Kadyshevsky Field Theory Relativistic πN, KN, and BB ...II. Kadyshevsky Formalism (KFT) , S-matrix,...

Kadyshevsky Field Theory

Relativistic πN, KN, and BB-interactions

Workshop NFQCD10, Yukawa Institute, Kyoto 2010

Th.A. Rijken

IMAPP, University of NijmegenThe Netherlands

University of Nijmegen, february 2010

– p. 1/103

Introduction

Contents: Topics Meson/baryon-Baryon Formalism

I. Flavor Nuclear Physics,QCD and MB- and BB-models

II. Kadyshevsky Formalism (KFT) ,S-matrix, Graphs and RulesSecond-quantization 4-momentum Quasi-particlesKadyshevsky Integral-equation

III. Jackiw-Gross R-product analysis:Analysis Frame-dependence in KFT, Derivative Int.’sElimination Non-causal and Contact termsEx: PV-, V-, ∆33Nπ-coupling

IV. BB̄-Pair-suppression:Momentum-space Vertex-descriptionLagrangian-description, Example

V. Takahashi-Umezawa (TU) Theory:S-matrix formulation TU-theoryExamples

VI. Discussion, Conclusion and Prospects

Th.A. Rijken University of Nijmegen NFQCD 2010: Kadyshevsky Formalism – p. 2/103

Particle and Flavor Nuclear PhysicsParticle and Flavor Nuclear Physics

Baryons: Quark flavors: u,d,s,c,b,t

Σ− Σ+

Ξ− Ξ0

n p

Σ0c Σ++c

Ξ+cc Ξ++cc

Ω+cc

Ω0c

Λ,Σ0

Λ+c ,Σ+c

Ξ0c Ξ+c

Qua

rks

Baryons

Flavornuclei

Aspects of flavored matter:

H.Bandō(1985) (paraphrase):”Extension NP to hyper/flavor NPis important/desirable to understandthe different aspects of Matter


Introduction: Modern Physics and QCDContemporary View on Low Energy Physics

Baryon-baryon Strong Interactions

• Concepts:

QCD: Colored quarks + gluons

Confinement SUc(3)

Strong coupling gQCD ≈ 1Lattice QCD: flux-tubes/strings

Flavor SU(N)-symmetry

Spontaneous CSB

Experiments:

πN- & KN-scattering,

NN- & YN-scattering,

Nuclei- & Hypernuclei-

production and decay,

Nuclear- & Hyperonic & Star

matter observation

Principle: "Experientia ac ratione"

(Christiaan Huijgens 1629-1695)

MB,BB-interaction models


QCD-world IQCD-world I: mesons and baryons

u d sc b t

gColored Quarks and Gluons⇔

⇐= Out-integration c,b,t

u d sg

EFT I, scale Λ >> 1 GeV/c2

weak-coupling g

QCD-world IIQCD-world II: Baryon/Meson-baryon Interactions

baryons,mesons

quarks, gluons

N,Λ,Σ,Ξπ,K, η, η′

Goldstone-boson exch.+ contact-termsChiral Pert. Models:Van Kolck, Epelbaum,Bonn-Jülich, Barcelona, etc.

Baryons,mesons,M ≤ 1.5 GeV/c2

Meson/baryon-exchangeNijmegen BB and MB models,Ehime, Jülich, etc.

Quark-ClusterRGM-approach

Quark-Gluon-+ OBE-exchangeTokyo, Kyoto-Niagatta,Tübingen, Salamanca,Nanjing, etc.

Parameters?

Physics output? cc?

Predictive power?


Gluon-Quark-ExchangeGluon-Quark-Exchange ⇔ Meson-Exchange

• Strong-coupling regime QQ-interaction: Multi-gluon exchange

P . . . . . . . . = PM

e.g.

a b

c d

=

a d

c b

⇔ Meson-exch.


Quark-Pair-Creation in QCDQuark-Pair-Creation in QCD ⇔ Flux-tube breaking

• Strong-coupling regime QQ-interaction: Multi-gluon exchange

Q

Q

Q̄

Q

QPC: 3P0-model:Micu, NP B10(1969);Carlitz & Kislinger, PR D2(1970),LeYaouanc et al, PR D8(1973).

QCD: Flux-tube/String-breaking⇒3 P0(QQ̄),

Isgur & Paton, PRD31(1985);Kokoski & Isgur, PRD35(1987)

⇒ Meson coupling constants !


Gluon-exchange ⇔ PomeronGluon-exchange QCD ⇔ Pomeron-exchange

• Gluon-exchange⇔ Pomeron-exchangeMultiple-gluon model: Low PR D12(1975),Nussinov PRL34(1975)Scalar Gluon-condensate: ITEP-school:〈0|g2Gaµν(0)Gaµν(0)|0〉 = Λ4c ,Λc ≈ 800 MeVLandshoff, Nachtmann, Donnachie,Z.Phys.C35(1987); NP B311(1988):〈0|g2T [Gaµν(x)Gaµν(0)]|0〉 =Λ4cf(x

2/a2), a ≈ 0.2− 0.3fmQCD-vacuuum: Copenhagen picture,Ambjorn & Olesen, NP B170(1980)

⇒ LE Pomeron-potential!

Quark-core (Pauli) repulsionOtsuki, Tamagaki & Wada, 1965.Oka, Shimizu & Yazaki, 1985. Etc.


Universal Three-body repulsion ⇔ PomeronUniversal Three-body repulsion ⇔ Pomeron-exchange

• Multiple Gluon-exchange⇔ Pomeron-exchange

Soft-core BB-models NSC97, ESC04:(i) nuclear saturation, (ii) EOS too softNishizaki,Takatsuka,Yamamoto,PTP 105(2001); ibid 108(2002): NTY-conjecture = universal repulsion in BB

Lagaris-Pandharipande NP A359(1981):medium effect→ TNIA,TNIRRijken-Yamamoto PRC73: TNR⇔ mV (ρ)

NTYR: Multiple-gluon-exchange↔Pomeron + Triple-pomeron!?


Introduction, AcknowledgementMeson-Baryon Soft-core Interactions

• Publications Kadyshevky theory, J-W.Wagenaar & Th.A.R:1. Pion-nucleon Scattering in Kadyshevsky Formalism:I. Meson Exchange Sector, Phys.Rev.C80:055204 (2009),2. Pion-nucleon Scattering in Kadyshevsky Formalism:II. Baryon Exchange Sector, Phys.Rev.C80:055205 (2009),3. Topics in Kadyshevsky Field Theory, e-Print: www.nn-online.org

• Soft-core Meson-baryon Model 2005:1. Dynamics: Meson- and Baryon-exchange2. Fit πN - and KN-scattering data3. Analysis Coupling-Constants: QPC, QCD-SR’s

• Publications SC-model:I, Pion-nucleon Interactions,

Polinder & Rijken, Phys.Rev. C72, 065210 (2005)II, Kaon-nucleon Interactions,

Polinder & Rijken, Phys.Rev. C72, 065211 (2005)—————————————————————————————–• Acknowledgement:

I am grateful for the important and stimulating collaborationwith my colleagues in Japan, prof.’s Y. Yamamoto, T. Motoba, and E. Hiyama.


IntroductionFlow: On the Covariantization Nijmegen Models

Nijmegen NN,YN,YY, and πN, KN models:(i) suppression E0↔ E

Introduction

Treatment Derivative Interactions:

(i) Removal frame-dependence and non-covariant terms(ii) Examples: pv-coupling πNN , gauge-invariant coupling ∆33Nπ

m

Application Gross-Jackiw technique, Nucl.Phys. B14 (1969)⇒ Corrections to the amplitudes.

m

Translation amplitude corrections into ∆Hint

m

Connection: TU= Takahashi-Umezawa Interaction Theory (1953):General Procedure Lint → HintS-matrix formulation with Tomonaga-Schwinger eq. for U-operatorTU-constructions not liable to Haag’s theorem (1955)!


IntroductionKFT: Developed Modern Theoretical Tools

Functional formulation KFT:Generating functional Z(J, sources) for KFT Wightman-functionsNeeded: Lagrangian for physical and quasi-particles!

m

Second-quantization: quasi-particle fields

⇓

Reduction formulas for Kadyshevky Theory

⊕Generating functional Z(J, η, η†) for generalized Wightman-functions(Green-functions = linear combinations Wightman-functions)Derivation Schwinger-Symanzik equation

⊕

Path-integral formulation Kadyshevsky Theory


IntroductionMotivation/goals for the Kadyshevsky Formalism:

(i) Manifest covariant formulation Nijmegen BB- and MB-models.(ii) Nijmegen models: work hypotheses = pair-suppression in

intermediate states, three-dimensional integral-equations.(iii) Relativistic few/many-body equations: no negative-energy intermediate states.

The usefulness of the Kadyshevsky formalism for our purpuses is twofold:(a) Particles, in particularly baryons, stay on-mass-shell. This makes it possibleto control pair-suppression in arbitrary complicated graph’s. This feature made itpossible to construct a covariant interaction theory where for matrix elementsthe positive-negative energy transitions are (absolutely) suppressed.(b) This formalism leads to 3-dimensional integral equations which partly justifiesand explains the succes of such equations, e.g. the Thompson-equation,in phenomenological low-energy hadron-scattering models(Polinder, thesis 2004, Polinder & Rijken, Phys.Rev. C 2005).(c) In this formalism we can treat the higher spin particles,both for mesons (J=1,2) and baryons (J=3/2).(d) Kadyshevsky-Field-Theory is covariant and equivalent to Feynman formalism.


Introductory remarksAdditional background information:We have developed a πN -model that is based on the formulation ofRelativistic Quantum Field Theory as developed by Kadyshevsky.Here, e.g. a formulation is given with absolute baryon-antibaryonpair-suppression in a Lorentz-invariant way.In contrast to the usual with Feynman graphs, the particlesin the Kadyshevsky-graphs remain on-mass-shell.This is exploited by introducing phenomenological vertex form factorswhich for the matrix-elements suppress the transitions between thepositive and the negative energy solutions in a covariant way.These kind of form factors are easily handled in the Kadyshevsky-formalism,and can rigourously be shown to be effective. This is impossible in theusual treatment using Feynman graphs. Therefore, pair-suppression can beintroduced phenomenologically and covariantly, and is accessible foran analysis using a fit to the meson-nucleon data.We applied these ideas to the πN -system and demonstrated explicitlythe covariance, frame-independence, crossing-symmetry,charge-conjugation invariance, and causal properties of the system,even with ’absolute pair-suppression’.We also examined the Kadyshevsky-integral-equation and showed that thesolutions are frame-independent etc. when the interaction-kernel is.


Kadyshevsky-formalism I

Kadyshevsky S-matrix theory

• Definition S-matrix: S = 1 + PSn, with

Sn =

„−i~

«n Z +∞

−∞

...

Z +∞

−∞

θ(x0n − x0n−1...θ(x02 − x01) ·

×HI(xn)HI(xn−1)...HI(x1) d4xn...d4x1 (A) .

• Kadyshevsky: Introduce nµ with n2 = n20 − n2 = 1, n0 > 0. ThenSn =

„−i~

«n Z +∞

−∞

...

Z +∞

−∞

θ[n · (xn − xn−1)]...θ[n · (x2 − x1)] ·

×HI(xn)HI(xn−1...HI(x1) d4xn...d4x1 (B) .

