Introduction to Effective Field Theories for Hadron Physics

Feng-Kun Guo

Institute of Theoretical Physics, CAS

ASEAN School of Hadron Physics

Feb. 29 – Mar. 04, 2016, Nakhon Ratchasima, Thailand

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 1 / 76 Outline

1 Why do we need effective field theories?

2 Effective field theory in a nut shell

3 Symmetries of QCD

4 Spontaneous breaking of chiral symmetry

5 Chiral effective Lagrangians Leading order chiral Lagrangian for Goldstone bosons Combining chiral symmetry and heavy quark symmetry CHPT at the next-to-leading order

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 2 / 76 References

Useful textbooks/monographs: J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model (1992) H. Georgi, Weak Interactions and Modern Particle Physics (2009) A.V. Manohar, M.B. Wise, Heavy Quark Physics (2000) S. Scherer, M.R. Schindler, A Primer for Chiral Perturbation Theory (2012)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 3 / 76 Elementary particles in the Standard Model

Mproton  (2mu + md)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 4 / 76 Quantum Electrodynamics (QED)

• Basic interaction vertex in QED

• Photon does not carry charge, photon-photon interaction only happens at higher orders

• Coupling constant is small, α ≈ 1/137 =⇒ great success of perturbation theory E.g., the electron magnetic moment

Exp.: g/2 = 1.001 159 652 180 73(28)[0.28 ppt] D. Hanneke et al, PRL100(2008)120801

Th.: g/2 = 1.001 159 652 181 78(77)[0.77 ppt] T. Aoyama et al, PRL109(2008)111807 (up to O α5
: 12672 Feynman diagrams!)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 5 / 76 QuantumChromoDynamics (QCD)

• QuantumChromoDynamics (QCD) is the theory of the strong interaction

X 1 g2θ L = q¯ (iD6 − m )q − Ga Gµν,a + s µνρσGa Ga QCD f f f 4 µν 64π2 µν ρσ u,d,s, f= c,b,t

a a
a a a b c Dµqf = ∂µ + igsAµλ /2 qf ,Gµν = ∂µAν − ∂ν Aµ − gsfabcAµAν

• Color gauge invariance under SU(3)c (quarks: red, blue and green): non-Abelian • Gluons: 8 colors, self-interacting

−10 • The θ-term breaks P and CP, θ is tiny: |θ| . 10

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 6 / 76 Two facets ofQCD

2 • Running of the coupling constant αs = gs /(4π)

• High energies asymptotic freedom, perturbative degrees of freedom: quarks and gluons • Low energies −1
nonperturbative, ΛQCD ∼ 300 MeV = O 1 fm color confinement, degrees of freedom: mesons and baryons ? theory of quarks and gluons −→ low-energy hadron spectrum

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 7 / 76 Two facets ofQCD

2 • Running of the coupling constant αs = gs /(4π)

• High energies asymptotic freedom, perturbative degrees of freedom: quarks and gluons • Low energies −1
nonperturbative, ΛQCD ∼ 300 MeV = O 1 fm color confinement, degrees of freedom: mesons and baryons ? theory of quarks and gluons −→ low-energy hadron spectrum

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 7 / 76 Theoretical tools for studying nonperturbative QCD

• Lattice QCD: numerical simulation in discretized Euclidean space-time finite volume (L should be large) finite lattice spacing (a should be small)

normally using mu,d larger than the physical values ⇒ chiral extrapolation

• Models (e.g., quark model, see lectures by Q. Zhao) • Low-energy effective field theories ofQCD: effective degrees of freedom can be mesons and baryons

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 8 / 76 Theoretical tools for studying nonperturbative QCD

• Lattice QCD: numerical simulation in discretized Euclidean space-time finite volume (L should be large) finite lattice spacing (a should be small)

normally using mu,d larger than the physical values ⇒ chiral extrapolation

• Models (e.g., quark model, see lectures by Q. Zhao) • Low-energy effective field theories ofQCD: effective degrees of freedom can be mesons and baryons

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 8 / 76 Theoretical tools for studying nonperturbative QCD

• Lattice QCD: numerical simulation in discretized Euclidean space-time finite volume (L should be large) finite lattice spacing (a should be small)

normally using mu,d larger than the physical values ⇒ chiral extrapolation

• Models (e.g., quark model, see lectures by Q. Zhao) • Low-energy effective field theories ofQCD: effective degrees of freedom can be mesons and baryons

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 8 / 76 Low-energy effective field theory (EFT)

S. Weinberg, “Phenomenological Lagrangians”, Physica 96A (1979) 327

Translation: The most general effective Lagrangian, up to a given order, consistent with the symmetries of the underlying theory ⇒ results consistent with the underlying theory! The degrees of freedom can be different from those of the underlining theory ⇒ we can work with hadrons directly for low-energy QCD

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 9 / 76 e2 1 2 2 8 sin θW MW − q e2  q2  = 2 1 + 2 + ... 8MW sin θW MW  2  GF q = √ + O 2 2 MW

Low-energy EFT (II) However, the “most general” means • an infinite number of parameters ⇒ intractable (?) • nonrenormalizable (in contrast to, e.g., QED and QCD) Solution: systematic expansion with a power counting • only a finite number of parameters at a given order, can be determined from experiments lattice calculations for QCD • renormalize order by order • existence of a small (dimensionless) quantity, e.g., separation of energy scales, E  Λ ⇒ expansion in powers of (E/Λ) − 2 2 Neutron decay (n → pe v¯e): weak interactions for |q |  MW (decoupling EFT)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 10 / 76 e2 1 2 2 8 sin θW MW − q e2  q2  = 2 1 + 2 + ... 8MW sin θW MW  2  GF q = √ + O 2 2 MW

Low-energy EFT (II) However, the “most general” means • an infinite number of parameters ⇒ intractable (?) • nonrenormalizable (in contrast to, e.g., QED and QCD) Solution: systematic expansion with a power counting • only a finite number of parameters at a given order, can be determined from experiments lattice calculations for QCD • renormalize order by order • existence of a small (dimensionless) quantity, e.g., separation of energy scales, E  Λ ⇒ expansion in powers of (E/Λ) − 2 2 Neutron decay (n → pe v¯e): weak interactions for |q |  MW (decoupling EFT)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 10 / 76 Low-energy EFT (II) However, the “most general” means • an infinite number of parameters ⇒ intractable (?) • nonrenormalizable (in contrast to, e.g., QED and QCD) Solution: systematic expansion with a power counting • only a finite number of parameters at a given order, can be determined from experiments lattice calculations for QCD • renormalize order by order • existence of a small (dimensionless) quantity, e.g., separation of energy scales, E  Λ ⇒ expansion in powers of (E/Λ) − 2 2 Neutron decay (n → pe v¯e): weak interactions for |q |  MW (decoupling EFT) e2 1 2 2 8 sin θW MW − q e2  q2  = 2 1 + 2 + ... 8MW sin θW MW  2  GF q = √ + O 2 2 MW Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 10 / 76 Low-energy EFT (III)

• Pro: model-independent, controlled uncertainty • Con: number of parameters increases fast when going to higher orders

Things need to be remembered for an EFT: • separation of energy scales: systematic expansion with a power counting • symmetry constraints from the full theory For low-energy QCD, we will consider (approximate) chiral symmetry of light quarks ⇒ CHiralPerturbationTheory (non-decoupling EFT) full theory ⇒ EFT via spontaneous symmetry breaking (SSB) generation of new light degrees of freedom heavy quark symmetry: spin and flavor

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 11 / 76 Low-energy EFT (III)

• Pro: model-independent, controlled uncertainty • Con: number of parameters increases fast when going to higher orders

Things need to be remembered for an EFT: • separation of energy scales: systematic expansion with a power counting • symmetry constraints from the full theory For low-energy QCD, we will consider (approximate) chiral symmetry of light quarks ⇒ CHiralPerturbationTheory (non-decoupling EFT) full theory ⇒ EFT via spontaneous symmetry breaking (SSB) generation of new light degrees of freedom heavy quark symmetry: spin and flavor

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 11 / 76 QCD symmetries (I)

X 1 g2θ L = q¯ (iD6 − m )q − Ga Gµν,a + s µνρσGa Ga QCD f f f 4 µν 64π2 µν ρσ u,d,s, f= c,b,t

• Exact: Lorentz-invariance, SU(3)c gauge, C (for θ = 0 with real mf , P and T as well)

• Hierachy of quark masses: mu,d,s  ΛQCD (∼ 300 MeV) mc,b,t

• For heavy quarks (charm, bottom) in a hadron, typical momentum transfer ΛQCD heavy quark flavor symmetry (HQFS) for any hadron containing one heavy quark: ∆p ΛQCD velocity remains unchanged in the limit mQ → ∞: ∆v = = mQ mQ ⇒ heavy quark is like a static color triplet source, mQ is irrelevant heavy quark spin symmetry (HQSS): see lectures by E.Eichten ~σ · B~ chromomag. interaction ∝ mQ spin of the heavy quark decouples

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 12 / 76 QCD symmetries (II): Heavy quark symmetry

• In the heavy quark limit mQ → ∞, consider the quark propagator

p + mQ m (v + 1) + k/ mQ → ∞ 1 + v i / Q / −−−−−−→ / i 2 2 = i 2 p − mQ + i 2mQv · k + k + i 2 v · k + i

here p = mQv + k, with residual momentum k ∼ ΛQCD. Velocity-dependent projector (in the rest frame ~v = 0 ) ⇒ particle component ! 1 + v/ 1 + γ0 1 0 = = (in Dirac basis) 2 2 0 0

• Decompose a heavy quark field into v-dep. fields −imQv·x Q(x) = e [Qv(x) + qv(x)] with: 1 + v/ 1 − v/ Q (x) = eimQv·x Q(x), q (x) = eimQv·x Q(x) v 2 v 2 1 ± v/ Exercise: Let P = , show P 2 = P , P P = 0, and ± 2 ± ± + − Q¯vγµQv = Q¯vvµQv

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 13 / 76 QCD symmetries (III): Heavy quark symmetry

• Heavy quark symmetry in the leading order term:

¯ ¯  −1 LQ = Q(iD/ − mQ)Q = Qv(iv · D)Qv + O mQ

• Let total angular momentum J~ = ~sQ + ~s`, ~sQ: heavy quark spin, ~s`: spin of the light degrees of freedom (including orbital angular momentum)

HQSS: s` is a good quantum number in heavy hadrons spin multiplets: ∗ P 1 − for singly heavy mesons, e.g. {D,D } with s` = 2 ,

MD∗ − MD ' 140 MeV, MB∗ − MB ' 46 MeV

for heavy quarkonia, e.g. {ηc, J/ψ} (no flavor symmetry for 2 or more heavy quarks)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 14 / 76 QCD symmetries (IV): Heavy quark symmetry

Examples of HQSS phenomenology: P + • Widths of the two D1 (J = 1 ) mesons +130 Γ[D1(2420)] = (27.4 ± 2.5) MeV  Γ[D1(2430)] = (384−110) MeV ~ P 1 + 3 + ~sl = ~sq + L ⇒ for P -wave charmed mesons: s` = 2 or 2 ∗ for decays D1 → D π: 1 + 1 − − 2 → 2 + 0 in S-wave ⇒ large width 3 + 1 − − 2 → 2 + 0 in D-wave ⇒ small width 3 1 thus, D1(2420): s` = 2 , D1(2430): s` = 2 3 + 1 − + − • Suppression of the S-wave production of 2 + 2 heavy meson pairs in e e annihilation X. Li, M. Voloshin, Phys. Rev. D 88 (2013) 034012 Exercise: Try to understand this as a consequence of HQSS.

