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PION PROPERTIES

IN CHIRAL PERTURBATION THEORY AT

NEXT-TO-LEADING ORDER

Judith Plenter

Tutor : Antonio Pich Zardoya

Curso académico 2015-2016

Abstract

Chiral Perturbation Theory is a low-energy effective field theory for the strong inter-action. It is based on the spontaneous breaking of the approximate chiral symmetry inQuantum Chromodynamics and allows to study the dynamics of strong interactions at longdistances in terms of the pseudoscalar mesons. Within this formalism, the mass, the decayconstant, the electromagnetic form factor and the charge radius of pions are calculated atnext-to-leading order and the low-energy constants Lr5 and Lr9 are determined.

Resumen

La Teorıa Quiral de Perturbaciones es una teorıa efectiva de las interacciones fuertes abaja energıa basada en la ruptura espontanea de la simetrıa quiral, que es una simetrıaaproximada del lagrangiano de la Cromodinamica Cuantica. Esta teorıa efectiva per-mite describir las interacciones fuertes a grandes distancias a traves de los mesones pseu-doescalares. Mediante este formalismo se calcula la masa, la constante de desintegracion,el factor de forma electromagnetico y el radio de carga del pion a segundo orden y sedeterminan los acoplamientos Lr5 y Lr9.

Contents

1. Chiral Perturbation Theory 11.1. Strong Interaction in the Standard Model . . . . . . . . . . . . . . . . . . . 11.2. Chiral Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3. Transformation Properties of Goldstone Bosons . . . . . . . . . . . . . . . . 31.4. Low-energy Effective Field Theory for the Strong Interaction . . . . . . . . 5

2. Derivation of Currents 92.1. Derivation of Operators at O(p4) in the Chiral Expansion . . . . . . . . . . 92.2. The Axialvector Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3. The Vector Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3. Self-energy: Mass and Wave-function Renormalization 173.1. Calculation of the Pion Self-energy . . . . . . . . . . . . . . . . . . . . . . . 183.2. Field-strength Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 213.3. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4. The Pion Decay Constant 234.1. Parametrization of Charged Pion Decay . . . . . . . . . . . . . . . . . . . . 234.2. The Pion Decay Constant at O(p4) . . . . . . . . . . . . . . . . . . . . . . . 244.3. Determination of Lr5(µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5. Electromagnetic Form Factor 295.1. Electromagnetic Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2. Calculation of the Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . 305.3. The Electromagnetic Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6. Conclusion 37

A. Expansions in Terms of the Meson Fields 39A.1. Expansion of U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2. Expansion of L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

B. Loop Integrals 43

iii

1. Chiral Perturbation Theory

Until the middle of the 20th century the most fundamental particles known were the con-stituents of the atoms, namely the electron, the proton and the neutron and additionallythe positron as well as the muon. Beginning with the discovery of the pion in 1947 theamount of discovered particles multiplied rapidly due to the advancements in acceleratortechnology. The multitude of particles created the need for a systematic explanation.

Since the discovery of the neutron as part of the atomic nuclei, it had become apparentthat the physical description in terms of only the electromagnetic interaction was insuffi-cient. Many of the newly discovered particles also appeared to be sensitive to this stronginteraction that was seemingly causing the binding of the nucleus. It was then possible togroup these so-called hadrons into multiplets with the same spin and same transformationproperties under parity. Thus those multiplets could be identified with the irreduciblerepresentations of the symmetry group SU(3). In this system, proposed by Gell-Mannand Ne’eman as the Eightfold Way, hadrons are built from the more fundamental quarks.Originally the quarks were introduced as fictitious constituents in order to make the group-theoretical classification of the hadron spectrum possible [GM64]. Only after the substruc-ture of the hadrons was experimentally probed were their components identified with thequarks (and the gluons mediating the interaction between them) [Sch03].

The implementation of the strong interaction between the quarks as a gauge theory iscalled Quantum Chromodynamics (QCD). Together with the unified theory of the elec-troweak interaction it makes up the Standard Model of particle physics. The elementaryparticles of the Standard Model besides the quarks are the leptons (electron, muon, tauand the corresponding neutrinos) and the gauge bosons mediating their interactions (pho-ton, gluon, Z0- and W±- boson) as well as the Higgs boson responsible for the masses offermions and gauge bosons.

