APPLICATIONS OF CHIRAL

APPLICATIONS OF CHIRAL

SYMMETRY AT HIGH ENERGIES

Johan Bijnens Lund University

bijnens@thep.lu.se

http://www.thep.lu.se/∼bijnens

Various ChPT: http://www.thep.lu.se/∼bijnens/chpt.html

Overview

Hadronic and Flavour Physics: why we do this Effective Field Theory

Chiral Perturbation Theor(y)(ies)

Hard Pion Chiral Perturbation Theory

Overview

Hadronic and Flavour Physics: why we do this Effective Field Theory

Chiral Perturbation Theor(y)(ies)

Hard Pion Chiral Perturbation Theory K_ℓ3 Flynn-Sachrajda, arXiv:0809.1229

K → ππ JB+ Alejandro Celis, arXiv:0906.0302

F_π^S and F_π^V JB + Ilaria Jemos, arXiv:1011.6531 a two-loop check

B, D → π JB + Ilaria Jemos, arXiv:1006.1197

B, D → π, K, η JB + Ilaria Jemos, arXiv:1011.6531

χ_c(J = 0, 2) → ππ, KK, ηη JB+Ilaria Jemos, arxiv:1109.5033

Some examples which do not have a chiral log prediction

Hadrons

Hadron: αδρoς (hadros: stout, thick)

Lepton: λǫπτ oς (leptos: small, thin, delicate) (_{ς = σ 6= ζ}) In those days we had n, p, π, ρ, K, ∆ and e, µ.

Hadrons: those particles that feel the strong force Leptons: those that don’t

Hadrons

Hadron: αδρoς (hadros: stout, thick)

Lepton: λǫπτ oς (leptos: small, thin, delicate) (_{ς = σ 6= ζ}) In those days we had n, p, π, ρ, K, ∆ and e, µ.

Hadrons: those particles that feel the strong force Leptons: those that don’t

But they are fundamentally different in other ways too:

Leptons are known point particles up to about 10⁻¹⁹m ∼ ~_{c/(1 TeV)}

Hadrons have a typical size of 10⁻¹⁵m, proton charge radius is 0.875 fm

Hadrons

Hadrons come in two types:

Fermions or half-integer spin: baryons (βαρυς barys, heavy)

Bosons or integer spin: mesons (µǫςoς mesos, intermediate)

Hadrons

Hadrons come in two types:

Fermions or half-integer spin: baryons (βαρυς barys, heavy)

Bosons or integer spin: mesons (µǫςoς mesos, intermediate)

Main constituents:

Baryons: three quarks or three anti-quarks Mesons: quark and anti-quark

Hadrons

Hadrons come in two types:

Fermions or half-integer spin: baryons (βαρυς barys, heavy)

Bosons or integer spin: mesons (µǫςoς mesos, intermediate)

Main constituents:

Baryons: three quarks or three anti-quarks Mesons: quark and anti-quark

Comments:

Quarks are as pointlike as leptons

Hadrons with different main constituents:

glueballs (no quarks), hybrids (with a basic gluon)

Hadron(ic) Physics

The study of the structure and interactions of hadrons

Flavour Physics

There are six types (Flavours) of quarks in three generations or families

up, down; strange, charm; bottom and top

The only (known) interaction that changes quarks into each other (violates the separate quark numbers) is the weak interaction

Violates also discrete symmetries: ^Charge conjugation, Parity and ^T ime reversal.

Flavour Physics

There are six types (Flavours) of quarks in three generations or families

up, down; strange, charm; bottom and top

The only (known) interaction that changes quarks into each other (violates the separate quark numbers) is the weak interaction

Violates also discrete symmetries: ^Charge conjugation, Parity and ^T ime reversal.

The study of quarks changing flavours (mainly) in decays

Flavour Physics

There are six types (Flavours) of quarks in three generations or families

up, down; strange, charm; bottom and top

The only (known) interaction that changes quarks into each other (violates the separate quark numbers) is the weak interaction

Violates also discrete symmetries: ^Charge conjugation, Parity and ^T ime reversal.