For more examples and applications, see the lectures by Q. Wang.

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 15 / 76 QCD symmetries (IV): Heavy quark symmetry

Examples of HQSS phenomenology: P + • Widths of the two D1 (J = 1 ) mesons +130 Γ[D1(2420)] = (27.4 ± 2.5) MeV  Γ[D1(2430)] = (384−110) MeV ~ P 1 + 3 + ~sl = ~sq + L ⇒ for P -wave charmed mesons: s` = 2 or 2 ∗ for decays D1 → D π: 1 + 1 − − 2 → 2 + 0 in S-wave ⇒ large width 3 + 1 − − 2 → 2 + 0 in D-wave ⇒ small width 3 1 thus, D1(2420): s` = 2 , D1(2430): s` = 2 3 + 1 − + − • Suppression of the S-wave production of 2 + 2 heavy meson pairs in e e annihilation X. Li, M. Voloshin, Phys. Rev. D 88 (2013) 034012 Exercise: Try to understand this as a consequence of HQSS.

For more examples and applications, see the lectures by Q. Wang.

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 15 / 76 QCD symmetries (V): Light flavor symmetry

Light meson SU(3) [u, d, s] multiplets (octet + singlet): see lectures by Q.Zhao • Vector mesons ds us meson quark content mass (MeV) ρ+/ρ− ud¯ /du¯ 775 √ ρ0 (uu¯ − dd¯)/ 2 775 du uu dd ud ss K∗+/K∗− us¯ /su¯ 892 K∗0/K¯ ∗0 ds¯ /sd¯ 896 √ ω (uu¯ + dd¯)/ 2 783 su sd φ ss¯ 1019

approximate SU(3) symmetry very good isospin SU(2) symmetry

mρ0 − mρ± = (−0.7 ± 0.8) MeV, mK∗0 − mK∗± = (6.7 ± 1.2) MeV

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 16 / 76 QCD symmetries (V): Light flavor symmetry

Light meson SU(3) [u, d, s] multiplets (octet + singlet): see lectures by Q.Zhao • Pseudoscalar mesons ds us meson quark content mass (MeV) π+/π− ud¯ /du¯ 140 √ π0 (uu¯ − dd¯)/ 2 135 du uu dd ud ss K+/K− us¯ /su¯ 494 K0/K¯ 0 ds¯ /sd¯ 498 √ η ∼ (uu¯ + dd¯− 2ss¯)/ 6 548 √ su sd η0 ∼ (uu¯ + dd¯+ ss¯)/ 3 958

very good isospin SU(2) symmetry

mπ± − mπ0 = (4.5936 ± 0.0005) MeV, mK0 − mK± = (3.937 ± 0.028) MeV

Why are the pions so light?

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 16 / 76 gluon mass

ψR ψR ψR ψL

QCD symmetries (VI): Chiral symmetry

• Masses of the three lightest quarks u, d, s are small ⇒ approximate chiral symmetry • Chiral decomposition of fermion fields: 1 1 ψ = (1 − γ )ψ + (1 + γ )ψ ≡ P ψ + P ψ = ψ + ψ 2 5 2 5 L R L R

2 2 PL = PL, PR = PR,

PLPR = PRPL = 0, PL + PR = 1

• For massless fermions, left-/right-handed fields do not interact with each other

¯ ¯ ¯ ¯
L[ψL,ψ R] = iψL D6 ψL + iψR D6 ψR − m ψLψR + ψRψL

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 17 / 76 QCD symmetries (VI): Chiral symmetry

• Masses of the three lightest quarks u, d, s are small ⇒ approximate chiral symmetry • Chiral decomposition of fermion fields: 1 1 ψ = (1 − γ )ψ + (1 + γ )ψ ≡ P ψ + P ψ = ψ + ψ 2 5 2 5 L R L R

2 2 PL = PL, PR = PR,

PLPR = PRPL = 0, PL + PR = 1

• For massless fermions, left-/right-handed fields do not interact with each other

¯ ¯ ¯ ¯
L[ψL,ψ R] = iψL D6 ψL + iψR D6 ψR − m ψLψR + ψRψL

gluon mass

ψR ψR ψR ψL

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 17 / 76 QCD symmetries (VII): Chiral symmetry

0 • Decompose LQCD into LQCD, the QCD Lagrangian in the 3-flavor chiral limit mu = md = ms = 0, and the light quark mass term:

0 T LQCD = L QCD − q¯Mq , q = (u, d, s) , M = diag(mu, md, ms)

0 •L QCD = iq¯L D6 qL + iq¯R D6 qR + . . .

is invariant underU (3)L×U(3)R transformations:

0 0 0 0 LQCD(Gµν , q ,Dµq ) = LQCD(Gµν , q, Dµq) 0 q = RP Rq + LP Lq = Rq R + Lq L

R ∈ U(3)R,L ∈ U(3)L

P • Parity: q(t, ~x) → γ0q(t, −~x) P 0 0 0 ⇒ qR(t, ~x) → PRγ q(t, −~x) = γ PLq(t, −~x) = γ qL(t, −~x) P 0 qL(t, ~x) → γ qR(t, −~x)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 18 / 76 QCD symmetries (VIII): Chiral symmetry

• Decompose U(3)L × U(3)R = SU(3)L × SU(3)R × U(1)V × U(1)A : 18 conserved (Noether) currents for massless QCD at the classical level λa J µ,a =q ¯ γµ q ,J µ,0 =q ¯ γµq , (λ : Gell-Mann matrices) L,R L,R 2 L,R L,R L,R L,R a • rewritten in terms of vector (V = L + R) and axial vector (A = R − L) currents λa λa V µ,a =qγ ¯ µ q , Aµ,a =qγ ¯ µγ q 2 5 2 µ,a µ,a ∂µV = 0 ,∂ µA = 0

• U(1)V : baryon or quark number conservation µ,0 µ,0 V =qγ ¯ µq,∂ µV = 0

• U(1)A: broken by quantum effects, anomaly

N g2 Aµ,0 =qγ ¯ γ q,∂ Aµ,0 = f s µνρσGa Ga µ 5 µ 32π2 µν ρσ

−10 ⇒ the QCD θ-term, |θ| . 10

• IsSU (3)L × SU(3)R realized in hadron spectrum?

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 19 / 76 QCD symmetries (VIII): Chiral symmetry

• Decompose U(3)L × U(3)R = SU(3)L × SU(3)R × U(1)V × U(1)A : 18 conserved (Noether) currents for massless QCD at the classical level λa J µ,a =q ¯ γµ q ,J µ,0 =q ¯ γµq , (λ : Gell-Mann matrices) L,R L,R 2 L,R L,R L,R L,R a • rewritten in terms of vector (V = L + R) and axial vector (A = R − L) currents λa λa V µ,a =qγ ¯ µ q , Aµ,a =qγ ¯ µγ q 2 5 2 µ,a µ,a ∂µV = 0 ,∂ µA = 0

• U(1)V : baryon or quark number conservation µ,0 µ,0 V =qγ ¯ µq,∂ µV = 0

• U(1)A: broken by quantum effects, anomaly

N g2 Aµ,0 =qγ ¯ γ q,∂ Aµ,0 = f s µνρσGa Ga µ 5 µ 32π2 µν ρσ

−10 ⇒ the QCD θ-term, |θ| . 10

• IsSU (3)L × SU(3)R realized in hadron spectrum?

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 19 / 76 QCD symmetries (VIII): Chiral symmetry

• Decompose U(3)L × U(3)R = SU(3)L × SU(3)R × U(1)V × U(1)A : 18 conserved (Noether) currents for massless QCD at the classical level λa J µ,a =q ¯ γµ q ,J µ,0 =q ¯ γµq , (λ : Gell-Mann matrices) L,R L,R 2 L,R L,R L,R L,R a • rewritten in terms of vector (V = L + R) and axial vector (A = R − L) currents λa λa V µ,a =qγ ¯ µ q , Aµ,a =qγ ¯ µγ q 2 5 2 µ,a µ,a ∂µV = 0 ,∂ µA = 0

• U(1)V : baryon or quark number conservation µ,0 µ,0 V =qγ ¯ µq,∂ µV = 0

• U(1)A: broken by quantum effects, anomaly

N g2 Aµ,0 =qγ ¯ γ q,∂ Aµ,0 = f s µνρσGa Ga µ 5 µ 32π2 µν ρσ

−10 ⇒ the QCD θ-term, |θ| . 10

• IsSU (3)L × SU(3)R realized in hadron spectrum?