• Since (B)=(A) Kadyshevsky formalism S-matrix = Feynman s-matrix.


Kadyshevsky-formalism I

II

IV

IV

IIII

III

III

x0 − y0

|x− y|

nµ


Kadyshevsky-formalism I• Kadyshevsky eT -products and Wick’s Theorem

We treat the free scalar fields. The Kadyshevsky eT -product we define aseT [φ(x)φ(y)] = θ[n · (x− y)] φ(x)φ(y) + θ[n · (y − x)] φ(y)φ(x) . (1)

We have the obvious identityeT [φ(x)φ(y)] = φ(y)φ(x) + θ[n · (x− y)] [φ(x), φ(y)] , (2)

and a similar expression for the ordinary T-product T [φ(x)φ(y)].For the difference we obtain

eT [φ(x)φ(y)]− T [φ(x)φ(y)] = +˘θ[n · (x− y)]− θ(x0 − y0)

¯[φ(x), φ(y)] =

˘θ[n · (x− y)]− θ(x0 − y0)

¯∆(x− y;m2) . (3)

In Fig. Minkowski the areas I and II are respectively the forward and backwardlight-cone, where (x− y)2 > 0. In the areas III and IV the distancesare space-like, i.e. (x− y)2 < 0.


Kadyshevsky-formalism INow, as seen in Fig. Minkowski in the arched area light-cone,

˘θ[n · (x− y)]− θ(x0 − y0)

¯6= 0 , (4)

which is outside the light-cone, where (x− y)2 < 0. But, for this region∆(x− y;m2) = 0. Therefore,

eT [φ(x)φ(y)] = T [φ(x)φ(y)] . (5)

For the interaction Lagrangian micro-causality reads

[LI(x),LI(y)] = 0 , for (x− y)2 < 0 . (6)

Therefore, from the above we can infer immediately thateT [LI(x)LI(y)] = T [LI(x)LI(y)] , (7)

a result that can be generalized immediately to a T-product of any number ofinteraction Lagrangians, i.e.

eT [LI(x1) . . . LI(xn)] = T [LI(x1) . . . LI(xn)] . (8)

From this one can conclude to the complete equivalence of the S-matrix using thesimple ’non-relativistic’ θ-functions and the similar Kadyshevsky definition using thecovariant, albeit frame-dependent, θ-functions. (Q.E.D.)


Kadyshevsky-formalism IFor the general case we could follow the usual proof, see [BD65], but we preferto exploit here Lorentz-transformation properties. Thereto, we consider theLorentz-transformation U(a) of the scalar field

U(a)φ(x)U−1(a) = φ(x′) where x′ = ax , (11)

leaving the vacuum invariant, i.e. U(a)|0〉 = |0〉. We now take a such thatnµ = aµν n̂

ν , where n̂µ = (1,0). Also, φ(x) = U(a)φ(x̂)U−1(a)and θ[n · (x− y)] = θ(x̂0 − ŷ0). Then,

eT (φ(x1) . . . φ(xn)) = U(a) {T (φ(x̂1) . . . φ(x̂n))}U−1(a) , (12)

and Wick’s Theorem for the Kadyshevsky eT -product follows linea recta fromthat for the ordinary T-product. (Q.E.D)The equivalence of the S-matrix expressions, the usual one using the standardT-product, and the

Kadyshevsky one enploying the eT -product, can now be seenagain also in the following way. From the Wick-expansion we see that theKadyshevskyeT -product for any number of fields is equivalent to the ordinary T-product,if the Kadyshevsky eT -product for two fields is the same.The latter has been demonstrated explicitly above. (Q.E.D.)


Kadyshevsky-formalism I• Kadyshevsky graph rules:

(i) Employ θ(n · x) = − 12πi

Z +∞

−∞

dκexp[−iκ(n · x)]

κ+ iǫ,

(ii) Wick-expansion for ordinary products of field operators. Vacuumcontractions: Instead of Feynman-propagators, Wightman 2-point functions:

〈0|φ(x)φ(y)|0〉 = ∆(+)(x− y;µ2)〈0|Aµ(x)Aν(y)|0〉 = D(+)µν (x− y)〈0|ψβ(x)ψ̄α(y)|0〉 = S(+)βα (x− y;m2) .

Second Quantization Quasi Particles

• Green function: with τ ′ = n · x′, and τ = n · x:

G(τ ′, τ) = iθ(n · x′ − n · x) = iθ(τ ′ − τ).

Then :

„i∂

∂τ+ iǫ

«G(τ ′, τ) = −δ(τ ′ − τ).


Kadyshevsky-formalism IIand

G(τ ′, τ) = −Z +∞

−∞

dκχκ(τ

′)χ∗κ(τ)

κ+ iǫ= iθ(τ ′ − τ) ,

χκ(τ) =1√2π

exp[−i(κ− iǫ)τ ].

For second-quantization formalism: introduce (auxiliary) K-fields

χ(τ) =

Zdκ

κ+ iǫa(κ)χκ(τ) , χ̄(τ) =

Zdκ

κ+ iǫa†(κ)χ∗κ(τ) ,

and from [a(κ′), a†(κ)] = κδ(κ′ − κ)ˆχ(τ ′), ¯χ(τ)

˜= −iθ(τ ′ − τ) .

Fock-space states:

|κ〉 = a†(κ)|0〉 , 〈0|χ(τ)|κ〉 = 1√2πe−i(κ−iǫ)τ .


Kadyshevsky-formalism II

Second Quantization Quasi Particles(cont.)

Addition to free Lagrangian:

LK = iχ̄(τ)·χ (τ) + iǫχ̄(τ)χ(τ) ,

·χ:= ∂χ/∂τ .

Interaction Hamiltonian: extra factor

HI → H̄I · {χ̄(n · x)χ(n · x)} ,

which produces in contractions between vertices the factor

〈0|χ(n · x′)χ̄(n · x)|0〉 = −iθ[n · (x′ − x)] .

Then, in κ-space

(κ′|S|κ) = S exp− i

~

Z +∞

−∞

H̄I(x)d4xff, with symmetrizer

S (HI(x1)...HI(xn)) =X

P

(HI(xπ1)...HI(xπn))


Kadyshevsky-formalism III

Kadyshevsky Rules in Momentum-space

• Rules computation −Mκ′,κ-amplitudes,

(κ′|S|κ) = 1κ′,κ − (2π)4iδ(Pf + κ′n− Pi − κn)Mκ′,κ :1. Draw Feynman graph G. Arbitrarily number its vertices and orienteach internal line from the vertex with the smaller number to the vertexwith the larger one, assigning a momentum p.2. Connect with dotted lines the 1st to the 2nd, the 2nd to the 3rd, etc.Orient them in the direction of increasing numbers and assigna (quasi) momentum κsn, where s = 1, 2, ..., n− 1, the number of the vertexleft by a given dotted line.Attach to vertex 1 an incoming dotted line with 4-momentum κin, and to the lastvertex n an outgoing external dotted line with 4-momentum κfn.3. For incoming (outgoing) particle lines identical rules as for Feynman-graphs.4. For each internal dotted line, momentum κn a factor G0(κ) = −(κ+ iǫ)−1.5. For each internal boson line, momentum q, a factor ∆(+)(q) = θ(q0)δ(q2 − µ2).


Kadyshevsky-formalism IV6. For each internal fermion line with momentum p, and positive energy, a factor

S(+)βα (p) = (p/+m) θ(p0)δ(p

2 −m2).For each internal fermion line with momentum p, and negative energy, a factor

S(−)βα (p) = (p/−m) θ(p0)δ(p2 −m2).

7. For each internal photon line, momentum and Feynman-gauge, a factor

D(+)µν (q) = −gµνθ(q0)δ(q2).

8. For each vertex (quasi) 4-momentum conservation (2π)4δ(....)9. Integration over internal particle and quasi-particle momenta.10. repeat these operations for all n! numberings of the vertices and sum.

• Example Kadyshevsky graphs:

q

p

q’

p’

P

K K’

(a)

q

p

q’

p’

P

K K’

(b)


Kadyshevsky Integral Equation I

pf ,Mf

qf ,mf

pi,Mi

qi,mi

nκ′ nκ=

pf ,Mf

qf ,mf

pi,Mi

qi,mi

nκ′ nκ+

4pn,Mn

qn,mn

nκn

pf ,Mf

qf ,mf

nκ′

pi,Mi

qi,mi

nκ

Figuur 2: graphical Integral EquationTh.A. Rijken University of Nijmegen NFQCD 2010: Kadyshevsky Formalism – p. 28/103

Quasi-Potential Equations

Kadyshevsky’s Quasi-potential Equation I

• Kadyshevsky relativistic formalism and equation:

Mκ′,κ(p′a, p

′b; pa, pb) = Iκ′,κ(p

′a, p

′b; pa, pb) +

Zd4p′′a

Zd4p′′b ·

×Zdκ′′Iκ′,κ′′(p

′a, p

′b; p

′′a , p

′′b ) Gκ′′(p

′′a , p

′′b ) ·

×Mκ′′,κ(p′′a , p′′b ; pa, pb)δ(4(p′′a + p′′b + κ′′ − pa − pb − κn) ,

• Quasi-particle momentum: κnµ, choosenµ = (pµa + p

µb )/√s = (p′µa + p

′µb )/√s′CM→ (1,0).

• Green function:G(pa, pb) =

−1(2π)2

δ+(p2a −M2a) δ+(p2b −M2b ) ·G0(κ)

• Quasi-particle Green function:G0(κ) = (1/2π) [1/ (κ− iǫ)] .


Quasi-Potential Equations

Kadyshevsky’s Quasi-potential Equation II• Kadyshevsky CM-integral equation (3-dimensional):

Mκ′,κ(p′,W ′;p,W ) = Iκ′,κ(p

′,W ′;p,W ) +

Zd3p

(2π)3·

×Iκ′,κ′′(p′,W ′;p′′,W ′′)„MaMbE′′aE

′′b

« −1√s+ κ−

√s′′·

×Mκ′′,κ(p′′,W ′′;p,W )W =

√s, W ′ =

√s′ =

√s+ κ− κ′, W ′′ =

√s′′ = E′′a + E

′′b

• ’Relativistic Lippmann-Schwinger equation’:T (p′,p;W ) = Mκ′,κ=0(p

′,W ′;p,W )

V (p′,p;W ) = Iκ′,κ(p′,W ′;p,W )

T (p′,p;W ) = V (p′,p;W ) +

Zd3p

(2π)3V (p′

′,p′′;W ) ·

×„MaMbE′′aE

′′b

«1√

s′′ −√sT (p′′,p,W )


Kadyshevsky Integral Equation IInµ-independence of Kadyshevsky Integral Equation

When generating Kadyshevsky diagrams to random order using the Kadyshevskyintegral equation, the (full) amplitude is identical to the one obtained in Feynmanformalism when the external quasi particle momenta are put to zero.It is therefore n-independent, i.e. frame independent.Since an approximation is used to solve the Kadyshevsky integral equation,namely tree level diagrams as driving terms, it is not clear whether the fullamplitude remains to be n-independent when the externalquasi particle momenta are put to zero.In examining the nµ-dependence of the amplitude we write the Kadyshevskyintegral equation schematically as

M00 = Mirr00 +

Zdκ M irr0κ G

′κ Mκ0 , (13)

Since n2 = 1, only variations in a space-like direction are unrestricted,i.e. n · δn = 0 [Gross69]. We therefore introduce the projection operator

Pαβ = gαβ − nαnβ , (14)

from which it follows that nαPαβ = 0. The n-dependence of the amplitude

Pαβ∂

∂nβM00 = P

αβ ∂Mirr00

∂nβ+ Pαβ

Zdκ

»∂M irr0κ∂nβ

G′κ Mκ0 +Mirr0κ G

′κ∂Mκ0∂nβ

–. (15)


Kadyshevsky Integral Equation IIIf both Kadyshevsky contributions are considered at second order in M00,then it is nµ-independent, since it yields the Feynman expression.As far as the second term above is concerned we observe the following

∂M irr0κ∂nβ

∝ κf(κ) , ∂Mκ0∂nβ

∝ κg(κ) , (16)

where f(κ) and g(κ) are functions that do not contain poles or zero’s at κ = 0.Therefore, the κ-integral above is of the formZ

dκ κ h(κ)G′κ . (17)

When performing the integral we decompose the G′κ as follows

G′κ ∝1

κ+ iε= P

1

κ− iπδ(κ) . (18)

As far as the δ(κ)-part is concerned we immediately see that it gives zerowhen used in the κ-integral. For the Principle valued integral, indicated in theP-integral figure by I, we close the integral by connecting the end point(κ = ±∞) via a (huge) semi-circle in the upper half, complex κ-plane(line II in the P-integral figure) and by connecting the points around zerovia a small semi circle also in the upper half plane, line III in the figureof the P-integral.


