Baryon Chiral Perturbation Theory

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Baryon Chiral Perturbation Theory

Bastian Kubis

Helmholtz-Institut f ¨ur Strahlen- und Kernphysik (Theorie) Universit ¨at Bonn, Germany

School on Flavour Physics, Benasque July 13–25, 2008

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Outline (1)

Part I: Basics

• construction of the meson-baryon Lagrangian

• power counting and its failure

• heavy-baryon ChPT and infrared regularisation

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Outline (1)

Part I: Basics

• construction of the meson-baryon Lagrangian

• power counting and its failure

• heavy-baryon ChPT and infrared regularisation

Part II: _{πN σ}-term and strangeness in the nucleon (1)

• quark mass dependence of the nucleon mass

• sigma term and πN scattering

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Outline (2)

Part III: Strangeness in the nucleon (2)

• Strangeness form factors and parity-violating _e⁻_p scattering

• Chiral perturbation theory: a failure

Part IV: Isospin-violating form factors

• Why? isospin violation and strangeness

• Chiral perturbation theory: a success

Part VI: Crimes and omissions

• The role of the ∆(1232) resonance

• Two- and more-nucleon systems

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Questions, anyone?

I reserve the use of irony without warning.

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Part I: Basics

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Chiral perturbation theory with baryons

• H. Leutwyler’s lectures: ChPT for (pseudo) Goldstone bosons

• perhaps the most important extension: inclusion of

nucleons [chiral SU(2)] / the baryon ground state octet [SU(3)]

• objective: understand the properties of nucleons

• long-term goal: put nuclear physics on a more systematic footing

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Chiral perturbation theory with baryons

• H. Leutwyler’s lectures: ChPT for (pseudo) Goldstone bosons

• perhaps the most important extension: inclusion of

nucleons [chiral SU(2)] / the baryon ground state octet [SU(3)]

• objective: understand the properties of nucleons

• long-term goal: put nuclear physics on a more systematic footing

• problem (as we shall see later):

nucleon mass is a new, heavy mass scale that does not vanish in the chiral limit,

mlimq→0 m_N ≈ m_N

• idea: view nucleons as (massive) matter fields coupled to pions and external sources

3-momenta ought to remain small ∼ M_π

• number of baryons conserved

no baryon–antibaryon creation / annihilation here: consider only processes with 1 baryon

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Construction of the meson–baryon Lagrangian (1)

• proceed as before:

⊲ choose suitable representation / transformation law for baryons under _SU(N₎_L _× _SU(N₎_R

⊲ construct effective Lagrangians according to increasing number of momenta

• remember: _U _7−→ _V_L_{U V}_R^† for the Goldstone boson fields

• introduce a new field _u according to _u² ₌ _U u 7−→

q

V_LU V_R^† = V_L uK^†(V_L, V_R, U)

= K(V_L, V_R, U)u V_R^†

K(V_L, V_R, U) ∈ SU(N) compensator field

• for SU_(N₎_V transformations (_V_L ₌ _V_R), reduces to K(V_L, V_R, U) = V_L = V_R

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Construction of the meson–baryon Lagrangian (2)

• particularly convenient representation for nucleons / baryons:

ψ =

p n

7−→ ψ^′ = Kψ [SU(2)]

B =





Σ⁰

√2 + ^√^Λ

6 Σ⁺ p

Σ⁻ −^√^Σ⁰

2 + ^√^Λ

6 n

Ξ⁻ Ξ⁰ −^√^2Λ

6



 7−→ B^′ = KBK⁻¹ [SU(3)]

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Construction of the meson–baryon Lagrangian (2)

ψ 7−→ ψ^′ = Kψ [SU(2)]

B 7−→ B^′ = KBK⁻¹ [SU(3)]

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Construction of the meson–baryon Lagrangian (2)

ψ 7−→ ψ^′ = Kψ [SU(2)]

B 7−→ B^′ = KBK⁻¹ [SU(3)]

• introduce a covariant derivative:

