Axions. Kerstin Helfrich. Seminar on Theoretical Particle Physics, / 31

Transcription

1 1 / 31 Axions Kerstin Helfrich Seminar on Theoretical Particle Physics,

2 2 / 31 Structure 1 Introduction 2 Repetition: Instantons Formulae The θ-vacuum 3 The U(1) and the strong CP problem The U(1) problem The strong CP problem 4 Solution: Axions With / without massless quarks Neutron dipole moment Introduction of the axion The axion as Goldstone boson The invisible axion 5 Conclusion

3 3 / 31 Introduction Instanton solutions create a new problem in QCD. A new symmetry is introduced, the axial U(1) PQ symmetry. This includes the introduction of a new dynamical pseudoscalar field and thus a particle, the axion a(x).

4 4 / 31 Formulae Mapping f : S 3 S 3 Winding number: n = 1 24π 2 dθ 1 dθ 2 dθ 3 tr ( ) ɛ ijk A i A j A k where A i = f 1 (x 0, x) i f (x 0, x) Consider L = 1 2g 2 tr (F µν F µν )

5 5 / 31 Formulae S E = d 4 xl should be finite s.t. for x : F µν (x) 0 and A µ (x) U 1 µ U. Instantons correspond to Us with nontrivial winding number! By comparison: d 4 x tr(f µν F µν ) = 1 2 d 4 x µ K µ with K µ = 4 3 ɛ µνλρtr [ (U + ν U)(U + λ U)(U + ρ U) ]

6 6 / 31 Formulae Thus: n = 1 16π 2 d 4 x tr (F µν µν F ) We have multiple vacua that are characterized by their winding number n. Instantons can be seen as the connection between vacuum states as they have winding numbers themselves. Consider G 1 n = n + 1, where G 1 is a gauge transformation of winding number 1.

7 Instantons - The θ-vacuum Transition amplitude: T = n e iht m J = [ [da] n m exp i ] (L + J A)d 4 x Semiclassically this leads to: [ ] T = exp [ S E ] exp 8π2 n g 2 7 / 31

8 8 / 31 Instantons - The θ-vacuum [G 1, H] = 0 because of the gauge invariance. Then: θ = n e inθ n s.t. G 1 θ = e iθ θ This θ-vacuum leads to an adequate formulation of the theory.

9 9 / 31 Instantons - The θ-vacuum Transition amplitude: T = θ e iht θ J = m,n e imθ e inθ m e iht n J = m,n e i(n m)θ e im(θ θ) [ [da] n m exp i ] (L + J A)d 4 x Result: L eff = L + θ ( 32π 2 tr F µν µν F )

10 10 / 31 The U(1) problem For two quark flavours and the chiral limit m u,d 0 the Lagrangian possesses the symmetry: SU(2) L SU(2) R U(1) V U(1) A - SU(2) L SU(2) R being the chiral symmetry - U(1) V leading to baryon number conservation via the current J B µ = uγ µ u + dγ µ d - U(1) A leading to a current J 5 µ = uγ µ γ 5 u + dγ µ γ 5 d

11 11 / 31 The U(1) problem An invisible axial symmetry means that it should be broken. Thus we expect a Goldstone boson. But: There is no fourth pion except the η which is much too heavy to be a Goldstone boson. This is the U(1) or η-mass problem. Solution: The instantons can break U(1) without leading to a Goldstone boson.

12 12 / 31 The strong CP problem The instanton solution to the U(1) problem generates a new problem, namely the strong CP problem. The θ-term in the effective Lagrangian violates P. Experimentally: no CP violation in the strong interaction has been found. Comparison experiment vs. theory: θ < 10 9 So why should there be such a strong bound on θ?

13 13 / 31 With massless quarks Remember: L eff = L + ( ) θ tr F µν F µν 32π 2 Consider an axial U(1)-transformation on the quark fields: Axial current: q e iαγ 5 q j 5 µ = q qγ µ γ 5 q = q [q R γ µ q R q L γ µ q L ] Yielding to µ j 5 µ = N f g 2 16π 2 tr ( F µν F µν ) with N f being the number of massless quarks.

14 With massless quarks Thus, for α = θ 2N f the θ-term of the effective Lagrangian can be removed. Therefore, in the massless quark case θ would be unphysical. 14 / 31

15 15 / 31 Without massless quarks Massive case: mqq mqe 2iαγ 5 q = m cos (2α)qq im sin (2α)qγ 5 q The second part clearly violates P and T invariance and the mass term for quarks can be written in the following way: L m = q Li M ij q Rj + q Ri M + ij q Lj Under U(1), M no longer is hermitean: M e 2iα M and M + e 2iα M +

16 16 / 31 Without massless quarks Thus: arg det M arg det M + 2αN Define θ = θ + 2αN as it is the quantity which is invariant under the transformation. θ still has to be zero to be CP non-violating.

