Why (light) meson decays are interesting · Workshop on meson decays at CLAS – Jeﬀerson Lab 5 August 2012. Why (light) meson decays are interesting Johan Bijnens Motivation Theoretical

Why (light)meson decaysare interesting

Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

1/28

Why (light) meson decays are interesting

Johan Bijnens

Lund University

[email protected]

http://thep.lu.se/∼bijnens

http://thep.lu.se/∼bijnens/chpt.html

Workshop on meson decays at CLAS – Jefferson Lab 5 August 2012


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

2/28

Outline

1 Motivation

2 Theoretical tools

3 Pseudo-scalars

4 Vectors

5 Axial-vectors

6 Others

7 Conclusions


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

3/28

The big picture

This talk: introduction and pep talk

Three main motivations:

Understanding QCD (almost 40 years old now)

LQCD = −1

4G aµνG aµν +

∑

q

qγµ(

∂µ − i

2gSλ

aG aµ

)

q

G aµν

= ∂µGaν− ∂νG

aµ− igS f

abcG bνG cν

Determining standard model parameters preciselyTesting/finding effects beyond the standard model

Some examples of all will be mentioned

Understanding QCD needed for the other two

I will not talk about D, B , J/ψ, Υ,. . .


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

3/28

The big picture




LQCD = −1

4G aµνG aµν +

∑

q

qγµ(

∂µ − i

2gSλ

aG aµ

)

q

G aµν


aµ− igS f

abcG bνG cν






Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

3/28

The big picture




LQCD = −1

4G aµνG aµν +

∑

q

qγµ(

∂µ − i

2gSλ

aG aµ

)

q

G aµν


aµ− igS f

abcG bνG cν






Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

3/28

The big picture




LQCD = −1

4G aµνG aµν +

∑

q

qγµ(

∂µ − i

2gSλ

aG aµ

)

q

G aµν


aµ− igS f

abcG bνG cν






Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

4/28

Mesons are simple

Made of a quark and anti-quark

So these we should really understand in detail

In order of difficulty:

Static properties: mass,. . .Dynamic properties: formfactorsDynamic properties: decays and scattering


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

4/28

Mesons are simple

Made of a quark and anti-quark

So these we should really understand in detail

In order of difficulty:

Static properties: mass,. . .Dynamic properties: formfactorsDynamic properties: decays and scattering


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

5/28

Mesons are simple

quark-antiquark

add gluons

Formfactor

Two body decay


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

5/28

Mesons are simple

quark-antiquark

add gluons

Formfactor

Two body decay


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

5/28

Mesons are simple

quark-antiquark

add gluons

Formfactor

Two body decay


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

5/28

Mesons are simple

quark-antiquark

add gluons

Formfactor

Two body decay


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

6/28

Which mesons?

Pseudo-scalar octet(nonet)

π±, π0

K±, K 0,K 0

η(, η′)

Vector nonet

ρ0, ρ±, ωK ∗, φ

Scalars

f0(500) or σ, f0(980), a0(980), K0(800) or κ

Axial-vectors

a1(1260), f1(1285), b1(1235), K1(1270), K1(1400),. . .

Others


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

7/28

Theoretical tools: some general comments

TheoryStart from first principlesIn principle improvable to any precisionBut always beware of assumptions, approximations,. . .An uncontrolled approximation turns a theory in a modelSometimes the theory has many free parameters (orfunctions)Example: Chiral Perturbation Theory or perturbative QCD

ModelsNeeded whenever cannot be done from the full theoryCan be useful to summarize/understand resultsCan be a first step towards finding a good theoryExample: Nambu-Jona-Lasinio model

In betweenTypically a theory with uncontrolled approximationsRequires a certain finesse to get reliable resultsExample: Schwinger-Dyson equations


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

7/28






Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

7/28






Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

8/28

Theoretical tools: QCD only

Perturbative QCD: not directly useful but input as constraintson other methods

Structure functions

Light-cone wave functions

Formfactors at large Q2

. . .