D^µ = ∂^µ + Γ^µ = ∂^µ + V ^µ Γ^µ chiral connection (vector)

Γ^µ = 1

2 u^†(∂^µ − i r^µ)u + u(∂^µ − i l^µ)u^† transforms according to

Γ^µ 7−→ KΓ^µK^† − (∂^µK)K^† such that

D^µψ 7−→ KD^µψ

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Construction of the meson–baryon Lagrangian (3)

• chiral vielbein (axial vector)

u^µ = i u^†(∂^µ − i r^µ)u − u(∂^µ − i l^µ)u^† transforms according to

u^µ 7−→ Ku^µK^†

• finally, rewrite quark mass term _χ _{= 2B(s} ₊ _{i p) = 2BM} ₊ _{. . .} χ₊ = u^†χu^† + uχ^†u

such that

χ₊ 7−→ Kχ₊K^†

⇒ everything transforms in the same way

• power counting: _Γ^µ_{, u}^µ ₌ _O(p), _χ₊ ₌ _O(p²₎

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The leading-order chiral meson-baryon Lagrangian

L⁽²⁾_ππ = F²

4 hu_µu^µ + χ₊i = F²

4 hD_µU D^µU^† + χU^† + U χ^†i

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L⁽²⁾_ππ = F²

4 hD_µU D^µU^† + χU^† + U χ^†i L⁽¹⁾_πN = ¯ψ

iγ_µD^µ − m + g_A

2 γ_µγ₅u^µ ψ L⁽¹⁾_φB = hB¯ (iγ_µD^µ − m)Bi

+ D

2 hBγ¯ _µγ₅{u^µ, B}i + F

2 hBγ¯ _µγ₅[u^µ, B]i

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The leading-order chiral meson-baryon Lagrangian

L⁽²⁾_ππ = F²

+ D

2 hBγ¯ _µγ₅[u^µ, B]i Remarks:

• iγ_µD^µ, m = O(p⁰) both individually large, but _iγ_µ_D^µ ₋ _m ₌ _O(p)

• in meson ChPT, Lagrangians came in even powers of momenta (Lorentz invariance)

• here, due to spin (Dirac structures), odd powers possible:

L_πN = L⁽¹⁾_πN + L⁽²⁾_πN + L⁽³⁾_πN + L⁽⁴⁾_πN + . . .

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L⁽²⁾_ππ = F²

+ D

2 hBγ¯ _µγ₅[u^µ, B]i new parameters:

• m: nucleon (baryon) mass in the chiral limit

• g_A: expand u_µ = 2a_µ + O(π) ⇒ axial vector coupling known from neutron beta decay, g_A = 1.26

• D/F: two axial vector couplings in SU(3), can be determined from semileptonic hyperon decays, SU(2) constraint D + F = g_A (_D _≈ _{0.80, F} _≈ _0.46)

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Goldberger-Treiman relation

• setting _v_µ ₌ _a_µ _{= 0}, the chiral vielbein is _u_µ ₌ ₋^∂_F^µ^π

π + O(π³) find the πN N vertex

L⁽¹⁾_πN → − g_A

2F_π ψγ¯ ^µγ₅∂_µπψ

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Goldberger-Treiman relation

L⁽¹⁾_πN → − g_A

• Feynman rule: ^q ^g^A

2F_π 6qγ₅~τ = V_{πN N}

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Goldberger-Treiman relation

L⁽¹⁾_πN → − g_A

• N → πN transition amplitude:

T_{πN N} = −i¯u(p^′)V_{πN N}u(p) = −ig_Am_N

F_π u(p¯ ^′)γ₅u(p)~τ

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Goldberger-Treiman relation

L⁽¹⁾_πN → − g_A

• N → πN transition amplitude:

T_{πN N} = −i¯u(p^′)V_{πN N}u(p) = −ig_Am_N

F_π u(p¯ ^′)γ₅u(p)~τ

• compared to the canonical amplitude ₋ _i_g_πN_u(p_¯ ^′_)γ₅_u(p)~τ : g_πN = g_Am_N