17 17 / 31 Neutron dipole moment In L, there is a contribution to the neutron dipole moment coming from the following graphs:

18 18 / 31 Neutron dipole moment Pion nucleon interaction: L πnn = g θ πnn Nτ a Nπ a Experiment: d n < e cm Calculation: d n = θ e cm Therefore: θ <

19 19 / 31 Introduction of the axion Solution proposed by Peccei and Quinn in 1977: make θ a dynamical field. Later on, a new pseudoscalar boson, the axion a(x), was added by Weinberg. Choose θ = a/f a with f a being the axion decay constant. + i L = 1 4g 2 tr (F µνf µν ) µa µ a+ q i (i/d m i )q i + a ( 32π 2 tr F µν F ) µν + θ ( f a 32π 2 tr µν F µν F )

20 20 / 31 Introduction of the axion Still to be shown: θ = 0 The θ vaccum is proportional to (1 cos θ). Thus θ = 0 would be the correct vacuum. Integrating out the effect of quarks and gluons in Euclidean space yields to: d 4 xv [0] d 4 xv [a] There must be a minimum at a = 0 or θ = 0 Thus the choice of θ = 0 is justified.

21 21 / 31 The axion as Goldstone boson Notation: - Q f L = (u f L, d f L ) are the SU(2) doublets - U f R, D f R denote the SU(2) singlets - ϕ is the doublet of the Higgs scalar fields and f labels the generations. The mass term comes from: with ψ = iτ 2 ϕ. L Y = Q f L X fg ϕd gr + Q f L Y fg ψu gr + h.c.

22 22 / 31 The axion as Goldstone boson 0 ϕ 0 = 1 2 ( 0 v 1 e iδ 1 0 ψ 0 = 1 2 ( v2 e iδ 2 0 ) for down-like quarks ) for up-like quarks Conditions in the SM: v 1 = v 2 and δ 1 = δ 2 ( ) ( ) 1 det M = det 2 v 1 e iδ 1 1 X det 2 v 2 e iδ 2 Y = e 3i(δ 1+δ 2 ) 1 8 (v 1v 2 ) 3 det (XY )

23 23 / 31 The axion as Goldstone boson Thus, we extend the SM and define two independent scalar doublets ϕ and ψ and get: θ θ + arg det M = 0 arg det M = 3(δ 1 + δ 2 ) + arg det(xy ) This is exactly what we needed. To still ensure the symmetry, suppose: ϕ e iαγ 1 ϕ, ψ e iαγ 2 ψ where Γ 1, Γ 2 are the Peccei-Quinn charges of ϕ, ψ

24 24 / 31 The axion as Goldstone boson Further transformations: Q L e iαγ Q, U L e iαγu, D L e iαγ d Invariance of L Y is given if: Γ 1 + Γ d = Γ Q, Γ 2 + Γ u = Γ Q, meaning Γ 1 + Γ 2 = 2Γ Q Γ u Γ d 0 For ϕ and ψ acquiring the former vevs, the SSB yields a Goldstone boson which is associated with the phase angle of these fields.

25 25 / 31 The axion as Goldstone boson Look at the neutral components of the scalar doublets: ϕ 0 = 1 2 (v 1 + ρ 1 (x)) e iθ 1(x)/v 1 ψ 0 = 1 2 (v 2 + ρ 2 (x)) e iθ 2(x)/v 2 Thus a(x) = [v 2 θ 1 (x) + v 1 θ 2 (x)] /v where v ( v v 2 2 ) 1/2

26 26 / 31 The axion as Goldstone boson The orthogonal second combination gets eaten and generates the Z-boson mass: χ(x) = [ v 1 θ 1 (x) + v 2 θ 2 (x)] /v This finally yields to the axion-quark Yukawa coupling: L a,q Y = ia(x) [ v2 m d dγ 5 d + v ] 1 m u uγ 5 u v v 1 v 2

27 27 / 31 The axion as Goldstone boson v 246 GeV from weak interaction data Axion mass: m a = ( v1 + v ) 2 74 kev v 2 v 1 Although many experiments tried to find this axion it was not found in any reaction so far. But this is not the end of the story.

28 28 / 31 The invisible axion Two models: i) There is the heavy quark theory. (KSVZ axion) ii) The PQ symmetry breaking scale can be separated from the electroweak breaking scale. (DFSZ axion) Breaking scale: 10 9 GeV f PQ GeV This yield to invisibility.

29 29 / 31 The invisible axion Possible decay processes: For m a > 2 m e : For m a < 2 m e : m a e e + m a γ γ This is a possible detection method.

30 30 / 31 Conclusion The axion is a very promising idea to solve the strong CP-violation problem. Even if it has not been detected this does not mean that it cannot exist as an invisible axion. Could be a candidate for the dark matter in the universe. Further experiments are conducted. e.g. the CAST-experiment at CERN is searching for solar axions.

31 31 / 31 Literature T. Cheng and L. Li: Gauge Theory of elementary particle physics (1988) J.E. Kim: Phys. Rep (1987) D. Bailin and A. Love: Introduction to Gauge Field Theory (1993) D. Bailin and A. Love: Cosmology in Gauge Field Theory and String Theory (2004) S. Coleman: Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman (1985) S. Weinberg, Phys. Rev. Lett. 40, 223 (1978) F. Wilczek, Phys. Rev. Lett. 40, 279 (1978) R. D. Peccei and H. Quinn, Phys. Rev. Lett. 38, 1440 (1977); Phys. Rev. D16, 279 (1978) J. E. Kim, Phys. Rev. Lett. 43, 103 (1979)