1. Axion Electrodynamics
Christopher T. Hill
Fermilab
U of Durham, UK, April 16, 2016

2. What is the axion?
( /f)

3. What is the axion?
( /f)
Kinetic Terms
Mass Terms

4. ?
mh’ = mp

5. ?
mh’ = mp
985 MeV
140 MeV
Way off!

6. ?
mh’ >> mp
Remedy I:
cf4 Det U + h.c.

7. ?
But! This can have a strong CP-phase:
cf4 eiq Det U + h.c.
p , h , h’
predicts neutron EDM much too large: g5
sin(q)
q < 10-12

8. cf4 ei(q - a/f) Det U + h.c. - c’f4 cos(q - a/f ) ?
Remedy II: The axion
cf4 ei(q - a/f) Det U + h.c.
The axion potential:
- c’f4 cos(q - a/f )
The axion mass:

9. Axions:
The axion is a pNGB associated with the spontaneous breaking of Peccei-Quinn symmetry.
Typically the PQ symmetry breaks at a high scale
At the QCD scale, instantons activate the U(1) axial
current anomaly.
The axion acquires a potential by mixing with
develops a VEV which cancels the QCD CP-violating phase .
Small oscillations about this minimum are associated with the axion mass and can constitute dark matter.
p , h , h’

10. Axions: The axion is an “angular variable” in the effective
action on scales much less than
It is useful to write axion expressions in terms of the angle
Variable:
The axion kinetic+mass term action can be written:

11. Axions:
The axion mass is controlled by mixing with the pseudoscalar nonet of mesons. The axion mass is then :
The prefactor, c, is where
and vanishes as
QCD: 2p
20.7

12. Cosmic Axions = Dark Matter?
Assume a cosmic axion field:
The axion energy density is:
Equate this to the galactic halo dark matter density:
Hence:

13. Axion Electrodynamics
The axion couples to the electromagnetic field via the U(1)
axial current anomaly:
Where: and = anomaly coefficient.

14. Axion Electrodynamics
The axion couples to the electromagnetic field via the U(1)
axial current anomaly:
Where: and = anomaly coefficient.
See the PDG article.

15. Axion Electrodynamics
The action for axion electrodynamics:
Note that is a total divergence in the limit
that
The axion anomaly can we written in two ways:
Gauge inv. but not chiral Chiral inv. but not gauge inv.

16. Axion Decoupling

17. Axion Decoupling
Display manifest gauge invariance:

18. Axion Decoupling Display manifest gauge invariance:
Integrate by parts;
Display manifest shift invariance:

19. Axion Decoupling
Display manifest gauge invariance:
Integrate by parts;
Display manifest shift invariance:
Since these differ only by a total divergence, both symmetries must be present in perturbation theory.

20. “The axion interactions induced perturbatively (ala Feynman diagrams) must always display derivative coupling.”

21. “The axion interactions induced perturbatively (ala Feynman diagrams) must always display derivative coupling.”
False!

22. “The axion interactions induced perturbatively (ala Feynman diagrams) must always display derivative coupling.”
False!
Since manifest gauge and shift symmetries differ only by a total divergence, both symmetries must be present in perturbation theory, but need not be simultaneously manifest.

23. Can an (oscillating) electric dipole moment
be generated from the anomaly perturbatively?

24. Can an (oscillating) electric dipole moment
be generated from the anomaly perturbatively?
We define a covariant OEDM for the electron:

25. Can an (oscillating) electric dipole moment
be generated from the anomaly perturbatively?
We define a covariant OEDM for the electron:
Integrate by parts:

26. Requires: Can an (oscillating) electric dipole moment
be generated from the anomaly perturbatively?
We define a covariant OEDM for the electron:
Integrate by parts:
Requires:

27. No! But generally: / Can an (oscillating) electric dipole moment
be generated from the anomaly perturbatively?
We define a covariant OEDM for the electron:
Integrate by parts:
No!
But generally: /

28. Do a simple calculation:
Electron
Magnetic
Moment
axion anomaly
axion

29. Do a simple calculation:

30. Do a simple calculation:

31. Do a simple calculation:

32. Do a simple calculation:
note decoupling

33. Do a simple calculation:

34. Do a simple calculation:
The resulting OEDM is:

35. Do a simple calculation:
The resulting OEDM is:

36. Callous sophisticates, US west coast:
Do a simple calculation:
The resulting OEDM is:
Callous sophisticates, US west coast:

37. No! Doesn’t Decouple! Callous sophisticates, US west coast:
Do a simple calculation:
The resulting OEDM is:
Callous sophisticates, US west coast:
No! Doesn’t Decouple!

38. What has happened to axion shift symmetry?:
Integrate by parts:
Decoupling requires:

39. What has happened to axion shift symmetry?:
Integrate by parts:
Decoupling requires:
Easy to check:

40. What has happened to axion shift symmetry?:
Integrate by parts:
Decoupling requires:
The effective action:

41. Do a less simple calculation:

42. Do a less simple calculation:

43. Do a less simple calculation:

44. Do a less simple calculation:

45. Do a less simple calculation:
The effective action:

46. YES! It Decouples! To the callous sophisticates:
Do a less simple calculation:
The effective action:
To the callous sophisticates:
YES! It Decouples!