F_π

remarkable for relating weak and strong interactions numerically: 13.1 . . .13.4 = 12.8

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Some other familiar Feynman rules

L⁽¹⁾_πN = ¯ψ

2 γ_µγ₅u^µ

ψ , D^µ = ∂^µ + Γ^µ

• expand Γ_µ, u_µ in powers of the pion fields:

iΓ_µ = v_µ + i

8F² [π, ∂_µπ] − 1 8F²

π, [π, v_µ]

+ . . . u_µ = 2a_µ − ∂_µπ

F + i

2F [v_µ, π] + . . .

ie6ǫ 1

2(1 + τ³) γ–N coupling (electric)

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Some other familiar Feynman rules

L⁽¹⁾_πN = ¯ψ

2 γ_µγ₅u^µ

ψ , D^µ = ∂^µ + Γ^µ

iΓ_µ = v_µ + i

8F² [π, ∂_µπ] − 1 8F²

π, [π, v_µ]

+ . . . u_µ = 2a_µ − ∂_µπ

F + i

2F [v_µ, π] + . . . q₂, b

q₁, a

1

4F² (6q₁+6q₂) ǫ^abcτ^c Weinberg–Tomozawa term

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L⁽¹⁾_πN = ¯ψ

2 γ_µγ₅u^µ

ψ , D^µ = ∂^µ + Γ^µ

iΓ_µ = v_µ + i

8F² [π, ∂_µπ] − 1 8F²

π, [π, v_µ]

+ . . . u_µ = 2a_µ−∂_µπ

F + i

2F [v_µ, π] + . . .

q, a g_A

2F 6q γ₅ τ^a (axial vector) πN N coupling

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L⁽¹⁾_πN = ¯ψ

2 γ_µγ₅u^µ

ψ , D^µ = ∂^µ + Γ^µ

iΓ_µ = v_µ + i

8F² [π, ∂_µπ] − 1 8F²

π, [π, v_µ]

+ . . . u_µ = 2a_µ − ∂_µπ

F + i

2F [v_µ, π] + . . .

a ieg_A

2F 6ǫ γ₅ ǫ^a3bτ^b Kroll–Rudermann term

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Weinberg’s power counting argument for pions (1)

• consider an arbitrary loop diagram based on L_eff = X

d

L^(d)

with L loops, I internal lines, V_d vertices of order d: A ∝

Z

(d⁴p)^L 1 (p²)^I

Y

d

(p^d)^V^d

• let _A be of chiral dimension _ν

ν = 4L − 2I + X

d

dV_d

• use topological identity for L to eliminate I L = I − X

d

V_d + 1 ⇒ ν = X

d

V_d(d − 2) + 2L + 2

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Weinberg’s power counting argument for pions (2)

ν = X

d

V_d(d − 2) + 2L + 2

• chiral Lagrangian starts with L⁽²⁾, i.e. d ≥ 2,

i.e. right-hand-side is a sum of non-negative terms

⇒ for fixed ν, there is only a finite number of combinations L, V_d

• +2L: each loop suppressed amplitude by two orders in the momentum expansion

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Weinberg’s power counting for the 1-baryon sector

• similar derivation as for the meson sector:

chiral dimension _ν of an arbitrary _L-loop diagram with _V_d^ππ meson–meson vertices of order _d

and _V_d^πN_′ meson–baryon vertices of order _d^′:

ν = 2L + 1 + X

d

V_d^ππ(d − 2) + X

d^′

V_d^πN′ (d^′ − 1)

note again: d ≥ 2, d^′ ≥ 1

• therefore:

O(p¹), O(p²): tree only

O(p³), O(p⁴): tree + one-loop

O(p⁵), O(p⁶): tree + one-loop + two-loop . . .

