Experimental verification of CPT symmetry in kaon physics

Wojciech Wiślicki, NCBJ

16 May 2014

HEP seminar, University of Warsaw - Caveats: i) we consider CPT only; no discussions of CP or T violation ii) restrict to neutral kaons

- C, P and T - the background, notations

- Approach: Bell-Steinberger relation CPT and anti-CPT theorems Decoherence

- CPT from Bell-Steinberger relation, world data, incl. KLOE

- CPT and Lorentz invariance, KLOE measurement

- Further discussions on meaning of the results and experimental perspectives Discrete symmetry transformations C, P and T: what they do?

Particle ↔ antiparticle

Space inversion

Time reversal

are real phases

Effective formalism for pairs of neutral mesons: the Hamiltonian and its symmetries

2x2, time-independent, in basis of flavour eigenstates ( s for K, c for D, b for B etc.)

, mass & decay matrices, both Hermitian Conservation of CPT requires

(CP requires, in addition )

Parametrize possible violation of CPT by

To remind: CP violation is traditionally parametrized by ε

From now on assume expecting possible

CPT-violating effects to be tiny (if any whatsoever ..)

Mass eigenstates are not the flavour eigenstates In pairs of electrically neutral flavoured mesons, flavour eigenstates mix via box diagrams, e.g. for kaons

ΔS=2 transitions

H eigenstates expressed via H eigenvalues (or CP- and CPT-violation parameters defined by them) and flavour eigenstates

CP violation parameter

CPT violation parameter ( means CPT not violated)

Bell-Steinberger formula (1965)

Relation between H eigenstates and decay amplitudes

Assumptions

- Initial states of K and final decay states belong to orthogonal subspaces -Time evolution is unitary (probability conserved)

Thesis

(sum over complete final set) Another form

Knowing amplitudes at rhs, kaon properties and ε, one determines Im(δ)

Note

Observation of BS failure entails breakdown of unitarity or incompletness of the set of final states CPT theorem (Schwinger, Lüders, Jost, Pauli, Bell: 1951-57)

Important theorem relating CPT invariance to fundamental properties of the theory Assumptions: - Hermitian H - Unitary evolution - Locality, i.e. fields (anti)commute for (half)integer spin fields at spacelike separations - Finite vacuum expectation value of field products - Lorentz invariance Thesis: The theory is CPT-invariant: [CPT,H]=0 Some phenomenological consequences of the CPT theorem (Lüders, Zumino, 1957)

- Particles and antiparticles have the same masses

- Particles and antiparticles have the same lifetimes (decay widths)

- Equality of partial decay widths of mutually CPT-coupled channels

- Equality of total reaction rates from initial set of particles and corresponding set of antiparticles with spins reversed

Further developments lead to inverse (anti-CPT) theorem (Greenberg 2002)

Under general assumptions (hermiticity, unitarity), in field theory of pointlike particles: violation of CPT invariance entails violation of Lorentz invariance

Numerous subtleties, depending on formulation of the CPT theorem

Clear hint for phenomenology: CPT tests probe also Lorentz invariance

Experimental test of Bell-Steinberger formula

Major advantage of neutral K system: only a few final states contribute significantly to rhs: main: 2π, much smaller: 2πγ, 3π, πe(μ)ν The rest is negligible

Rewrite rhs of BS relation to the form

The α's can be always expressed by amplitude ratios η and branching fractions

Phases for η should be known (interferometric measurements)

The major data contributors (after 1990)

- CPLEAR, CERN: proton-antiproton anihilation at rest, closed

- NA48, CERN: KL and KS decays in the same detector, closed neutral kaon program

- KTeV, Fermilab: similar principle, closed

- KLOE, Frascati: φ(1020) → KL KS, finished previous runs, uptaking with upgraded apparatus

Recent contributions only from KLOE 2π non-radiative decays

World average data

2πγ decays

Inner bremsstrahlung

Direct emission

And much smaller for neutral pions

Interferometric measurement of E773 (FNAL) gives

Combining with BR

Small compared to 2π

3π decays

For neutral-pion mode recent improvement from KLOE (factor 5 in BR and accuracy on η, Phys.Lett. B723(2013)54)

Semileptonic decays

General parametrization of Kl3 matrix elements

parametrize CPT

parametrize ΔS═ΔQ

where

determined by KLOE from

and determined by CPLEAR from

In practice, solution of BS eqn.

where

The numbers

Sole CPLEAR

BS

At 95% confidence level

Much worse knowledge on Re(δ), base only on

CPLEAR semi-leptonic data

CPT vs. Lorentz symmetry: Intuition support (vague)

- x-dependent couplings in Lagrangian density, slowly varying in large scale (e.g. the Galaxy); derivatives select direction in space-time

- vector fields with degenerated vacuum

Lowest-energy configuration requires non-zero

The vacuum contains intrinsic direction violating rotational invariance and thus Lorentz symmetry More specific proposal: extension of the Standard Model by CPT violating terms

Kostelecky, Colladay, late 1990's

Lagrangian contains additional CPT-violating terms

Modified Dirac equation for quark or lepton field

CPT-violating terms of SME; couplings a, b, .. are phenomenological at this stage, to be constrained by experiment

For experiment's parametrizations consider the simplest case: nonzero first term of SME

with

4-vector quantifying CPT violation, depending on flavour (quark's or lepton's); perhaps more fundamental physics behind

Fundamental theory still unknown but enables some experimentally testable predictions

Sidereal-time variations of observables

For system of neutral mesons, e.g.