47. This result is subtle and echoes the behavior of the anomaly itself:

48. This result is subtle and echoes the behavior of the anomaly itself:

49. This result is subtle and echoes the behavior of the anomaly itself:

50. Magnetic monopoles acquire electric charge in presence of a nonzero q angle, “Witten Effect.”

51. Magnetic monopoles acquire electric charge in presence of a nonzero q angle, “Witten Effect.”
A magnetic monopole–antimonopole pair will have a magnetic dipole; this becomes an electric dipole moment
For nonzero q angle. Oscillating axion field implies a nonzero oscillating q angle.

52. Magnetic Dipole
An infinitesimally small dipole is indistinguishable from a monopole-anti-monopole pair.

53. Electric Dipole

54. Electric Dipole

55. Theorem:
A space-time filling coherent oscillating axion field will cause any magnetic N-pole to become an oscillating electric N-pole through the anomalous coupling of the axion to electromagnetic fields.

56. Axion in a Source-free Magnetic Field:

57. Axion in a Source-free Magnetic Field:
Non - propagating

58. Axionic Electrodynamics
RF Cavity Energetics
Conducting wall r = R

59. Axionic Electrodynamics
RF Cavity Energetics
Conducting wall r = R
Particular Solution:
The particular solution
doesn’t satisfy the conducting
boundary condition:

60. Axionic Electrodynamics
RF Cavity Energetics
Conducting wall r = R
Particular Solution:
Homogeneous Solution:

61. Axionic Electrodynamics
RF Cavity Energetics
Conducting wall r = R
Full Solution:

62. Axionic Electrodynamics
RF Cavity Energetics
Conducting wall r = R
Impose boundary condition:
Cavity solution.

63. Axionic Electrodynamics
RF Cavity Energetics
Finite Q:

64. Axionic Electrodynamics
RF Cavity Energetics
Finite Q:
Only need B in the wall!!!

65. Axionic Electrodynamics
RF Cavity Energetics
At Large Q

66. Axionic Electrodynamics
RF Cavity Energetics

67. How does the axion induce a physical signal?
Vector potential
Vacuum
Ohm’s Law Conductor

68. How does the axion induce a physical signal?

69. How does the axion induce a physical signal?
We only need B in the conductor

70. Radiation from axion induced OEDM
Dipole source
Dipole “plume”

71. Radiation from axion induced OEDM
Dipole “plume”

72. Radiation from axion induced OEDM
Dipole source

73. Radiation from axion induced OEDM
Near zone:

74. Radiation from axion induced OEDM
Far zone:
Electric Dipole Radiation:

75. Radiation from axion induced OEDM

76. Radiation from axion induced OEDM

77. Radiation from axion induced OEDM

78. Radiation from axion induced OEDM

79. Radiation from axion induced OEDM

80. Radiation from axion induced OEDM

81. Radiation from axion induced OEDM

82. Radiation from axion induced OEDM

83. Radiation from axion induced OEDM

84. Radiation from axion induced OEDM
Summary:
Agrees with
classical calculation

85. Radiation from axion induced OEDM

86. Estimates 2

87. Power Radiated by a single electron
in the axion cosmic field

88. Power from coherent assemblage of electrons

91. Slab Radiator Power Output

92. A Simple Experiment: Slab Radiator Power Output
(3s) 0.1K

93. Possible Advantages of Magnetic Arrays:
Non-resonant, broadband radiator
Enhanced configurations, eg. scalable , X or XYZ ?
Signal can be modulated by multiple arrays, eg XY + X’Y’
Power levels comparable to cavities; may have physical advantages in axion ``sweet-spot’’

94. Axion electrodynamics in a nut-shell:
Axial anomaly a g g
In background strong magnetic field
induce non-propagating
induces a propagating and in a conducting material or source
Detectable Poynting Vector x

95. The axion serves an essential role in QCD
Axion is a compelling Dark Matter Candidate
Essentially defined by two parameters: faxion
and ganomaly
Detection is challenging but may be doable:
Broadband Radiator
Cavity Experiment

96. END

98. My Idiosyncratic system of units:

104. Result:

107. Result:

108. Result:
Consistent with Pauli-Schroedinger Result:

110. Estimate of the RF Cavity Energetics

111. Estimate of the RF Cavity Energetics2

112. Power from coherent assemblage of electrons

113. Estimate of the RF Cavity Energetics

114. This result is subtle and echoes the behavior of the anomaly itself:

115. This result is subtle and echoes the behavior of the anomaly itself:

117. Power Radiated by a single electron
Immersed in the axion cosmic field

118. Power from coherent assemblage of electrons

119. Power from coherent assemblage of electrons