Kadyshevsky Integral Equation IISince every single (tree level) amplitude is proportional to 1/(κ+ A+ iε),where κ is related to the momentum of the incoming or outgoingquasi particle and A some positive or negative number, the poles willalways be in the lower half plane and not within the contour.Therefore, the contour integral is zero.

ℜ(κ)

ℑ(κ)

I I

II

III

Figuur 3: Principle value integral


Kadyshevsky-formalism IIISince we have added integrals (II and III in the P-integral figure, we need to knowwhat their contributions is. The easiest part is integral III. Its contributionis half the residue at κ = 0 and since the only remaining integrandpart h(κ) in the κ-integral doesn’t contain a pole at zero it is zero.If we want the contribution of integral II to be zero, than the integrand shouldat least be of order O( 1

κ2). Unfortunately, this is not (always) the case as we will

see in sections with OBE-graphs and paper 2. To this end we introduce aphenomenological "form factor"

F (κ) =

„Λ2κ

Λ2κ − κ2 − iǫ(κ)ε

«Nκ, (19)

where Λκ is large and Nκ is some positive integer. Also, ε is real, positive,though small and ǫ(κ) = θ(κ)− θ(−κ).The effect of the function F (κ) on the original integrand is little,since for large Λκ it is close to unity. However, including this function inthe integrand makes sure that it is at least of order O( 1

κ2), so that integral II

gives zero contribution. The −iǫ(κ)ε part ensures that there are now poleson or within the closed contour, since they are always in the lower half plane,as indicated by the dots in the figure of the P-integral.


Kadyshevsky-formalism V

Derivative Interactions, Gross-Jackiw Analysis

• Derivative interactions: HI = −LI + ∆HI ,where in general ∆HI contains non-covariant "contact terms"(c.t.).Cancellation contributions c.t.’s against non-covariant terms from

S′ =

∞X

n=0

„i

~

«n Z +∞

−∞

d4x1...

Z +∞

−∞

d4xn T [LI(x1)...LI(xn)] .

• Gross-Jackiw(66): Covariant frame-, i.e. nµ-, independent T ∗-product:

S =

∞X

n=0

„i

~

«n Z +∞

−∞

d4x1...

Z +∞

−∞

d4xn T∗ [LI(x1)...LI(xn)] .

In short hand notation : T ∗(x1, ..., xn) = T (x1, ..., xn) + τ(x1, ..., xn) .

Require T ∗-product frame, i.e. nµ, independent, ensuring a Lorentz-invariantS-matrix: consider variations δnµ such that δn2 = n · δn = 0, giving

Pαβδ

δnβT ∗(x1, ..., xn) = P

αβ δ

δnβT (x1, ..., xn) + P

αβ δ

δnβτ(x1, ..., xn) = 0 .

where Pαβ = gαβ − nαnβ .


Kadyshevsky-formalism IV• Consider S(2):

S(2) = −(1/2!)Zd4x

Zd4y T [LI(x)LI(y)] ,

T [x− y;n] = θ[n · (x− y)]LI(x)LI(y) + θ[−n · (x− y)]LI(y)LI(x),

Pαβ∂

∂nβT [x− y;n] = Pαβ(x− y)βδ [n · (x− y)] [LI(x),LI(y)] .

In general local and Schwinger terms:δ [n · (x− y)] [LI(x),LI(y)] = C(n)δ4(x− y) + PαβSα(n)∂βδ4(x− y)....

(i) IF Sα(n) = 0, etc, i.e. no-derivatives r.h.s.:Pαβ δ

δnβT (x− y;n) = 0⇒ τ(x− y;n) = 0.

(ii) IF Sα(n) 6= 0, rest =0: ⇒ τ(x− y;n) = (S · n)δ4(x− y) + τ0(x− y).

Scheme: With T ∗,LI ⇒ Covariant, non-covariant + τ terms⇒Lorentz-invariancy.

• Jackiw-Gross(69) scheme equivalent with Takahashi-Umezawa(53)!!


Kadyshevsky-formalism IV

Frame-independent Kadyshevsky Formalism

• Adaption Gross-Jackiw method:Pαβ

δ

δnβθ [n · (x− y)]LI(x)LI(y) = Pαβ(x− y)βδ [n · (x− y)]LI(x)LI(y)

Observation/conjecture:

δ[n · (x− y)]LI(x)LI(y) = N [LI(x)LI(y)] |0 + 12

nC(n) + Si(n)∂i

oδ(x− y) ,

δ[n · (y − x)]LI(y)LI(x) = N [LI(x)LI(y)] |0 − 12

C(n)− 1

2Si(n)∂i

ffδ(x− y) .

For example C(n) = −2δ(x− y) and Sm(n) = 0 for the caseφ(x)

·

φ (y) = N [φ(x)·

φ (y)]− ∂0∆(+)(x− y;m2) ,·

φ (y)φ(x) = N [φ(x)·

φ (y)] + ∂0∆(+)(x− y;m2),

and for ∂µφ(x)∂νφ(y) = N [∂µφ(x)∂νφ(y)] + ∂µ∂ν∆(+)(x− y;m2),

one has



δ[n · (x− y)]∂µφ(x)∂νφ(y) = N [∂µφ(x)∂νφ(y)]|0− (δµmδν0 + δµ0 δνm) ∂mx δ(x− y) ,

δ[n · (y − x)]∂νφ(y)∂µφ(x) = N [∂µφ(x)∂νφ(y)]|0 ++(δµmδ

ν0 + δ

µ0 δνm) ∂

mx δ(x− y) ,

which gives C(n) = 0, Sm(n) = 2 (δµmδν0 + δ

µ0 δνm)



Frame-independent Kadyshevsky Formalism (cont)

• Adaption Kadyshevsky formalism (~ = 1):

S(n) = inZd4xn...

Zd4x1 R [LI(xn)...LI(x1)] ,

R [LI(xn)...LI(x1)] = θ[n · (xn − xn−1)]...θ[n · (x2 − x1)] · LI(xn)...LI(x1).Introduce the R∗ − product : R∗(xn, ..., x1;n) = R(xn, ..., x1;n) + ρ(xn, ..., x1;n).

Frame− independence R∗ − product : Pαβ δδnβ

R(x1, ..., xn) + Pαβ δ

δnβρ(x1, ..., xn) = 0 ,

which gives for two-point functions

Pαβδ

δnβρ(x− y;n) = 1

2

hPαβSβ(n) + P

αβP γδ (Qβγ +Qγβ) ∂δiδ4(x− y) ,

from which it follows that ρ(x− y;n) = τ(x− y;n)/2. Similarly as in the Feynmanformalism, the introduction of the R∗-product in the Kadyshevsky formalism yields a covariantand frame independent S-matrix, and S(Kadyshevsky)=S(Feynman)for on energy shell initial and final states.


Example derivative Interaction, 1

Pseudo-vector Pion-Nucleon Interaction

Lpv(x) = (f/mπ)ψ̄(x)γ5γµψ(x)∂µφ(x) .

The non-zero ETC and ETAC relations:nψa(x), ψ

†b(y)

o|0 = δabδ(x− y) ,

»φ(x),

·

φ (y)

–|0 = δ(x− y) .

The ETC for the interaction Lagrangian becomes after some algebra

[LI(x),LI(y)] |0 ⇒ (f/mπ)2“ψ†Γµψ

” “ψ†Γνψ

”[∂µφ(x), ∂νφ(y)] |0

+i(f/mπ)2

hψ†Γmψ · ψ†Γ0ψ + ψ†Γ0ψ · ψ†Γmψ

i|0∂mx δ(x− y) .

The r.h.s. contains a Schwinger-term, giving for nµ = (1,0)

Sµ(n) = i(f/mπ)2

hψ†Γmψ · ψ†(Γ · n)ψ + ψ†(Γ · n)ψ · ψ†Γmψ

i|0

This leads to the right τ(x− y;n) to cancel the non-covariant termproduced by the T-product in 2nd order:

τ(x− y;n) = (S.n)δ4(x− y) = 2i(f/mπ)2(ψ†γ5ψ)2δ4(x− y) .


Example derivative Interaction, 2

Vector-exchange in pion-Nucleon

The interaction Lagrangian we take in this case is

LI(x) = gψ̄γµψ V µ + fφ†↔

∂ µ φ Vµ ≡ L(1)I (x) + L

(2)I (x) (20)

For the vector field we have the commutator and the ETC

[V µ(x), V ν(y)] = i

„−gµν − ∂

µ∂ν

m2V

«∆(x− y;m2V ) ⇒ (21)

[V µ(x), V ν(y)] |0 = im2V

(δµmδν0 + δ

µ0 δν0 ) ∂

mδ(x− y) . (22)

This gives for the relevant Lagrangian commutator for πNhL(1)I (x),L

(2)I (y)

i|0 = i fg

m2V

hψ̄γmψ · φ†

↔

∂ 0 φ+ ψ̄γ0ψ · φ†↔

∂m φi∂mδ(x− y) .

This leads for nµ = (1,0) to the Schwinger term

Sα = ifg

m2V

hψ̄γαψ · φ†

↔

∂ 0 φ+ ψ̄γ0ψ · φ†↔

∂ α φi⇒ (23)

τ(x− y;n) = (S · n) = 2i fgm2V

ˆψ̄(γ · n)ψ

˜ “φ†(

↔

∂ ·n)φ”δ4(x− y) . (24)

This is indeed the correct term to cancel the second-order non-covariant piece.