1.1. Strong Interaction in the Standard Model

The quarks in the Standard Model are fermions that appear in six different flavors (up,down, charm, strange, top, bottom). The detected mesons are identified with the boundstate of a quark and an antiquark while the (anti-)baryons correspond to a state of three(anti-)quarks. Applying this concept to all of the hadronic spectrum one needs to introducea new quantum number, the color charge, in order to not violate Fermi-Dirac statisticsin the baryonic sector. Color itself cannot be observed since all asymptotic states arecolor singlets leading to the first experimental requirement for QCD: Due to their colorcharge, quarks can only appear in color-neutral bound states. This concept is calledConfinement. From measuring the ratio between the hadronic and leptonic τ decay widthsthe number of colors can be determined to be three. Further it has been experimentallyestablished through the measurement of the proton form factors in deep inelastic scatteringexperiments that quarks behave as nearly free particles at very short distances, leading toanother experimental requirement which is called Asymptotic Freedom [Pic12] [Pic99].

Since all asymptotic states are color singlets the most relevant symmetry of QCD is the

1

invariance under rotations in the three-dimensional color space. Particularly this SU(3)csymmetry, acting on the quark vector in color space, should also hold when being promotedto a local one

q −→ exp

−igs θa(x)

λac2

q (1.1)

and can thus be used to construct the gauge theory of Quantum Chromodynamics. Hereθa(x) is an arbitrary function depending on space-time, the T ac = λac/2 are the generatorsof SU(3)c with the λac being the 8 Gell-Mann matrices and gs is the strong couplingconstant. Starting with the Dirac Lagrangian and requiring the invariance under thegauge symmetry, the QCD Lagrangian becomes

LQCD =∑

flavor

qf(i /D −m

)qf −

1

4Gµνa Gaµν , (1.2)

where the covariant derivative is given by

Dµq =

[∂µ − igs

λac2Gµa(x)

]q, (1.3)

with the gluon fields Gµa(x) and the gluon field-strength tensor Gµνa [Pic99].Due to the fact that SU(3)c is a non-abelian Lie group, LQCD generates gluon self-

interaction diagrams leading to the strong scale dependence of the QCD running couplingwhich to leading order is given by

αs(µ2) =

αs(µ20)

1− β12αs(µ20)π ln

(µ2

µ20

)+ ...

. (1.4)

The constant β1 is negative leading to the observed asymptotic behaviour of the QCDcoupling: Increasing the energy the strength of the strong interaction decreases, resultingin the phenomenon of asymptotic freedom and making the application of perturbationtheory possible. In this perturbative region the predictions of QCD are consistent withall experimental data. In the low-energy regime the coupling increases thus confiningthe quarks and gluons inside color-neutral hadrons. Since in this regime QCD is non-perturbative the direct connection between QCD and the relevant hadronic degrees offreedom (mesons and baryons) cannot be analytically calculated [Pic99] [Sch03] [Kub07].

A description of strong interaction in the low-energy regime can be achieved either bylattice computations or through the construction of an effective field theory depending onthe particles appearing at this energy scale. This can generally be done by finding themost general effective Lagrangian that respects the same symmetries as the fundamentalLQCD. With the quantum number color not showing up explicitly in any hadron, a theorybased on them will automatically be invariant under rotations in color space. However,besides the local gauge symmetry QCD has more global symmetries that need to berespected. On the one hand QCD is invariant under separate U(1) phase transformationsfor each flavor leading to flavor (and baryon) number conservation. On the other handQCD has an approximate symmetry that will be the basis for the construction of ChiralPerturbation Theory: Ignoring the quark mass term LQCD decouples into separate partsfor the left-handed and right-handed quarks qL/R = 1

2(1∓ γ5)q as

LQCD = qLi /DqL + qRi /DqR −1

4Gµνa Gaµν +O(m) = L0

QCD +O(m). (1.5)