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 19 / 76 Wigner–Weyl v.s. Nambu–Goldstone

• Noether’s theorem: continuous symmetry ⇒ conserved currents a a R 3 a,0 a,µ Let Q be symmetry charges: Q = d ~xJ (t, ~x), ∂µJ = 0 a a • Qa is the symmetry generator: U = eiα Q , H: Hamiltonian, thus

UHU −1 = H ⇒ [Qa,H] = 0,

[Qa,H]|0i = QaH|0i − HQa|0i = 0 | {z } =0 • Wigner–Weyl mode: Qa|0i = 0 or equivalently U|0i = |0i degeneracy in mass spectrum • Nambu–Goldstone mode: U|0i= 6 |0i, spontaneously broken (hidden) Qa|0i= 6 |0i: new states degenerate with vacuum, massless Goldstone bosons spontaneously broken continuous global symmetry ⇒ massless GBs the same quantum numbers as Qa|0i ⇒ spinless #(GBs) = #(broken generators)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 20 / 76 Fate of SU(3)L×SU(3)R (I)

• Vector and axial charges:

a a a a a a a 0 QV = QL + QR,QA = QR − QL, [QV,A,HQCD] = 0 0 here HQCD: QCD Hamiltonian in the chiral limit. 0 0 Under parity transformation: qR → γ qL, qL → γ qR

µ,a a µ,a a JL → JR,µ,JR → JL,µ

a −1 a a −1 a ⇒ PQV P = QV ,PQAP = −QA 0 0 For an eigenstate of HQCD: HQCD|ψi = E|ψi with P |ψi = ηP |ψi, then

0 a a 0 a HQCDQA|ψi = QAHQCD|ψi = EQA|ψi, a a −1 a PQA|ψi = PQAP P |ψi = −ηP QA|ψi

a Parity doubling: QA|ψi has the same mass but opposite parity

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 21 / 76 Fate of SU(3)L×SU(3)R (II)

• But, in the hadron spectrum:

mNucleon,P =+ = 939 MeV  mN ∗(1535),P =− = 1535 MeV,

mπ, P =− = 139 MeV  ma0(980),P =+ = 980 MeV

• No parity doubling in hadron spectrum ⇒

a a a iα QA QA|0i= 6 0, or e |0i= 6 |0i Nambu–Goldstone mode(hidden symmetry):

In QCD, SU(3)L×SU(3)R is spontaneously broken down to SU(3)V

• GBs should have J P = 0−

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 22 / 76 SSB in linear σ model – 1

• Linear σ model with an O(4) symmetry 1 L = ∂ ΦT ∂µΦ − V (Φ), LSM 2 µ λ 2 V (Φ) = ΦT Φ − v2
, with ΦT = (σ, π , π , π ) 4 1 2 3 • σ = πa = 0 is not a minimum of V (Φ) T 2 There is a continuum of degenerate vacua: ΦminΦmin = v

Choose Φmin = (v, 0, 0, 0), vacuum is only invariant under O(3) rotations

spontaneous symmetry breaking: O(4) → O(3)

0 T • Perturb around Φmin = (v,~0), let Φ = Φmin + (σ , π1, π2, π3) , then 1 1 λ 2 L = ∂ π ∂µπ + ∂ σ0∂µσ0 − σ0 2 + π π + 2vσ0
LSM 2 µ a a 2 µ 4 a a

Goldstone bosons (GBs): πa’s are massless 0 2 2 σ is massive: mσ0 = 2λv

mσ0 and 5 interaction terms described by only 2 parameters

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 23 / 76 SSB in linear σ model – 1

• Linear σ model with an O(4) symmetry 1 L = ∂ ΦT ∂µΦ − V (Φ), LSM 2 µ λ 2 V (Φ) = ΦT Φ − v2
, with ΦT = (σ, π , π , π ) 4 1 2 3 • σ = πa = 0 is not a minimum of V (Φ) T 2 There is a continuum of degenerate vacua: ΦminΦmin = v

Choose Φmin = (v, 0, 0, 0), vacuum is only invariant under O(3) rotations

spontaneous symmetry breaking: O(4) → O(3)

0 T • Perturb around Φmin = (v,~0), let Φ = Φmin + (σ , π1, π2, π3) , then 1 1 λ 2 L = ∂ π ∂µπ + ∂ σ0∂µσ0 − σ0 2 + π π + 2vσ0
LSM 2 µ a a 2 µ 4 a a

Goldstone bosons (GBs): πa’s are massless 0 2 2 σ is massive: mσ0 = 2λv

mσ0 and 5 interaction terms described by only 2 parameters

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 23 / 76 SSB in linear σ model – 2

1 1 λ 2 L = ∂ π ∂µπ + ∂ σ0∂µσ0 − σ0 2 + π π + 2vσ0
LSM 2 µ a a 2 µ 4 a a Only 2 parameters, important for cancellation! Examples: tree-level scattering amplitudes

• π3(p1)π3(p2) → π3(p3)π3(p4) (individually large terms!)

iA = + + +

i(−2iλv)2 i(−2iλv)2 i(−2iλv)2 = −i6λ + + + s − (2λv2) t − (2λv2) u − (2λv2)    4   4  i 2 s + t + u pπ pπ = −i6λ + 2 (2λv) 3 + 2 + O 4 = O 4 2λv 2λv mσ mσ

pπ: a generic momentum of GBs 2 2 2 Mandelstam variables: s = (p1 + p2) , t = (p1 − p3) , u = (p1 − p4) , P 2 s + t + u = i pi

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 24 / 76 SSB in linear σ model – 2

1 1 λ 2 L = ∂ π ∂µπ + ∂ σ0∂µσ0 − σ0 2 + π π + 2vσ0
LSM 2 µ a a 2 µ 4 a a Only 2 parameters, important for cancellation! Examples: tree-level scattering amplitudes 0 0 • π3σ → π3σ Exercise: Show that at tree-level  2  0 0 pπ A(π3σ → π3σ ) = O 2 mσ

Lessons from the linear σ model:

SSB happens when there is a degeneracy of the vacuum

nonvanishing VEV of some Hermitian operator, here hσi = v

SSB ⇒ massless GBs( πa), # = dim(O(4)) − dim(O(3))=3 GBs decouple at vanishing momenta!

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 24 / 76 SSB in linear σ model – 2

1 1 λ 2 L = ∂ π ∂µπ + ∂ σ0∂µσ0 − σ0 2 + π π + 2vσ0
LSM 2 µ a a 2 µ 4 a a Only 2 parameters, important for cancellation! Examples: tree-level scattering amplitudes 0 0 • π3σ → π3σ Exercise: Show that at tree-level  2  0 0 pπ A(π3σ → π3σ ) = O 2 mσ

Lessons from the linear σ model:

SSB happens when there is a degeneracy of the vacuum

nonvanishing VEV of some Hermitian operator, here hσi = v

SSB ⇒ massless GBs( πa), # = dim(O(4)) − dim(O(3))=3 GBs decouple at vanishing momenta!

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 24 / 76 a Aµ

a Aµ πa ψ ψ ψ1 ψ2 1 2 T

Derivative coupling

Symmetry implies a derivative coupling for GBs, i.e.,

GBs do not interact at vanishing momenta

a a a R 3 a a • Consider GB π : hπ |QA|0i = d x hπ |A0(x)|0i= 6 0 a a Lorentz invariance ⇒ π (q)|Aµ(0)|0 = −iqµFπ • Consider the matrix element

a hψ1|Aµ(0)|ψ2i = + 1 = Ra + F qµ T a µ π q2 µ a Current conservation ⇒ q Aµ = 0, thus

µ a a a q Rµ + FπT = 0 ⇒ lim T = 0 qµ→0

•⇒ GBs couple in a derivative form !!

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 25 / 76 Derivative coupling

Symmetry implies a derivative coupling for GBs, i.e.,

GBs do not interact at vanishing momenta

a a a R 3 a a • Consider GB π : hπ |QA|0i = d x hπ |A0(x)|0i= 6 0 a a Lorentz invariance ⇒ π (q)|Aµ(0)|0 = −iqµFπ

a • Consider the matrix element Aµ

a Aµ πa ψ ψ ψ1 ψ2 a 1 2 T hψ1|Aµ(0)|ψ2i = + 1 = Ra + F qµ T a µ π q2 µ a Current conservation ⇒ q Aµ = 0, thus

µ a a a q Rµ + FπT = 0 ⇒ lim T = 0 qµ→0

•⇒ GBs couple in a derivative form !!

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 25 / 76 SSB in QCD

• Hamiltonian invariant under a group G = SU(Nf )L × SU(Nf )R,

vacuum invariant under its vector subgroup H = SU(Nf )V .

a a QV |0i = 0,QA|0i= 6 0 • SSB ⇒ massless pseudoscalar Goldstone bosons 2 #(GBs) = dim(G) − dim(H) = Nf − 1 ± 0 ± 0 0 for Nf = 3, 8 GBs: π , π ,K ,K , K¯ , η ± 0 for Nf = 2, 3 GBs: π , π Pions get a small mass due to explicit symmetry

breaking by tiny mu,d (a few MeV)

Mπ  Mother hadron

also, ms  mu,d ⇒ MK  Mπ

• Mechanism for SU(Nf )L × SU(Nf )R → SU(Nf )V in QCD not well understood

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 26 / 76 Chirl perturbation theory (CHPT) as low-energy EFT for QCD

CHPT: a low energy EFT for QCD:

• an example for a non-decoupling EFT: degrees of freedom are different from those of the underlying theory

• a theory for the Goldstone bosons ofSU (Nf )L × SU(Nf )R → SU(Nf )V • most general Lagrangian with the same global symmetries as QCD

How do the Goldstone bosons transform under SU(Nf )L × SU(Nf )R?

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 27 / 76 Nonlinear realization of chiral symmetry (I)

Weinberg (1968); Coleman, Wess, Zumino (1969); Callan, Coleman, Wess, Zumino (1969)

To study the transformation properties of the Goldstone bosons (GBs) SSB • Assume G −→ H, we have GBs

Φ = (φi, . . . , φn),n = dim(G) − dim(H)

• Left coset of H with respect to g ∈ G: gH = {gh|h ∈ H} coset space G/H: set of all left cosets {gH|g ∈ G}

dim(G/H) = dim(G) − dim(H) • GBs can be parametrized as coordinates of G/H:

choosing a representative element in each coset, e.g., for h1,2 ∈ H,

gh1H = gh2H

we can choose either gh1 or gh2 as the parameterization

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 28 / 76 Nonlinear realization of chiral symmetry (II)

• Representation invariance: free to choose the set of representative elements / set of coordinates on G/H

• Transformation properties of the GBs uniquely determined: Parameterizing GBs by u ∈ G/H transformation under g ∈ G

g u = u0 h(g, u)

since any element of the coset {u0h(g, u)|h(g, u) ∈ H} can be used ⇒ Nonlinear transformation of GBs

g∈G u → u0 = g u h−1(g, u)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 29 / 76 Nonlinear realization of chiral symmetry (II)

• Representation invariance: free to choose the set of representative elements / set of coordinates on G/H

• Transformation properties of the GBs uniquely determined: Parameterizing GBs by u ∈ G/H transformation under g ∈ G

g u = u0 h(g, u)

since any element of the coset {u0h(g, u)|h(g, u) ∈ H} can be used ⇒ Nonlinear transformation of GBs

g∈G u → u0 = g u h−1(g, u)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 29 / 76 Application to QCD (I)

SSB For QCD, G = SU(Nf )L × SU(Nf )R −→ H = SU(Nf )V

• g = (gL, gR) ∈ SU(Nf )L × SU(Nf )R for gL ∈ SU(Nf )L , gR ∈ SU(Nf )R

g1g2 = (gL1 , gR1 )(gL2 , gR2 ) = (gL1 gL2 , gR1 gR2 )