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

9/28


Lattice gauge theory:

Discretize space time and integrate out the quarks andgluons numerically

From first principles (in principle)

Extrapolations needed (see talks next week)

Static properties: good

Formfactors of stable particle: starting

Decays and scattering of stable particles: difficult

Resonances: very difficult

Main reason for all the difficulties: everything is inEuclidean space (imaginary time)

Why: Minkowski space: integrands are oscillatory:∫

dt e iωt rather than∫

dt e−ωt


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

10/28


Light cone quantization Brodsky, Pauli, Phys.Rept. 301 (1998) 299-486

[hep-ph/9705477]

Only physical degrees of freedom (ie no ghosts . . . )

Wave functions are expanded in Fock states: partonsdirectly visible

The perturbative vacuum is the physical vacuum

In principle allows for a competing numericalnonperturbative method

Was a very active field 1990s

Main (unsolved) difficulty: dealing with the zero modeThis is where all the trouble of spontaneous symmetrybreaking and confinement hides in this approach

Note: this not quark models on the light cone


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

11/28


QCD, Finite Energy Sum Rules, . . .

All rely on analyticity and Cauchy’s theorem1

2πi

∮

C

dz f (z) =∑

poles

residues

a typical curvex

q2Im q2

Re q2

Circle and residue points: perturbative QCD

Axis: data and/or resonance saturation


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

12/28

Theoretical tools: other theory

Dispersion relations and unitarity

Again Cauchy’s theorem

But now choose f (z) e.g. a decay or scattering amplitude

s, t, u: moire parameters

Unitarity 1 = S†S = 1 + T †T + i(T − T †)

Due to the cuts: phases provide constraints

Integral equations for the amplitudes

Questions: subtraction constants, experimental input forphases, asymptotic behaviour


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

13/28


Chiral Perturbation Theory

see my talk tomorrow

Well defined effective field theory

Might or might not converge

Often many parameters (Low-Energy-Constants)

All (in principle) measurable

In addition models can/must be used to estimateparameters

State of the art: 2-loop (most needed processes done)

For other than pseudo-scalars: quite limited


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

14/28


Schwinger-Dyson equations

Idea (φ3): = +

Full three-point function involves full four-point function

Four involves five, . . .

An infinite set of consistency equations

Need to truncate

Need for a starting ansatz to make life bearable (usually afull gluon propagator)

Usually kept at the “quenched” approximation


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

15/28

Models with “Quarks”

Nonrelativistic constituent quark models: understandingthe spectrum (fill up octets and nonets)

Chiral quark model: quarks plus pseudo-scalars, noconfinement

Nambu-Jona-Lasinio models: Quarks with a four quarkinteraction

Has spontaneous chiral symmetry breakingProduces a constituent quark mass from a gap equation:

= +

mesons from a bubble sum

Mesons but no confinement


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

16/28

Models with “Quarks”

Add vector-like four quark interactions =⇒ vectors andaxial-vertex

Add a ‘t Hooft vertex to get η′ better (or a variation onthat vertex)

make the vertex non-local

Add Polyakov loop

Many more variations possible

Usually large Nc or tree level at the “meson” level

Some attempts to go beyond that: many difficulties andnot clear if it ever yielded something useful


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

17/28

Effective Lagrangians

Basic degrees of freedom: hadron fields

Beware of product (mis)labelling

Chiral Perturbation TheoryChiral Effective TheoryAre very popular names and “de vlag dekt niet altijd delading” since they are not protected names (free flagdoesn’t make free bottom)

Note field redefinitions: same Lagrangian can look verydifferent

Hope: find a simple Lagrangian and then refine it

A full classification attempt: Resonance chiral theory(RχT), also attempts to go to one-loop.


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

18/28

π-decays

Lightest hadrons

π0: main decay electromagnetic

→ γγ: test of the anomaly→ e+e−: existing discrepancy (KTeV) with standardmodel? extra contributions?→ γe+e−: FV

→ νν: looking for new stuff

π+ (or π− to the charge conjugate state)

main decay: weak leptonic→ µ+ν(γ): Fπ

→ e+νγ: FV and FA: CVC→ e+ν: lepton universality→ π0e+ν: Vud


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

19/28

π0 → e+e−

KTeV: BR = (7.48 ± 0.29± 0.25) · 10−8

π0

e+

e−

Fπγγ(q21 , q

22)

Fπγγ(q21 , q

22) is an object we also like to know for g − 2 of

muon.