Example derivative Interaction, 3a

Gauge-invariant ∆33Nπ interaction

The gauge-invariant ∆33Nπ interaction Lagrangian reads

LI = gǫµναβˆ`∂µψ̄ν

´Γαψ · ∂βφ+ ψ̄Γα (∂µψν) · ∂βφ

˜

≡ L(1)I + L(2)I , (25)

with, Γα = γ5γα. The equal-time (anti)commutation relations for the nucleon and pionfields aren

ψa(x), ψ†b(y)

o|0 = δabδ(x− y) ,

hφ(x), φ̇(y)

i|0 = iδ(x− y) . (26a)

For the (free) Rarita-Schwinger field [RS41] field one has [Car71]nψνa(x), ψ̄

λb (y)

o= (−i)Λνλ(−i∂) (iγ · ∂ +M)∆(x− y;M) ,

Λνλ(−i∂) = gνλ − 13γνγλ +

2

3M2∂ν∂λ +

i

3M(γν∂λ − γλ∂ν) . (27a)


Example derivative Interaction, 3bFor πN -scattering in second order in g, we get only contributions from the cross-term

commutators [L(2)I (x),L(1)I (y)]. and [L

(1)I (x),L

(2)I (y)], henceforth referred to as the

21- and the 12-commutator respectively. The 21-commutator giveshL(2)I (x),L

(1)I (y)

i= − i

3g2ǫµναβǫρλγδ (∂βφ(x) ∂δφ(y)) ·

×ψ̄(x)γαˆ(gλν + γλγν) (i∂/−M∆) ∂xµ∂xρ∆(x− y;M2∆)

˜γγψ(y) . (28)

Taking now equal times, i.e. x0 = y0, then only the time-derivatives operating on theinvariant ∆(x− y;M2∆)-function survive. There will be four contributions: (i) iγ0∂0from the Dirac-operator, (ii) δ0µ∂0∂ρ=r, (iii) δ

0ρ∂0∂µ=m, and (iv) for µ = ρ = 0 giving

iγ0∂0 · ∂20 = iγ0∂0 ·Pk=1,3 ∂k∂

k −M2∆) iγ0∂0 ·Pk=1,3 ∂k∂

k. In the last step we

skipped the M2∆-term since it will not give a Schwinger term and therefore can beignored.


Example derivative Interaction, 3cTo write the spatial derivatives in a covariant form we use the nµ-vector, and write∂m = Pµκ∂

κ. We find, after rearranging terms in order to exhibit the dipole- andquadrupole-type Schwinger terms,

δ [n.(x− y)]hL(2)I (x),L

(1)I (y)

i= − i

3g2ǫµναβǫρλγδ · (∂βφ(x) ∂δφ(y)) ·

»ψ̄(x)γα (gλν + γλγν) ·

M∆ (nµPρσ + nρPµσ) ∂

σδ4(x− y)

−i(γ · n) (PµκPρσ − nµnρPκσ) ∂κ∂σδ4(x− y)

−iγτPτω (nµPρσ + nρPµσ) ∂ω∂σδ4(x− y)ff· γγψ(y)

–. (29)

Recalling the formulas

S(2) =(+i)2

2!

Zd4x

Zd4y T [LI(x)LI(y)] ,

T [x− y;n] = θ[n · (x− y)] LI(x)LI(y) + θ[−n · (x− y)] LI(y)LI(x),

Pαβδ

δnβT (x− y;n) = Pαβ(x− y)βδ [n · (x− y)] [LI(x),LI(y)] .


Example derivative Interaction, 3dWe identify

Pωκδ

δnκS(2) ⇐ − 1

2!

Zd4x

Zd4y Pωκ(x− y)κ · δ [n.(x− y)]

hL(2)I (x),L

(1)I (y)

i

≡ Pωκ δδnκ

S(2)c.t. . (31)

Now, the strategy is to modify the T-product by including τ -terms, such as to cancelthe so called ’contact terms’. Let us define

Pωκδ

δnκτ (2) ⇐ + 1

2!

Zd4x

Zd4y Pωκ(x− y)κ · δ [n.(x− y)]

hL(2)I (x),L

(1)I (y)

i.(32)

So, specializing to πN , the τ -terms have to satisfy

Pωκδ

δnκτ(p′, q′; p, q) = −Pωκ δ

δnκS(2)(p′, q′; p, q) (33)

and so they can be obtained from (29). Moreover, the factor 1/2! can be droppedbecause there is an identical contribution form the ’12-commutator’.


Example derivative Interaction, 3eWe split the contributions to the τ(x− y;n)-function into two parts:

τ(x− y;n) = τ1(x− y;n) + τ2(x− y;n) , (34)

where now the τ1,2-functions satisfy

Pωσ∂

∂στ1(x− y;n) = + i

3M∆g

2ǫµναβǫρλγδ · (∂βφ(x) ∂δφ(y)) . ·

×ˆψ̄(x)γα (gλν + γλγν) γγψ(y)

˜·

× (nµPρω + nρPµω) δ4(x− y), (35)

Pωσ∂

∂στ2(x− y;n) = +1

3g2ǫµναβǫρλγδ · (∂βφ(x) ∂δφ(y)) ·

×ˆψ̄(x)γα (gλν + γλγν) γ

ǫγγψ(y)˜·

×

(PǫωPρσ + PǫσPρω)nµ + (PǫωPµσ + PǫσPµω)nρ

+(PµωPρσ + PµσPρω − 2nµnρPωσ)nǫff∂σδ4(x− y) .(36)


Example derivative Interaction, 3fNext, for the contribution to the πN -matrix elements we evaluate the derivative ∂σ.There are two contributions:

(i) ∂σˆψ̄(x)ψ(y)

˜→ − i

2(p′ + p)σ , (∂βφ(x) ∂δφ(y))→

`qβq

′δ + q

′βqδ

´,

(ii) ∂σ (∂βφ(x) ∂δφ(y))→ − i2(q′ + q)σ

`qβq

′δ − q′βqδ

´.

We now write〈p′q′|τ (2)c.t.|p, q〉 = (2π)4δ(p′ + q′ − p− q) τ̄(p′, q′; p, q) withτ̄(p′, q′; p, q) = τ̄1(p

′, q′; p, q) + τ̄2(p′, q′; p, q) , (37)

and find from (EX.34) and (EX.35) that

Pωσδ

δnστ̄1(p

′, q′; p, q) = +2i

3M∆g

2ǫµναβǫρλγδ`qβq

′δ + q

′βqδ

´·

×ˆū(p′)γα (gλν + γλγν) γγu(p)

˜· (nµPρω + nρPµω) , (38)

Pωσδ

δnστ̄2(p

′, q′; p, q) = − i3g2ǫµναβǫρλγδ ·

ˆ`qβq

′δ + q

′βqδ

´Qσ −

`qβq

′δ − q′βqδ

´P σ

˜·

×ˆū(p′)γα (gλν + γλγν) γ

ǫγγu(p)˜·

×

(PǫωPρσ + PǫσPρω)nµ + (PǫωPµσ + PǫσPµω)nρ

+ (PµωPρσ + PµσPρω − 2nµnρPωσ)nǫff∂σδ4(x− y) , (39)


Example derivative Interaction, 3g

(i) Using FORM [Vermaseren] we obtain for τ̄1:

Pµνδ

δnντ̄1(p

′, q′; p, q) = +i

3M∆g

2

»8(n · q)(n · q′) nµ − 4(n · q′)qµ − 4(n · q)q′µ

–.(40)

To solve this we consider the equationsPµνa

ν ≡ 8(n · q)(n · q′) nµ − 4(n · q′)qµ − 4(n · q)q′µ , whereaµ = a1 qµ + a2 q

′µ + a3 nµ ,

which has as solutiona1 = −4(n · q′) , a2 = −4(n · q) , a3 = undetermined .

So,aµ = −4(n · q′)qµ − 4(n · q)q′µ + a3nµ ,

which gives as solution for τ̄1

τ̄1(p′, q′; p, q) = −4i(n · q′)(n · q)× 1

3g2M∆ . (41)


Example derivative Interaction, 3h

(ii) Using FORM [Vermaseren] we obtain for τ̄2:

Pµνδ

δnντ̄2(p

′, q′; p, q) = +i

3g2

»8n/

„(n · q)(q′ · P ) + (n · q′)(q · P )

− 4(n · q)(n · q′)(n · P )«

+ P/(n · q′)(n · q)–nµ

+

»n/

„8(n · q′)(n · P )− 4(q′ · P )

«− P/(n · q′)

–qµ

+

»n/

„8(n · q)(n · P )− 4(q · P )

«− P/(n · q)

–q′µ + 8n/(n · q′)(n · q) Pµ

+

»8(n · q′)(n · q)(n · P )− 4(n · q′)(q · P )− 4(n · q)(q′ · P )

–γµ

ff(42)

Repeating the procedure above, we write

Pµνδτ̄2/δnµ ∝ Pµνbν ≡. . .

ffwith

bµ = b1 qµ + b2 q′µ + b3 Pµ + b4γµ + b5nµ .


Example derivative Interaction, 3iThen, we obtain the coefficients

b1 = 8n/(n · q′)(n · P )− 4n/(q′ · P )− 4P/(n · q′) ,b2 = 8n/(n · q)(n · P )− 4n/(q · P )− 4P/(n · q′) ,b3 = 8n/(n · q)(n · q′) ,b4 = 8(n · q)(n · q′)(·P )− 4(n · q)(q′ · P )− 4(n · q′)(q · P ) ,b5 = undetermined .

From this we get as a solution for τ̄2:

τ̄2(p′, q′; p, q) = +

i

3g2

»− 4P/(n · q′)(n · q)− 4n/(n · q)(P · q′)− 4n/(n · q′)(P · q)

+ 8n/(n · q′)(n · q)(n · P )–. (43)

Again, these τ -functions cancel the non-covariant, i.c. the ’frame-dependent’, termsproduced by the T-product in second order. Also, they are the right corrections to theKadyshevsky amplitudes such as to give agreement with the Feynman-amplitudeswhen κ = κ′ = 0.


Relativistic Pair-suppression

Relativistic BB̄-Pair-suppression

• Is that possible? (Thomson-limit electron-compton scattering!?)• Academic example: Dense medium with all anti-baryon states filled,i.e. at T = 0: pF = 0, p̄F =∞Ground state |Ω〉, and SF (x− y) = −i〈Ω|T [ψ(x)ψ̄(y)]|Ω〉In momentum space (se e.g. Bowers & Zimmermann, Phys.Rev. D7 (1973))

SF (p; pF , p̄F ) =p/+M

2Ep

1− nF (p)p0 − Ep + iǫ

+nF (p)

p0 − Ep − iǫ

− 1− n̄F (p)p0 + Ep − iǫ

− n̄F (p)p0 + Ep + iǫ

ff

nF =0,n̄F =1=⇒ p/+M2Ep

1

p0 − Ep + iǫ− 1p0 + Ep + iǫ

ff

= (p/+M) /ˆp2 −M2 + i0p0

˜,

i.e. covariant and of a purely ’retarded’ type.



