![Page 1: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/1.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protected axions in a clockworkgauge symmetry model
Quentin Bonnefoy
based on arXiv:1804.01112in collaboration with E. Dudas and S. Pokorski
Centre de Physique Théorique - École Polytechnique
StringPheno 2018Warsaw, July 4th 2018
![Page 2: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/2.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axions: pseudo Nambu-Goldstone bosons (pNGB)
Two examples of limitations in axion model building:
• low masses are not automatically consistent withUV completions in a quantum theory of gravity
• some axion models require intermediate scale(∼ 109−11 GeV) or super-Planckian dynamics
![Page 3: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/3.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axions: pseudo Nambu-Goldstone bosons (pNGB)
Two examples of limitations in axion model building:
• low masses are not automatically consistent withUV completions in a quantum theory of gravity
• some axion models require intermediate scale(∼ 109−11 GeV) or super-Planckian dynamics
![Page 4: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/4.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axions: pseudo Nambu-Goldstone bosons (pNGB)
Two examples of limitations in axion model building:
• low masses are not automatically consistent withUV completions in a quantum theory of gravity
• some axion models require intermediate scale(∼ 109−11 GeV) or super-Planckian dynamics
![Page 5: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/5.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axions: pseudo Nambu-Goldstone bosons (pNGB)
Two examples of limitations in axion model building:
• low masses are not automatically consistent withUV completions in a quantum theory of gravity
• some axion models require intermediate scale(∼ 109−11 GeV) or super-Planckian dynamics
![Page 6: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/6.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Mass range in axion models:
Small masses: need for a controlled explicit breaking ofthe axionic symmetry
However: global symmetries expected to be broken byquantum gravity effects
![Page 7: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/7.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Mass range in axion models:
Small masses: need for a controlled explicit breaking ofthe axionic symmetry
However: global symmetries expected to be broken byquantum gravity effects
![Page 8: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/8.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Mass range in axion models:
Small masses: need for a controlled explicit breaking ofthe axionic symmetry
However: global symmetries expected to be broken byquantum gravity effects
![Page 9: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/9.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protection of global symmetries: make them accidental.
Famous gauge protection: NGB from extra dimensions
Example: 5d abelian gauge theory compactified on S1/Z2, withDirichlet boundary conditions for Aµ → leaves in the spectruma massless scalar (zero mode of A5) with a shift symmetry
Breaking of the shift symmetry: requires non-localsuppressed effects
![Page 10: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/10.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protection of global symmetries: make them accidental.
Famous gauge protection: NGB from extra dimensions
Example: 5d abelian gauge theory compactified on S1/Z2, withDirichlet boundary conditions for Aµ → leaves in the spectruma massless scalar (zero mode of A5) with a shift symmetry
Breaking of the shift symmetry: requires non-localsuppressed effects
![Page 11: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/11.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protection of global symmetries: make them accidental.
Famous gauge protection: NGB from extra dimensions
Example: 5d abelian gauge theory compactified on S1/Z2, withDirichlet boundary conditions for Aµ
→ leaves in the spectruma massless scalar (zero mode of A5) with a shift symmetry
Breaking of the shift symmetry: requires non-localsuppressed effects
![Page 12: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/12.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protection of global symmetries: make them accidental.
Famous gauge protection: NGB from extra dimensions
Example: 5d abelian gauge theory compactified on S1/Z2, withDirichlet boundary conditions for Aµ → leaves in the spectruma massless scalar (zero mode of A5) with a shift symmetry
Breaking of the shift symmetry: requires non-localsuppressed effects
![Page 13: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/13.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protection of global symmetries: make them accidental.
Famous gauge protection: NGB from extra dimensions
Example: 5d abelian gauge theory compactified on S1/Z2, withDirichlet boundary conditions for Aµ → leaves in the spectruma massless scalar (zero mode of A5) with a shift symmetry
Breaking of the shift symmetry: requires non-localsuppressed effects
![Page 14: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/14.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axion decay constant range in axion models:
L ⊃ gaγγfa
aFµνFµν + ...
QCD axion: fa ∼ 109−11 GeVQuintessence/inflation/relaxion: fa &MP
fa naturally obtained from known scale?