• Choice of a representative element inside each left coset is free † 1 † 1 (gL, gR)H = (gLgR, )(gR, gR) H = (gLgR, ) H | {z } ∈H=SU(Nf )V

Goldstone bosons can be parameterized by a unitary matrix  φ  U = g g† = exp i L R F 0 P8 here, φ = a=1 λaφa for SU(3), and ~τ · ~π for SU(2)

λa: Gell-Mann matrices, τi(i = 1, 2, 3): Pauli matrices

φa (πi) : Goldstone boson fields F 0 : dimensionful constant (to be determined later)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 30 / 76 Application to QCD (I)

SSB For QCD, G = SU(Nf )L × SU(Nf )R −→ H = SU(Nf )V

• g = (gL, gR) ∈ SU(Nf )L × SU(Nf )R for gL ∈ SU(Nf )L , gR ∈ SU(Nf )R

g1g2 = (gL1 , gR1 )(gL2 , gR2 ) = (gL1 gL2 , gR1 gR2 )

• Choice of a representative element inside each left coset is free † 1 † 1 (gL, gR)H = (gLgR, )(gR, gR) H = (gLgR, ) H | {z } ∈H=SU(Nf )V

Goldstone bosons can be parameterized by a unitary matrix  φ  U = g g† = exp i L R F 0 P8 here, φ = a=1 λaφa for SU(3), and ~τ · ~π for SU(2)

λa: Gell-Mann matrices, τi(i = 1, 2, 3): Pauli matrices

φa (πi) : Goldstone boson fields F 0 : dimensionful constant (to be determined later)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 30 / 76 Application to QCD (II)

• Acting g = (L, R) ∈ G on the coset (U, 1)H

g (U, 1)H = (L U, R)H = (LUR† R,R)H = (LUR†, 1)(R,R) H | {z } ∈SU(Nf )V ⇒ transformation property of U:

g U → LUR† g One can also parametrize the GBs such that U → RUL†. Any one is okay if used consistently.

• For g ∈ H = SU(Nf )V , we have R = L U → LUL† ⇒ φ → L φ L†

i.e., GB fields transform linearly under SU(Nf )V

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 31 / 76 Intermediate summary – 1

• Heavy quark spin and flavor symmetry for hadrons containing a single heavy quark

• QCD in the chiral limit mu = md = ms = 0 has a global symmetry

SU(3)L × SU(3)R × U(1)V

U(1)A is anomalously broken by quantum effects • Strong evidence for chiral symmetry spontaneous breaking SSB SU(Nf )L × SU(Nf )R −→ SU(Nf )V 3 (8) pseudoscalar Goldstone bosons: π±, π0 (, K±, K0, K¯ 0, η) SSB • For G −→ H, Goldstone bosons can be identified with G/H   √φ3 + √φ8 π+ K+ √ 2 6 iφ  φ φ  U = exp , φ = 2  π− − √3 + √8 K0  for SU(3) F 0  2 6  K− K¯ 0 − 2√φ8 6

g∈G nonlinear transformation: U −→ LUR †

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 32 / 76 Construction of the effective Lagrangian for GBs (I)

Aim: reproduce low-energy structure of QCD

• Effective Lagrangian invariant under G = SU(Nf )L × SU(Nf )R (and C, P ) U →G LUR†,U →C U T ,U →P U † • What does "low-energy" mean here? Goldstone boson fields (contained in U) as the only degrees of freedom ⇒ energy range restricted to well below 1 GeV (separation of energy scales, more see later)

• Low energies: expand in powers of momenta ( = number of derivatives)

• Lorentz invariance ⇒ only even number of derivatives are allowed L = L(0) + L(2) + L(4) + ...

•L (0)? U is unitary, UU † = 1, hence

† n h(UU ) i = const. , h...i ≡ Trflavor[...]

⇒ Leading non-trivial term is L(2)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 33 / 76 Construction of the effective Lagrangian for GBs (I)

Aim: reproduce low-energy structure of QCD

• Effective Lagrangian invariant under G = SU(Nf )L × SU(Nf )R (and C, P ) U →G LUR†,U →C U T ,U →P U † • What does "low-energy" mean here? Goldstone boson fields (contained in U) as the only degrees of freedom ⇒ energy range restricted to well below 1 GeV (separation of energy scales, more see later)

• Low energies: expand in powers of momenta ( = number of derivatives)

• Lorentz invariance ⇒ only even number of derivatives are allowed L = L(0) + L(2) + L(4) + ...

•L (0)? U is unitary, UU † = 1, hence

† n h(UU ) i = const. , h...i ≡ Trflavor[...]

⇒ Leading non-trivial term is L(2)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 33 / 76 Construction of the effective Lagrangian for GBs (I)

Aim: reproduce low-energy structure of QCD

• Effective Lagrangian invariant under G = SU(Nf )L × SU(Nf )R (and C, P ) U →G LUR†,U →C U T ,U →P U † • What does "low-energy" mean here? Goldstone boson fields (contained in U) as the only degrees of freedom ⇒ energy range restricted to well below 1 GeV (separation of energy scales, more see later)

• Low energies: expand in powers of momenta ( = number of derivatives)

• Lorentz invariance ⇒ only even number of derivatives are allowed L = L(0) + L(2) + L(4) + ...

•L (0)? U is unitary, UU † = 1, hence

† n h(UU ) i = const. , h...i ≡ Trflavor[...]

⇒ Leading non-trivial term is L(2)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 33 / 76 Construction of the effective Lagrangian for GBs (II)

• Leading term of the effective Lagrangian is L(2) just one single term (nonlinear σ model): F 2 L(2) = h∂ U∂µU †i with U = exp iφ 4 µ F 0

 φ3 φ8 + +  √ ! √ + √ π K π0 2π+ √ 2 6  − φ3 φ8 0  φ SU(2) = √ , φ SU(3) = 2  π − √ + √ K  2π− −π0  2 6  K− K¯ 0 − 2√φ8 6 (2) •L is invariant under SU(Nf )L × SU(Nf )R:

µ † 0 µ 0† h∂µU∂ U i → h∂µU ∂ U i † µ † † µ † = hL∂µUR R∂ U L i = h∂µU∂ U i

µ † † µ † Exercise: Show that U∂ U = 0, therefore U∂µU U∂ U is not present.

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 34 / 76 The low-energy constant F

iφ φ2 • Expand U in powers of φ, U = 1 + F 0 − 2F 02 + ... (2) + µ − + µ − ⇒ canonical kinetic terms L = ∂µπ ∂ π + ∂µK ∂ K + ... for F 0 = F

µ µ (2) • Calculate the Noether currents La , Ra from L ⇒ F 2 V µ = Rµ + Lµ = i hλ ∂µU, U †i a a a 4 a F 2 Aµ = Rµ − Lµ = i hλ ∂µU, U † i a a a 4 a

µ µ 3
• Expand the currents in powers of φ, Aa = −F ∂ φa + O φ

µ µ −ip·x h0|Aa (x)|φb(p)i = ip F e δab ⇒ F is the pion decay constant in the chiral limit

F ≈ Fπ + + Fπ = 92.2 MeV measured in the leptonic decay of the pion π → ` ν`

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 35 / 76 PCAC

So far, only considered chiral limit mu = md = ms = 0. For non-zero quark masses, • the singlet vector current still conserved (baryon number conservation) µ ∂ Vµ = 0 a 1 • vector currents Vµ conserved when mu = md = ms= m (i.e., M = m ) Gell-Mann, Ne’eman, The Eightfold Way  λa  ∂µV a = i q¯ M, q µ 2 a • Aµ not conserved any more: Partially Conserved Axial Current (PCAC)  λa  ∂µAa = i q¯ M, q µ 2

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 36 / 76 Explicit symmetry breaking: quark masses

• In the chiral limit mu = md = ms = 0 (2) F 2 µ † 2 2 L = 4 ∂µU∂ U contains no terms ∝ M φ theory for massless Goldstone bosons, but pions have nonvanishing masses

• In nature, u, d, s quark masses are small (mu,d,s  ΛQCD), but non-zero chiral symmetry explicitly broken

Lm = −q¯R m qL − q¯L m qR if symmetry breaking is weak ⇒ a perturbative expansion in the quark masses

• Effective Lagrangian is still an appropriate tool to systematically derive all symmetry relations

CHiral Perturbation Theory (CHPT): double expansion in low momenta and quark masses

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 37 / 76 Explicit symmetry breaking: quark masses

• In the chiral limit mu = md = ms = 0 (2) F 2 µ † 2 2 L = 4 ∂µU∂ U contains no terms ∝ M φ theory for massless Goldstone bosons, but pions have nonvanishing masses

• In nature, u, d, s quark masses are small (mu,d,s  ΛQCD), but non-zero chiral symmetry explicitly broken

Lm = −q¯R m qL − q¯L m qR if symmetry breaking is weak ⇒ a perturbative expansion in the quark masses

• Effective Lagrangian is still an appropriate tool to systematically derive all symmetry relations

CHiral Perturbation Theory (CHPT): double expansion in low momenta and quark masses

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 37 / 76 Explicit symmetry breaking: quark masses

• In the chiral limit mu = md = ms = 0 (2) F 2 µ † 2 2 L = 4 ∂µU∂ U contains no terms ∝ M φ theory for massless Goldstone bosons, but pions have nonvanishing masses

• In nature, u, d, s quark masses are small (mu,d,s  ΛQCD), but non-zero chiral symmetry explicitly broken

Lm = −q¯R m qL − q¯L m qR if symmetry breaking is weak ⇒ a perturbative expansion in the quark masses

• Effective Lagrangian is still an appropriate tool to systematically derive all symmetry relations

CHiral Perturbation Theory (CHPT): double expansion in low momenta and quark masses

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 37 / 76 Explicit symmetry breaking: the spurion trick (I)

Spurion in 3 steps: very useful trick for explicit symmetry breaking

1. Introduce a spurion field (e.g. quark mass, electric charge, γµ,... ) with a transformation property so that the symmetry breaking term in the full theory is invariant 2. Write down invariant operators in EFT including the spurion field 3. Set the spurion field to the value which it should take

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 38 / 76 Explicit symmetry breaking: the spurion trick (II)

• Apply the spurion trick to quark masses: 0 † L QCD = L QCD−q¯LMqR − q¯RM qL

Treat M as a complex spurion field M → M0 = LMR†

Then construct Lagrangian invariant under SU(Nf )L × SU(Nf )R

2 L eff = L eff(U, ∂U, ∂ U, . . . , M)

This procedure guarantees that chiral symmetry is broken in exactly the same way in the effective theory as it is in QCD • The spurion trick is very useful to construct EFT operators with a given symmetry transformation property

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 39 / 76 LO chiral Lagrangian with the mass term (I)