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Weinberg’s power counting for the 1-baryon sector

• similar derivation as for the meson sector:

chiral dimension _ν of an arbitrary _L-loop diagram with _V_d^ππ meson–meson vertices of order _d

and _V_d^πN_′ meson–baryon vertices of order _d^′:

ν = 2L + 1 + X

d

V_d^ππ(d − 2) + X

d^′

V_d^πN′ (d^′ − 1)

note again: d ≥ 2, d^′ ≥ 1

• therefore:

O(p¹), O(p²): tree only

O(p³), O(p⁴): tree + one-loop

O(p⁵), O(p⁶): tree + one-loop + two-loop . . .

• and the problem is: it doesn’t work . . .

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The failure of power counting in meson–baryon ChPT

• loop integrals cover all energy scales

• Goldstone boson sector: all mass scales "small"

in a mass-independent regularisation scheme (like dimensional regularisation), naive power counting has to work

• with baryons: new mass scale m_N ≈ Λ_χ ≈ 1 GeV loop integration picks up momenta p ∼ m_N

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The failure of power counting in meson–baryon ChPT

• loop integrals cover all energy scales

• Goldstone boson sector: all mass scales "small"

in a mass-independent regularisation scheme (like dimensional regularisation), naive power counting has to work

• with baryons: new mass scale m_N ≈ Λ_χ ≈ 1 GeV loop integration picks up momenta p ∼ m_N

• schematically: Gasser, Sainio, Švarc 1988

0 1 2 3

ππ, # loops 0

2 4 6 8

D T_ππ ~ q^D

0 1 2 3

πN, # loops 0

2 4 6 8

D T_πN ~ q^D

⇒ higher-order loops renormalise lower-order couplings

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Remedies (1): Heavy-baryon ChPT

Jenkins, Manohar 1991 Bernard, Kaiser, Meißner 1995

• in close analogy to heavy-quark EFT:

decompose baryon momentum according to p_µ = m_Nv_µ

| {z }

large

+ l_µ

|{z}

residual

, v² = 1 , v · l ≪ m_N

• nucleon propagator in the heavy-baryon limit:

1

p² − m²_N → 1 2m_N

1

v · l + O 1/m²_N

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Remedies (1): Heavy-baryon ChPT

| {z }

large

+ l_µ

|{z}

residual

, v² = 1 , v · l ≪ m_N

= + + + . . .

1/m_N 1/m_N 1/m_N

(34)

Remedies (1): Heavy-baryon ChPT

| {z }

large

+ l_µ

|{z}

residual

, v² = 1 , v · l ≪ m_N

1

p² − m²_N → 1 2m_N

1

v · l + O 1/m²_N

• eliminates mass scale m_N from propagator re-enters as parametrical suppression factor

• two-fold expansion

p Λ_χ

n

,

p m_N

n

(_m_N _≈ _Λ_χ)

(35)

Heavy-baryon ChPT (2)

• decompose _ψ into velocity eigenstates:

"big": H_v(x) = e^im^N^v^·^xP_v⁺ψ(x)

"small": h_v(x) = e^im^N^v^·^xP_v⁻ψ(x)

• projectors P_v^± = ¹₂(1± 6v)

• exponential "rotates away" the large mass term from time evolution of the field _H_v

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Heavy-baryon ChPT (2)

• decompose _ψ into velocity eigenstates:

"big": H_v(x) = e^im^N^v^·^xP_v⁺ψ(x)

"small": h_v(x) = e^im^N^v^·^xP_v⁻ψ(x)

• projectors P_v^± = ¹₂(1± 6v)

• exponential "rotates away" the large mass term from time evolution of the field _H_v

• L⁽¹⁾_πN becomes:

L⁽¹⁾_πN = ¯H_v iv · D + g_AS · u

H_v + O 1/m_N Pauli-Lubanski spin vector S_µ = ₂ⁱγ₅σ_µνv^ν

⇒ nucleon mass gone from _L⁽¹⁾_πN

⇒ Dirac structure massively simplified

• 1/m_N corrections can be constructed systematically on Lagrangian level à la Foldy–Wouthuysen

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Remedies (2): Infrared regularisation