![Page 15: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/15.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axion decay constant range in axion models:
L ⊃ gaγγfa
aFµνFµν + ...
QCD axion: fa ∼ 109−11 GeVQuintessence/inflation/relaxion: fa &MP
fa naturally obtained from known scale?
![Page 16: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/16.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Axion decay constant range in axion models:
L ⊃ gaγγfa
aFµνFµν + ...
QCD axion: fa ∼ 109−11 GeVQuintessence/inflation/relaxion: fa &MP
fa naturally obtained from known scale?
![Page 17: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/17.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Clockwork models:
Example of the scalar clockwork:
L = −∑N
k=0(|∂µφk|2 + V (|φk|2))− (
∑N−1k=0 εkφ
qkφ∗k+1 + h.c.)
→ Clockwork Goldstone boson acl. ∼ θ0qN
+ ...+θN−1
q + θN→ and effective decay constants feff ∼ qif
Obtained from 5D with linear dilaton geometry.
Scalar:
L5D = gMN∂Mφ∂Nφ(xµ, y) with ds2 = e4k|y|/3(dx2 + dy2)
![Page 18: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/18.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Clockwork models:
Example of the scalar clockwork:
L = −∑N
k=0(|∂µφk|2 + V (|φk|2))− (
∑N−1k=0 εkφ
qkφ∗k+1 + h.c.)
→ Clockwork Goldstone boson acl. ∼ θ0qN
+ ...+θN−1
q + θN→ and effective decay constants feff ∼ qif
Obtained from 5D with linear dilaton geometry.
Scalar:
L5D = gMN∂Mφ∂Nφ(xµ, y) with ds2 = e4k|y|/3(dx2 + dy2)
![Page 19: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/19.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Clockwork models:
Example of the scalar clockwork:
L = −∑N
k=0(|∂µφk|2 + V (|φk|2))− (
∑N−1k=0 εkφ
qkφ∗k+1 + h.c.)
→ Clockwork Goldstone boson acl. ∼ θ0qN
+ ...+θN−1
q + θN→ and effective decay constants feff ∼ qif
Obtained from 5D with linear dilaton geometry. Scalar:L5D = gMN∂Mφ∂Nφ(xµ, y) with ds2 = e4k|y|/3(dx2 + dy2)
![Page 20: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/20.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Clockwork models:
Example of the scalar clockwork:
L = −∑N
k=0(|∂µφk|2 + V (|φk|2))− (
∑N−1k=0 εkφ
qkφ∗k+1 + h.c.)
→ Clockwork Goldstone boson acl. ∼ θ0qN
+ ...+θN−1
q + θN→ and effective decay constants feff ∼ qif
Obtained from 5D with linear dilaton geometry.
Scalar:
L5D = gMN∂Mφ∂Nφ(xµ, y) with ds2 = e4k|y|/3(dx2 + dy2)
![Page 21: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/21.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
This work:
Use a clockwork-inspired gauge group toprotect an axion, and study its phenomenology
![Page 22: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/22.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Outline
A gauge theory with a pGB
SM couplings
Application: FDM
![Page 23: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/23.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
A gauge theory with a pGB
![Page 24: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/24.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
A gauge theory: 4D UV-completion of a deconstructedabelian gauge theory on a linear dilaton background
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
Ahmed & Dillon (2017), Coy Frigerio & Ibe (2017), Choi Im & Shin (2017)
![Page 25: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/25.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
A gauge theory: 4D UV-completion of a deconstructedabelian gauge theory on a linear dilaton background
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
Ahmed & Dillon (2017), Coy Frigerio & Ibe (2017), Choi Im & Shin (2017)
![Page 26: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/26.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
with one global accidental U(1): φk → eiqkαφk
In the spectrum for a complete spontaneous breaking,Goldstone boson a:
a ∼ 1qNf0
θ0 + 1qN−1f1
θ1 + ...+ 1fNθN
![Page 27: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/27.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
with one global accidental U(1): φk → eiqkαφk
In the spectrum for a complete spontaneous breaking,Goldstone boson a:
a ∼ 1qNf0
θ0 + 1qN−1f1
θ1 + ...+ 1fNθN
![Page 28: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/28.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
with one global accidental U(1): φk → eiqkαφk
In the spectrum for a complete spontaneous breaking,Goldstone boson a
a ∼ 1qNf0
θ0 + 1qN−1f1
θ1 + ...→ Site-dependent couplings?