F 2 L(2) = h∂ U∂µU †i +2 BhMU † + M†Ui 4 µ

• Read off mass terms for GBs [SU(3)]: GMOR relation on lattice:

2 Mπ± = B(mu + md) 2 MK± = B(mu + ms) 2 MK0 = B(md + ms)

• Gell-Mann–Oakes–Renner (GMOR) relation: 2 MGB ∝ mq CHPT can be used to extrapolate lattice re-

sults from large to the physical values of mu,d (or equivalently pion masses) ETM Col., JHEP08(2010)097

• Unified power counting for derivative and quark mass expansions:

2
mq = O p

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 40 / 76 LO chiral Lagrangian with the mass term (II)

F 2 L(2) = h∂ U∂µU †i +2 BhMU † + M†Ui 4 µ

• Flavor-neutral states φ3, φ8 are mixed: !T ! ! m + m √1 (m − m ) B φ3 u d 3 u d φ3 2 φ √1 (m − m ) 1 (m + m + 4m ) φ 8 3 u d 3 u d s 8

• Diagonalize with ! ! ! π0 cos  sin  φ = 3 η − sin  cos  φ8 √ ! 0 1 3 md − mu 1 • π η mixing angle:  = arctan , mˆ = (mu + md) 2 2 ms − mˆ 2

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 41 / 76 LO chiral Lagrangian with the mass term (III)

• Mass eigenvalues:

2 2
Mπ0 = B(mu + md) − O (mu − md) B M 2 = (m + m + 4m ) + O (m − m )2
η 3 u d s u d

• In the isospin limit mu = md:

2 2 2 2 Mπ± = Mπ0 ,MK± = MK0 (of course!)

• Gell-Mann–Okubo (GMO) mass formula for pseudoscalars:

2 2 2 4MK = 3Mη + Mπ

⇒ fulfilled in nature at 7% accuracy

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 42 / 76 Representation invariance

• Freedom to choose coordinates on coset space G/H

• Haag’s theorem on field redefinition: Haag (1958); Coleman, Wess, Zumino (1969)

If fields φ and χ are related nonlinearly by a local function as

φ = χ F [χ] with F [0] = 1,

then the same physical observables (on-shell S-matrices) can be obtained using either field φ with Lagrangian L[φ] or χ with L[χF [χ]].

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 43 / 76 Representation invariance: ππ scattering (I)

F 2 In the chiral limit, L(2) = h∂ U∂µU †i 4 µ Amplitude for π+π− → π0π0: • Exponential representation: iφ U = exp F with φ = ~τ · ~π, π0 = π , π± = √1 (π ∓ iπ ) for SU(2). 3 2 1 2 s A = F 2 parameter-free prediction in accordance with Goldstone theorem: vanishes at zero-momentum

• Square-root representation:

1 p  U = F 2 − ~π0 2 + i~τ · ~π0 F

calculating with ~π0 as the pion fields gives the same scattering amplitude.

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 44 / 76 Representation invariance: ππ scattering (I)

F 2 In the chiral limit, L(2) = h∂ U∂µU †i 4 µ Amplitude for π+π− → π0π0: • Exponential representation: iφ U = exp F with φ = ~τ · ~π, π0 = π , π± = √1 (π ∓ iπ ) for SU(2). 3 2 1 2 s A = F 2 parameter-free prediction in accordance with Goldstone theorem: vanishes at zero-momentum

• Square-root representation:

1 p  U = F 2 − ~π0 2 + i~τ · ~π0 F

calculating with ~π0 as the pion fields gives the same scattering amplitude.

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 44 / 76 Representation invariance: ππ scattering (II)

Exercise: Calculate the amplitude and show that the two representations are related by the following field redefinition: F |~π| √ ~π0 = ~π sin = ~π + ..., here |~π| ≡ ~π2 |~π| F Exercise: Show that the amplitude with the quark mass term included reads

2 s − Mπ A(s, t, u) = 2 Fπ Weinberg (1966)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 45 / 76 Including matter fields

• So far, EFT for (pseudo) Goldstone bosons • Matter fields (fields which are not Goldstone bosons) can be included as well, e.g. baryon CHPT: nucleons [SU(2)] / baryon ground state octet [SU(3)] heavy-hadron CHPT: heavy-flavor (charm, bottom) mesons and baryons

• At low-energies, 3-momenta remain small ∼ Mπ, derivative expansion is feasible

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 46 / 76 Transformation of matter fields

• Proceed as before

need to know how matter fields transform under SU(Nf )L × SU(Nf )R construct effective Lagrangians according to increasing number of momenta • Transformation properties of matter fields:

well-defined transformation rule under the unbroken SU(Nf )V not necessarily form representations of SU(Nf )L × SU(Nf )R: transformation of matter fields under SU(Nf )L × SU(Nf )R not uniquely defined, related by field redefinition (again: representation invariance) • Example: heavy-meson CHPT

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 47 / 76 Transformation of matter fields

• Proceed as before

need to know how matter fields transform under SU(Nf )L × SU(Nf )R construct effective Lagrangians according to increasing number of momenta • Transformation properties of matter fields:

well-defined transformation rule under the unbroken SU(Nf )V not necessarily form representations of SU(Nf )L × SU(Nf )R: transformation of matter fields under SU(Nf )L × SU(Nf )R not uniquely defined, related by field redefinition (again: representation invariance) • Example: heavy-meson CHPT

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 47 / 76 Bispinor fields for heavy mesons

• Recall HQSS for heavy quarks (charm, bottom); 1 2 (1 + v/) projects onto the particle component of the heavy quark spinor. • Convenient to introduce heavy mesons as bispinors: 1 + v/ H = P ∗µγ − P γ  , H¯ = γ H†γ a 2 a µ a 5 a 0 a 0 P = {Qu,¯ Qd,¯ Qs¯}: pseudoscalar heavy mesons, P ∗: vector heavy mesons • Charge conjugation:

Ha destroys mesons containing a Q, but does not create mesons with a Q¯ Free to choose the phase convention for charge conjugation. If we use, e.g.,

C (Q¯) ∗ C ∗(Q¯) Pa → +Pa ,Pa,µ →− Pa,µ , then the fields annihilating mesons containing a Q¯ is (C = iγ2γ0)  T ¯ 1 + v/  ¯ ¯  H(Q) = C − P ∗(Q) γµ− P (Q) γ C−1 a 2 a,µ a 5 | {z } | {z } C ∗ C P → Pa,µ→ a

 ¯ ¯  1 − v/ = + P ∗(Q)γµ− P (Q)γ a,µ a 5 2

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 48 / 76 Transformations of heavy meson fields (I)

Consider the SU(3) case, the charmed meson ground state anti-triplet:

0 + +
 ¯
 Ha = D ,D ,Ds a cu,¯ cd, cs¯ transform under the global unbroken SU(3)V as

V ∈SU(3) H −−−−−−→V HV † • Representation invariance:

free to choose how H transforms under SU(3)L × SU(3)R as long as it reduces to the above under SU(3)V • Example:

describe the heavy mesons by H1 or H2, under g = (L, R) ∈ SU(3)L × SU(3)R g † g † H1 → H1 L , H2 → H2 R

both transform as an anti-triplet under (V,V ) ∈ SU(3)V

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 49 / 76 Transformations of heavy meson fields (I)

Consider the SU(3) case, the charmed meson ground state anti-triplet:

0 + +
 ¯
 Ha = D ,D ,Ds a cu,¯ cd, cs¯ transform under the global unbroken SU(3)V as

V ∈SU(3) H −−−−−−→V HV † • Representation invariance:

free to choose how H transforms under SU(3)L × SU(3)R as long as it reduces to the above under SU(3)V • Example:

describe the heavy mesons by H1 or H2, under g = (L, R) ∈ SU(3)L × SU(3)R g † g † H1 → H1 L , H2 → H2 R

both transform as an anti-triplet under (V,V ) ∈ SU(3)V

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 49 / 76 Transformations of heavy meson fields (I)

Consider the SU(3) case, the charmed meson ground state anti-triplet:

0 + +
 ¯
 Ha = D ,D ,Ds a cu,¯ cd, cs¯ transform under the global unbroken SU(3)V as

V ∈SU(3) H −−−−−−→V HV † • Representation invariance:

free to choose how H transforms under SU(3)L × SU(3)R as long as it reduces to the above under SU(3)V • Example:

describe the heavy mesons by H1 or H2, under g = (L, R) ∈ SU(3)L × SU(3)R g † g † H1 → H1 L , H2 → H2 R

both transform as an anti-triplet under (V,V ) ∈ SU(3)V

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 49 / 76 Transformations of heavy meson fields (II)

• Example:

describe the heavy mesons by H1 or H2, under g = (L, R) ∈ SU(3)L × SU(3)R g † g † H1 → H1 L , H2 → H2 R

both transform as an anti-triplet under (V,V ) ∈ SU(3)V • Related to each other through field redefinition:

i  i  g H = H U = H + H φ + ...,U = exp φ → LUR † 2 1 1 F 1 F

• But H1,2 are inconvenient: complicated parity transformation (P ) L(L†) needs to be replaced by R(R†) under parity ⇒

P 0 0 g 0 0 † H1,a(t, ~x) → γ H1,b(t, −~x)γ Uba → γ H1,b(t, −~x)γ UbaR

P [recall for a spinor: ψ(t, ~x) → γ0ψ(t, −~x)]

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 50 / 76 Transformations of heavy meson fields (II)

• Example:

describe the heavy mesons by H1 or H2, under g = (L, R) ∈ SU(3)L × SU(3)R g † g † H1 → H1 L , H2 → H2 R

both transform as an anti-triplet under (V,V ) ∈ SU(3)V • Related to each other through field redefinition:

i  i  g H = H U = H + H φ + ...,U = exp φ → LUR † 2 1 1 F 1 F

• But H1,2 are inconvenient: complicated parity transformation (P ) L(L†) needs to be replaced by R(R†) under parity ⇒

P 0 0 g 0 0 † H1,a(t, ~x) → γ H1,b(t, −~x)γ Uba → γ H1,b(t, −~x)γ UbaR

P [recall for a spinor: ψ(t, ~x) → γ0ψ(t, −~x)]

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 50 / 76 Transformations of heavy meson fields (III)

• An elegant/convenient way: introduce  iφ  u = exp or u2 = U 2F

g † it transforms under g ∈ SU(3)L × SU(3)R as (recall U → LUR ) g u → L uh †(L, R, φ)= h(L, R, φ) u R†

h(L, R, φ): space-time dependent nonlinear function, called compensator field

For for SU(3)V transformations (L = R = V ), reduces to h(L, R, φ) = V † • We can construct H = H1u or H = H2u , it transforms as g H → H h† P Exercise: Show that under parity transformation h(t, ~x) → h(t, −~x), and P H(t, ~x) → γ0H(t, −~x)γ0 .