2

This means that for vanishing quark masses (the “chiral limit”) the QCD Lagrangian isinvariant under separate SU(3) transformations in flavor space. Since the masses of the upand down quark, and also to a lesser degree the mass of the strange quark, are very smallcompared to the typical hadronic scale of about 1 GeV it is reasonable to treat the massesof these light quarks as a perturbation. This implies that (1.5) may be used for the lightquark sector, with exemplary the left-handed quark vector in flavor space transforming asa (3, 1) multiplet under the chiral symmetry group G = SU(3)L×SU(3)R [Sch03] [Pic98]:

qL =

uds

L

G−−→ gL qL, gL = exp

−iθa

λaL2

∈ SU(3)L. (1.6)

Due to the unitarity of the elements of SU(3), LQCD is invariant under chiral transforma-tions when the quark masses can be neglected.

1.2. Chiral Symmetry Breaking

Chiral symmetry is certainly explicitly broken through the non-vanishing quark masses butthis is expected to be a small effect and can thus be treated as a perturbation. Conceptuallymore relevant is on the other hand that the doubled SU(3) symmetry is not representedin the hadronic spectrum. The linear combination T aV = T aR + T aL generates the invariantsubgroup H ∈ G which is realized in the Wigner-Weyl mode meaning that the irreduciblerepresentations of H can be related to the physical states: The mesonic sector of thehadron spectrum contains an octet of pseudoscalar mesons with the pions, kaons andη in a mass range of (135–548) MeV and quantum numbers JP = 0−. The remaininggenerators of G, which are given by T aA = T aR − T aL, should create degenerate multiplets(same mass, same spin) of opposite parity if one assumes the axial generators T aA to beunbroken. Those multiplets have not been observed, leading to the assumption that theground state is indeed not invariant under the full symmetry group G but instead onlyunder the subgroup H. The full symmetry group is thus spontaneously broken as

G = SU(3)L × SU(3)RSSB−−−−→ SU(3)V . (1.7)

According to the Goldstone theorem this spontaneous symmetry breaking produces onemassless Goldstone boson for each broken generator. The bosons’ transformation proper-ties under parity must be the same as those of the first component of the current associ-ated with the generator, meaning that due to the T aA producing an axial vector currentthe Goldstone bosons must be pseudoscalars. Although there exists no octet of masslesspseudoscalars the Goldstone bosons may be associated with the octet formed by the pions,kaons and the η since their masses are very small compared to the typical hadronic scale.Especially the pions, for whom the effect of the explicit symmetry breaking due to thequark masses is the smallest, are with less than 140 MeV by far the lightest particles inthe hadronic spectrum.

1.3. Transformation Properties of Goldstone Bosons

The effective theory in terms of the Goldstone bosons is built based on the spontaneousbreaking of the chiral symmetry. Therefore it is imperative to determine the transfor-mation properties of the meson fields under the elements of the chiral group, which aredefined only for the quark fields, to construct the effective Lagrangian invariant under it.

3

Defining a vector of the Goldstone fields as ~φ = (φ1, ..., φ8) where every element is a realscalar field φa : M4 → R set in Minkowski spacetime M4, the action of the symmetrygroup on ~φ is generally given by a mapping

~φG−−→ ~φ′ = ~f(g, ~φ), (1.8)

depending both on the field vector and the group element g ∈ G. Although this mappingis generally not a representation of the group G the action of the identity e should begiven by

~f(e, ~φ) = ~φ (1.9)

and the composition law (group-homomorphism property)

~f(g1, ~f(g2, ~φ)) = ~f(g1g2, ~φ) (1.10)

can be required.The origin of the field vector ~φ = ~0 may be identified with the ground state of the

system, which has to be invariant under the unbroken subgroup H. Therefore the imageof the origin with respect to elements of h is given by ~f(h,~0) = ~0 leading to the relationfor the mapping induced by a general group element:

~f(gh,~0) = ~f(g,~0) ∀g ∈ G ∀h ∈ H. (1.11)

This relation means in particular that the function ~f maps the origin onto the same vectorin the Goldstone space for a specific group element of G combined with any element ofH, which is just the definiton of the (left-) coset of g: gH = gh | g ∈ G. Cosets eithercompletely overlap or are completely disjoint such that the set of all distinct cosets isdenoted by G/H and called the quotient group. Due to H being an invariant subgroup,G/H = gH | g ∈ G can be seen as a group itself and is called coset space.