![Page 29: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/29.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
with one global accidental U(1): φk → eiqkαφk
How much approximate? Gauge invariant operators:
|φk|2 and φ0φq1...φ
qN
N
→ exponential increase of the order of the breakingoperators with q and N
![Page 30: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/30.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
SM couplings
![Page 31: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/31.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Couplings to SM fields:
L ⊃ gaγγfa
aFµνFµν +
iga,EDM
faaNγµνγ
5NFµν
+gaNNfa
∂µaNγµγ5N +
gaeefa
∂µaeγµγ5e
![Page 32: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/32.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
KSVZ model:
L ⊃ φQLQR + h.c.Q triangle loop−−−−−−−−−→ a
fF F
Gauge-anomalous now. Need more fermions:
L ⊃ φ0QL,0QR,0 + h.c.
U(1)1 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 + h.c.
U(1)2 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 +
q2∑i=1
φ2QiL,2QiR,2 + h.c.
U(1)3 anom.−−−−−−−−→ ...
![Page 33: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/33.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
KSVZ model:
L ⊃ φQLQR + h.c.Q triangle loop−−−−−−−−−→ a
fF F
Gauge-anomalous now. Need more fermions:
L ⊃ φ0QL,0QR,0 + h.c.
U(1)1 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 + h.c.
U(1)2 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 +
q2∑i=1
φ2QiL,2QiR,2 + h.c.
U(1)3 anom.−−−−−−−−→ ...
![Page 34: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/34.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
KSVZ model:
L ⊃ φQLQR + h.c.Q triangle loop−−−−−−−−−→ a
fF F
Gauge-anomalous now. Need more fermions:
L ⊃ φ0QL,0QR,0 + h.c.
U(1)1 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 + h.c.
U(1)2 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 +
q2∑i=1
φ2QiL,2QiR,2 + h.c.
U(1)3 anom.−−−−−−−−→ ...
![Page 35: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/35.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
KSVZ model:
L ⊃ φQLQR + h.c.Q triangle loop−−−−−−−−−→ a
fF F
Gauge-anomalous now. Need more fermions:
L ⊃ φ0QL,0QR,0 + h.c.
U(1)1 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 + h.c.
U(1)2 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 +
q2∑i=1
φ2QiL,2QiR,2 + h.c.
U(1)3 anom.−−−−−−−−→ ...
![Page 36: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/36.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
KSVZ model:
L ⊃ φQLQR + h.c.Q triangle loop−−−−−−−−−→ a
fF F
Gauge-anomalous now. Need more fermions:
L ⊃ φ0QL,0QR,0 + h.c.
U(1)1 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 + h.c.
U(1)2 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 +
q2∑i=1
φ2QiL,2QiR,2 + h.c.
U(1)3 anom.−−−−−−−−→ ...
![Page 37: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/37.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
KSVZ model:
L ⊃ φQLQR + h.c.Q triangle loop−−−−−−−−−→ a
fF F
Gauge-anomalous now. Need more fermions:
L ⊃ φ0QL,0QR,0 + h.c.
U(1)1 anom.−−−−−−−−→ L ⊃ φ0QL,0QR,0 +
q∑i=1
φ1QiL,1QiR,1 + h.c.+
q2∑i=1
φ2QiL,2QiR,2
U(1)2 anom.−−−−−−−−→ ...