Meson four-velocity in the observer's frame

Difference between SME couplings of meson's valence quarks

CPT observables are expected to vary with magnitude and orientation of meson momentum

Direct calculation gives

Since laboratory rotates together with the Earth, there is a sidereal-time („star time”) dependence of the observables

Sidereal time is shorter by ~4 mins/24 hrs than solar time, due to the orbital motion of the Earth around Sun

Working with sidereal time ensures variations on daily (and not on seasonal) basis

At noon, observer on Earth has Sun and a distant star in zenith

After 24 hrs Earth moved on its orbit around Sun

Observer looking at the same star needs another 4 mins to catch up and let Earth self-rotate to see Sun in zenith again

Pict. courtesy Wikipedia

Time-dependence of δ in rotating frame

Ω is Earth sidereal rotational frequency χ is angle between z Lab axis and Earth's rotation axis

Θ, φ – K angles in LAB Neutral kaons produced in coherent entangled state

Observable: double-differential spectrum of decay-time difference

Event reconstruction in KLOE

Two-pion K decays considered

K decay vertices reconstructed from π tracks

K decay lengths and Φ vertex positions were left free in fits

KLOE result, published recently PL B730(2014)89

(all in GeV)

Main contributions to systematics: selection cuts, background, uncertainty in experiment's orientation

High sensitivity of the method was demonstrated, close to K-Kbar mass difference Other results using the same methodology

Babar, PRL 100(2008)131802

FOCUS (FNAL), PLB 556(2003)7

Final remarks on the meaning of results and experimental perspectives

Currently, using neutral mesons one can test CPT with sensitivity around

We are close to mass scale given by combination

If there are effects related to new physics at Planck scale, one might touch them, provided experimental sensitivity and accuracy are improved

Sensitivity of interferometric method depends on how accurate one measures the region of first interference peak; subsequent peaks are quenched by exponential terms

Experimental resolutions in time are crucial

Time resolution in KLOE was >~ 100 ps (of the order of KS lifetime) – that is not good enough!

Time resolution in KLOE-2 is going to be ~25 ps - better but still not good enough..

LHCb spectrometer gives overwhelmingly better time resolution

This is fine but should be exploited with K, and not B, because frequency of oscillations is higher for heavier quarks

For B, better resolution is partially compensated by shorter period of oscillations Should be studied with K at LHCb Needs dedicated trigger for decays

Other CPT invariance tests

Backups

Effect of discrete symmetries

2πγ, determination of form-factors and ratios

KTeV data for KL, formulas by Sehgal

KTeV measurements are done with cut

For small γ energies, the inner bremsstrahlung amplitude becomes singular

At E→0, behavior of IB amplitudes is given by F.Low's theorem (1958)

Amplitudes of radiative processes are singular in the limit of low energy. However, they can be expressed in terms of non-radiative amplitudes and their derivatives over energy and photon's angles

Then

Amplitudes' ratios behave thus

CPT violation as a dissipative process

Two motivations, rooted in basics of quantum mechanics and, again, search for background effects

1. If quantum-mechanical system of entangled fields propagates in a medium, quantum coherence tends to fade away (decoherence effect)

2. In gravitational background of micro black holes decoherence entails loss of information (Hawking 1976)

Loss of coherence in medium is expected: interaction (with a medium) destroys interference terms in density matrix (except for special cases, not applicable to KLOE) Evolution of pure into mixed state entails loss of information and irreversibility, hence inverse scattering matrix does not exist in general.

Evolution of pure state into mixed state violates CPT invariance ( Theorem proved by R. Wald, Phys. Rev. D21(1980)2742)

Far-reaching consequence: if there exists indelible background in microscale (e.g. quantum gravity), decoherence should be

always with us For system of neutral mesons this can be parametrized

„Decoherence parameter”

ζ can be determined experimentally using decay time spectra in pairs of mesons

This parametrization is not really meaningful physically (e.g. depends on basis); more sophisticated approaches exist but for CPT just (non)existence of decoherence

is crucial

Isolated quantum system evolves according to M equation and pure quantum states remain pure forever

In dissipative medium, evolution is governed by Liouville equation where dissipative term gives decoherence

meaning that pure quantum states become mixed states and interference disappears

For 2-meson system, dissipative term can be represented as 4x4 matrix and parametrized (formalism developed by A. Kossakowski 1972 and G. Lindblad 1976 for general dissipative systems)

Decay-time difference spectrum I(Δt) is modified by 3 real parameters α, β, γ quantifying (roughly speaking) dissipative contributions to the pure and interference terms

KLOE result, 2007 (all in GeV)

There also exists result for B system, by Belle Collaboration, but with worse accuracy

What next so ..

KLOE is going to improve results (~ order of magnitude in total accuracy) mainly by: a) highier luminosity due to modified beam geometry (statistics increase) b) 4 times better time resolution due to the GEM-based inner tracker detector (just installed)

KTeV at FNAL