Effective Vertex Pair-suppression

q

p

p’

q’

P_a

K

K’

(a)

q

p

p’

q’

P_b

K

K’(b)

Consider the baryon-exchange K-graph with a matrix element of the formM (b)(pf , pi) ∼ ū(p′) [Γ0 (−P/+MB) Γ0]u(p) , (44)

which comes from the S(−)(P )-propagator. We write

−P/+MB = −2MBX

s

V (P, s) V̄ (P, s) (45)

where V (P, s) denotes the Dirac-spinor for baryon B.The inclusion of ’strong-pair-suppression’ is effectuated by making the substitution

Γ0 → Γ(pf , pi) = g2

nexp

hea

“p/f −Mf

”iΓ0 exp

ˆea

`p/i −Mi

´˜

+exph−ea

“p/f +Mf

”iΓ0 exp

ˆ−ea

`p/i +Mi

´˜o, ea = a

2M> 0 .

where the subscripts i and f denote respectively the incoming and outgoingbaryon at the vertex, and the ’bare-vertex’ satisfies γ0Γ†0γ0 = Γ0.



Effective Vertex Pair-suppression (cont.)Then, one gets for on-shell fermions, and large a > 0 pair-suppression

ū(pf ) Γ(pf , pi) u(pi) =g

2

`1 + e−2a

´ū(pf ) Γ0(pf , pi) u(pi) ,

v̄(pf ) Γ(pf , pi) v(pi) =g

2

`1 + e−2a

´v̄(pf ) Γ0(pf , pi) v(pi) ,

v̄(pf ) Γ(pf , pi) u(pi) = g e−a v̄(pf ) Γ0(pf , pi) u(pi) ,

ū(pf ) Γ(pf , pi) v(pi) = g e−a ū(pf ) Γ0(pf , pi) v(pi) . (46)

For the matrix element M (b)(pf , pi) we get

M (b)(pf , pi) ∼ e−2a [ū(p, sf )Γ0V (P, s)]ˆV̄ (P, s)Γ0u(P, si)

˜. (47)

ie. the result is simply a strong pair-suppression factor for large positive a for thecontribution of graph (b).



Effective Vertex Pair-suppression (cont.)From the perspective of charge-conjugation invariance it is easily seen thatthe mentioned effective vertex is a good choice.Under charge-conjugation C one has

C : ψ → C ēψ , ψ̄ → − eψ C−1 , C−1γµ C = −fγµ . (48)

Then, one has that

ψ̄f p/ ψi → − eψf C−1p/ C ēψi = +ψ̄i p/ ψf ,

ψ̄f Γ0 p/ ψi → − eψf C−1Γ0 p/ C ēψi = +ψ̄i p/ (Γ0)c ψf , (49)

with (Γ0)c = ˜C−1Γ0C From this it is clear that our chosen Γ(pf , pi) gives C-invariance.



Effective Vertex Pair-suppression (cont.)For a proper representation in x-space we have to split the fermion field ina positive and negative frequency part. We write

ψ(x) = ψ(+)(x) + ψ(−)(x) , (50)

which is invariant under orthochronous Lorentz transformations L↑+. Then,

exp (eap/)→ ψ(x|a) ≡ exp [ + iea∂/ ]ψ(+)(x) + exp [ − iea∂/ ]ψ(−)(x) . (51)

So, with this prescription we can write down an effective interaction Lagrangiancorresponding to the vertex above. We write

LI(x) = g2

ˆψ̄(x|a)Γ0 ψ(x|a) + ψ̄(x| − a)Γ0 ψ(x| − a)

˜· e−a . (52)

Now, in perturbation theory the fields are free fields, ïn- or öut-fields with mass M , sothat for e.g. absolute pair-suppression, i.e. a =∞,

LI(x)⇒ g2

hψ(+)(x)Γ0 ψ

(+)(x) + ψ(−)(x)Γ0 ψ(−)(x)

i, (53)

where ψ(±)(x) ≡ ψ(±)†(x)γ0.



Strong/Not-absolute pair-suppression

For a 0.For the rest of these notes we restrict ourselves in working only out the consequencesfor the interaction with absolute pair-suppression.


Pair-interactions ESC-models, 1

Pair-interactions ESC

∑i,j,...

Ri, Hj, etc.

M1

M2

⇒∑

pairsM1

M2

MPEFiguur 4: Meson-pair description and low-energy approximation.


Pair-interactions ESC-models, 2

Pair-interactions ESC and Negative-energy

M1

M2

⇒∑

pairsM1

M2

MPEFiguur 5: Meson-pair description and Z-diagrams.


Baryon-resonance

Example: JP = 12

+- Resonance, PS-coupling

q

p

q’

p’

P

K K’

(a)

q

p

q’

p’

P

K K’

(b)

Consider baryons like N(940) and the P11-resonance N∗(1470), i.e. the Roperresonance. The two Kadyshevsky graphs are shown above. The ps-vertex means thatΓ0 = γ5 Then, the Kadyshevsky rules give for graph (a) and graph (b) (see e.g.Wagenaar & Rijken 2008)

M(a)κ′,κ = −

g2

4

`1 + e−2a

´2ū(p′)

»−1

2(Mf +Mi) +MB

ff−Q/+

+ {∆E,s −As}n/]u(p)»1

2(κ′ + κ) + ∆E,s − As + iǫ

–−1/2As ,

M(b)

κ′,κ = −g2e−2a ū(p′)

»−1

2(Mf +Mi) +MB

ff−Q/+

+ {∆E,s +As}n/]u(p) ·»1

2(κ′ + κ)−∆E,s −As + iǫ

–/2As . (55)


Baryon-resonanceHere,

∆s =1

2

ˆ(p′ + p) + (q′ + q)

˜, ∆E,s = ∆s · n . (56)

and

As =˘M2B −∆2s + (∆s · n)2

¯1/2(57)

• Absolute Pair-suppression, a→∞:In this case one has for the on-energy-shell amplitudes, i.e. κ′ = κ = 0, M (b)0,0 = 0, and

M(a)0,0 becomes

M(a)0,0 ⇒ −

1

4g2 ū(p′)

»(MB −M)−Q/

2As (∆s · n−As)+

n/

2As

–u(p) . (58)

In order to make this amplitude frame-independent, i.e. nµ independent, we subtract

from M (a)0,0 the piece

∆M (a) = −14g2 ū(p′)

[(MB −M)−Q/ ] ∆s · n

2As (s−M2B)+

n/

2As

ffu(p) . (59)


Baryon-resonanceThis gives

M̄(a)0,0 = M

(a)0,0 −∆M (a) = −

1

4g2 ū(p′) [(MB −M)−Q/]u(p) ·

× 12As

1

∆s · n−As− ∆s · ns−M2B

ff

= −18g2 ū(p′) [(MB −M)−Q/ ]u(p) 1

s−M2B. (60)

Here we used the identity s−M2B = (∆s · n)2 −A2s.One sees that by rescaling the coupling g → 2

√2g, one obtains the same amplitudes as usual !!

In the following, we derive the subtraction terms using:1. The Jackiw-Gross scheme, by an adaption of the R-product in the Kadyshevskyfield theory.(Gross & Jackiw, Nucl.Phys. B14 (1969) 269)2. The Takahashi-Umezawa theory of interaction.(Takahashi & Umezawa, Progr. Theor. Phys. 9, 501 (1953))


Jackiw-Gross

Jackiw-Gross R-product analysis, PS-coupling

With pair-suppression, the R-product formulation is the right starting pointin perturbation theory. For example in case of ’absolute pair-suppression’, thecontractions in the Wick-expansion do not lead to SF (x, y)-functions.This can be seen as follows: the Feynman Green function isSF (x− y)βα = −iθ(x0 − y0)〈0|ψβ(x)ψ̄α(y)|0〉+ iθ(y0 − x0)〈0|ψ̄α(y)ψβ(x)|0〉, (61)

where 〈0|ψβ(x)ψ̄α(y)|0〉 = 〈0|ψ(+)β (x)ψ(+)α(y)|0〉 ,

〈0|ψ̄α(y)ψβ(x)|0〉 = 〈0|ψ(−)α(y)ψ(−)β (x)|0〉 . (62)

But, the negative-frequency terms never occur in the Wick-expansionfor ’absolute pair-suppression.For the analysis of the frame-dependent terms in the Kadyshevsky formalismwe consider

Pαβ∂

∂nβθ[n · (x− y)]LI(x)LI(y) = Pαβ(x− y)βδ[n · (x− y)]LI(x)LI(y) (63)


Jackiw-GrossFor πN we have that

LI(x) = g2

„¯ψ(+)(x)Γ0ψ

(+)(x) + ¯ψ(−)(x)Γ0ψ(−)(x)

«φ(x)

∼ g2

h¯ψ(+)Γ0ψ

(+)(x)iφ(x) . (64)

Then, for the evaluation of the nµ-dependence we getδ[n · (x− y)]LI(x)LI(y) ∼ δ(x0 − y0) ψ(+)α(x)ψ(+)β (x) · ψ(+)γ(y)ψ

(+)δ (y) =

δ(x0 − y0)N». . .

–+ δ(x0 − y0)ψ(+)α(x)

nψ

(+)β (x), ψ

(+)γ(y)

oψ

(+)δ (y) =

δ(x0 − y0)N». . .

–+ δ(x0 − y0)ψ(+)α(x) ·

`+i∂/x +M

´βγ

∆(+)(x− y) · ψ(+)δ (y) ,

where, e.g. [NO90],

∆(+)(x− y) = 12

»i∆(x− y) + ∆(1)(x− y)

–, (65)

with ∂0∆(x− y)˛̨x0=y0 = −δ(x− y) , ∂0∆(1)(x− y)

˛̨x0=y0 = 0 . (66)


Jackiw-GrossTherefore,

δ(x0 − y0)`+i∂/x +M

´βγ

∆(+)(x− y) =1

2δβγδ

4(x− y) + 12δ(x0 − y0)

`+i∂/x +M

´βγ

∆(1)(x− y) (67)

From this we get, for Γ0 = γ5, thatZd4x

Zd4y Pαβ(x− y)βδ[n · (x− y)]LI(x)LI(y)⇒ +g

2

8

Zd4x

Zd4y ·

×Pαβ(x− y)βnψ(+)(x) ·

`−i∂/x +M

´∆(1)(x− y) · ψ(+)(y)

oδ(x0 − y0) =

Pαβ∂

∂nβ

+g2

8

Zd4x

Zd4y θ[n · (x− y)]·

× ψ(+)(x)h`−i∂/x +M

´∆(1)(x− y)

i· ψ(+)(y)

o. (68)

Notice the appearance of the non-causal ∆(1)(x− y)-function. So, such terms haveto beremoved from the M (2)-matrix. These non-causal terms are typical for non-localinteractionsas we analyze here in connection with pair-suppression.