triangle loops−−−−−−−−→√
1+q2+...+q2N
f aF F
![Page 38: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/38.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
Exponentially reduced effective decay constant:
feff ∼ fqN
Still requires ∼ qN additional fermions
Ex: to get feff ∼ 1012 GeV from f ∼MP , needs 106 fermions (atthe Planck scale)
VS 1012 if q = 1
Due to the number of additional fermions AND the highaccidental global charges:
feff =f√∑
fermions(charges)∼ f√
qNqN
![Page 39: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/39.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
Exponentially reduced effective decay constant:
feff ∼ fqN
Still requires ∼ qN additional fermions
Ex: to get feff ∼ 1012 GeV from f ∼MP , needs 106 fermions (atthe Planck scale)
VS 1012 if q = 1
Due to the number of additional fermions AND the highaccidental global charges:
feff =f√∑
fermions(charges)∼ f√
qNqN
![Page 40: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/40.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
Exponentially reduced effective decay constant:
feff ∼ fqN
Still requires ∼ qN additional fermions
Ex: to get feff ∼ 1012 GeV from f ∼MP , needs 106 fermions (atthe Planck scale)
VS 1012 if q = 1
Due to the number of additional fermions AND the highaccidental global charges:
feff =f√∑
fermions(charges)∼ f√
qNqN
![Page 41: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/41.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
Exponentially reduced effective decay constant:
feff ∼ fqN
Still requires ∼ qN additional fermions
Ex: to get feff ∼ 1012 GeV from f ∼MP , needs 106 fermions (atthe Planck scale) VS 1012 if q = 1
Due to the number of additional fermions AND the highaccidental global charges:
feff =f√∑
fermions(charges)∼ f√
qNqN
![Page 42: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/42.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
Exponentially reduced effective decay constant:
feff ∼ fqN
Still requires ∼ qN additional fermions
Ex: to get feff ∼ 1012 GeV from f ∼MP , needs 106 fermions (atthe Planck scale) VS 1012 if q = 1
Due to the number of additional fermions AND the highaccidental global charges:
feff =f√∑
fermions(charges)∼ f√
qNqN
![Page 43: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/43.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν+ gaNNfa
∂µaNγµγ5N +...
Example: coupling to the first SM generation
L ⊃ − 1
MP
(uRHφiYuQL + dR(Hφi)
∗YdQL + eR(Hφi)∗YeLL
)+ h.c.
chiral redef.−−−−−−−→ L ⊃ −iqi∂µa2√
1 + ...+ q2Nf(uγ5γ
µu+ dγ5γµd+ eγ5γ
µe)
Site-dependent coupling to the spins derived in minimalsetup
![Page 44: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/44.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν+ gaNNfa
∂µaNγµγ5N +...
Example: coupling to the first SM generation
L ⊃ − 1
MP
(uRHφiYuQL + dR(Hφi)
∗YdQL + eR(Hφi)∗YeLL
)+ h.c.
chiral redef.−−−−−−−→ L ⊃ −iqi∂µa2√
1 + ...+ q2Nf(uγ5γ
µu+ dγ5γµd+ eγ5γ
µe)
Site-dependent coupling to the spins derived in minimalsetup
![Page 45: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/45.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν+ gaNNfa
∂µaNγµγ5N +...
Example: coupling to the first SM generation
L ⊃ − 1
MP
(uRHφiYuQL + dR(Hφi)
∗YdQL + eR(Hφi)∗YeLL
)+ h.c.
chiral redef.−−−−−−−→ L ⊃ −iqi∂µa2√
1 + ...+ q2Nf(uγ5γ
µu+ dγ5γµd+ eγ5γ
µe)
Site-dependent coupling to the spins derived in minimalsetup
![Page 46: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/46.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν+ gaNNfa
∂µaNγµγ5N +...
Example: coupling to the first SM generation
L ⊃ − 1
MP
(uRHφiYuQL + dR(Hφi)
∗YdQL + eR(Hφi)∗YeLL
)+ h.c.
chiral redef.−−−−−−−→ L ⊃ −iqi∂µa2√
1 + ...+ q2Nf(uγ5γ
µu+ dγ5γµd+ eγ5γ
µe)
Site-dependent coupling to the spins derived in minimalsetup
![Page 47: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/47.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν + gaNNfa
∂µaNγµγ5N +...
Axion field a = θ0+qθ1+...+qNθN√1+q2+...+q2N
In the effective theory:
√1+q2+...+q2N
f aF Fqi∂µa√
1+q2+...+q2NfNγµγ5N
Scales: f fagaγγ∼ f
qNfagaee∼ f
qi−N
![Page 48: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/48.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν + gaNNfa
∂µaNγµγ5N +...
