Gravitational Anomalies, Dark Matter and Leptogenesis
Transcript of Gravitational Anomalies, Dark Matter and Leptogenesis
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Gravitational Anomalies, Dark Matter and Leptogenesis
Sarben SarkarDept. of Physics, Kingโs College London, UK
References: M de Cesare, N E Mavromatos, S Sarkar, Eur. Phys. J C 75 514 (2015)T Bossingham, N E Mavromatos, S Sarkar, Eur Phys J C 78 113 (2018)T Bossingham, N E Mavromatos and S Sarkar, Eur. Phys. J. C 79 50 (2019)N E Mavromatos and S Sarkar, Universe 5, no. 1:5 https://doi.org/10.3390/5010005N E Mavromatos and S Sarkar, Eur. Phys. C 80 558 (2020)J Ellis, N E Mavromatos and S Sarkar, Phys. Lett. B 725 407 (2013)
Tie together Bary(Lepto)ogenesis (BAU), dark matter and neutrino mass from a string-inspired perspective
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D
A
M
S
Our recipe:
D Dark matter( gravitational axion)
A Anomalies (gravitational + gauge)
M ordinary matter(SM)
S strings(gravity with torsion)
Components of the Universeโs energy density: l ๐๐ component energy densityl ๐๐ critical energy density
l ๐ = ๐๐๐๐, ๐๐๐๐๐๐๐ = ๐. ๐๐ ยฑ ๐. ๐๐, ๐๐ฉ = ๐. ๐๐๐ ยฑ ๐. ๐๐๐
l ๐๐ฉ is the ratio for baryonic matter
l ๐๐ฉ baryon number density, ๐๐ธ photon number density
l๐๐ฉ๐๐ธ= ๐. ๐ ยฑ ๐. ๐ร๐๐,๐๐ ๐๐ญ ๐~๐๐ฎ๐๐ฝ
l If Universe had been matter-antimatter symmetric ๐๐ฉ๐๐ธ=
๐๐ฉ๐๐ธโ ๐๐,๐๐
l Leptogenesis: creation of primordial lepton-antilepton asymmetry (M Fukugita and T Yanagida, Phys. Lett. B174 45 (1986))
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Sakharovโs Baryogenesis recipe ( A D Sakharov, JETP Lett. 5 24 (1967))
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Baryon number violating processes(sphaleron) in Standard Model
C and CP violation
CPT invariance assumed
Out of equilibrium: first orderelectroweak phase transition
\
ฮ XโY +b( )โ ฮ XโY +b( )XโY +b ฮB =1 XโY +b ฮB = โ1
ฮ ๐ โ ๐ + ๐โ ฮ ๐ + ๐ โ ๐
CPT symmetry
C is charge conjugationP is parity (space reflection)T is time-reversal
๐ฏ = ๐ช๐ท๐ป
CPT theorem:For any Lorentz invariant Hermitian local Lagrangian
ฮL x( )ฮโ1 = Lโ โx( )
Anomalies here there and everywhere:( literally the spice of life)
l Electroweak interactions chiral
l ๐ฑ๐๐ฉ =๐๐โ๐ ๐๐๐ณ๐ธ๐๐๐๐ณ + ๐๐น๐ธ๐๐๐น + ๐ ๐น๐ธ๐๐ ๐น
๐ฑ๐๐ณ = โ๐ ๐๐๐ณ๐ธ๐๐๐๐ณ + ๐๐๐น๐ธ๐๐๐๐น
l Triangle (ABJ) anomaly ( S L Adler, Phys. Rev 177 2426 (1969))
l ๐๐ ๐ฑ๐๐ณ = ๐๐ ๐ฑ๐๐ฉ =๐ต๐๐๐๐ ๐
โ๐๐๐พ๐๐5๐พ๐๐ + ๐,๐๐ฉ๐๐7๐ฉ๐๐
where the ๐พ๐๐ and ๐ฉ๐๐ are the ๐บ๐ผ ๐ and ๐ผ(๐) gauge field two forms
l ๐ฉ โ ๐ณ is conserved Leptogenesis leads to baryogenesis
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l Sakharov recipe difficulties: insufficient CP violation in the Standard Model, electroweak phase transition required to be 1st order