Two arbitrary elements of the coset space g1H and g2H are either equivalent or aremapped onto different vectors in the Goldstone space, which can be seen by assuming theopposite and mapping both ~f(gi,~0) and ~f(gj ,~0) with respect to g−1

i leading to ~f(e,~0) =~0 = ~f(g−1

i gj ,~0). Since only the elements of H leave the ground state invariant it wouldfollow that g−1

i gj ∈ H and therefore gj ∈ giH which is contradictory to the assumptionof two disjoint cosets. Therefore the mapping is isomorphic and invertible leading to theinterpretation that the elements of the quotient space (the disjoint sets of cosets) producea mapping of the meson ground state onto different vectors in the Goldstone space leadingto the possibility of identifying the Goldstone fields with the elements of G/H.

Specifically for each coset gH and therefore for each Goldstone field an arbitrary groupelement r = ghr ∈ gH is chosen to be the coset representative. Through the action in themapping ~f the effect of the group elements on the Goldstone bosons can be seen:

~φ = ~f(r,~0)G−−→ ~f(g, ~φ) = ~f(gghr,~0) = ~f(g′hr,~0) (1.12)

So to obtain the transformed ~φ′ from a given ~φ one needs to multiply the coset gH rep-resenting the original field by the group element g to obtain the coset representing thetransformed field. Although the transformed field is sufficiently identified by any memberof the new coset, the found one is not necessarily from the set of chosen coset represen-tatives, making a correcting transformation necessary such that the coset representativeidentifying the transformed meson field is given by r′ = gghrh

−1 = grh−1.

4

In the chiral symmetry the transformation is produced by the separate left- and righthandedgroup elements g = (gL, gR). The convention for the choice of the coset representativeis to rewrite g = (E, gRg−1

L )(gL, gL) characterising each Goldstone boson by the unitarymatrix defined through the representatives of the left- and righthanded coset

U(φ) = rRr−1L = exp

(iφaλa

f

)(1.13)

transforming linearly under the chiral group as [Kub07] [Sch03] [Jon90]

U(φ)→ (gRrRh−1)(gLrLh

−1)−1 = gRU(Φ)g−1L . (1.14)

1.4. Low-energy Effective Field Theory for the StrongInteraction

In the formalism of the effective field theory called Chiral Perturbation Theory (ChPT)the Goldstone fields are parametrized in flavor space as

U(Φ) = ei√2f

ΦΦ(x) =

1√2φaλa =

1√2π0 + 1√

6η8 π+ K+

π− − 1√2π0 + 1√

6η8 K0

K− K0 − 2√6η8

,

(1.15)

where f is a dimensionful constant, that can be related to the pion decay constant, andthe fields of the pseudoscalar mesons are linear combinations of the real fields φa.

From this matrix the most general Lagrangian invariant under the chiral symmetry,parity and continuous Lorentz transformations has to be constructed. Being an effectivefield theory the Lagrangian can be written as an expansion in terms of powers of themomenta, which appear in the Lagrangian through derivatives. Since the derivative of afield ∂µφ is odd under parity only terms with an even number of derivatives may appear.A term without any derivatives can only be constant since UU † = 1 and is therefore of nointerest, leading to the general expansion

LChPT(U) =∑n

L2n = L2 + L4 +O(p6). (1.16)

An isolated theory of pseudoscalar mesons is only useful for calculating pure meson pro-cesses. Much more interesting is the study of the hadronic part of the Standard Modelwhich includes also couplings to the electroweak part. At the same time it is not rea-sonable to completely neglect the explicit breaking of the chiral symmetry through thenon-vanishing quark masses considering that especially the kaons and the η masses can-not be considered a very small perturbation. Both of these concepts can be incorporatedinto the theory by introducing couplings to external classical fields, both in LQCD and inLChPT, in the most general (and therefore equivalent) way. Organizing potential externalfields into groups based on their properties under Lorentz transformations one can intro-duce the generalized external fields vµ = 1

2(rµ + lµ), aµ = 12(rµ − lµ), s and p which are

Hermitian matrices coupling to the quark fields as

LQCD = L0QCD + qγµ (vµ + γ5aµ) q − q (s− iγ5p) q. (1.17)

5

Here the electromagnetic field enters for example as vµ ⊃ eQAµ where the quark-chargematrix Q = 1

3diag(2,−1,−1) introduces an explicit breaking of chiral symmetry. Thequark-mass matrix appears in s ⊃M = diag(mu,md,ms).