Jackiw-GrossApplication to πN -scattering, and restricting ourselves to the s-channelresonance contribution, we have to evaluate the matrix element

(2π)4δ`p′ + q′ − p− q

´∆M

(2)0,0 = 〈|∆HI |〉 ≡ −

i

8g2 〈p′q′|

Zd4x

Zd4y ·

×θ[n · (x− y)] (φ(x)φ(y)) · ψ(+)(x)h`−i∂/x +M

´∆(1)(x− y)

i· ψ(+)(y)|pq〉

= − i8g2

Zd4x

Zd4y · − 1

2πi

Zdκ

exp [−iκn · (x− y)]κ+ iǫ

ei(p′+q′)·x e−i(p+q)·y ·

×ū(p′)»Z

d4P

(2π)3(−P/+M) δ

`P 2 −M2

´e−iP ·(x−y)

–u(p)

= (2π)4δ`p′ + q′ − p− q

´· g

2

8

Zdκ

κ+ iǫ

ˆū(p′) (−P/+M)u(p)

˜δ

`P 2 −M2

´,(69)

where P = ∆s − κn , ∆s = 12(p′ + q′ + p+ q) . (70)

The solutions for κ are

κ± = −∆s · n±As , P±0 = ∆s − κ± . (71)


Jackiw-GrossThis gives

δ`P 2 −M2

´⇒ δ

`(∆s − κn)2 −M2

´=

1

2As

»δ(κ− κ+) + δ(κ− κ−)

–. (72)

Performing now the κ-integral, we get for ∆M (2)0,0

∆M(2)0,0 = +

g2

8ū(p′)

»

1

2As

( `−∆/s + (∆s · n−As)n/+M

´

−∆s · n+As+

`−∆/s + (∆s · n+As)n/+M

´

−∆s · n− As

)#u(p) =

g2

16Asū(p′)

[(MB −M)−Q/ ]

»1

−∆s · n+As− 1

∆s · n+As

–− 2n/

ffu(p) =

g2

4ū(p′)

[(MB −M)−Q/ ] (∆s · n)

2As (M2B − s)− n/

2As

ffu(p) . (73)

This indeed explains the ’compensation terms’ (3.5) we had to introducein order to make the Kadyshevsky amplitude for ’absolute pair-suppression’Lorentz invariant1We note that for baryon-exchange the analysis runs analogously.


Jackiw-GrossIn view of the results so far in this section, we can now identify

the effective extra Hamiltonian. This ∆HI has to be such that(2π)4δ(Pf − Pi)∆M (2)0,0 = +〈 |∆HI | 〉.Then, from ∆M (2)0,0 we infer that the corrective interaction Hamiltonian has to be

∆HI ≡ −i g2

8

Zd4x

Zd4y θ[n · (x− y)] · φ(x)φ(y)

× ¯ψ(+)(x)h`−i∂/x +M

´∆(1)(x− y)

i· ψ(+)(y). (74)

Of course a similar term has to be added for πN̄ -scattering.We note that, although non-local as expected, the extra Hamiltonian exhibits again’absolute pair-suppression’!

Off-energy-shell subtraction

The explicit form of ∆HI enables us to analyze properly the off-energy-shell behaviorof the corrections to the πN matrix elements. We will demonstrate that for the spin-1/2baryon graphs, and subsequently extract the rule to derive the result by a rathersimple substitution prescription for the ’covariantization’ of the second-orderamplitudes.


Jackiw-GrossAs an example, we treat here again the spin-1/2 resonance amplitude with

PS-coupling. Then, the off-energy-shell matrix πN matrix elements of ∆HI , are

(2π)4δ`p′ + q′ + κ′n− p− q − κn

´∆M

(2)

κ′,κ ≡− i

8g2 〈p′q′κ′|

Zd4x

Zd4y θ[n · (x− y)] · φ(x)φ(y)

× ¯ψ(+)(x)h`−i∂/x +M

´∆(1)(x− y);M2B

i· ψ(+)(y)|pqκ〉 =

(2π)4δ (. . .) · g2

8

Zdκ1

κ1 + iǫ

ˆū(p′) (−P/+MB)u(p)

˜δ

`P 2 −M2B

´, (75)

Here P = 12(κ′ + κ) + ∆s − κ1n, and ∆s = 12 (p′ + q′ + p+ q) which is the same as for

κ′ = κ = 0.The solutions for κ1 are

κ± =1

2(κ′ + κ) + ∆s ·n±As , P (κ(±)) = ∆s− κ±n = ∆s− [(∆s · n)±As]n . (76)

Again using

δ`P 2 −M2B

´⇒ δ

`(∆s − κ1n)2 −M2B

´=

1

2As

»δ(κ1 − κ+) + δ(κ1 − κ−)

–,(77)

we can perform the κ1-integral.


Jackiw-GrossWith the short-hand notation κ̄ = (κ′ + κ)/2, we get

∆M(2)κ′,κ = (g

2/8)ū(p′)

»

1

2As

( `−∆/s + (∆s · n+As)n/+MB

´12(κ′ + κ) + ∆s · n+As

+

`−∆/s + (∆s · n−As)n/+MB

´12(κ′ + κ) + ∆s · n− As

)#u(p) =

g2

16Asū(p′)

[(MB −M)−Q/ ]

»1

κ̄+ ∆s · n+As+

1

κ̄+ ∆s · n−As

–

+

»(∆s · n+As)κ̄+ ∆s · n+As

+(∆s · n− As)κ̄+ ∆s · n−As

–n/

ffu(p) (78)

Reminding that for a =∞ the off-energy-shell matrix element isM

(a)

κ′,κ = −(g2/4) ū(p′) {[(MB −M)−Q/] + ((∆s · n)−As)n/}u(p) ·

× [κ̄+ (∆s · n)−As + iǫ]−1 /2As , Then, (79)

M̄(2)κ′,κ ≡M

(a)κ′,κ + ∆M

(2)κ′,κ = −(g

2/8) · ū(p′)

[(MB −M)−Q/ ]− κ̄n/ffu(p) ·

×ˆ(κ̄+ ∆s · n)2 − A2s

˜−1,

which in the limit κ′, κ→ 0 agrees with M̄ (a)0,0 given before.This settles the off-energy-shell behavior of the amplitudes!


Jackiw-Gross

Prescription ’covariantization’ 2nd order

From the foregoing analysis we have found for ’absolute pair-suppression’ thefollowing recipe for the ’covariantization’ of the Kadyshevsky amplitudes:Subtraction Non-causal parts: Make in the amplitude from graph’s of type (a) thesubstitution ∆(+)(x− y;M2)→ i

2∆(x− y;M2) (80)

With this substitution the amplitude M (2)κ′,κ contains the propagator factor

θ[n · (x− y)] ∆(x− y;M2) = −∆ret(x− y;M2) =Z

d4p

(2π)4e−ip·(x−y)

p2 −M2 + i0p0

=

Zd4p

(2π)41

2Ep

»1

p0 − Ep + iǫ− 1p0 + Ep + iǫ

–, (81)

which is of the same form as in our ’academic example’!!Sofar, we have seen that in the cases where the interaction LI does not containderivatives of the fermion fields this substitution works for πN . Here, we have tosubtract out the non-causal part to arrive at Lorentz invariant expressions forpair-suppression. When dealing with the JP = (3/2)+-resonance, using thegauge-invariant coupling, this will not be the complete story. Because,in that case apart from the non-causal parts we also have toremove the non-invariant ’contact terms’.


Meson-exchange,1

Scalar-Meson-Exchange I

Here, we proceed with the discussion of the meson exchange processes. We give theamplitudes for meson-baryon scattering or pion-nucleon scattering, specifically,meaning that we take equal initial and final states (Mf = Mi = M and mf = mi = m,where M and m are the masses of the nucleon and pion, respectively). The resultsfor general meson-baryon initial and final states are presented in appendix A1 ofpaper 2. For the description of the scalar meson exchange processes at tree level,graphically shown in figure 6, we use the interaction Lagrangians (?? ) and (?? ), whichlead to the vertices

ΓPPS = gPSS , ΓS = gS .

q

p

q′

p′

Pa

κ

κ′

κ1

(a)

q

p

q′

p′

Pb

κ

κ′

κ1

(b)

Figuur 6: Scalar meson exchangeTh.A. Rijken University of Nijmegen NFQCD 2010: Kadyshevsky Formalism – p. 71/103

Meson-exchange,2

Scalar-Meson-Exchange II

Applying the Kadyshevsky rules the amplitudes read

Ma,bκ′κ = gPSSgS

Zdκ1

κ1 + iε

ˆū(p′s′)u(ps)

˜θ(P 0a,b)δ(P

2a,b −M2S) , (82)

where Pa,b = ±∆t + 12 (κ′ + κ)− nκ1 (here a corresponds to the + sign and b to the −sign) and ∆t = 12 (p

′ − p− q′ + q). For the κ1 integration we consider the δ-function in(82)

(a) : δ(P 2a −M2S) =1

|κ+1 − κ−1 |`δ(κ1 − κ+1 ) + δ(κ1 − κ−1 )

´,

κ±1 = ∆t · n+1

2

`κ′ + κ

´± At ,

(b) : δ(P 2b −M2S) =1

|κ+1 − κ−1 |`δ(κ1 − κ+1 ) + δ(κ1 − κ−1 )

´,

κ±1 = −∆t · n+1

2

`κ′ + κ

´± At , (83)

where At =p

(n ·∆t)2 −∆2t +M2S . In both cases θ(P 0a,b) selects the κ−1 solution.Therefore,

Pa = ∆t − (∆t · n)n+Atn , Pb = −∆t + (∆t · n)n+ Atn . (84)


Meson-exchange,3

Scalar-Meson-Exchange III

With these expression we find for the amplitudes

M(a)

κ′κ = gPSSgSû̄(p′s′)u(ps)

˜ 12At· 1∆t · n+ κ̄− At + iε

,

M(b)κ′κ = gPSSgS

û̄(p′s′)u(ps)

˜ 12At· 1−∆t · n+ κ̄− At + iε

, (85)

where κ̄ = 12(κ′ + κ). Adding the two together and putting κ′ = κ = 0 we get

M00 = gPSSgSû̄(p′s′)u(ps)

˜ 1t−M2S + iε

, (86)

which is Feynman result [Pol05].


Meson-exchange,4

Meson-Exchange IV

We discussed the n-dependence of the Kadyshevsky integral equation. In order to dothat we need to know the n-dependence of the amplitude (15)

M(a+b)0κ = M

(a)0κ +M

(b)0κ ,

= gPSSgSû̄(p′s′)u(ps)

˜ At − κ2At

1

(∆t · n)2 − (κ2 −At)2 + iε,

∂Ma+b0κ∂nβ

= κ gPSSgSû̄(p′s′)u(ps)

˜

×n ·∆t(∆t)β2A3t

(n ·∆t)2 − 3A2t − κ2

4+ 2κAt“

(n ·∆t)2 −Àt − κ2

´2+ iε

”2 . (87)

If we would only consider scalar meson exchange in the Kadyshevsky integralequation the integrand would be of the form (17), where h(κ) would by itself be oforder O( 1

κ2) as can be seen from (87). Therefore, the phenomenological "form

factor"(19) would not be needed.


Vector-Meson-exchange,1

Vector-Meson-Exchange, Example I

Before we go on with real vector meson exchange, we consider simplified vectormeson exchange. We use this as an example to illustrate seaming problems thatmight occur in the results in the Kadyshevsky formalism, especially when compared tothose in the Feynman formalism. We stress that although we consider the example ofsimplified vector meson exchange, these peculiarities are generally present wheninteraction Lagrangians containing derivatives and/or higher spin fields (s ≥ 1) areconsidered.In order to study simplified vector meson exchange we take again the interactionLagrangian, without the σµν -term

LI = g φai←→∂µφb · φµ + g ψ̄γµψ · φµ , (88)

For the various components of the diagrams we take the following functions

Dµν =

„−gµν + P

µP ν

M2V

«1

P 2 −M2V + iε,

Γψ̄ψµ = g γµ , Γφφµ = g

`q′ + q

´µ, (89)

where we obtained the vertex functions via LI = −HI → −Γ.