l DAMS ingredients beyond SM
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Bosonic gravitational multiplet of strings consists of a โข graviton, ๐๐๐โข spin 0 scalar field ๐ฝ the dilatonโข spin 1 antisymmetric gauge field ๐ฉ๐๐ with gauge invariant
field strength
๐,๐๐ฝ๐บ๐๐๐๐๐ฏ๐๐๐ = ๐๐๐๐ ๐ = ๐๐ฉ๐ ๐
โข ๐ต right-handed sterile neutrino with large Majorana mass โข A CPTV background with non-zero ๐ฉ๐ ๐
Hยตฮฝฯ = โ ยตโกโฃBฮฝฯโคโฆ
axion
Strings Gravity with torsionl The Kalb-Ramond field tensor in the low energy string
gravitational action appears as:
where -๐น is the curvature for the affine connection
(D J Gross and J H Sloan, Nucl. Phys. B 291 41 (1987) contorsion
l Torsion tensor ๐ป ๐๐๐ฟ = ๐ช๐๐๐ฟ โ ๐ช๐๐๐ฟ
l In a generlisation of Einsteinโs theory: add torsion to get Einstein-Cartan theory
l Torsion is necessary to incorporate fermions
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S = d4x โg 12ฮบ 2 Rโ
16HยตฮฝฯH
ยตฮฝฯโโโ
โโ โโซ = d4x โgโซ
12ฮบ 2 R
โโโ
โโ โ
ฮฮฝฯยต = ฮฮฝฯ
ยต + ฮบ3Hฮฝฯ
ยต โ ฮฯฮฝยต
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โ = โ๐บ๐ด +i ๐ต๐ธ๐๐๐๐ตโ๐๐ต๐ ๐ต๐๐ต+๐ต๐ต๐ -
๐ต๐ธ๐๐ฉ๐๐ธ๐๐ตโ โ๐๐๐๐ณ๐7๐๐ต + ๐. ๐.
DAMS Leptogenesis effective Lagrangian
CPTV and LIV background from torsion
โข Model leads to acceptable leptogenesis at thetree level, owing to Majorana nature of ๐ต which is taken to be very heavyโข Freeze out of N at ๐ป = ๐ป๐ซ.โข B-L conservation converts ๐ซ๐ณ to ๐ซ๐ฉโข ๐ฉ๐ = ๐๐๐โข Non-zero ๐ฉ๐ backgroundโข The underlying torsion connects this model
to dark matter
Details of leptogenesis
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โข Stick to ๐ = ๐ for simplicityโข Background:
๐ฉ๐ = ๐, ๐ = ๐, ๐, ๐๐ฉ๐ ๐ = ๐ฝ๐ ๐
where ๐ = ๐๐ต๐ป, and ๐ ๐ = ๐ or ๐ ๐ = ๐,๐.
โข Gravitational background๐๐๐ = ๐, ๐๐๐= โ๐๐ ๐ ๐น๐๐, space Cartesianโข Generation of lepton asymmetry due to CP and CPTV
tree-level decays:โข ๐ต โ ๐,๐8, ๐๐๐ channel 1โข ๐ต โ ๐8๐,, ๐๐๐ channel 2where ๐ยฑ are charged leptons, ๐ ๐ are active (anti)neutrinosโข For ๐ฝ โ ๐ the decay rates of ๐ต differ for channels 1
and 2 leading to a lepton asymmetry which freezes out at ๐ป๐ซ.
Results for leptogenesis
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๐ซ๐ณ๐๐๐
๐โ. ๐๐๐ ๐ฉ๐
๐๐ตat freezeout temperature ๐ป = ๐ป๐ซ
๐๐ต๐ป๐ซ
โ ๐. ๐
For ๐~๐๐,๐ and ๐๐ต = ๐ถ ๐๐๐ ๐ป๐๐ฝ and ๐ฉ๐~๐. ๐๐ด๐๐ฝ forphenomenologically acceptable leptogenesis to occur at
๐ป๐ซ~๐๐๐ป๐๐ฝ
Gravitational axions as Dark Matter I
l Peccei and Quinn solution of strong CP problemโ โ ๐ ๐ ๐ถ๐
๐๐ ๐ป๐ ๐ฎ โง ๐ฎ
where the ๐ vacuum parameter becomes a field and ๐ฎ is the QCD gauge field 2-form (R D Peccei and H R. Quinn, Phys Rev. Lett 38 1440 (1977))
Standard procedure: select a vacuum with ๐ ๐ = ๐ with a fluctuation a ๐