The study of quarks changing flavours (mainly) in decays Experimental research typically done at flavour/hadron factories

Hadron Physics: WASA@COSY

Flavour Physics: DAΦNE in Frascati

Flavour Physics: KEK B in Tsukuba

Flavour Physics: NA48/62 at CERN

Flavour Physics

The Standard Model Lagrangian has four parts:

L^H(φ)

| {z } Higgs

+ L^G(W, Z, G)

| {z } Gauge X

ψ=fermions

ψiD¯ / ψ

| {z }

gauge-fermion

+ X

ψ,ψ^′=fermions

g_ψψ^′ψφψ¯ ^′

| {z }

Yukawa

Last piece: weak interaction and mass eigenstates different

Many extensions: much more complicated flavour changing sector

Flavour Physics

Experiments in flavour physics often very precise

New effects start competing with the weak scale: can be very visible

If it changes flavour: limits often very good

Flavour Physics

Experiments in flavour physics often very precise

New effects start competing with the weak scale: can be very visible

If it changes flavour: limits often very good

s d

u, c t

W

γ, g, Z

Heavy particles can contribute in loop

Flavour Physics

Experiments in flavour physics often very precise

New effects start competing with the weak scale: can be very visible

If it changes flavour: limits often very good

s d

u, c t

W

γ, g, Z

Heavy particles can contribute in loop

Sometimes need a precise prediction for the standard model effect

Flavour Physics

A weak decay:

Hadron: 1 fm

W-boson: 10⁻³ fm ^s

f

u d u

Flavour Physics

A weak decay:

Hadron: 1 fm

W-boson: 10⁻³ fm ^s

f

u d u

Flavour Physics

Flavour and Hadron Physics: need structure of hadrons Why is this so difficult?

Flavour Physics

Flavour and Hadron Physics: need structure of hadrons Why is this so difficult?

QED _{L = ψγµ} (∂^µ − ieA^µ) ψ − ¹₄F_µνF^µν QCD: _{L = qγµ} ∂^µ − i^g₂G^µ

q − ¹₈tr (G_µνG^µν) G_µ = G^a_µλ^a is a matrix

Flavour Physics

Flavour and Hadron Physics: need structure of hadrons Why is this so difficult?

QED _{L = ψγµ} (∂^µ − ieA^µ) ψ − ¹₄F_µνF^µν QCD: _{L = qγµ} ∂^µ − i^g₂G^µ

q − ¹₈tr (G_µνG^µν) G_µ = G^a_µλ^a is a matrix

F_µν = ∂_µA_ν − ∂^νA_µ

G_µν = ∂_µG_ν − ∂^νG_µ − ig (G^µG_ν − G^νG_µ) gluons interact with themselves

e(µ) smaller for smaller µ, g(µ) larger for smaller µ QCD: low scales no perturbation theory possible

Comments

Same problem appears for other strongly interacting theories

What to do:

Give up: well not really what we want to do

Comments

Same problem appears for other strongly interacting theories

What to do:

Give up: well not really what we want to do

Brute force: do full functional integral numerically Lattice Gauge Theory:

discretize space-time

quarks and gluons: 8 × 2 + 3 × 4 d.o.f. per point Do the resulting (very high dimensional) integral numerically

Large field with many successes

Comments

Same problem appears for other strongly interacting theories

What to do:

Give up: well not really what we want to do

Brute force: do full functional integral numerically Lattice Gauge Theory:

discretize space-time

quarks and gluons: 8 × 2 + 3 × 4 d.o.f. per point Do the resulting (very high dimensional) integral numerically

Large field with many successes Not applicable to all observables

Need to extrapolate to small enough quark masses

Comments

Same problem appears for other strongly interacting theories

What to do:

Give up: well not really what we want to do

Brute force: do full functional integral numerically Lattice Gauge Theory:

discretize space-time

quarks and gluons: 8 × 2 + 3 × 4 d.o.f. per point Do the resulting (very high dimensional) integral numerically

Large field with many successes Not applicable to all observables

Need to extrapolate to small enough quark masses Be less ambitious: try to solve some parts only: EFT

Wikipedia

http://en.wikipedia.org/wiki/

Effective field theory

In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the

appropriate degrees of freedom to describe physical

phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter

distances (or, equivalently, higher energies).

Effective Field Theory (EFT)

Main Ideas:

Use right degrees of freedom : essence of (most) physics

If mass-gap in the excitation spectrum: neglect degrees of freedom above the gap.

