Other phenomena?
Dilaton stars and clumps? (maybe seed primordial black holes)
Dilatonic hair? (hard)
Dilatonic radiation from BH annihilation? (final burst)

40. 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?

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

42. 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.

43. 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? “

44. 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? “

45. 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.

46. 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.

47. 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)

48. 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.

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

50. “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.

51. 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.

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

53. Coleman Weinberg Symmetry Breaking

54. Coleman Weinberg Symmetry Breaking

55. Coleman Weinberg Symmetry Breaking

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

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

58. 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

59. 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

60. 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:

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

62. Compute conventional CW Potential:

63. Compute a Weyl Invariant CW Potential:

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

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

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

67. 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 :

68. 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.

69. 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?

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