As expected the explicit symmetry breaking has the effect that after having introducedthe external fields LQCD turns out to not be invariant any more under the chiral symmetrygroup. This can be fixed if one defines the external fields to also transform under a localSU(3)L × SU(3)R transformation as

s+ ip −→ gR(s+ ip)g†L,

lµ −→ gLlµg†L + igL∂µg

†L, (1.18)

and rµ −→ gRrµg†R + igR∂µg

†R.

The left- and righthanded external currents can be introduced into the effective Lagrangianby defining the covariant derivative

DµU = ∂µU − irµU + iUlµ (1.19)

and the field-strength tensors FµνL/R as

FµνL = ∂µlν − ∂ν lµ − i[lµ, lν ]. (1.20)

In leading order the fields aµ and vµ contribute to the effective Lagrangian in L2 onlythrough the covariant derivatives since the field-strength tensors contracted with deriva-tives of U would automatically be of O(p4). The most general Lagrangian invariant underthe assumed symmetries is then given by

L2 =f2

4〈DµU

†DµU + U †χ+ χ†U〉 where χ = 2B0(s+ ip). (1.21)

The trace 〈...〉 is necessary to obtain a scalar Lagrangian from the field and mass matricesand the two appearing constants f and B0 can be related to the pion decay constant andthe quark condensate [Pic98].

For any given order in p2 the matrix U needs to be expanded in order to find theLagrangian in terms of the meson fields. In O(p2) an expansion up to order φ4 in thefields is sufficient for finding kinetic, mass and interaction terms. To find these terms,while considering the explicit symmetry breaking through the quark masses but onlymeson-meson interactions, one sets s =M and rµ = lµ = p = 0. Inserting the expansionof U (see appendix A.1) one finds that

L02 =

f2

4〈∂µU †∂µU〉+

f2

2B0〈M(U † + U)〉

= f2B0(mu +md +ms) +1

2〈∂µΦ∂µΦ〉 −B0

⟨MΦ2

⟩+

1

12f2

⟨(Φ↔∂µ Φ

)(Φ↔∂µ Φ

)⟩+B0

1

6f2

⟨MΦ4

⟩+O

(Φ6

f4

). (1.22)

While the first term is an irrelevant constant, the second one delivers the kinetic terms asexpected for the mesons as real or complex Klein-Gordon fields

1

2〈∂µΦ∂µΦ〉 =

1

2∂µπ

0∂µπ0 + ∂µπ−∂µπ+ + ∂µK

0∂µK0 + ∂µK−∂µK+ +

1

2∂µη8∂

µη8

(1.23)

6

and the second one determines the meson masses as functions of the quark masses :

−B0〈MΦ2〉 = − 2B0m︸ ︷︷ ︸M2π+

π−π+ −B0(md +ms)︸ ︷︷ ︸M2K0

K0K0 −B0(mu +ms)︸ ︷︷ ︸M2K+

K−K+

−B0m (π0)2 −B01

3(m+ 2ms)η

28 −B0

mu −md√3

π0η8, (1.24)

where m ≡ (mu + md)/2 has been defined. The interaction terms can be found in theappendix A.2.