• Transformation under HQSS: Ha → SHa, S does not commute with γ-matrices !

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 51 / 76 Transformations of heavy meson fields (III)

• An elegant/convenient way: introduce  iφ  u = exp or u2 = U 2F

g † it transforms under g ∈ SU(3)L × SU(3)R as (recall U → LUR ) g u → L uh †(L, R, φ)= h(L, R, φ) u R†

h(L, R, φ): space-time dependent nonlinear function, called compensator field

For for SU(3)V transformations (L = R = V ), reduces to h(L, R, φ) = V † • We can construct H = H1u or H = H2u , it transforms as g H → H h† P Exercise: Show that under parity transformation h(t, ~x) → h(t, −~x), and P H(t, ~x) → γ0H(t, −~x)γ0 .

• Transformation under HQSS: Ha → SHa, S does not commute with γ-matrices !

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 51 / 76 Building blocks of the chiral Lagrangian

• Useful to introduce combinations of u whose transformations only involve h: 1 Γµ = u†(∂µ − i lµ)u + u(∂µ − i rµ)u† 2 uµ = i u†(∂µ − i lµ)u − u(∂µ − i rµ)u†

Γµ: chiral connection, vector; uµ: chiral vielbein, axial vector g g Γµ → h Γµh† + h ∂µh†,u µ → h uµh† • Introduce a covariant derivative: DµH = ∂µH − H Γµ

which transform the same way as H under SU(Nf )L × SU(Nf )R g DµH → DµH h† • Include the quark mass term χ = 2B(s + i p) = 2BM + ... by introducing † † † † χ+ = u χu + uχ u,χ + → h χ+h All fields transform in terms of h, convenient to construct the effective Lagrangians

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 52 / 76 Building blocks of the chiral Lagrangian

• Useful to introduce combinations of u whose transformations only involve h: 1 Γµ = u†(∂µ − i lµ)u + u(∂µ − i rµ)u† 2 uµ = i u†(∂µ − i lµ)u − u(∂µ − i rµ)u†

Γµ: chiral connection, vector; uµ: chiral vielbein, axial vector g g Γµ → h Γµh† + h ∂µh†,u µ → h uµh† • Introduce a covariant derivative: DµH = ∂µH − H Γµ

which transform the same way as H under SU(Nf )L × SU(Nf )R g DµH → DµH h† • Include the quark mass term χ = 2B(s + i p) = 2BM + ... by introducing † † † † χ+ = u χu + uχ u,χ + → h χ+h All fields transform in terms of h, convenient to construct the effective Lagrangians

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 52 / 76 Building blocks of the chiral Lagrangian

• Useful to introduce combinations of u whose transformations only involve h: 1 Γµ = u†(∂µ − i lµ)u + u(∂µ − i rµ)u† 2 uµ = i u†(∂µ − i lµ)u − u(∂µ − i rµ)u†

Γµ: chiral connection, vector; uµ: chiral vielbein, axial vector g g Γµ → h Γµh† + h ∂µh†,u µ → h uµh† • Introduce a covariant derivative: DµH = ∂µH − H Γµ

which transform the same way as H under SU(Nf )L × SU(Nf )R g DµH → DµH h† • Include the quark mass term χ = 2B(s + i p) = 2BM + ... by introducing † † † † χ+ = u χu + uχ u,χ + → h χ+h All fields transform in terms of h, convenient to construct the effective Lagrangians

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 52 / 76 Heavy meson CHPT at LO

• LO Lagrangian [O (p)]: g L(1) = −i Tr H¯ v (DµH)  + Tr H¯ H γ γ  uµ HM a µ a 2 a b µ 5 ba | {z } | {z } kinetic term+Dπ scattering+... terms for D∗→Dπ+...

invariant under Lorentz transformation, chiral symmetry, parity Tr: trace in the spinor space, a, b: indices in the light flavor space • Notice that the mass dimension of H is 3/2. √a rel. Nonrelativistic normalization: Ha ' MH Ha D-meson propagator: i i (p = M v + k) ' MH × 2 2 H 2v · k + i p − MH + i | {z } i 2 2 = [1+O(k /MH )] 2MH v · k + i

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 53 / 76 Simplified two-component notation (I)

The superfield for pseudoscalar and vector heavy mesons: 1 + v/ H = [P ∗µγ − P γ ] a 2 a µ a 5 In the rest frame of heavy meson, vµ = (1,~0). We take the Dirac basis ! ! ! 1 0 0 σi 0 1 γ0 = , γi = , γ5 . 0 −1 −σi 0 1 0 ! 1 + v/ 1 + γ0 1 0 Simplifications: = = 2 2 0 0      ~ ∗ 0 − Pa + Pa · ~σ 0 0 H = , H¯ =   a   a  † ~ ∗†  0 0 Pa + Pa · ~σ 0

Thus, it is convenient to simply use the two-component notation ~ ∗ Ha = Pa + Pa · ~σ

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 54 / 76 Simplified two-component notation (II)

• In the simplified two-component notation:

 ¯ µ   † 0 0
 −i Tr Havµ(D H)a ⇒ i Tr Ha ∂ Ha − HbΓba −i   = 2i P †∂0P + P ∗i †∂0P ∗i
+ P †P + P~ ∗† · P~ ∗ φ, ∂0φ + ... a a a a 4F 2 a b a b ba | {z } scattering between (D,D∗,B,¯ B¯∗) and GBs(π,K,η)

The LO, O (p), scattering amplitudes are completely determined by chiral symmetry! k 1 k • The axial coupling term (u = − F ∂ φ + ...): g g Tr H¯ H γ γ  uµ ⇒ − Tr H†H σi ui 2 a b µ 5 ba 2 a b ba g h   i = − Tr P ∗i †σi + P †
P ∗jσj + P † σk uk 2 a a b b ba 1 1 i = P †P ∗i∂iφ + P ∗i †P ∂iφ + ijkP ∗i †P ∗j∂kφ + ... F a b ba F a b ba F a b ba | {z } term for D∗→Dπ

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 55 / 76 Determination of g

∗ • Measured D widths: PDG2015

Γ(D∗0) < 2.1 MeV, Γ(D∗±) = (83.4 ± 1.8) keV B(D∗+ → D0π+) = (67.7 ± 0.5)%, B(D∗+ → D+π0) = (30.7 ± 0.5)%

• Amplitude from the LO Lagrangian: √ ∗+ 0 + i 2g p A(D → D π ) = ~εD∗ · ~qπ MD∗ MD F | {z } and the two-body decay width accounts for NR normalization

∗+ 0 + 1 |~qπ| 1 2 Γ(D → D π ) = 2 |A| ⇒ g ' 0.57 8π MD∗ 3

• HQFS: g should be approximately the same in bottom sector with   a relative uncertainty of ∼O ΛQCD − ΛQCD ∼ O (0.2) mc mb Lattice QCD results:

gb = 0.492 ± 0.029 ALPHA Collaboration, Phys. Lett. B 740 (2015) 278 gb = 0.56 ± 0.03 ± 0.07 RBC and UKQCD Collaborations, Phys. Rev. D 93 (2016) 014510

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 56 / 76 Mass splittings among heavy mesons (I)

• Spin-dependent term ⇒ mass difference between vector and pseudoscalar µν i µ ν mesons (σ = 2 [γ , γ ])

λ2  ¯ µν  2-comp. 2λ2  † i i L∆ = Tr Haσµν Haσ ⇒ − Tr Haσ Haσ mQ mQ 4λ2  ~ ∗ † ~ ∗ †  = Pa · Pa − 3Pa Pa mQ

8λ2 ⇒ M ∗ − M = − P Pa a mQ • Thus, we expect M ∗ − M m B B ' c ' 0.3 MD∗ − MD mb measured values:

MD∗ − MD ' 140 MeV, MB∗ − MB ' 46 MeV

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 57 / 76 Mass splittings among heavy mesons (II)

• Light quark mass-dependent terms in two-component notation:  †  0  †  Lχ = −λ1Tr HaHb χ+,ba − λ1Tr HaHa χ+,bb B here, χ = u†χu† + uχ†u =4 BM− φ, {φ, M} + ... + 2F 2 SU(3) mass differences: 2 2
M + − M + = 4λ1B(ms − md) = 4λ1 M ± − M ± Ds D K π −1 ⇒ λ1 ' 0.11 GeV

Isospin splitting induced by md − mu:

(MD0 − MD+ )quark mass = 4λ1B(mu − md) 2 2 2 2
= 4λ1 MK± − MK0 − Mπ± + Mπ0 = −2.3 MeV

Exp. value: MD0 − MD+ = −(4.77 ± 0.08) MeV = M − M
+ M − M
D0 D+ quark mass D0 D+ e.m. •L χ also contributes to scattering between a heavy meson and the lightest pseudoscalar mesons (GBs)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 58 / 76 Intermediate summary — 2

• Lagrangians need to respect the global symmetries of QCD, explicit symmetry breaking can be included using the spurion technique

• The LO mesonic chiral Lagrangian invariant under SU(Nf )L × SU(Nf )R F 2 L(2) = h∂ U∂µU †i + 2BhMU † + M†Ui with U = eiφ/F 4 µ 2 GMOR relation: Mπ ∝ mq • Representation invariance: physical observables do not change under field redefinition φ = χ F [χ] with F [0] = 1

• Under SU(Nf )L × SU(Nf )R, matter fields do not have a unique transformation law, convenient to introduce the compensator field h heavy-meson CHPT as an example

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 59 / 76 Unitarity of S-matrix

• Probability conservation ⇒ unitarity of the S-matrix

SS† = S†S = 1

T -matrix: S = 1 + iT ⇒ T − T † = i T T †

thus, unitarity dictates a relation for the partial-wave scattering amplitude t`:

2 Im t`(s) = σ(s)|t`(s)|

here σ(s): two-body phase space factor s − M 2 • From the LO CHPT, the ππ scattering amplitude A(s, t, u) = π , F 2 no imaginary part! ⇒ unitarity is broken

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 60 / 76 Unitarity of S-matrix

• Probability conservation ⇒ unitarity of the S-matrix

SS† = S†S = 1

T -matrix: S = 1 + iT ⇒ T − T † = i T T †

thus, unitarity dictates a relation for the partial-wave scattering amplitude t`:

2 Im t`(s) = σ(s)|t`(s)|

here σ(s): two-body phase space factor s − M 2 • From the LO CHPT, the ππ scattering amplitude A(s, t, u) = π , F 2 no imaginary part! ⇒ unitarity is broken

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 60 / 76 Going to higher orders