Becher, Leutwyler 1999

consider (relativistic) nucleon self-energy graph (at threshold):

= Γ(2 − ^d₂) (4π)^d/2(d − 3)

m^d_N⁻³ + M_π^d⁻³ m_N + M_π

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Remedies (2): Infrared regularisation

Becher, Leutwyler 1999

consider (relativistic) nucleon self-energy graph (at threshold):

= Γ(2 − ^d₂) (4π)^d/2(d − 3)

m^d_N⁻³ + M_π^d⁻³ m_N + M_π

"regular" part

• fractional powers in m_N, regular in M_π, p

• violates naive power counting

• can be expanded as polynomial in M_π, p

⇒ can be absorbed by re- definition of contact terms

"infrared" part

• fractional powers in M_π, p

• obeys power counting rules

• non-analytic terms, imaginary parts . . .

⇒ all "interesting" loop contributions

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Infrared regularisation (2)

P k

P − k P

a = M_π² − k² − iǫ , b = m²_N − (P − k)² − iǫ

H = 1 i

Z d^dk (2π)^d

1 ab =

Z 1 0

dz1 i

Z d^dk (2π)^d

1

[(1 − z)a + zb]²

• Landau equations: singularity structure z = _m ^M^π

N+Mπ leading (pinch) singularity ↔ P² = (m_N + M_π)² z = 0 endpoint singularity _↔ _M_π² _{= 0}

z = 1 endpoint singularity _↔ _m²_N _{= 0}

• z = 1 not a "low-energy" singularity

• obtain infrared part by avoiding the z = 1 endpoint singularity:

IR =

Z ^∞

0

dz1 i

Z d^dk (2π)^d

1

[(1 − z)a + zb]²

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Relation infrared regularisation ↔ heavy-baryon

• IR = + + + . . .

⇒ resummation of all 1/m_N corrections of a certain diagram

(41)

Relation infrared regularisation ↔ heavy-baryon

• IR = + + + . . .

• wait — isn’t this just what we did for the heavy-baryon propagator in the first place?

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Relation infrared regularisation ↔ heavy-baryon

• IR = + + + . . .

• wait — isn’t this just what we did for the heavy-baryon propagator in the first place?

• yes —

but we interchanged summation and (highly irregular) integration the difference is the "regular" part

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Part II: πN σ -term and

strangeness in the nucleon (1)

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Quark mass dependence of the nucleon mass

• calculate m_N up to O(p³):

L⁽²⁾_πN = c₁ψhχ¯ ₊iψ + . . . (6 more terms) + one-pion loop:

+

m_N = m − 4c₁M_π² − 3g_A² M_π³

32πF_π² + O(M_π⁴)

• pion loop yields non-analytic term _∝ _M_π³ _∝ _m_ˆ ^3/2

• leading correction term comes with unknown low-energy constant _c₁

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The πN σ -term

• define scalar form factor of the nucleon:

hN(p^′)|m(¯ˆ uu + ¯dd)|N(p)i = σ(t)u(p¯ ^′)u(p) σ ≡ σ(0) = mˆ

2m_N hN|¯uu + ¯dd|Ni

(46)

The πN σ -term

• Feynman–Hellman theorem:

σ = ˆm∂m_N

∂mˆ = −4c₁M_π² − 9g_A² M_π³

64πF_π² + O(M_π⁴)

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The πN σ -term

• relate _σ to strangeness content of the nucleon!

σ = mˆ 2m_N

hN|¯uu + ¯dd − 2¯ss|Ni

1 − y , y = 2hN|¯ss|Ni hN|uu¯ + ¯dd|Ni now (m_s − m)(¯ˆ uu + ¯dd − 2¯ss) is the part of L_QCD that produces SU(3) mass splittings:

σ = σˆ

1 − y , σˆ = mˆ

m_s − mˆ m_Ξ + m_Σ − 2m_N

≃ 26 MeV higher-order corrections: _σ_ˆ _→ ₍₃₆ _± ₇₎ MeV

• if we know σ, we know y ⇒ strangeness in the nucleon

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The σ -term and πN scattering (1)