Examples:











Solid state physics: conductors: neglect the empty bands above the partially filled one

Atomic physics: Blue sky: neglect atomic structure

EFT: Power Counting

➠ gap in the spectrum _=⇒ separation of scales

➠ with the lower degrees of freedom, build the most general effective Lagrangian

EFT: Power Counting

➠ gap in the spectrum _=⇒ separation of scales

➠ with the lower degrees of freedom, build the most general effective Lagrangian

➠ _∞# parameters

➠ Where did my predictivity go ?

EFT: Power Counting

➠ gap in the spectrum _=⇒ separation of scales

➠ with the lower degrees of freedom, build the most general effective Lagrangian

➠ _∞# parameters

➠ Where did my predictivity go ?

= ⇒

EFT: Power Counting

➠ gap in the spectrum _=⇒ separation of scales

➠ with the lower degrees of freedom, build the most general effective Lagrangian

➠ _∞# parameters

➠ Where did my predictivity go ?

= ⇒

➠ Taylor series expansion does not work (convergence radius is zero when massless modes are present)

➠ Continuum of excitation states need to be taken into account

Example: Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

Example: Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

L^A = Φ^†_v∂_tΦ_v + . . . L^γA = GF_µν² Φ^†_vΦ_v + . . . Units with h/ = c = 1: G energy dimension ₋₃:

Example: Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

L^A = Φ^†_v∂_tΦ_v + . . . L^γA = GF_µν² Φ^†_vΦ_v + . . . Units with h/ = c = 1: G energy dimension ₋₃:

σ ≈ G²E_γ⁴

Example: Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

L^A = Φ^†_v∂_tΦ_v + . . . L^γA = GF_µν² Φ^†_vΦ_v + . . . Units with h/ = c = 1: G energy dimension ₋₃:

σ ≈ G²E_γ⁴

blue light scatters a lot more than red

=⇒ red sunsets

=⇒ blue sky Higher orders suppressed by 1 Å/λ_γ.

References

A. Manohar, Effective Field Theories (Schladming lectures), hep-ph/9606222

I. Rothstein, Lectures on Effective Field Theories (TASI lectures), hep-ph/0308266

G. Ecker, Effective field theories, Encyclopedia of Mathematical Physics, hep-ph/0507056

D.B. Kaplan, Five lectures on effective field theory, nucl-th/0510023

A. Pich, Les Houches Lectures, hep-ph/9806303

S. Scherer, Introduction to chiral perturbation theory, hep-ph/0210398

J. Donoghue, Introduction to the Effective Field Theory

Chiral Perturbation Theory

Exploring the consequences of the chiral symmetry of QCD and its spontaneous breaking using effective field theory techniques

Chiral Perturbation Theory

Exploring the consequences of the chiral symmetry of QCD and its spontaneous breaking using effective field theory techniques

Derivation from QCD:

H. Leutwyler, On The Foundations Of Chiral Perturbation Theory, Ann. Phys. 235 (1994) 165 [hep-ph/9311274]

The mass gap: Goldstone Modes

UNBROKEN: V (φ)

Only massive modes around lowest energy state (=vacuum)

BROKEN: V (φ)

Need to pick a vacuum hφi 6= 0: Breaks symmetry No parity doublets

Massless mode along bottom For more complicated symmetries: need to describe the bottom mathematically: G → H =⇒ G/H

Some clarifications

φ(x): orientation of vacuum in every space-time point Examples: spin waves, phonons

Nonlinear: acting by a broken symmetry operator changes the vacuum, φ(x) → φ(x) + α

The precise form of φ is not important but it must describe the space of vacua (field transformations possible)

In gauge theories: the local symmetry allows the vacua to be different in every point, hence the Goldstone

Boson might not be observable as a massless degree of freedom.

The power counting

Very important:

Low energy theorems: Goldstone bosons do not interact at zero momentum

Heuristic proof:

Which vacuum does not matter, choices related by symmetry

φ(x) → φ(x) + α should not matter

Each term in _L must contain at least one ∂_µφ

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown

Power counting: Dimensional counting

Expected breakdown scale: Resonances, so M_ρ or higher depending on the channel

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown

Power counting: Dimensional counting

Expected breakdown scale: Resonances, so M_ρ or higher depending on the channel

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU (3)_V But _LQCD = X

q=u,d,s

[i¯q_LD/ q_L + i¯q_RD/ q_R − m^q (¯q_Rq_L + ¯q_Lq_R)]

So if m_q = 0 then SU (3)_L × SU(3)^R.