l Move away from Goldstone axion a ๐ฅ ๐ญ๐จ gravitational axion b ๐ whose couplings are determined by the Planck scale
l Classical torsionful spacetime manifold characterised by:torsion 2-from
๐ ๐๐ +๐ ๐๐ โง ๐๐ = โ๐, curvature 2-from
๐ ๐ ๐๐ +๐ ๐
๐ โง ๐ ๐๐ = ๐น ๐
๐
where ๐๐ is the vierbein and ๐ ๐๐ is the spin-connection 1-forms
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๐๐ = ๐๐๐ ๐ ๐๐
๐๐๐ = ๐๐๐๐๐ ๐๐
Gravitational axions as Dark Matter II
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The action of fermions of SM on a torsionful manifold is given by
๐บ = ๐บ๐๐๐๐๐๐๐ โ๐๐โ๐โซ ๐ฟ๐๐ธ โงโ ๐๐ฟ๐ โ๐๐ฟ๐ โงโ ๐ธ๐ฟ๐ -๐
๐โซโฑ โงโโฑ-โซ๐ป๐ ๐ฎ โงโ ๐ฎ
where for any fermionic field ๐ฟ๐บ๐๐๐๐๐๐๐=
๐๐๐ฟ๐ โซ๐บ๐๐๐๐ ๐ก
๐๐ โง ๐๐ โง ๐๐
๐๐ฟ = ๐ ๐ฟ+๐๐๐๐๐๐ธ๐๐๐ฟ+ ๐๐๐๐ฟ+ig๐ ๐ฟ, ๐ฟ๐= ๐๐ ๐ด๐ท๐
๐
with ๐ and ๐ the gauge connection 1-forms for ๐ผ ๐ ๐๐ and SU(3)c,๐ธ = ๐ธ๐๐๐ and ๐ธ๐๐=
๐๐๐ธ๐, ๐ธ๐ .
Equation of motion with respect to ๐๐๐ gives that the torsion satisfies๐ฃ๐๐๐ = โ ๐ฟ๐
๐๐บ๐๐๐๐ โ๐ ๐ฑ ๐๐ ๐
where ๐ฑ ๐๐ ๐ = ๐๐ฟ๐๐ธ๐ ๐ธ๐๐ฟ๐. (See also R Utiyama, Phys Rev 101, 1597 (1956); T W B Kibble, J Math Phys 2 212(1961);D W Sciama, Monthly Not. Roy, Astr. Soc. 113 34 (1953); F W Hehl, Gen Rel Grav 4 333 (1973); E Cartan, Riemannian Geometry in an Orthogoanl Frame (World Scientific, 2001), O Casillo-Feliso;a et al. Phys Rev D 91 085017 (2015) )
Anomalies(cf S Basilakos et al. PRD 101 045001 (2020)
M J Duncan, N Kaloper, K A Olive, Nucl. Phys. B387 215 (1992))
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Writing ๐ฑ๐ = โ๐ ๐ฑ๐๐ ๐๐๐ as a 1-form the axial anomaly implies
๐ โ ๐ฑ๐=โ ๐ถ๐๐๐ธ๐
๐ ๐ โง ๐-๐ถ๐๐ต๐
๐๐ ๐ป๐ ๐ฎ โง ๐ฎ โ ๐ต๐
๐๐ ๐\๐น๐๐ โง \๐น๐๐
which leads to the following quantised torsion-field contribution to the action
โ ๐ถ๐๐๐ธ๐
๐ ๐๐โซ๐ ๐ โง ๐ - ๐
๐โซ๐ b โงโ ๐ ๐ โ๐๐๐ ๐ โซ ๐ฏ + ๐ต๐
๐๐๐ \๐น๐๐ โง \๐น๐๐ โ
๐ถ๐๐๐ โซ ๐ฝ + ๐ต๐
๐๐๐ ๐ป๐ ๐ฎ โง ๐ฎ
The parameter ๐๐ = ๐ฟ:๐ ๐/๐ = ๐ร๐๐๐๐ GeV.
๐ฏ and ๐ฝ terms label Pontryagin densities, are allowed in the theory, play therole of counterterms and do not affect the equations of motion.
From the last term see that b solves the strong CP problem and ๐๐~๐๐:๐๐๐๐ฝand so can be regarded as the QCD axion.
Concluding remarks
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โข The gravitational anomaly (GA) may not be cancelledin the early universe in a quantum theory of gravity.
โข The gauge anomaly though can be designed to cancel for asuitable GUT theory. Anomaly cancellation is necessary for the theories to be renormalizable.
โข GA implies breakdown of diffeomorphism lnvarianceand hence local Lorentz invariance (LIV)
โข LIV allows a non-zero ๐ฝrequired for our leptogenesis model
โข The Kalb-Ramond axion could play the role of a QCD axionand so be axionic dark matter (M Lattanzi and S Mercuri, Phys Rev D81 125015 (2010))