• remember: learnt about M_π² from ππ scattering

• isospin even / odd _πN scattering amplitudes:

T_πN^± = 1 2

T(π⁻p → π⁻p) ± T(π⁺p → π⁺p)

• decompose _T^± further (spin-flip / non-flip amplitudes . . . )

⇒ consider specific combinations D^¯^±

• show in Chiral Perturbation Theory:

Σ ≡ F_π²D¯⁺ s = u = m²_N, t = 2M_π²

| {z }

"Cheng-Dashen point"

= σ(2M_π²) + ∆_R

∆_R = O(M_π⁴) is very small, _∆_R . 2 MeV

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The σ -term and πN scattering (2)

Procedure:

πN scattering _−→⁽¹⁾ _Σ ₌ _F_π²_D^¯_CD⁺ _−→⁽²⁾ _σ(2M_π²₎ _−→⁽³⁾ _σ(0) _−→⁽⁴⁾ _y

(50)

The σ -term and πN scattering (2)

Procedure:

πN scattering _−→⁽¹⁾ _Σ ₌ _F_π²_D^¯_CD⁺ _−→⁽²⁾ _σ(2M_π²₎ _−→⁽³⁾ _σ(0) _−→⁽⁴⁾ _y (1)

u^-channel s^-channel

⁼ ⁰ s ⁼ ⁽m⁺M⁾² u⁼ ⁽m⁺M⁾²

t⁼ ⁴M²

*

threshold CD−point

• Cheng–Dashen point:

s = u = m²_N, t = 2M_π²

• physical region:

s ≥ (m_N + M_π)², _t _≤ ₀

• extrapolate using dis- persion relations

Σ = 60 ± 7 MeV

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Procedure:

u^-channel s^-channel

⁼ ⁰ s ⁼ ⁽m⁺M⁾² u⁼ ⁽m⁺M⁾²

t⁼ ⁴M²

*

• Cheng–Dashen point:

s = u = m²_N, t = 2M_π²

• physical region:

s ≥ (m_N + M_π)², _t _≤ ₀

• extrapolate using dis- persion relations

Σ = 60 ± 7 MeV

(2) Σ = σ(2M_π²) + ∆_R , ∆_R ≤ 2 MeV → okay

(52)

The σ -term and πN scattering (2)

Procedure:

• σ(2M_π²) = σ(0) + ∆σ

⇒ understand the t-dependence of the scalar form factor σ(t)!

• crude estimate:

⊲ σ(t) = σ(0)n

1 + 1

6hr²i_σ t + . . .o

⊲ assume _hr²_i_σ _{≃ hr}²_i_EM _{= 0.8} fm² ⇒ ∆σ ≈ 3.5 MeV

• one-loop ChPT: _∆σ _{= 4.6} MeV

σ ≈ 55 MeV

(53)

Procedure:

• y = 1 − σˆ

σ = 1 − 35 MeV

55 MeV ≈ 0.4 !!

• "σ term puzzle":

300 MeV of nucleon mass due to strange quarks??

(54)

The σ -term and πN scattering (2)

Procedure:

• scalar form factor beyond one loop:

isoscalar _s-wave _ππ scattering _⇒ strong!

• dispersive analysis, _hr²_i_σ _≈ _2hr²_i_EM large curvature _⇒ _∆σ _≈ ₁₅ MeV

σ ≈ 45 MeV

Gasser, Leutwyler, Sainio 1991

(55)

Procedure:

• y = 1 − σˆ

σ = 1 − 35 MeV

45 MeV ≈ 0.2

• strangeness in the nucleon sizeable, but not outrageous hN|m_sss|N¯ i ≃ 130 MeV

(56)

πN scattering lengths

u

^-channel

s

^-channel

⁼ ⁰

s

⁼ ⁽

m

⁺

M

⁾²

u

⁼ ⁽

m

⁺

M

⁾²

t

⁼ ⁴

M

²

*

Büttiker, Meißner 1999

(57)