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown

Power counting: Dimensional counting

Expected breakdown scale: Resonances, so M_ρ or higher depending on the channel

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU (3)_V But _LQCD = X

q=u,d,s

[i¯q_LD/ q_L + i¯q_RD/ q_R − m^q (¯q_Rq_L + ¯q_Lq_R)]

So if m_q = 0 then SU (3)_L × SU(3)^R.

Can also see that via ^{v < c}, m_q 6= 0 =⇒

Chiral Perturbation Theory

h¯qqi = h¯q^Lq_R + ¯q_Rq_Li 6= 0

SU (3)_L × SU(3)^R broken spontaneously to SU (3)_V

8 generators broken _=⇒ 8 massless degrees of freedom and interaction vanishes at zero momentum

We have 8 candidates that are light compared to the other hadrons: π⁰, π⁺, π⁻, K⁺, K⁻, K⁰, K⁰, η

Chiral Perturbation Theory

h¯qqi = h¯q^Lq_R + ¯q_Rq_Li 6= 0

SU (3)_L × SU(3)^R broken spontaneously to SU (3)_V

8 generators broken _=⇒ 8 massless degrees of freedom and interaction vanishes at zero momentum

Power counting in momenta (all lines soft):

p²

1/p²

R _{4 4}

(p²)² (1/p²)² p⁴ = p⁴

(p²) (1/p²) p⁴ = p⁴

Chiral Perturbation Theories

Baryons

Heavy Quarks

Vector Mesons (and other resonances)

Structure Functions and Related Quantities Light Pseudoscalar Mesons

Two or Three (or even more) Flavours

Strong interaction and couplings to external currents/densities

Including electromagnetism

Including weak nonleptonic interactions Treating kaon as heavy

Many similarities with strongly interacting Higgs

Hard pion ChPT?

In Meson ChPT: the powercounting is from all lines in Feynman diagrams having soft momenta

thus powercounting = (naive) dimensional counting

Hard pion ChPT?

In Meson ChPT: the powercounting is from all lines in Feynman diagrams having soft momenta

thus powercounting = (naive) dimensional counting

Baryon and Heavy Meson ChPT: p, n, . . . B, B^∗ or D, D^∗ p = M_Bv + k

Everything else soft

Works because baryon or b or c number conserved so the non soft line is continuous

p π

Hard pion ChPT?

In Meson ChPT: the powercounting is from all lines in Feynman diagrams having soft momenta

thus powercounting = (naive) dimensional counting

Baryon and Heavy Meson ChPT: p, n, . . . B, B^∗ or D, D^∗ p = M_Bv + k

Everything else soft

Works because baryon or b or c number conserved so the non soft line is continuous

Decay constant works: takes away all heavy momentum

General idea: M_p dependence can always be

reabsorbed in LECs, is analytic in the other parts k.

Hard pion ChPT?

(Heavy) (Vector or other) Meson ChPT:

(Vector) Meson: p = M_V v + k

Everyone else soft or p = M_V v + k

Hard pion ChPT?

(Heavy) (Vector or other) Meson ChPT:

(Vector) Meson: p = M_V v + k

Everyone else soft or p = M_V v + k

But (Heavy) (Vector) Meson ChPT decays strongly

ρ ρ

π

π

Hard pion ChPT?

(Heavy) (Vector or other) Meson ChPT:

(Vector) Meson: p = M_V v + k

Everyone else soft or p = M_V v + k

But (Heavy) (Vector) Meson ChPT decays strongly First: keep diagrams where vectors always present Applied to masses and decay constants

Decay constant works: takes away all heavy momentum

It was argued that this could be done, the

nonanalytic parts of diagrams with pions at large momenta are reproduced correctly JB-Gosdzinsky-Talavera

Done both in relativistic and heavy meson formalism General idea: M_V dependence can always be

reabsorbed in LECs, is analytic in the other parts k.