• Perturbative unitarity: imaginary part given by loops (2) (4) (2) 2 Im t` (s) = 0, Im t` (s) = σ(s)|t` (s)| ,... • Symmetries do not forbid higher order terms in effective Lagrangians: more derivatives, more insertion of quark masses • More derivatives ⇒ non-renormalizable . . . , if we include in the Lagrangian all of the infinite number of interactions allowed by symmetries, then there will be a counterterm available to cancel every ultraviolet divergence. In this sense, . . . , non-renormalizable theories are just as renormalizable as renormalizable theories, as long as we include all possible terms in the Lagrangian. S.Weinberg, The Quantum Theory of Fields, Vol.1

• Q: How should we deal with the infinite number of terms? A: Power counting is needed: organize the infinite number of terms according to power of the expansion parameter, finite number of terms up to a given order

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 61 / 76 Going to higher orders

• Perturbative unitarity: imaginary part given by loops (2) (4) (2) 2 Im t` (s) = 0, Im t` (s) = σ(s)|t` (s)| ,... • Symmetries do not forbid higher order terms in effective Lagrangians: more derivatives, more insertion of quark masses • More derivatives ⇒ non-renormalizable . . . , if we include in the Lagrangian all of the infinite number of interactions allowed by symmetries, then there will be a counterterm available to cancel every ultraviolet divergence. In this sense, . . . , non-renormalizable theories are just as renormalizable as renormalizable theories, as long as we include all possible terms in the Lagrangian. S.Weinberg, The Quantum Theory of Fields, Vol.1

• Q: How should we deal with the infinite number of terms? A: Power counting is needed: organize the infinite number of terms according to power of the expansion parameter, finite number of terms up to a given order

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 61 / 76 Going to higher orders

• Perturbative unitarity: imaginary part given by loops (2) (4) (2) 2 Im t` (s) = 0, Im t` (s) = σ(s)|t` (s)| ,... • Symmetries do not forbid higher order terms in effective Lagrangians: more derivatives, more insertion of quark masses • More derivatives ⇒ non-renormalizable . . . , if we include in the Lagrangian all of the infinite number of interactions allowed by symmetries, then there will be a counterterm available to cancel every ultraviolet divergence. In this sense, . . . , non-renormalizable theories are just as renormalizable as renormalizable theories, as long as we include all possible terms in the Lagrangian. S.Weinberg, The Quantum Theory of Fields, Vol.1

• Q: How should we deal with the infinite number of terms? A: Power counting is needed: organize the infinite number of terms according to power of the expansion parameter, finite number of terms up to a given order

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 61 / 76 Necessity of higher order Lagrangian from renormalization

Consider the example of ππ scattering

• At LO, O p2
: two derivatives or one quark mass insertion

= O p2

• One-loop with two O p2
vertices:

Z 4 q1q2q3q4 = d q 2 2 2 2 (q − Mπ)[(p − q) − Mπ] = O p4+4−4
= O p4

• The loop is divergent, divergence absorbed by the counterterms in the O p4
Lagrangian

How can we construct the higher order Lagrangian?

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 62 / 76 Building blocks

• Powerful to include external fields ⇒ incorporate electroweak interactions, quark masses:

0 µ LQCD = LQCD +qγ ¯ (vµ + γ5aµ) q − q¯(s + iγ5p) q 0 µ µ = LQCD +q ¯Lγ lµqL +q ¯Rγ rµqR − q¯L(s + ip)qR − q¯R(s − ip)qL

lµ = vµ − aµ, rµ = vµ + aµ left-/right-handed sources µν µ ν ν µ µ ν µν µ ν ν µ µ ν FR = ∂ r − ∂ r − i[r , r ], FL = ∂ l − ∂ l − i[l , l ] χ = 2B(s + ip) = 2BM + ... scalar/pseudoscalar sources

• Transformation laws for the building blocks:

U → LUR†,χ → L χ R†, µν µν † µν µν † FL → LFL L ,F R → RFR R

µν µν µν † † µν • Another set of building blocks: uµ, χ± and f± . f± = uFL u ± u FR u They all transform as O → h O h†

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 63 / 76 Building blocks

• Powerful to include external fields ⇒ incorporate electroweak interactions, quark masses:

0 µ LQCD = LQCD +qγ ¯ (vµ + γ5aµ) q − q¯(s + iγ5p) q 0 µ µ = LQCD +q ¯Lγ lµqL +q ¯Rγ rµqR − q¯L(s + ip)qR − q¯R(s − ip)qL

lµ = vµ − aµ, rµ = vµ + aµ left-/right-handed sources µν µ ν ν µ µ ν µν µ ν ν µ µ ν FR = ∂ r − ∂ r − i[r , r ], FL = ∂ l − ∂ l − i[l , l ] χ = 2B(s + ip) = 2BM + ... scalar/pseudoscalar sources

• Transformation laws for the building blocks:

U → LUR†,χ → L χ R†, µν µν † µν µν † FL → LFL L ,F R → RFR R

µν µν µν † † µν • Another set of building blocks: uµ, χ± and f± . f± = uFL u ± u FR u They all transform as O → h O h†

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 63 / 76 Construction of higher order chiral Lagrangian

• We need a counting scheme (more see later): U ∼O p0
small momentum / derivative ∼O (p) 2
2 light quark masses χ ∼O p ⇐ MGB ∼ mq

external fields lµ, rµ ∼O (p) ⇐ DµU = ∂µU − ilµU + iUrµ

• At a given order, write down the most general Lagrangian allowed by symmetries (for θ = 0): chiral symmetry, P , C and T most general ⇒ be able to absorb all divergences at the same order

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 64 / 76 Construction of higher order chiral Lagrangian

• We need a counting scheme (more see later): U ∼O p0
small momentum / derivative ∼O (p) 2
2 light quark masses χ ∼O p ⇐ MGB ∼ mq

external fields lµ, rµ ∼O (p) ⇐ DµU = ∂µU − ilµU + iUrµ

• At a given order, write down the most general Lagrangian allowed by symmetries (for θ = 0): chiral symmetry, P , C and T most general ⇒ be able to absorb all divergences at the same order

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 64 / 76 SU(3) chiral Lagrangian at NLO

4
• SU(3) chiral Lagrangian at O p Gasser, Leutwyler (1985)

(4) † µ 2 † µ † ν L = L1hDµU D Ui + L2hDµU Dν UihD U D Ui † µ † ν † µ † † +L3hDµU D UDν U D Ui + L4hDµU D Uihχ U + χU i † µ † † † † 2 +L5hDµU D U(χ U + χU )i + L6hχ U + χU i † † 2 † † † † +L7hχ U − χU i + L8hχ Uχ U + χU χU i µν † µν † † µν −iL 9hFL DµUDν U + FR DµU Dν Ui + L10hU FL UFRµν i +2 contact terms

• L1,...,10: low-energy constants (LECs)

• NLO SU(2) chiral Lagrangian contains 7 terms: `1,...,7 Gasser, Leutwyler (1984)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 65 / 76 Low-energy constants

• One loop diagrams with vertices from L(2) are of O p4
, divergences should be absorbed by counterterms in L(4) • Low-energy constants generally contain two parts:

r Li = Li (µ)+Γ i × divergence

r renormalized LECs Li (µ) are finite, scale-dependent • Scale dependence of LECs cancel the one from loop integrals ⇒ physical observables are scale-independent ! r • Li ’s are independent of light quark masses by construction, parameterize the short-distance physics Values not fixed by chiral symmetry: extracted using experimental data estimated with models such as resonance saturation Ecker et al (1989) using lattice simulations Flavour Lattice Averaging Group (FLAG) (2013)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 66 / 76 Low-energy constants

• One loop diagrams with vertices from L(2) are of O p4
, divergences should be absorbed by counterterms in L(4) • Low-energy constants generally contain two parts:

r Li = Li (µ)+Γ i × divergence

r renormalized LECs Li (µ) are finite, scale-dependent • Scale dependence of LECs cancel the one from loop integrals ⇒ physical observables are scale-independent ! r • Li ’s are independent of light quark masses by construction, parameterize the short-distance physics Values not fixed by chiral symmetry: extracted using experimental data estimated with models such as resonance saturation Ecker et al (1989) using lattice simulations Flavour Lattice Averaging Group (FLAG) (2013)

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 66 / 76 Chiral Lagrangian at higher orders

(2) (4) (6) Leff = L + L + L + ...

• Number of terms allowed by the symmetries increases very fast • How many LECs are contained in these for SU(N), N = (2, 3)? L(2) contains (2,2) constants ( F , B) (4) L contains (7, 10) constants Gasser, Leutwyler (1984, 1985) (6) L contains (53, 90) constants Bijnens, Colangelo, Ecker (1999)

• Why different for SU(2)/ SU(3)? same most general SU(N) Lagrangian, but matrix-trace relations [Cayley–Hamilton] render some of the structures redundant

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 67 / 76 Weinberg’s power counting

Weinberg (1979) • Consider an arbitrary Feynman diagram with L

loops, I internal lines, Vd vertices of order d:

Z 1 Y A ∝ (d4p)L (pd)Vd (p2)I d • The chiral dimension of A X D = 4L − 2I + dVd d P • Use topological identity for L to eliminate I, L = I − dVd + 1 X D = Vd(d − 2)+2L + 2 d

Lowest order is O p2
, i.e. d ≥ 2 ⇒ rhs is a sum of non-negative numbers

For a given order D, there is only a finite number of combinations of L and Vd Each loop is suppressed by two orders in the momentum expansion

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 68 / 76 Weinberg’s power counting

Weinberg (1979) • Consider an arbitrary Feynman diagram with L

loops, I internal lines, Vd vertices of order d:

Z 1 Y A ∝ (d4p)L (pd)Vd (p2)I d • The chiral dimension of A X D = 4L − 2I + dVd d P • Use topological identity for L to eliminate I, L = I − dVd + 1 X D = Vd(d − 2)+2L + 2 d

Lowest order is O p2
, i.e. d ≥ 2 ⇒ rhs is a sum of non-negative numbers

For a given order D, there is only a finite number of combinations of L and Vd Each loop is suppressed by two orders in the momentum expansion

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 68 / 76 Power counting – example: ππ scattering

X D = Vd(d − 2) + 2L + 2 d

• D = 2 L = 0, lowest order tree-level diagram only

• D = 4 L = 0, tree-level diagram with one insertion from L(4)

V4 = 1,Vd>4 = 0

L = 1, one-loop diagram with only d = 2 vertices

Vd>2 = 0

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 69 / 76 Power counting – example: ππ scattering

X D = Vd(d − 2) + 2L + 2 d

• D = 2 L = 0, lowest order tree-level diagram only

• D = 4 L = 0, tree-level diagram with one insertion from L(4)