Since the π0 and the η8 have the same quantum numbers a mixing between the inter-action eigenstates takes place so that their masses cannot be immediately read off theLagrangian. Since finally the mixing is small this effect will be neglected in the following.Further, since the difference between the up and down quark mass is small compared tothe difference to the mass of the strange quark, most calculations will be done in theisospin limit mu = md = m. In this sense quark masses appearing in O(p4) results will besubstituted by the following expressions for the meson masses:

2B0m = M2π B0(m+ms) = M2

K (1.25)

The particle η8 appearing in ChPT is not a physical particle since it mixes with the η1

stemming from the pseudoscalar singlet1 forming the physical particles η and η′. Sincethis process is not negligible with a considerably big mixing angle of at least θ = −11.5

[B+12, p. 200] it is not justified to use the mass of the η meson. Instead the parameterMη8 may be written in terms of the pion and kaon masses as

M2η8 ≈

2

3B0 (m+ 2ms) =

4M2K −M2

π

3, (1.26)

which is just the Gell-Mann-Okubo mass relation.For calculations at O(p4) the next-order chiral Lagrangian is necessary, which is given

in terms of ten measurable low-energy constants as [Pic98]

L4 = L1〈DµU†DµU〉2 + L2〈DµU

†DνU〉〈DµU †DνU〉+ L3〈DµU

†DµUDνU†DνU〉+ L4〈DµU

†DµU〉〈U †χ+ χ†U〉+ L5〈DµU

†DµU(U †χ+ χ†U)〉+ L6〈U †χ+ χ†U〉2 (1.27)

+ L7〈U †χ− χ†U〉2 + L8〈χ†Uχ†U + U †χU †χ〉− iL9〈FµνR DµUDνU

† + FµνL DµU†DνU〉+ L10〈U †FµνR UFLµν〉

+H1〈FRµνFµνR + FLµνFµνL 〉+H2〈χ†χ〉.

Since the last two terms in this Lagrangian do not contain pseudoscalar fields they arenot directly measurable.

Of these constants the renormalized couplings Lr5(µ) and Lr9(µ) will be determined inthe following through the calculation of the pion decay constant and the pion vector formfactor, respectively.

A general amplitude at O(p4) will include tree-level contributions from both L2 and L4

as well as loop-diagrams from L2. The divergences contained in these expressions haveto be absorbed by redefinitions of the low-energy couplings appearing in L4, which is thetypical procedure for an effective field theory: Every order in the momentum expansion isrenormalized entirely within itself.

1An effective field theory containing the η1 is given by a version of ChPT which includes an expansion inthe inverse of the number of colors 1/Nc to extend SU(3)R × SU(3)L × U(1)V to U(3)L × U(3)R.

7

2. Derivation of Currents

Chiral Perturbation Theory is not an isolated theory of pseudoscalar mesons instead itis a low-energy effective field theory of the strongly interacting part of the StandardModel. Therefore the ability to calculate processes involving only those mesons cannotbe sufficient. The interactions of the non-hadronic part of the Standard Model with thequarks are clearly given by LSM, although the part involving quarks and gluons cannot becalculated for low energies. The relevant Green’s functions involving both non-hadronicfields as well as quark and gluon fields can be separated into the solvable electroweak partand the strong part. This strong part will generally be the vacuum expectation valueof an operator written in terms of quark and gluon fields, which cannot be calculatedperturbatively in QCD. In order to calculate this expression in ChPT the action of theoperator on the ChPT-vacuum must be known and therefore the form of the operator interms of the Goldstone fields needs to be found. This can be systematically done usingthe path integral formalism.

2.1. Derivation of Operators at O(p4) in the Chiral Expansion

The general expression for the derivation of n-point correlation functions using the pathintegral formalism is [PS95]

〈Ω|T [φ(x1)...φ(xn)] |Ω〉 =

∫Dφ φ(x1)...φ(xn) exp

i∫

d4x L∫

Dφ expi∫

d4x L

=δ

δij(x1)...

δ

δij(xn)iZ[j]

∣∣∣∣j=0

, (2.1)

where the generating functional is defined as

W [j] = eiZ[j] =

∫Dφ exp

i

∫d4x (L+ jφ)

. (2.2)

In the case of strong interaction the generating functional can be written both in termsof the fundamental QCD as well as using the effective theory:

W [vµ, aµ, s, p] = eiZ[vµ,aµ,s,p]

=

∫Dq Dq DGµ exp

i

∫d4x

(L0

QCD + qγµ(vµ + γ5aµ)q − q(s− iγ5p)q)

(2.3)

=

∫DU exp

i

∫d4x (L2 + L4 + ...)