Toy model: one heavy one light scalar

JB-Gosdzinsky-Talavera 97

p ≈ -M⋅ v p ≈ 0 p ≈ M⋅ v

Momentum space

soft, particle on-shell, anti-particle on-shell

Toy model: one heavy one light scalar

JB-Gosdzinsky-Talavera 97

p ≈ -M⋅ v p ≈ 0 p ≈ M⋅ v

Momentum space

soft, particle on-shell, anti-particle on-shell

(a) (b) (c)

(d) (e) (f)

relativistic theory

heavy-meson theory

Toy model: one heavy one light scalar

Self-energy or mass corrections

Φ π Φ

π

Π^F_φ = 2λ²M² i 16π²

 1

ǫ + log 4π − γ^e − log M² µ²



+ 4λ²m² i 16π²



1 − log −m²

µ² + log M² µ²



+ O(1/M²)

Toy model: one heavy one light scalar

Self-energy or mass corrections

Φ π Φ

π

Π^F_φ = 2λ²M² i 16π²

 1

ǫ + log 4π − γ^e − log M² µ²



+ 4λ²m² i 16π²



1 − log −m²

µ² + log M² µ²



+ O(1/M²)

Φ π Φ

Π^E_φ = −4iλ² m² 16π²



−1

ǫ + γ_e − log 4π − 1 + log m² µ²



+O(1/M²)

log(m²) terms are the same

Toy model: one heavy one light scalar

Scalar formfactor:

(a) (b)

(b):

8iλ² (4π)²

 1

ǫ − γ^e + log(4π) − Z ₁

0

log m² − q²x(1 − x) − iε µ²

 dx



(a):

I = 8iλ²M² (4π)²

Z ₁

0

ydxdy

(−m² + y(1 − y)[Q² − q²x] + [xy − (xy)²]q² + iε)

Toy model: one heavy one light scalar

Scalar formfactor:

(a) (b)

(b):

8iλ² (4π)²

 1

ǫ − γ^e + log(4π) − Z ₁

0

log m² − q²x(1 − x) − iε µ²

 dx



(a):

I = 8iλ²M² (4π)²

Z ₁

0

ydxdy

(−m² + y(1 − y)[Q² − q²x] + [xy − (xy)²]q² + iε) 1/M: away from y ≈ 0, y ≈ 1 expand in 1/M²; in (1 − y) and y near 1 and 0.

I ≈ 8iλ²M² (4π)²

Z ₁

0

dx

(Z ₁

1−δ

dy

(−m² + (1 − y)Q² + x(1 − x)q² + iǫ) + Z _α

0

ydy

−m² + yQ² + iε )

= 8iλ²M² (4π)²

Z ₁

0

dx



− 1

M² log    

m² − x(1 − x)q² M²

− i π Z _δ

0

dz δ(m² − x(1 − x)q² − zQ²)



full agreement in nonanalytic dependence on m² and q²

Hard pion ChPT?

Heavy Kaon ChPT:

p = M_Kv + k

First: only keep diagrams where Kaon goes through Applied to masses and πK scattering and decay

constant Roessl,Allton et al.,. . .

Applied to K_ℓ3 at q_max² Flynn-Sachrajda

Works like all the previous heavy ChPT

Hard pion ChPT?

Heavy Kaon ChPT:

p = M_Kv + k

First: only keep diagrams where Kaon goes through Applied to masses and πK scattering and decay

constant Roessl,Allton et al.,. . .

Applied to K_ℓ3 at q_max² Flynn-Sachrajda

Flynn-Sachrajda argued K_ℓ3 also for q² away from q_max² .

JB-Celis Argument generalizes to other processes with hard/fast pions and applied to _{K → ππ}

JB Jemos B, D → D, π, K, η vector formfactors, charmonium decays and a two-loop check

General idea: heavy/fast dependence can always be reabsorbed in LECs, is analytic in the other parts k.

Hard pion ChPT?

nonanalyticities in the light masses come from soft lines soft pion couplings are constrained by current algebra

q→0limhπ^k(q)α|O|βi = − i

F_π hα| 

Q^k₅, O

|βi ,

Hard pion ChPT?

nonanalyticities in the light masses come from soft lines soft pion couplings are constrained by current algebra

q→0limhπ^k(q)α|O|βi = − i

F_π hα| 

Q^k₅, O

|βi ,

Nothing prevents hard pions to be in the states α or β So by heavily using current algebra I should be able to get the light quark mass nonanalytic dependence

Hard pion ChPT?