Axion field a = θ0+qθ1+...+qNθN√1+q2+...+q2N
In the effective theory:
√1+q2+...+q2N
f aF Fqi∂µa√
1+q2+...+q2NfNγµγ5N
Scales: f fagaγγ∼ f
qNfagaee∼ f
qi−N
![Page 49: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/49.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Application: FDM
![Page 50: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/50.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
For an ALP dark matter candidate:
Masses can be as low as m ∼ 10−22 eV ("fuzzy" dark mattercandidate)
Can be perturbative or non-perturbative
Focus here on perturbative gravitational origin and onmisalignment mechanism (with pre-inflationnary breaking):
V = − φ0φq1...φ
qN
N
M1+q+...+qN−4P
⊃ −(
fMP
)1+q+...qNM4P cos
(afa
)and
< ainit >=random
![Page 51: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/51.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
For an ALP dark matter candidate:
Masses can be as low as m ∼ 10−22 eV ("fuzzy" dark mattercandidate)
Can be perturbative or non-perturbative
Focus here on perturbative gravitational origin and onmisalignment mechanism (with pre-inflationnary breaking):
V = − φ0φq1...φ
qN
N
M1+q+...+qN−4P
⊃ −(
fMP
)1+q+...qNM4P cos
(afa
)and
< ainit >=random
![Page 52: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/52.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
For an ALP dark matter candidate:
Masses can be as low as m ∼ 10−22 eV ("fuzzy" dark mattercandidate)
Can be perturbative or non-perturbative
Focus here on perturbative gravitational origin and onmisalignment mechanism (with pre-inflationnary breaking):
V = − φ0φq1...φ
qN
N
M1+q+...+qN−4P
⊃ −(
fMP
)1+q+...qNM4P cos
(afa
)and
< ainit >=random
![Page 53: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/53.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Ωah2 = 0.12 when:
![Page 54: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/54.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Detection of spin precession (with ∂µafaNγµγ5N): Coupled
at site 0
![Page 55: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/55.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Detection of spin precession (with ∂µafaNγµγ5N): Coupled
at site 0
![Page 56: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/56.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Detection of spin precession (with ∂µafaNγµγ5N): Coupled
at site N
![Page 57: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/57.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Detection of spin precession (with ∂µafaNγµγ5N): Coupled
at site 0
![Page 58: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/58.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Conclusion
![Page 59: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/59.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
We considered a pGB protected against (gravitational)breaking effects. Its mass is easily very small, even with fewadditional gauge groups.
It can have all usual axion couplings, associated to scaleswhich display the clockwork charges of the scalars. Whileaxion-spin couplings are generated in minimal setups, anomalouscouplings to gauge fields require an additional fermion sector.
In the minimal setup, the (unavoidable) gravity contributionis sufficient to provide the correct DM density, andspin-precession-based searches can detect such a particle.
![Page 60: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/60.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
We considered a pGB protected against (gravitational)breaking effects. Its mass is easily very small, even with fewadditional gauge groups.
It can have all usual axion couplings, associated to scaleswhich display the clockwork charges of the scalars. Whileaxion-spin couplings are generated in minimal setups, anomalouscouplings to gauge fields require an additional fermion sector.
In the minimal setup, the (unavoidable) gravity contributionis sufficient to provide the correct DM density, andspin-precession-based searches can detect such a particle.
![Page 61: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/61.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
We considered a pGB protected against (gravitational)breaking effects. Its mass is easily very small, even with fewadditional gauge groups.
It can have all usual axion couplings, associated to scaleswhich display the clockwork charges of the scalars. Whileaxion-spin couplings are generated in minimal setups, anomalouscouplings to gauge fields require an additional fermion sector.
In the minimal setup, the (unavoidable) gravity contributionis sufficient to provide the correct DM density, andspin-precession-based searches can detect such a particle.
![Page 62: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/62.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Thank you!