.

Thus operators and amplitudes from the the two formalisms can be related, with J

9

being the appropriate combination of functional derivatives:1

〈Ω| O(q, q, Gµ) |Ω〉 =

∫Dq Dq DGµ O(q, q, Gµ) exp

i∫

d4x LQCD

∫Dq Dq DGµ exp

i∫

d4x LQCD

=

δ

δiJZ[vµ, aµ, s, p]

∣∣∣∣j=j0

=

∫DU δ

δiJ expi∫

d4x (L2 + L4 + ...)∫

DU expi∫

d4x (L2 + L4 + ...) ∣∣∣∣∣

j=j0

(2.4)

=

∫DU O(Φ) exp

i∫

d4x (L2 + L4 + ...)∫

DU expi∫

d4x (L2 + L4 + ...)

= 〈0| O(Φ) |0〉 .

Here and in the following j = j0 is used as an abbreviation for vµ = aµ = p = 0 ands =M.

Due to the nature of the ChPT Lagrangian as an expansion in the momentum theoperator O(Φ) will generally be an infinite sum of products of Goldstone fields and theirderivatives. Since the ChPT vacuum is perturbative the vacuum expectation value of theoperator in terms of the Goldstone fields can be determined by expanding around theclassical solution, which involves solving the path integrals as of O(p4).As another possibility the operator can be inserted into transition amplitudes directly andevaluated including loop contributions depending on the desired degree of accuracy. Thissecond method will be pursued in the following.Up to O(p4) the necessary terms of the operator are

O(Φ) ≈∫

d4x

[δL2

δJ+δL4

δJ

]≡ O2(Φ) +O4(Φ). (2.5)

Of these, O2(Φ) needs to be derived up to an order in the fields that allows loop calculationswhile for O4(Φ) the leading order contribution is sufficient.

2.2. The Axialvector Current

The axial vector current is the operator derived from the generating functional by takingthe functional derivative with respect to the external field aµ. Using the QCD Lagrangian(1.17) it takes the simple form qγµγ5q. Using the formalism described above its shape interms of the Goldstone fields up to O(p4) is derived through derivatives of the first twoterms of the ChPT Lagrangian:

1The interacting vacuum of ChPT will be denoted as |0〉, while the non-perturbativity of the QCD vacuumwill be emphasized by the use of |Ω〉.

10

〈Ω| qγµγ5q |Ω〉 =δ

δiaµZ[vµ, aµ, s, p]

∣∣∣∣j=j0

=

∫DU δ

δiaµexp

i∫

d4x (L2 + L4 + ...)∫

DU expi∫

d4x (L2 + L4 + ...) ∣∣∣∣∣

j=j0

(2.6)

=

∫DU Aµ(Φ) exp

i∫

d4x (L2 + L4 + ...)∫

DU expi∫

d4x (L2 + L4 + ...)

= 〈0|Aµ(Φ) |0〉 .

This results in the expression

Aµ(Φ) = Aµ2 (Φ) +Aµ4 (Φ) + ... =

∫d4x

[δL2

δaµ+δL4

δaµ+ ...

]j=j0

. (2.7)

Specifically only the component [Aµ(Φ)]12 will be given here in terms of the Goldstonefields since it will be used in the calculation of the pion decay constant later on.

2.2.1. L2 Contribution

The external field aµ enters the chiral Lagrangian through the covariant derivative:

DµU |vµ=0 = ∂µU − iaµU − iUaµ DµU†∣∣∣vµ=0

= ∂µU† + iaµU

† + iU †aµ (2.8)

Taking into account that aµ is a matrix, the functional derivative of these expressionsreads

δ (DνU(x))αβ (x)

δaµ(y)=δ (DνU)αβδ(aµ)ρσ

= −iδνµ (δαρUσβ + δβσUαρ) δ(x− y) (2.9)

andδ(DνU

†(x))αβ

(x)

δaµ(y)= iδνµ

(δαρU

†σβ + δβσU

†αρ

)δ(x− y).