Vector-Meson-Exchange, Example II

The Kadyshevsky diagrams for the (simplified) vector meson exchange are shown infigure 7.

Pa ↓ κ1

p p′

q q′

(a)

κ

κ′

Pb ↑ κ1

p p′

q q′

(b)κ

κ′

Figuur 7: Vector meson exchange in Kadyshevsky formalism.



Vector-Meson-Exchange III

Applying the Kadyshevsky rules we get the following amplitudes

M(a,b)

κ′κ = −g2

Zdκ1

κ1 + iε

ˆū(p′s′)γµu(ps)

˜ „gµν −

Pµa,bPνa,b

M2V

«

× θ(P 0a,b)δ(P 2a,b −M2V )`q′ + q

´ν, (90)

The κ1-integral is discussed in (83) and (84). We, therefore, give the result immediately

M(a)

κ′κ = −g2 ū(p′s′)

»2Q/− 1

M2V

„(Mf −Mi) + 1

2n/(κ′ − κ)− (∆t · n−At)n/

«

×„

1

4(sp′q′ − spq) +

1

4(upq′ − up′q)− (m2f −m2i )

− 2(∆t · n− At)n ·Q«–

u(ps)

× 12At

1

∆t · n+ 12 (κ′ + κ)− At + iε,

(91)



Vector-Meson-Exchange IV

M(b)κ′κ = −g

2 ū(p′s′)

»2Q/− 1

M2V

„(Mf −Mi) + 1

2n/(κ′ − κ)− (∆t · n+At)n/

«

×„

1

4(sp′q′ − spq) +

1

4(upq′ − up′q)− (m2f −m2i )

− 2(∆t · n+At)n ·Q«–

u(ps)

× 12At

1

−∆t · n+ 12 (κ′ + κ)− At + iε. (92)

Adding the two together and putting κ′ = κ = 0 we should get back the Feynmanexpression

M00 = M(a)00 +M

(b)00

= −g2ū(p′s′)»2Q/+

(Mf −Mi)M2V

(m2f −m2i )–u(ps)

1

t−M2V + iε

−g2ū(p′s′) [n/]u(ps) 2Q · nM2V

. (93)



Vector-Meson-Exchange V

From (93) we see that the first term on the rhs is indeed the Feynman result. However,the second term on the rhs is an unwanted, n-dependent, contact term.As mentioned before, similar discrepancies are obtained when couplings containinghigher spin fields (s ≥1) are used. Therefore, it seems that the Kadyshevskyformalism doesn’t yield the same results in these cases as the Feynman formalismwhen κ′ and κ are put to zero. Since the real difference between Feynman formalismand Kadyshevsky formalism lies in the treatment of the TOP or θ-function also thedifference in results should find its origin in this treatment.In Feynman formalism derivatives are taken out of the TOP in order to get Feynmanfunctions, which may yield extra terms. This is also the case in the above example a

T [φµ(x)φν(y)] = −»gµν +

∂µ∂ν

M2V

–i∆F (x− y)− iδµ0 δν0 δ4(x− y) ,

Sfi = (−i)2g2Zd4xd4y

ˆψ̄γµψ

˜xT [φµ(x)φν(y)]

hφa←→i∂νφb

iy,

⇒Mextra = −g2ū(p′s′) [n/]u(ps) 2Q · nM2V

. (94)

aOf course that’s why we have chosen such an example.



Vector-Meson-Exchange VI

If we include the nµ-vector in the θ-function of the TOP, which would not make adifference, then we can make the replacement δµ0 → nµ. This, to make the result moregeneral.If we include the extra term of (94) on the Feynman side we see that both formalismsyield the same result. So, that’s cured.Although we have exact equivalence between the two formalisms, the result, thoughcovariant, is still n-dependent, i.e. frame-dependent. Of course this is not what wewant. As it will turn out there is another source of extra terms exactly cancelling forinstance the one that pops-up in our example ((93), (94)). There are two methods forgetting these extra terms cancelling the one in (93) and (94): the Takahashi-Umezawa(TU) method is more fundamental and the Gross-Jackiw (GJ) method is moresystematic and pragmatic. Next we introduce the Takahashi-Umezawa.


Takahashi-Umezawa,1

Takahashi & Umezawa Solution I

In order to demonstrate the TU method [Tak53a,Tak53b,Ume56] we start with arewritten version of the Yang-Feldman (YF) equations [Yang50] for a generalinteraction

Φα(x) = Φα(x)−Zd4y Rαβ(∂) Da(y) ∆ret(x− y) · jβ;a(y), (95)

where Φα(x) and Φα(x) are fields in the Heisenberg Representation (H.R.) and InteractionRepresentation (I.R.), respectively. In the following ΦI.R. ≡ Φin, which is in line ofthe later link with the S-matrix formulation.Furthermore, the vectors Da(x) and jα;a(x) are defined to be

Da(x) ≡ (1, ∂µ1 , ∂µ1∂µ2 , . . .) ,

jα;a(x) ≡„− ∂LI∂Φα(x)

, − ∂LI∂ (∂µ1Φα(x))

, − ∂LI∂ (∂µ1∂µ2Φα(x))

, . . .

«, (96)

Next, a free auxiliary field Φα(x, σ) is introduced, where σ is a space-like surface andx does not necessarily lie on σ. We pose that it has the following form

Φα(x, σ) ≡ Φα(x) +Z σ

−∞

d4y Rαβ(∂)Da(y) ∆(x− y) · jβ;a(y) , (97)


Takahashi-Umezawa,2

Takahashi & Umezawa Solution II

Combining (97) with (95) leads to

Φα(x) = Φα(x/σ) +1

2

Zd4y

»Rαβ(∂)Da(y), ǫ(x− y)

–∆(x− y) · jβ;a(y). (98)

This equation will be used to express the fields in the H.R. in terms of fields in the I.R.In section [BMP] it is explained that the auxiliary fields and the fields in the I.R. arerelated by a unitary operator using the BMP theory. Also it is shown how theinteraction Hamiltonian should be deduced.Applying these concepts to our example of the simplified vector meson exchangewhere we studied the interaction Lagrangian

LI = g φai←→∂µφb · φµ + g ψ̄γµψ · φµ , (99)

we determine the "currents"via (96)jφa,a = (−g i∂µφb · φ

µ, ig φb · φµ) ,jφb,a = (g i∂µφa · φ

µ,−ig φa · φµ) ,jψ,a = (−g γµψ · φµ, 0) ,

jφµ,a =“−g φa

←→i∂µφb − g ψ̄γµψ, 0

”. (100)


Takahashi-Umezawa,3

Takahashi & Umezawa Solution III

Using (98) we can express the fields in the H.R. in terms of fields in the I.R., i.e. freefields

φa(x) = φa(x/σ) , φb(x) = φb(x/σ) ,

∂µφa(x) = [∂µφa(x, σ)]x/σ +1

2

Zd4y

ˆ∂xµ∂

yν , ǫ(x− y)

˜∆(x− y) (igφb · φν)y

= [∂µφa(x, σ)]x/σ + ignµφb n · φ ,

∂µφb(x) = [∂µφb(x, σ)]x/σ +1

2

Zd4y

ˆ∂xµ∂

yν , ǫ(x− y)

˜∆(x− y) (−igφa · φν)y

= [∂µφb(x, σ)]x/σ − ignµφa n · φ ,ψ(x) = ψ(x/σ) ,

φµ(x) = φµ(x/σ) +

1

2

Zd4y

»„−gµν − ∂

µ∂ν

M2V

«, ǫ(x− y)

–∆(x− y)

×“−gφa

←→i∂νφb − gψ̄γνψ

”y

= φµ(x/σ)− g nµ

M2V

“φan ·

←→i∂ φb + ψ̄n/ψ

”. (101)


Takahashi-Umezawa,4

Takahashi & Umezawa Solution IV

As can be seen from (98) the first term on the rhs is a free field and the second termcontains the current expressed in terms of fields in the H.R., which on their turn areexpanded similarly. Therefore, one gets coupled equations. In solving these equationswe assumed that the coupling constant is small and therefore considered only termsup to first order in the coupling constant in the expansion of the fields in the H.R.Practically speaking, the currents on the rhs of (101) are expressed in terms of freefields. These expansions (101) are used in the commutation relations of the fields withthe interaction Hamiltonian

[φa(x),HI(y)] = iU−1(σ)∆(x− y)h−g i∂µφb · φµ + g

←−i∂µφb · φµ

iyU(σ)

= i∆(x− y)»−g ←→i∂µφb · φµ + g

2

M2Vn · ←→i∂ φb

“φan ·


”− g2 φa(n · φ)2

–

y

[ψ(x),HI(y)] = iU−1(σ)(i∂/+M)∆(x− y) [−g γµψ · φµ]y U(σ)

= i(i∂/+M)∆(x− y)×»−g γµψ · φµ + g

2

M2Vn/ψ

“φan ·


”–

y

,

(102)


Takahashi-Umezawa,5

Takahashi & Umezawa Solution V

[φµ(x),HI(y)] = iU−1(σ)„−gµν − ∂

µ∂ν

M2V

«∆(x− y)

×h−g φa

←→i∂νφb − g ψ̄γνψ

iyU(σ)

= i

„−gµν − ∂

µ∂ν

M2V

«∆(x− y)

h−g φa

←→i∂νφb − g ψ̄γνψ

− g2 nν φ2an · φ− g2 nν φ2bn · φiy. (103)

As stated below these are the fundamental TU-equations from which the interactionHamiltonian can be determined

HI = −g φa←→i∂µφb · φµ − g ψ̄γµψ · φµ − g

2

2φ2a(n · φ)2 −

g2

2φ2b(n · φ)2

+g2

2M2V

ˆψ̄n/ψ

˜2+

g2

M2V

ˆψ̄n/ψ

˜ hφan ·

←→i∂ φb

i+

g2

2M2V

hφan ·

←→i∂ φb

i2+O(g3) . . . .

If equation (101) was solved completely, then the rhs of (101) would contain higherorders in the coupling constant and therefore also the interaction Hamiltonian (104).These terms are indicated by the ellipsis.


Takahashi-Umezawa,6

Takahashi & Umezawa Solution VI

Defining the θ-function to be 1 in its origin we assure that all terms in the interactionHamiltonian (104) relevant to πN -scattering are quadratic in the external quasi fields,even higher order terms in the coupling constant.The only term of order g2 in (104) that gives a contribution to the first order in theS-matrix describing πN -scattering is the first term on the last line of (104). Itscontribution to the first order in the S-matrix is

S(1)fi = −i

Zd4xHI(x) = −ig

2

M2V

Zd4x

ˆψ̄n/ψ

˜ hφan ·

←→i∂ φb

ix

=−ig2M2V

ū(p′s′)n/u(ps)n · (q′ + q) ⇒Mcanc = g2 ū(p′s′)n/u(ps)2n ·QM2V

. (104)

Indeed we see that this term (104) cancels the extra term in (93).


