The strong CP-Problem

Jonathan Steinberg

Universität Siegen

Peccei-Quinn symmetry and 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

The standard model of is extremely successful, but: How to integrate non-zero neutrino masses? SM fails What are dark matter and dark energy? How to explain the - 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 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 fields Gµ(x) (2) the fields qj(x) with j ∈ {0, ..., Nf } 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 ∈ {0, ..., Nf } 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) 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 ∈ M2(C) with

U −1 = U † and det(U) = 1

Generators are hermitian, traceless matrices ⇒ 3 generators

Temporal gauge 1 classical ground state is

a Fµν(x) = 0 ∀a ∈ {1, 2, 3}

a vector potential Aµ is gauge transformation of zero i A (x) = Aa (a)T a = U∂ U † 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 † = U(~x) U(~x)† = 0 0 g 0 gc ∂t

Boundary conditions In order to give θ-term a constribution: Impose boundary conditions on U for |x| → ∞ U(~x) should approach particular matrix for |~x| → ∞ 3 U : R ∪ {∞} → 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 ∪ {∞} ? (2) Tool : Alexandroff one-point compactification (3) From this we get:

3 3 4 R ∪ {∞} ' 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 †)(U∂ U †)(U∂ U †) 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 ∈ 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 n∈Z

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 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 EDM

mumd 1 dn(θ) ∝ eθ 2 mu + md mn

∼ 10−16θ ecm

chiral perturbation theory: −16 |dn| > 1.1 × 10 θ ecm Lattice calc. : −16 |dn| ∼ 3.6(2)(9) × 10 θ ecm −26 Exp.: |dn| < 2.4 × 10 ecm

=⇒ |θ| < 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 of the 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. Sikivie, Phys. Today 49, 21-28 Jonathan Steinberg The strong CP-Problem 23 / 38 The axionic solution of the strong CP problem The Peccei-Quinn symmetry

There could be a mechanism, which make QCD automatically CP invariant This mechanism must be dynamical, i.e., θ must be a dynamical variable Mechanism pull θ to zero ones the model has been arranged

Jonathan Steinberg The strong CP-Problem 24 / 38 The axionic solution of the strong CP problem The Peccei-Quinn symmetry

Until now, no experimental evidence for the axion switch on of nonperturbative QCD effects coherent axion field oscillations Device for experimental test: high quality oscillator

Jonathan Steinberg The strong CP-Problem 25 / 38 The axionic solution of the strong CP problem The Peccei-Quinn symmetry

Idea: Make θ dynamical3, that is θ → θ(x) =: a(x) a(x) should be a spin zero field, called axion field a(x) should respect a non-linearly realized U(1)PQ-symmetry, that is

a(x) 7→ a(x) + κfa

This symmetry is broken by coupling to gluonic topological charge density α L ⊃ −a(x)Γ(x) where Γ(x) = s Ga (x)G˜aµν(x) 8π µν

3R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (25) Jonathan Steinberg The strong CP-Problem 26 / 38 The axionic solution of the strong CP problem The Peccei-Quinn symmetry

Consequence: Eliminate the QCD parameter θ by defining κ = −θ i.e., a(x) 7→ a(x) − θ α L ⊃ − s [θ + a(x)] Ga G˜aµν 8π µν

All we need to solve the strong CP problem Use anomaly to remove θ-term from L Choose κ = −θ one can add terms to L which do not spoil solution

Jonathan Steinberg The strong CP-Problem 27 / 38 The axionic solution of the strong CP problem The axion Lagrangian

Building block of PQ solution: effective operator aGG˜ Consider: 2-flavor QCD i.e., qT = (u, d)

1 2 a gs µν ˜ ∂µa 0 µ La = (∂µa) + 2 G Gµν + qcqγ γ5q 2 fa 32π 2fa

a gem ˜µν + FµνF − qLeMqqR + h.c. fa 8π

0 0 0 with Mq = diag(mu, md) and cq = diag(cu, cd).

Jonathan Steinberg The strong CP-Problem 28 / 38 The axionic solution of the strong CP problem The axion Lagrangian

Now: Eliminate aGG˜ by an field-dependent axial transformation

a i γ5Qa q 7→ e 2fa q

Qa is a matrix in flavor space with Tr(Qa) = 1 This generates a term like

2 Tr(Qa) a ˜ ∆ = −αs 2 GG 8π fa

Since Tr(Qa) = 1 =⇒ ∆ cancels axion-gluon term

Jonathan Steinberg The strong CP-Problem 29 / 38 The axionic solution of the strong CP problem The axion Lagrangian Effective potential at energies below ΛQCD: has absolute minimum at θ = 0 (Vafa, Witten) vanishing VEV ha(x)i = 0 No strong CP violation in vacuum One can compute V (θ) also explic- itly (Ch. Per. Th.) q  2 2  mu + md + 2mumd cos(θ) V (θ) = Σ(mu + md) 1 −  mu + md

where Σ = −huui = −hddi

Jonathan Steinberg The strong CP-Problem 30 / 38 The axionic solution of the strong CP problem The axion Lagrangian

corresponding particle excitation → axion has a mass ⇒ pseudo NG

109GeV ma = 5.691(51) meV fa

yields a very light boson could play role of dark matter

Jonathan Steinberg The strong CP-Problem 31 / 38 Experimental search

Two types of axion searches: Dark matter independent Dark matter dependent

Different types of experiments are sensitive to specific mass scales

Jonathan Steinberg The strong CP-Problem 32 / 38 Experimental search

Depending of the experiment they look for axion couplings to EM-field (ADMX,ABRACADABRA)

a µν L ⊃ FµνF˜ fa

Fermions (Atom interferometry, stellar cooling)

∂µa µ L ⊃ ψγ γ5ψ fa

Gluon fields (CASPEr-electric)

a µν L ⊃ GµνG˜ fa

Jonathan Steinberg The strong CP-Problem 33 / 38 Experimental search

Coupling to γ:

there exists an axion-photon coupling gaγγ gaγγ is model dependent KSVZ DFSZ Expected axion-photon coupling: O(10−17 − 10−12GeV−1)

3C.Grupen, Astroparticle Physics, Springer, 2005 Jonathan Steinberg The strong CP-Problem 34 / 38 Experimental search

The mass of the axion Can be estimated from cosmological considerations

−6 −2 10 eV < ma < 10 eV

numerical studies from QCD: 1 − 100µeV

promising technique in favored mass range: axion haloscope Earth-based instruments immersed in the Milky Way’s halo

Jonathan Steinberg The strong CP-Problem 35 / 38 Experimental search The ADMX experiment

3L. J. Rosenberg et al., PNAS, 112(40), 2015 Jonathan Steinberg The strong CP-Problem 36 / 38 Conclusion

We have seen that: ¨ there is a topological θ term in LQCD → ¨CP

make θ dynamical field a(x) respecting U(1)PC symmetry corresponding particle excitation → axion axion could be a dark matter candidate various experiments for different mass ranges experiments must be extremely sensitive

Jonathan Steinberg The strong CP-Problem 37 / 38 Further reading

There are (at least) two nice reviews: (1) Theory and Phenomenology of CP Violation, T. Mannel, Nucl. Phys. B 167 (2015) (2) The landscape of QCD axion models, Luca di Lutio et al., Phys. Rep. 870 (2020) There are (at least) two nice lecture series at youtube: (1) Andreas Ringwald, Axions and Axion Like Particles, https://www.youtube.com/watch?v=iWG5twaFaKo (2) The strong CP puzzle and axions, https://indico.cern.ch/event/703481/

Jonathan Steinberg The strong CP-Problem 38 / 38