Field Theory: a process at given external momenta

Take a diagram with a particular internal momentum configuration

Identify the soft lines and cut them

The result part is analytic in the soft stuff

So should be describably by an effective Lagrangian with coupling constants dependent on the external given momenta (Weinberg’s folklore theorem)

Envisage this effective Lagrangian as a Lagrangian in hadron fields but all possible orders of the momenta included.

Hard pion ChPT?

⇒ ⇒ ⇒

This procedure works at one loop level, matching at tree level, nonanalytic dependence at one loop:

Toy models and vector meson ChPT JB, Gosdzinsky, Talavera

Recent work on relativistic baryon ChPT Gegelia, Scherer et al.

Extra terms kept in many of our calculations: a one-loop check

Some two-loop checks

Hard pion ChPT?

This effective Lagrangian as a Lagrangian in hadron fields but all possible orders of the momenta included:

possibly an infinite number of terms

If symmetries present, Lagrangian should respect them but my powercounting is gone

Hard pion ChPT?

This effective Lagrangian as a Lagrangian in hadron fields but all possible orders of the momenta included:

possibly an infinite number of terms

If symmetries present, Lagrangian should respect them In some cases we can prove that up to a certain order in the expansion in light masses, not momenta, matrix elements of higher order operators are reducible to

those of lowest order.

Lagrangian should be complete in neighbourhood of original process

Loop diagrams with this effective Lagrangian should reproduce the nonanalyticities in the light masses Crucial part of the argument

The main technical trick

For getting soft singularities in an integral we need the meson close to on-shell

This only happens in an area of order m⁴ So typically R

d⁴p 1/(p² − m²) ∼ m⁴/m² but if ∂_µφ on that propagator we get an extra factor of m.

So extra derivatives are only at same order if they hit hard lines

and then they are part of the hard part which can be expanded around

K → 2π in SU(2) ChPT

Add K = K⁺ K⁰

!

Roessl

L^(2)ππ = F²

4 (hu^µu^µi + hχ⁺i) ,

L⁽¹⁾_πK = ∇^µK^†∇^µK − M²KK^†K ,

L⁽²⁾_πK = A₁hu^µu^µiK^†K + A₂hu^µu^νi∇^µK^†∇^νK + A₃K^†χ₊K + · · · Add a spurion for the weak interaction ∆I = 1/2, ∆I = 3/2

JB,Celis

t^ij_k −→ tⁱ

′j^′

k^′ = t^ij_k (g_L)_k^′k(g_L^† )_i^i′(g_L^† ) ^j

′

j

tⁱ_1/2 −→ tⁱ_1/2^′ = tⁱ_1/2(g_L^† )_i^i′.

K → 2π in SU(2) ChPT

The ∆I = 1/2 terms: τ_1/2 = t_1/2u^†

L1/2 = iE₁ τ_1/2K + E₂ τ_1/2u^µ∇^µK + iE₃hu^µu^µiτ1/2K +iE₄τ_1/2χ₊K + iE₅hχ⁺iτ1/2K + E₆τ_1/2χ₋K

+E₇hχ⁻iτ1/2K + iE₈hu^µu_νiτ1/2∇^µ∇^νK + · · · + h.c. . Note: higher order terms kept in both _L1/2 and _L⁽²⁾_πK to

check the arguments

Using partial integration,. . . : hπ(p¹)π(p₂)|O|K(p^K)i =

f (M²_K)hπ(p¹)π(p₂)|τ1/2K|K(p^K)i + λM² + O(M⁴)

K → ππ: Tree level

(a) (b)

A^LO₀ =

√3i 2F²



−1

2E₁ + (E₂ − 4E³) M²_K + 2E₈M⁴_K + A₁E₁



A^LO₂ =

r3 2

i F²

h(−2D¹ + D₂) M²_Ki

K → ππ: One loop

(a) (b) (c) (d)

(e) (f)

K → ππ: One loop

Diagram A₀ A₂

Z −^2F₃²A^LO₀ −^2F₃²A^LO₂

(a) √

3i

−¹₃E₁ + ²₃E₂M²_K q

3 2i

−²₃D₂M²_K

(b) √

3i

−₉₆⁵ E₁ − ₄₈⁷ E₂ + ²⁵₁₂E₃
M²_K + ²⁵₂₄E₈M⁴_K q

3

2i −⁶¹₁₂D₁ + ⁷⁷₂₄D₂
M²_K

(e) √

3i₁₆³ A₁E₁

(f) √

3i ¹₈E₁ + ¹₃A₁E₁

The coefficients of A(M_π²)/F⁴ in the contributions to A₀ and A₂. Z denotes the part from wave-function renormalization.

