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The Strong CP-Problem The strong CP-Problem Jonathan Steinberg Universität Siegen Peccei-Quinn symmetry and axion models 18.01.2020 Jonathan Steinberg The strong CP-Problem 1 / 38 Outline of the talk (1) Theta vacuum in QCD (2) The strong CP problem (3) The axionic solution of the strong CP problem (4) Experimental tests for the axion Jonathan Steinberg The strong CP-Problem 2 / 38 Beyond the standard model The standard model of particle physics is extremely successful, but: How to integrate non-zero neutrino masses? SM fails What are dark matter and dark energy? How to explain the baryon- Cosmology antibaryon asymmetry of the universe? Why is the Higgs particle so light? Why is CP conserved in QCD? Fine tuning Why is the vacuum energy so tiny? Jonathan Steinberg The strong CP-Problem 3 / 38 The dark sector of SM A dark sector beyond the standard model is strongly motivated by SM might be complex with several constituents Axions and axion-like particles are (pseudo)scalars strongly motivated by theory maybe the most elegant solution to the strong CP problem might be showing up in astro (particle)physics already 1. 1E. Aprile et al, Excess electronic recoil events in XENON1T, Phys. Rev. D 102, 072004 Jonathan Steinberg The strong CP-Problem 4 / 38 At the beginning of QCD 2: Z 1 S = d4x q(iγ Dµ − M )q − Ga Gaµν QCD µ q 4 µν where Mq = diag(mu; md; :::) The vacuum structure of QCD Quantum chromodynamics The basic objects are a (1) the gluon fields Gµ(x) (2) the quark fields qj(x) with j 2 f0; :::; Nf g flavor index (3) the gauge group SU(3) ! non-abelian 2D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (26), 1343 Jonathan Steinberg The strong CP-Problem 5 / 38 The vacuum structure of QCD Quantum chromodynamics The basic objects are a (1) the gluon fields Gµ(x) (2) the quark fields qj(x) with j 2 f0; :::; Nf g flavor index (3) the gauge group SU(3) ! non-abelian At the beginning of QCD 2: Z 1 S = d4x q(iγ Dµ − M )q − Ga Gaµν QCD µ q 4 µν where Mq = diag(mu; md; :::) 2D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (26), 1343 Jonathan Steinberg The strong CP-Problem 5 / 38 The vacuum structure of QCD There is a further term Z 1 α S = d4x q(iγ Dµ − M )q − Ga Gaµν − s θGa G~aµν QCD µ q 4 µν 8π µν The new θ-term is gauge-invariant is renormalizable has a topological origin Jonathan Steinberg The strong CP-Problem 6 / 38 The vacuum structure of QCD Why the θ term is often not considered Term can be written as a total derivative one assumes compactly supported fields Consequence: negligible in perturbation theory How to see, that it is a total derivative? 1 Ga G~a = @ J µ with J µ = µνρσAa F a − f abcAb Ac µν µν µ CS CS ν ρσ 3 ρ σ µ where JCS Chern-Simons current As we will see: Z 4 µ d x @µJCS = 0; ±1; ±2; ::: Jonathan Steinberg The strong CP-Problem 7 / 38 The vacuum structure of CQD θ-term could have a contribution in nonperturbative processes How can we calculate the contribution Needs some topological argument Related to non-trivial vacuum structure of QCD To get intuition: Consider SU(2) gauge theory with gauge fields only Jonathan Steinberg The strong CP-Problem 8 / 38 The vacuum structure of QCD Reminder: Lie group SU(2) group of special unitary matrices, i.e, U 2 M2(C) with U −1 = U y and det(U) = 1 Generators are hermitian, traceless matrices ) 3 generators Temporal gauge 1 classical ground state is a Fµν(x) = 0 8a 2 f1; 2; 3g a vector potential Aµ is gauge transformation of zero i A (x) = Aa (a)T a = U@ U y µ µ g µ Jonathan Steinberg The strong CP-Problem 9 / 38 The vacuum structure of QCD Temporal gauge 2 Fix gauge transformation to be time independent U(x) = U(x) This leads to temporal gauge i.e., A0 = 0, since i i @ A = U@ U y = U(~x) U(~x)y = 0 0 g 0 gc @t Boundary conditions In order to give θ-term a constribution: Impose boundary conditions on U for jxj ! 1 U(~x) should approach particular matrix for j~xj ! 