TU-method

Takahashi-Umezawa method: S-matrix formulation

Takahashi-Umezawa [Tak53a,Ume56] introduced the auxiliary σ-field

χ[x, σ] ≡ χ(x) +Z σ

−∞

d4x′ ∆(x− x′) j(x′) , (105)

satisfying the KG-equation with the free-field mass m. According to Haag’s theorem ingeneral there does not exist a unitary transformation which relates the in(out)-fieldsφas(x) to the Heisenberg fields Φ(x). On the other hand there is no objection againstthe existence of a unitary U [σ] relating the TU-auxiliary and the asymptotic fields

Φ[x, σ] = U−1[σ] φas(x) U [σ] , (106)with U [σ] ≡ U(σ,−∞). This because both fields are free fields with equal mass.We establish these matters, deriving the relation between the TU Hint, the U [σ],and a perturbative formula of the S-matrix in the context of the axiomatic S-matrixtheory. We follow here Bogoliubov and collaborators [BMP58,BS59,BLT75],see also [Schweb64], section 18b.We refer in the following to this as the BMP-theory.• Haag, Mat. Fys. Medde. Dan. Vid. Selsk. 29, No. 12, 1955• Bogoliubov & Logunov & Todorov Shirkov, Intr. to Axomatic Quantum Field Theory, 1975• Schweber, Relativistic Quantum Fields, 1962; Roman, Introduction to Quantum Field Theory,1969


TU-method

Asymptotic Completeness, Yang-Feldman Equations

In the context of axiomatic field theory, Lehmann, Symanzik, and Zimmermannformulated an asymptotic condition utilizing the notion of weakconvergence in the Hilbert space of state vectors. See e.g. [BD65] for a detailedexposition of the LSZ-formalism. Here, the Heisenberg field operator Φ(x) andthe asymptotic fields φas with as = −in,−out satisfy the equations a

(∂2 +m2)Φ(x) = j(x) , (∂2 +m2)φas = 0 , (107)

where j(x) is the Heisenberg current, and the asymtotic fields are free fields.Then, the relation between these fields is given by the Yang-Feldman (YF)equations

Φ(x) = φin,out(x) +

Z∆ret,adv(x− y) j(y) d4y. (Y F ) (108a)

Upon subtraction we get the relation

φout(x) = φin(x)−Z

∆(x− y) j(y) d4y . (109)Next one assumes the completeness of both the asymtotic fields, i.e. for thecorresponding Hilbert spaces and vacuum states one has

Hin = Hout = H , |0〉in = |0〉out = |0〉 . (110)It can be shown that on a certain dense set of Has the LSZ asymptoticconditions and the YF-equations are valid, cfrm. (BLT75), chapter 14.

a j( )


TU-method

Scattering Matrix, Unitarity, Microcausality

In order to bypass the use of a unitary operator U as a mediator between theasymptotic and the (interacting) Heisenberg fields, we follow here the method ofBogoliubov and collaborators, see [BS59,BLT75].Assume that the S-operator is a functional of the asymptotic fields χρ(x). In thefollowing we use in-fields, i.e. χρ(x) = φin,ρ(x). Then, the assumption ofasymptotic completeness implies the existence of the expansion

S = 1 +∞X

n=1

Zd4x1 . . . d

4xn Sn(x1α1, . . . , xnαn) : χα1(x1) . . . χαn(xn) : . (S)(111)

This is no restriction, because any e.g. ordinary product of fields can beexpanded in normal-ordered ones by Wick’s theorem. Also, the use of the normal-ordered products here ensures the stability of the vacuum, i.e. 〈0|S|0〉 = 11. Then, functional derivatives of the S-operator have the form

δS

δχα(x)=

X

n

X

j

δααj

Zd4x1 . . . d̂4xj . . . d

4xn ·

Sn (x1α1, . . . , xjαj = xα, . . . xnαn) : χα1(x1) . . . χ̂αj (xj) . . . χαn(xn) : .(112)

2. Unitarity S† S = 1 gives upon functional differentiationδS†

δχρ(x)S = −S† δS

δχρ(x), (A) (113)


TU-methodand a similar relation starting from S S† = 1.

3. The Heisenberg current is defined as

jρ(x) = iS† δS

δχρ(x)(B) (114)

4. Then microcausality takes the form, see (BS59), chapter 17δjρ(x)

δχλ(y)= 0 , for x ≤ y . (c) (115)

Theorem 0.1 Let H(x) be a (local) function of the asymptotic fieldsχα(x), which is defined even when χα(x) do not satisfy free field equations.Let the S-operator be defined as the time-ordered exponential

S = T

»exp

− i

Zd4x HI(x)

ff–. (D) (116)

Then, the microcausility conditionδ

δχβ(y)

S†

δS

δχα(x)

ff= 0 for x ≤ y . (117)


TU-methodFurthermore, it follows from (B) and (D)

jρ(x) =∂HI(x)

∂χρ(x). (118)

For example in QED HI(x) = eψ̄(x)γµψ(x)Aµ(x), and we get that theelectromagnetic current operator is jµ(x) = eψ̄(x)γµψ(x),which is correct [BD65].5. It can be shown that with the current (B) the asymptotic fields χin/out,ρ(x)satisfy the Yang-Feldman equation

χout,ρ(x) = χin,ρ(x)−Zd4y ∆(x− y) jρ(y) . (119)


TU-method

Correspondence with LSZ-Theory

The correspondence is obtained by the identification

jρ(x) = iS† δS

δχρ(x)≡

`∂2 +m2

´χρ(x) , (120)

where χρ(x) denotes the (interacting) Heisenberg field. We notethat for a hermitean field χρ(x) the current is also hermitean, due to the relation

j†ρ(x) = −iδS†

δχρ(x)S = jρ(x) . (121)

Functionally differentiating the current gives the equationsδjρ(x)

δχσ(y)= iS†

δ2S

δχσ(y)δχρ(x)+ ijσ(y) jρ(x) , (122a)

δjσ(y)

δχρ(x)= iS†

δ2S

δχρ(x)δχσ(y)+ ijρ(x) jσ(y) . (F ) (122b)

Subtraction givesδjρ(x)

δχσ(y)− δjσ(y)δχρ(x)

= −i»jρ(x) , jσ(y)

–. (E) (123)

Note that for spacelike separations, i.e. (x− y)2 < 0, causality and (E) imply that thecurrent commutators vanish.


TU-method

Takahashi-Umezawa: ’Fundamental relation’ HintTo determine Hint we follow Takahashi-Umezawa [Tak53a,Ume56].Using their auxiliary σ-field

χ[x, σ] ≡ χ(x) +Z σ

−∞

d4x′ ∆(x− x′) j(x′) , (J) (124)

which like χ(x) ≡ χin(x) satisfies the free KG-equation with mass m.First, one can prove the following theoremConjecture 0.1

»χ[x, σ], χ[y, σ]

–=

»χ(x), χ(y)

–= i∆(x− y;m) . (125)


TU-method

Proof: 0.1 Using (J) gives»χ[x, σ], χ[y, σ]

–−

»χ(x), χ(y)

–= +

Z σ

−∞

d4y′ ∆(y − y′)»χ(x), j(y′)

–

−Z σ

−∞

d4x′ ∆(x− x′)»χ(y), j(x′)

–

+

Z σ

−∞

Z σ

−∞

d4x′d4y′ ∆(x− x′)∆(y − y′)»j(x′), j(y′)

–.

Now, we use (E) and (C1) to rewrite the above expression:»χ[x, σ], χ[y, σ]

–−

»χ(x), χ(y)

–= −i

Z σ

−∞

d4y′Z ∞

−∞

d4x′∆(x− x′)∆(y − y′)

× δj(y′)

δj(x′)+ i

Z σ

−∞

d4x′Z ∞

−∞

d4y′ ∆(x− x′)∆(y − y′)δj(x′)

δj(y′)

−iZ σ

−∞

d4x′Z σ

−∞

d4y′ ∆(x− x′)∆(y − y′)„δj(x′)

δj(y′)− δj(y

′)

δj(x′)

«

⇒ 0 ,which follows from the microcausality condition that annihilates the possiblycontributions from the difference in boundary values in the integrals. (Q.E.D.)


TU-methodThis justifies the conjecture of the existence of a unitary operator U [σ] such that

χ[x, σ] = U−1[σ] χ(x) U [σ] . (K) (126)

Functional differentiation of (J) and (K) gives

δχ[x, σ]

δσ(x′)= U−1[σ]

»χ(x),

δU [σ]

δσ(x′)U−1[σ]

–U [σ] = ∆(x− x′) j(x′) . (127)

defining the interaction Hamiltonian density by the TS-equation

iδU [σ]

δσ(x′)= HI(x, σ) U [σ] , (128)

the solution is the time ordered exponential

U [σ] = T

exp

»−i

Z σ

−∞

d4x HI(x, n)–ff

. (129)

This gives the TU-fundamental equation for the determination of Hint:»χ(x),HI(x, n)

–= i∆(x− x′) U [σ]j(x)U−1[σ] . (TU) (130)


TU-methodFinally, we notice that the field χ[x, σ] provides a free field interpolationbetween χin(x) and χout(x): see (YF) and (J). Therefore

χout = χ[x,∞] = U−1[∞] χin U [∞] , (131)with χin(x) ≡ χ(x). One sees that

S = U [+∞] = U [+∞,−∞] , U [−∞] = 1 , (132)which brings the familiar relation:

χout(x) = S−1 χin(x) S . (133)

This establishes the relation of the U -matrix, and ipso facto of the interactionHamiltonian, to the S-matrix.We conclude that we have demonstrated/established the Takahashi-Unezawa-methodin the framework of the axiomatic S-matrix theory. Therefore, it is not affected by theHaag-theorem.


TU-method

Application TU-scheme in BMP-theory

It is clear from above that in the BMP-theory the TU-equations for the Heisenbergfields (eq:T.16), and the determining equation for the interaction Hamiltonian(eq:T.23) are valid. Next we give two examples of application of the formalismdescribed above.1. PV-Coupling Nucleons: Then, starting with the interaction Lagrangian

Lpv(x) = fmπ

ψ̄(x)γµγ5ψ(x) · ∂µφ(x) . (134)

Then, like in equation (eq:T.28) one gets that»φ(x) , HI [x′ : n]

–= −i(f/mπ) U [σ] ∂′µ

`ψ̄(x′)γµγ5 ψ(x

′)´U−1[σ] ∆(x− x′) ,

»ψ(x) , HI [x′ : n]

–= −(f/mπ) γµγ5 U [σ] ψ(x′) · ∂′µφ(x′) U−1[σ] S(x− x′) ,(135)

and similarly to (eq:T.29)

ψ(x) = ψ[x/σ] , φ(x) = φ[x/σ] , (136a)

∂µφ(x) = (∂µφ[x, σ])x/σ +f

mπ

„ψ̄[x/σ]γνγ5ψ[x/σ)

«nµ(x) nν(x) . (136b)


TU-methodSo, in this case there was no problem with the handling of the operators

involving the Heisenberg fields ψ and φ, which occur in

U [σ] j(x′) U−1[σ] .

This, because of the identity of the U [σ]-transformed free fields ψ[x/σ] and φ[x/σ] asexpressed in the first two identities in (eq:U.73). Therefore, the construction of theinteraction Hamiltonian can be done and is the same as that foundpreviously in this case: expression (eq:T.32).

2. Vector-Coupling Nucleons:


TU-method

BMP-theory and Haag-theorem

There still seems to be a problem: considering the PV-coupling example

Kadyshevsky Field Theory Relativistic πN, KN, and BB ...II. Kadyshevsky Formalism (KFT) , S-matrix,...

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