A(M_π²) = −_16π^Mπ22 log ^M_µ2^π2

Kπ intermediate state does not contribute, but did for

Flynn-Sachrajda

K → ππ: One-loop

A^{N LO}₀ = A^LO₀



1 + 3

8F² A(M²)



+ λ₀M² + O(M⁴) , A^{N LO}₂ = A^LO₂



1 + 15

8F² A(M²)



+ λ₂M² + O(M⁴) .

K → ππ: One-loop

A^{N LO}₀ = A^LO₀



1 + 3

8F² A(M²)



+ λ₀M² + O(M⁴) , A^{N LO}₂ = A^LO₂



1 + 15

8F² A(M²)



+ λ₂M² + O(M⁴) . Match with three flavour SU (3) calculation Kambor, Missimer, Wyler; JB, Pallante, Prades

A^(3)LO₀ = −i√

6CF₀⁴ F_KF²



G₈ + 1 9G₂₇



M²_K , A^(3)LO₂ = −i10√

3CF₀⁴

9F_KF² G₂₇M²_K ,

When using ^Fπ = F 

1 + _F¹₂A(M²) + ^M_F2²l^r₄

, F_K = F_K 

1 + _8F³₂A(M²) + · · · ,

logarithms at one-loop agree with above

Hard Pion ChPT: A two-loop check

Similar arguments to JB-Celis, Flynn-Sachrajda work for the pion vector and scalar formfactor ^JB-Jemos

Therefore at any t the chiral log correction must go like the one-loop calculation.

But note the one-loop log chiral log is with t >> m²_π Predicts

F_V (t, M²) = F_V (t, 0) 

1 − _16π^M22_F² ln ^M_µ2² + O(M²) F_S(t, M²) = F_S(t, 0) 

1 − ⁵_{2 16π}^M22_F² ln ^M_µ2² + O(M²)

Note that F_V,S(t, 0) is now a coupling constant and can be complex

Hard Pion ChPT: A two-loop check

Take the full two-loop ChPT calculation

JB,Colangelo,Talavera, valid for t, m²_π ≪ Λ²χ

Expand this for _{t ≫ m}²_π

t² ln t, . . . terms go in F_S,V (t, 0)

But the one-loop for F_V (t, 0) is known and HPChPT predicts how the chiral log m² log m² adds to this

F_V = 1 + x₂

"

1

6(s − 4) ¯J(s) + s



−l6^r − 1

6L − 1 18N

#

+ x²₂



P_V⁽²⁾ + U_V⁽²⁾



+ O(x³2) .

U_V⁽²⁾ = J(s)¯

"

1

3l^r₁(−s² + 4s) + 1

6l^r₂(s² − 4s) + 1

3l₄^r(s − 4) + 1

6l₆^r(−s² + 4s) − 1

36L(s² + 8s − 48)

+ 1 N

 7

108s² − 97

108s + 3 4

# + 1

9K₁(s) + 1

9K₂(s) 1

8s² − s + 4

 + 1

6K₃(s)



s − 1 3



− 5

3K₄(s) .

A two-loop check

Full two-loop ChPT JB,Colangelo,Talavera, expand in t >> m²_π: F_V (t, M²) = F_V (t, 0) 

1 − _16π^M22_F² ln ^M_µ2² + O(M²) F_S(t, M²) = F_S(t, 0) 

1 − ⁵_{2 16π}^M22_F² ln ^M_µ2² + O(M²) with

F_V (t, 0) = 1 + _16π^t2F²

 5

18 − 16π²l₆^r + ^iπ₆ − ¹₆ ln _µ^t2



F_S(t, 0) = 1 + _16π^t2F²

1 + 16π²l₄^r + iπ − ln _µ^t2 

The needed coupling constants are complex Both calculations have two-loop diagrams with overlapping divergences

The chiral logs should be valid for any t where a

Electromagnetic formfactors

F_V^π(s) = F_V^πχ(s)



1 + 1

F² A(m²_π) + 1

2F² A(m²_K) + O(m²_L)

 ,

F_V^K(s) = F_V^Kχ(s)



1 + 1

2F² A(m²_π) + 1

F² A(m²_K) + O(m²_L)

 .