V4 = 1,Vd>4 = 0

L = 1, one-loop diagram with only d = 2 vertices

Vd>2 = 0

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 69 / 76 Power counting – example: ππ scattering

X D = Vd(d − 2) + 2L + 2 d

• D = 2 L = 0, lowest order tree-level diagram only

• D = 6 L = 0, tree-level diagram with one insertion from L(6) L = 0, tree-level diagram with two insertions from L(4) L = 1, one-loop diagram with one insertion from L(4)

L = 2, two-loop diagram with Vd = 2 vertices

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 69 / 76 Chiral symmetry breaking scale Λχ

2   F µ † † † 1 ˜(4) 1 ˜(6) L eff = DµUD U + χU + χ U + 2 L + 4 L + ... 4 Λχ Λχ

• What is the scale Λχ? hard energy scale where the momentum expansion definitely fails 2 2 uncertainty estimate: higher-order corrections are suppressed by ∼ p /Λχ ⇒ how big? • Estimate with resonance masses: The only dynamical degrees of freedom are the GBs, no resonances ⇒ CHPT must fail once the energy reaches the resonance region: a perturbative momentum expansion cannot generate a pole

Λχ ≈ M res

Lowest narrow resonance Mρ ≈ 770 MeV,

typically M res ∼ 1 GeV

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 70 / 76 Chiral symmetry breaking scale Λχ

2   F µ † † † 1 ˜(4) 1 ˜(6) L eff = DµUD U + χU + χ U + 2 L + 4 L + ... 4 Λχ Λχ

• What is the scale Λχ? hard energy scale where the momentum expansion definitely fails 2 2 uncertainty estimate: higher-order corrections are suppressed by ∼ p /Λχ ⇒ how big? • Estimate with resonance masses: The only dynamical degrees of freedom are the GBs, no resonances ⇒ CHPT must fail once the energy reaches the resonance region: a perturbative momentum expansion cannot generate a pole

Λχ ≈ M res

Lowest narrow resonance Mρ ≈ 770 MeV,

typically M res ∼ 1 GeV

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 70 / 76 Chiral symmetry breaking scale Λχ

2   F µ † † † 1 ˜(4) 1 ˜(6) L eff = DµUD U + χU + χ U + 2 L + 4 L + ... 4 Λχ Λχ

• Naturalness argument: Manohar, Georgi (1984) Compare

2 2 4 4  p  Z d q 1 dim.reg 1 p ∝ ∝ log µ F 2 (2π)4 (q2)2 (4π)2 F 4

2 4 F p ˜ ∝ 2 4 `i(µ) Λχ F ˜ • Scale-dependent LEC `i(µ) compensates for log µ dependence of the loop graph ˜ • If no accidental fine-tuning (naturalness), `i(µ) should not be smaller than shift induced by change in scale µ ⇒

Λχ ≈ 4πF ≈ 1.2 GeV r (4) −3 Exercise: Using the naturalness assumption, estimate the size of Li in L (A: 10 )

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 70 / 76 Meson masses at NLO – 1 Take the pion mass as an example of NLO calculations. • Mass: pole in the two point correlation function 2 2 At LO, Mπ = M ≡ B(mu + md) • Higher orders: self-energy Z ab 4 −ipx  a b  i δ ∆π(p) = d xe 0 T π (x)π (0) 0

= i i i = + [−iΣ(p2)] + ... p2 − M 2 + i p2 − M 2 + i p2 − M 2 + i i Zπ = 2 2 2 = 2 2 + non-singular terms p − M − Σ(p )+ i p − Mπ + i Exercise: show that the wave function renormalization constant is 1 Zπ = 0 2 1 − Σ (Mπ)

Mπ: physical pion mass, solution of the equation 2 2 2 Mπ − M − Σ(Mπ)=0

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 71 / 76 Meson masses at NLO – 2

• Self-energy contains two parts: π, K, η tadpole loops and counterterms in L(4)

Assuming isospin symmetry (mu = md =ˆm), the pion mass at NLO:

 I(M 2) I(M 2 ) 8M 2 24M 2  M 2 = M 2 1 + − η,2 + (2L − L ) + η,2 (2L − L ) π 2F 2 2F 2 F 2 8 5 F 2 6 4

2 2 2 here M = 2Bm,ˆ Mη,2 = 3 B(2m ˆ + ms) • Loop is divergent, in dimensional regularization (good for preserving symmetry)

Z ddl 1 M 2  M 2  I(M 2)= iµ4−d = λ + log (2π)d l2 − M 2 + i 16F 2 µ2 2 λ = − [log(4π) + Γ0(1) + 1], divergent for d = 4 ! d − 4 Γi On the other hand, L ’s are divergent either L = Lr + λ i i i 32π2 1 3 11 5 Γ = , Γ = , Γ = , Γ = 4 8 5 8 6 144 8 48

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 72 / 76 Meson masses at NLO – 2

• Self-energy contains two parts: π, K, η tadpole loops and counterterms in L(4)

Assuming isospin symmetry (mu = md =ˆm), the pion mass at NLO:

 I(M 2) I(M 2 ) 8M 2 24M 2  M 2 = M 2 1 + − η,2 + (2L − L ) + η,2 (2L − L ) π 2F 2 2F 2 F 2 8 5 F 2 6 4

2 2 2 here M = 2Bm,ˆ Mη,2 = 3 B(2m ˆ + ms) • Loop is divergent, in dimensional regularization (good for preserving symmetry)

Z ddl 1 M 2  M 2  I(M 2)= iµ4−d = λ + log (2π)d l2 − M 2 + i 16F 2 µ2 2 λ = − [log(4π) + Γ0(1) + 1], divergent for d = 4 ! d − 4 Γi On the other hand, L ’s are divergent either L = Lr + λ i i i 32π2 1 3 11 5 Γ = , Γ = , Γ = , Γ = 4 8 5 8 6 144 8 48

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 72 / 76 Meson masses at NLO – 2

• Self-energy contains two parts: π, K, η tadpole loops and counterterms in L(4)

Assuming isospin symmetry (mu = md =ˆm), the pion mass at NLO:  I(M 2) I(M 2 ) 8M 2 24M 2  M 2 = M 2 1 + − η,2 + (2L − L ) + η,2 (2L − L ) π 2F 2 2F 2 F 2 8 5 F 2 6 4

2 2 2 here M = 2Bm,ˆ Mη,2 = 3 B(2m ˆ + ms) • Renormalization: divergences from LECs and loops cancel out (as well as the µ-dependence) ⇒ the pion mass is finite and µ-independent

 1 M 2 1 M 2 M 2 = M 2 1 + M 2 log − M 2 log η,2 π 32π2F 2 µ2 96π2F 2 η,2 µ2 8M 2 24M 2  + (2Lr − Lr) + η,2 (2Lr − Lr) F 2 8 5 F 2 6 4

2 • Mπ vanishes in chiral limit, loop correction does not generate a non-zero mass • Chiral logarithm: non-analytic in the light quark masses

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 72 / 76 Quark mass ratios revisited – 1

• Calculate pion/kaon masses beyond leading order: 2 h i Mπ+ = B(mu + md) 1 + O (m, ˆ ms)

2 h i MK+ = B(mu + ms) 1 + O (m, ˆ ms) • Form dimensionless ratios: 2 MK ms +m ˆ h 2
i 2 = 1 +∆ M + O mq Mπ mu + md 2 2 (MK0 − MK+ )strong md − mu h 2
i 2 2 = 1 +∆ M + O mq MK − Mπ ms − mˆ 2 2 8(MK − Mπ) r r ∆M = 2 (2L8 − L5) + chiral logs Fπ

• Double ratio Q2 particularly stable:

2 2 2 2 2 2 ms − mˆ MK MK − Mπ h 2
i Q = 2 2 = 2 2 2 1 + O mq) md − mu Mπ (MK0 − MK+ )strong

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 73 / 76 Quark mass ratios revisited – 1

• Calculate pion/kaon masses beyond leading order: 2 h i Mπ+ = B(mu + md) 1 + O (m, ˆ ms)

2 h i MK+ = B(mu + ms) 1 + O (m, ˆ ms) • Form dimensionless ratios: 2 MK ms +m ˆ h 2
i 2 = 1 +∆ M + O mq Mπ mu + md 2 2 (MK0 − MK+ )strong md − mu h 2
i 2 2 = 1 +∆ M + O mq MK − Mπ ms − mˆ 2 2 8(MK − Mπ) r r ∆M = 2 (2L8 − L5) + chiral logs Fπ

• Double ratio Q2 particularly stable:

2 2 2 2 2 2 ms − mˆ MK MK − Mπ h 2
i Q = 2 2 = 2 2 2 1 + O mq) md − mu Mπ (MK0 − MK+ )strong

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 73 / 76 Quark mass ratios revisited – 2

• Leutwyler’s ellipse: Leutwyler (1996)

 2  2 mu 1 ms + 2 = 1 md Q md

• Remember Dashen’s theorem:

2 2 2 2 2
(MK+ − MK0 )em = (Mπ+ − Mπ0 )em + O e mq

QDashen = 24.2

• Value extracted from lattice simulations FLAG (2013)

Nf = 2 + 1: 22.6 ± 0.7 ± 0.6

Nf = 2: 24.3 ± 1.4 ± 0.6

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 74 / 76 ππ scattering beyond LO

• S-wave scattering lengths:

0 2 a0 a0

LO 0.16 −0.045 Weinberg (1966)

NLO (one-loop) 0.20 ± 0.01 −0.042 ± 0.002 Gasser, Leutwyler (1983)

NNLO (two-loop) 0.217 ± ... −0.041 ± ... Bijnens et al. (1996)

NNLO + Roy eq. 0.220 ± 0.005 −0.0444 ± 0.0010 Colangelo et al. (2001)

• Compare again with the modern data from NA48/2 ± 0 0 ± (Ke4 decays & K → π π π ) B.Bloch-Devaux, Kaon09

0 a0 = 0.2210 ± 0.0047stat ± 0.0015syst 2 a0 = −0.0429 ± 0.0044stat ± 0.0016syst

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 75 / 76 Summary – 3

• Symmetries allow terms with more derivatives ⇒ non-renormalizable in the traditional sense • EFT has a power counting to organize terms in the Lagrangian. Lagrangian contains the most general terms allowed by symmetries at a given order, thus EFT can be renormalized order by order

• Scale separation: p, M  Λχ ∼ 1 GeV ⇒ allows for an uncertainty estimate ⇒ chiral Lagrangians should not be naively used for p & Λχ

Feng-Kun Guo (ITP) EFT for Hadron Physics 03. 2016 76 / 76