1 3 U : R [ f1g ! U2(C) Jonathan Steinberg The strong CP-Problem 10 / 38 The vacuum structure of QCD Now there are two questions: (1) How can we implement this structure ? (2) How can we compute this ? Idea: Use topology Jonathan Steinberg The strong CP-Problem 11 / 38 The vacuum structure of QCD Now there are two questions: (1) How can we implement this structure ? (2) How can we compute this ? Idea: Use topology Jonathan Steinberg The strong CP-Problem 11 / 38 The vacuum structure of QCD Now there are two questions: (1) How can we implement this structure ? (2) How can we compute this ? Idea: Use topology Jonathan Steinberg The strong CP-Problem 11 / 38 The vacuum structure of QCD Can every U(~x) can be smoothly deformed into every other U(~x)? If yes: All these field configurations are gauge equivalent =) single quantum vacuum state If no: All these field configurations are not gauge equivalent =) More then only one quantum vacuum state Jonathan Steinberg The strong CP-Problem 12 / 38 The vacuum structure of QCD Remembering topology 3 (1) How can we think about R [ f1g ? (2) Tool : Alexandroff one-point compactification (3) From this we get: 3 3 4 R [ f1g ' S ⊂ R Jonathan Steinberg The strong CP-Problem 13 / 38 The vacuum structure of QCD We can now interprete S3 ! SU(2) Use spherical coordinates for S3, i.e., set of angles The map from S3 to SU(2) do not need to be one-to-one ) winding number measures this failure Maps with a specific winding number Consider maps S3 ! SU(2) where 3 spatial / vacuum S specified by zµ = (~z; z4) with zµzµ = 1 This gives two polar angles and one azimuthal a map of winding number n is then map polar angle to polar angle map spatial sphere azimuthal angle to n times the vacuum azimuthal Jonathan Steinberg The strong CP-Problem 14 / 38 The vacuum structure of QCD Finally one can compute 1 Z h i 3 n = − d3x ijk Tr (U@ U y)(U@ U y)(U@ U y) Z 24π2 i j k Question: How one can prove this claim? (1) Take the above constructed map and check, that it gives n indeed (2) Show that RHS is invariant under smooth deformations of U(~x) Use infinitesimal deformations U 7! U + δU One finds: Un(~x)Uk(~x) = Un+k(~x) For the experts: We have shown Π3(SU(2)) = (Z; +) Jonathan Steinberg The strong CP-Problem 15 / 38 The vacuum structure of QCD Consequences for QCD SU(3) gauge theory has an infinite number of classical field conf. of zero energy configurations can be labeled by integer n 2 Z different configurations are separated by energy barriers Jonathan Steinberg The strong CP-Problem 16 / 38 The vacuum structure of QCD Vacuum energy of QCD How to calculate vacuum amplitude Z? Z Z = [dG][dq][dq] exp (−S[G; q; q]) To get most general Lagrangian, we have to add Lθ-term One can expand path integral for the vacuum to vacuum amplitude X iθn Z(θ) = e Zn n2Z Zn path integral over definite topological sector Zn describes boundary condition for the field Jonathan Steinberg The strong CP-Problem 17 / 38 The vacuum structure of QCD Vacuum energy of QCD Vacuum energy as a function of θ in QCD is 1 Z(θ) E (θ) = − ln 0 V Z(0) The Vafa-Witten theorem If Z is a Fourier series with positive coefficients, then E0(θ) has an absolute minimum at θ = 0 Consequence: If θ = θ(x) would be a dynamical quantity hθ(x)i = 0 Jonathan Steinberg The strong CP-Problem 18 / 38 The vacuum structure of QCD Vacuum energy of QCD To sum up: If one of the quarks has zero mass ! no θ dependence If mu; md 6= 0: QCD has additional parameter θ Question: What physical effects can θ yield ? Jonathan Steinberg The strong CP-Problem 19 / 38 The strong CP problem Consequences of the θ-term Consider again θ-term in QCD a aµν ~ a ~ a GµνG / E · B where Ea is the chromoelectric field Ba is the chromomagnetic field We expect: θ-term violated CP θ-term will probably lead to CP violation in flavour conserving interactions in addition to CKM Jonathan Steinberg The strong CP-Problem 20 / 38 The strong CP problem Consequences of the θ-term Estimate neutrons EDM mumd 1 dn(θ) / eθ 2 mu + md mn ∼ 10−16θ ecm chiral perturbation theory: −16 jdnj > 1:1 × 10 θ ecm Lattice calc. : −16 jdnj ∼ 3:6(2)(9) × 10 θ ecm −26 Exp.: jdnj < 2:4 × 10 ecm =) jθj < 7 × 10−11 Jonathan Steinberg The strong CP-Problem 21 / 38 The strong CP problem The strong CP problem Why is the value of θ is so small? Why there is no CP violation in flavor conserving interactions? Why is the electric dipole moment of the neutron so small ? Jonathan Steinberg The strong CP-Problem 22 / 38 The axionic solution of the strong CP problem The Peccei-Quinn symmetry In QCD: P and CP is a good symmetry In SM: P and CP are broken But QCD is a part of SM The deviation from being symmetric is measured by angle θ 2P.
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