πN scattering lengths in ChPT

• at threshold: _πN scattering lengths a^± = 1

4π(1 + M_π/m_N)T^± s = (m_N + M_π)²

• in ChPT:

a⁻ = M_π

8π(1 + M_π/m_N)F_π² + O M_π³

a⁺ = 0 + M_π²(−g_A² + 8m_N(−2c₁ + c₂ + c₃))

16πm_N(1 + M_π/m_N)F_π² + O M_π³

• a⁻ = 8.0 × 10⁻²M_π⁻¹ + small corrections

• a⁺ vanishes at leading order, several LECs, bad convergence

⇒ the helpful guy for the σ term barely known

⇒ chiral series very different for a^±

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πN scattering lengths from pionic atoms (1)

• π⁻p, _π⁻_d systems, bound by electromagnetism

calculate energy levels as in quantum mechanics for the hydrogen atom!

• energy levels perturbed by strong interactions:

ground state not stable, decays: A_π⁻_p → π⁰n, γn, . . .

• ground state strong energy level shift ∆E^str and width Γ related to _πN scattering amplitudes at threshold:

∆E^str = − 2α³µ²_cA(π⁻p → π⁻p) + . . . ∝ a⁺ + a⁻ + . . . Γ = 8α³µ²_cp^∗|A(π⁻p → π⁰n)|² + . . . ∝ − a⁻ + . . . µ_c: reduced mass, _p^∗: CMS-momentum of _π⁰

Deser, Goldberg, Baumann, Thirring 1954 Gasser, Lyubovitskij, Rusetsky 2008

• π⁻d: additional information due to different isospin combinations

(59)

πN scattering lengths from pionic atoms (2)

from pionic hydrogen and deuterium measurements: PSI 1995-2006

-0.1 -0.09 -0.08

-0.01 0 0.01

Deuteron, isospin breaking at O(p ) 2

Hydrogen energy, isospin breaking at O(p ) 2

Hydrogen width, isospin breaking at O(p )2

- a (M )^- _π^-1 a (M )+ π-1

a⁻ = (8.52 ± 0.18) × 10⁻²M_π⁻¹ a⁺ = (0.15 ± 0.22) × 10⁻²M_π⁻¹

Meißner, Raha, Rusetsky 2006

a⁺ very small, quite sensitive to isospin breaking corrections

(60)

Spares

(61)

Heavy-baryon vs. infrared regularisation (1)

• consider the electromagnetic form factors of the nucleon:

hN(p^′)|J_µ^em|N(p)i = eu(p¯ ^′)

γ_µF₁^N(t) + iσ_µνq^ν

2m_N F₂^N(t)

u(p)

• important contribution to the spectral function _Im_F₁^v_(t) [F₁^v(t) = F₁^p(t) − F₁ⁿ(t)] from so-called triangle graph:

• "normal" threshold at t = 4M_π²

• anomalous threshold at t = 4M_π² − ^M

4 π

m²_N

HB= 4M_π² + O(M_π⁴)

⇒ analytic structure distorted

ImF₁^v(t) =^IR g_A²

192πF_π²(4m²_N − M_π²)

1 − 4M_π² t

3/2

+ . . . ^p-wave

ImF₁^v(t) ^HB= g_A²

96πF² (5t − 8M_π²)

1 − 4M_π² t

^1/2

+ . . . ^wrong!

(62)

Heavy-baryon vs. infrared regularisation (2)

• resummation of nucleon "recoil effects"

(⇒ correct relativistic kinematics) sometimes helps:

Gⁿ_E(t) = F₁ⁿ(t) + t

4m²_N F₂ⁿ(t) , Q² = −t

0 0.1 0.2 0.3 0.4

Q² [GeV²]

−0.05 0 0.05 0.1 0.15

G E

n (Q2 )

blue: O(q³) red: _O(q⁴₎ full: IR

dashed: HB

(dot-dashed: disp.th.)

BK, Meißner 2000