B, D → π, K, η

P_f(p_f) 

_qiγµqf

 _Pi(pi)

= (p_i + p_f)_µf₊(q²) + (p_i − pf)_µf₋(q²) f_+B→M(t) = f_+B→M^χ (t)F_B→M

f_−B→M(t) = f_−B→M^χ (t)F_B→M F_B→M is always the same for f₊, f₋ and f₀

This is not heavy quark symmetry: not valid at endpoint and valid also for _{K → π}.

Not like Low’s theorem, depends on more than just the external legs

LEET: in this limit the two formfactors are related

B, D → π, K, η

F_K→π = 1 + 3

8F² A(m²_π) (2 − flavour) F_B→π = 1 + 3

8 + 9

8g² A(m²_π)

F² +  1

4 + 3 4g²

 A(m²_K)

F² +  1

24 + 1 8g²

 A(m²_η) F² , F_B→K = 1 + 9

8g² A(m²_π)

F² +  1

2 + 3 4g²

 A(m²_K)

F² +  1

6 + 1 8g²

 A(m²_η) F² ,

F_B→η = 1 + 3

8 + 9

8g² A(m²_π)

F² +  1

4 + 3 4g²

 A(m²_K)

F² +  1

24 + 1 8g²

 A(m²_η) F² ,

F_Bs→K = 1 + 3 8

A(m²_π)

F² +  1

4 + 3 2g²

 A(m²_K)

F² +  1

24 + 1 2g²

 A(m²_η) F² , F_Bs→η = 1 + 1

2 + 3 2g²

 A(m²_K)

F² +  1

6 + 1 2g²

 A(m²_η) F² .

F_Bs→π vanishes due to the possible flavour quantum numbers.

Note: F_B→π = F_B→η

Experimental check

CLEO data onf₊(q²)|V^cq| for _{D → π} and _{D → K} with

|V^cd| = 0.2253, _{|Vcs| = 0.9743}

0.7 0.8 0.9 1 1.1 1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 f +(q2 )

q² [GeV²] f_{+ D→π}

f_{+ D→K}

Experimental check

CLEO data onf₊(q²)|V^cq| for _{D → π} and _{D → K} with

|V^cd| = 0.2253, _{|Vcs| = 0.9743}

0.7 0.8 0.9 1 1.1 1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 f +(q2 )

q² [GeV²] f_{+ D→π}

f_{+ D→K}

0.7 0.8 0.9 1 1.1 1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 f +(q2 )

q² [GeV²] f_{+ D→π}F_D→K/F_D→π

f_{+ D→K}

f_+D→π = f_+D→KF_D→π/F_D→K

Applications to charmonium

We look at decays χ_c0, χ_c2 → ππ, KK, ηη

J/ψ, ψ(nS), χ_c1 decays to the same final state break isospin or U-spin or V -spin, they thus proceed via electromagnetism or quark mass differences: more difficult. (Some comments later)

So construct a Lagrangian with a chiral singlet scalar and tensor field.

L^χc = E₁F₀² χ₀ hu^µu_µi + E²F₀² χ^µν₂ hu^µu_νi .

Applications to charmonium

We look at decays χ_c0, χ_c2 → ππ, KK, ηη

J/ψ, ψ(nS), χ_c1 decays to the same final state break isospin or U-spin or V -spin, they thus proceed via electromagnetism or quark mass differences: more difficult. (Some comments later)

So construct a Lagrangian with a chiral singlet scalar and tensor field.

L^χc = E₁F₀² χ₀ hu^µu_µi + E²F₀² χ^µν₂ hu^µu_νi . No chiral logarithm corrections

Expanding the energy-momentum tensor result

Donoghue-Leutwyler at large q² agrees.

These decays should have small SU (3)_V breaking