Unitarity: on-shell γγ BR ≥ (4.75 ± 0.02) · 10−8

Typical VMD 6.4 · 10−8ChPT : fix coefficient from η → µ+µ−, compatible (signambiguity)

Dorokhov-Ivanov BR = (6.2 ± 0.1) · 10−8 (3.3 σ)

(Reasonable?) assumption Fπγγ(t, t) = 1/(1 + t/s1)


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

20/28

K -decays

Main decay: weak (nonleptonic, semileptonic, leptonic)

Interplay of weak and strong interaction

K → ππ: ∆I = 1/2 enhancement

K → 3π:

Dynamics: Dalitzplot distributionsCP-violationππ-scattering lengths

K → ℓν: Lepton universality, FK

K → πℓν: Vus , formfactors

K → ππℓν: formfactors (L− i r ), ππ-scattering lengths

K → πℓ+ℓ−, K → πνν: get VCKM fully from K system,strong constraints on new physics models


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

21/28

η-decays

η → 3π: mu −md or Q: Lanz on Tuesday

η → γ∗γ∗: anomaly, formfactors

η → π0γγ dominated by high orders in ChPT

η → π+π−γ∗: anomaly, formfactors


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

22/28

Vectors: Theoretical tools/questions

Effective Lagrangians: does there exist a consistentpower-counting?Is there a preferred way to describe vectors?Massive Yang-Mills, Hidden Local Symmetry,Antisymmetric tensor,. . . ?Does there exist something like Vector Meson Dominancebeyond the pion electromagnetic formfactor?Large Nc says that a tree level Lagrangian should exist,but not that it must be a simple one: how well does thisreally work?Lattice has difficulties with resonances, can be done butnot very accurate at present (if quark masses such thatresonace is above threshold)For static properties and decay constants ChPT can bedoneKSRF relation between mV , gV and fπ


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

23/28

Examples of decays

ω → 3π

Can this be modelled via ω → ρπ → 3π?

ρ-ω-φ mixing

ρ0-ρ+ mass difference: e+e− and τ , also in otherproduction modes?

Radiative decays

ρ, ω → πγ, φ→ ηγ, η′ → ργ η′ → ωγSame but with the photon off-shell, i.e. to e+e−, µ+µ−

Are these related via (naive) VMD (no from NA60)


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

24/28

ω → π0µ+µ−

Fω =1

1−m2µµ/Λ2

ω

F expω

6= 1

1−m2µµ/m2

V

Terschlusen-Leupold

F expω

=1 +m2

µµ/m2

V

1−m2µµ/m2

V


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

25/28

ω → π0µ+µ−

100 (P1)(P2)

stand. VMD

corresponding differential

decay rate:

6

7

8

9

[10-6

GeV

-1]

param. set (P1)param. set (P2)

stand. VMD NA60

Terschlusen-Leupold achieve this with lowest order terms in theantisymmetric tensor representation, other representations needhigher derivative interactions


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

26/28

Axial vectors

Are they the chiral partners of the ρ?

How do the axial nonets mix in the strange and singletsectors?Relevant question for muon g − 2

a1 → 3π via a1 → ρπ or additional contributions

a1 → πγ: this vanishes in almost simple theoretical models

Mass: Weinberg sum rules predict ma1 ≈√2mρ

Width (PDG) 250 to 600 MeV our estimate

is there a problem with continuum/resonance separationdue to the large width?


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

27/28

Others

Many more resonances

Questions similar:

Does it really existmasswidthMain decay modesMixing with other particles

On the theory side

nature of the particleWhat is the best way to describe itIf used in resonance saturation models we need itscouplings


Johan Bijnens

Motivation

Theoreticaltools

Pseudo-scalars

Vectors

Axial-vectors

Others

Conclusions

28/28

An overview of theoretical tools

A few examples of questions and decays

Documents

Why (light) meson decays are interesting · Workshop on meson decays at CLAS – Jeﬀerson Lab 5 August 2012. Why (light) meson decays are interesting Johan Bijnens Motivation Theoretical