![Page 63: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/63.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Backups
![Page 64: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/64.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Example of a Peccei-Quinn symmetry:
Peccei-Quinn symmetry:
Explains why the CP "θ-term" LQCD ⊃θQCD32π2 ε
µνρσFµνFρσverifies θQCD < 10−10
Postulates a global symmetry with a SU(3)2 × U(1)PQanomaly → makes θQCD dynamical (axion) and
stabilizes it at θQCD = 0
![Page 65: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/65.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
PQ symmetry: Global symmetry with a SU(3)2 × U(1)PQanomaly + axion→ θQCD = 0
Specific realization: KSVZ model with
LPQ ⊃ φQLQR + h.c.− V (|φ|2), φU(1)PQ−−−−−→ eiαφ and φ = f+r√
2ei af .
Then:
QCD anom. + instantons→ L ⊃ m2πf
2π
√mumd
mu+mdcos(af − θQCD
)Possible correction: LPQ ⊃
φn
Mn−4P
+ h.c.→ destabilizes
θ < 10−10 if n < 10 (if f & 109 GeV). Indeed:
φn
Mn−4P
term→ L ⊃( f√
2MP
)nM4P cos
(naf
)
![Page 66: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/66.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protection of global symmetries: make them accidental.
• Ex: B, L in the renormalizable standard model lagrangian• U(1)PQ protection: Barr and Seckel (1992)
Fields1
1φ1 φ2 Qi=1...q
L Qi=1...pL Qi=1...p+q
R
SU(3) 1 1 3 3 3U(1) p q p −q 0U(1)PQ q −p q p 0
where gcd(p, q) = 1 and p+ q ≥ 10
L ⊃ φ1QLY QR + φ∗2QLY QR︸ ︷︷ ︸LPQ
+φq1φ
∗p2
Mp+q−4P︸ ︷︷ ︸LPQ
+h.c.
![Page 67: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/67.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L = −14
∑Ni=1 Fµν,iF
µνi −
∑Nk=0 |Dµφk|2 − V (|φ0|2, |φ1|2, ...)
with one global accidental U(1): φk → eiqkαφk
Gravitational breaking under control:
L ⊃ φ0φq1...φ
qN
N
M1+...−4P
→ m(grav)a =
(f√2MP
) q+...+qN−12 √
1+q2+...+q2NMP
![Page 68: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/68.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Mass suppression with few additional gauge groups:
![Page 69: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/69.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
L ⊃ gaγγfaaFµνF
µν +gaNNfa
∂µaNγµγ5N + ...
Number of fermions ∼ qN : growing with protection quality.General feature:
L ⊃ −∑i
(Oiψi,Lψi,R)triangle loops−−−−−−−−→ i
32π2log
(∏i
Oi
)FF
![Page 70: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/70.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
For a QCD axion:
Protection efficient if m(grav)a < 10−5
(m
(QCD)a ∼ mπfπ
2fa
)(with fa defined by the axion-gluons coupling: L ⊃ a
faGG)
In our setup:
fa =f√
1 + q2 + ...+ q2N
![Page 71: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/71.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
θQCD < 10−10 if:[m
(QCD)a ∼ mπfπ
2fa
]> 105
[m
(grav)a ∼
(fMP
) q+...+qN−12 f
faMP
]
![Page 72: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/72.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Protected QCD axion with f ∼ 1011 GeV and q = 3, N = 2→13 additional colored Dirac fermions (+13 additional singletDirac fermions)
No Landau pole for QCD below the Planck mass
![Page 73: Protected axions in a clockwork gauge symmetry modelsp18.fuw.edu.pl/wp-content/uploads/participants-database/bonnefoy... · Introduction A gauge theory with a pGB SM couplings Application:](https://reader036.vdocuments.us/reader036/viewer/2022071006/5fc321d1bd1ab974af5db071/html5/thumbnails/73.jpg)
Introduction A gauge theory with a pGB SM couplings Application: FDM Conclusion
Stability of the DM ALP’s?
No anomaly: no ALP-photon conversion via usualL ⊃ a
faFF
Instead: derivative interactions + tiny mass → longlifetime
Example: coupling to a heavy anomaly-free set of electricallycharged fermions:
L ⊃ y1φiψR,1ψL,1 + y2φiψL,2ψR,2 + h.c. .
fermions integr.−−−−−−−−−→ Leff ⊃e2
48π2qif
( 1
m21
− 1
m22
)(aF F−1
2∂µaFνη∂
ηFµν)
Lifetimes for the FDM: ∼ 10300s