Thus the L2 contribution of the axial vector current becomes

Aµ2 (Φ) =f2

4

∫d4x

δ

δaµ

⟨DνU

†DνU⟩∣∣∣∣j=j0

=f2

4

∫d4x

[δ(DνU

†)αβ

δaµ(DνU)βα +

(DνU †

)βα

δ (DνU)αβδaµ

]j=j0

= if2

4

[(δαρU

†σβ + δβσU

†αρ

)(∂µU)βα −

(∂µU †

)βα

(δαρUσβ + δβσUαρ)

](2.10)

= if2

4

[U † · ∂µU + ∂µU · U † − U · ∂µU † − ∂µU † · U

]σρ

= if2

2

[∂µU · U † − ∂µU † · U

]This expressions can be written in terms of the Goldstone fields using the expansion ofthe matrix U as given in the equations (A.1):

Aµ2 (Φ) = −√

2f

[∂µΦ− 1

3f2

(∂µΦ Φ2 − 2Φ ∂µΦ Φ + Φ2 ∂µΦ

)+O

(Φ

f

)5]

(2.11)

11

Inserting the expression for Φ (1.15) the relevant component of the axial vector currentreads

[Aµ2 (Φ)]12

= −√

2f

[1− 1

3f2

(2π0π0 + 2π+π− +K+K− +K0K0

)]∂µπ+

+

√2

3fπ+(K0∂µK0 − 2K0∂µK0 +K−∂µK+ − 2K+∂µK− − 2π0∂µπ0 − 2π+∂µπ−

)− 1√

3f(K+η∂µK0 + K0η∂µK+ − 2K+K0∂µη) +

1

fπ0(K+∂µK0 − K0∂µK+

).

(2.12)

2.2.2. L4 Contribution

Of all the terms in the L4 Lagrangian only a few can contribute. To find out which onesit is necessary to consider that with respect to the expansion of the effective Lagrangianin momenta the leading order contribution of L4 is of the same order as the next-to-leading order contribution of L2. Since Aµ2 was derived until cubic order in the fields,allowing for one-loop diagrams when contracted with a single external field, the comparableAµ4 expression need only be of first order in the fields to deliver corresponding tree-levelamplitudes.

Another general observation is that only Lagrangian terms of first order in the externalfield aµ can contribute to the axial vector current since after the functional derivative allexternal fields except for the mass matrix are set to zero. This directly means that theterm proportional to L10 starting at order a2 does not contribute.

The first three terms of L4 all contain the covariant derivative DµU(†) four times. The

only terms that are of first order in aµ contain three derivatives of the meson matrix∂µU

(†) and are hence already of order φ3, meaning that they will not contribute to theleading-order L4 contribution to the axial vector current. On the other hand the termsproportional to the couplings L6, L7 and L8 do not have any covariant derivatives, arethus independent of aµ and do not contribute at any order.

The term L9 contains the aµ already in the field strength tensor so in any potentialcontribution the covariant derivatives reduces to the actual derivatives ∂µU

(†). Sincethere are two of them any contribution from L9 would at least be of quadratic order inthe fields (or considering parity in case of the axial vector current of cubic order) and cantherefore also be neglected in a tree-level calculation.

The terms that remain are the ones proportional to the low-energy constants L4 andL5, whose contribution to the axial vector current is calculated very similarly to the onefrom L2 seen before. For the L4 part the functional derivative gives

Aµ4,L4(Φ) =

δ

δiaµ

∫d4x iL4

⟨DνU

†DνU⟩⟨

U †χ+ χ†U⟩∣∣∣∣j=j0

= 2B0L4

∫d4x

δ

δaµ

⟨DνU

†DνU⟩

︸ ︷︷ ︸4/f2 Aµ0

⟨M(U † + U)

⟩∣∣∣vµ=aµ=0

=8B0L4

f2

⟨M(U † + U)

⟩ [Aµ0 +O

(Φ2

f2

)], (2.13)

where Aµ0 = −√

2f∂µΦ is introduced as a shorthand for the leading-order contribution toAµ2 (Φ). The remaining trace is, apart from the prefactors, the mass-dependent term of

12

the chiral Lagrangian at O(p2):

⟨M(U † + U)

⟩= 2(mu +md +ms)−

2

f2

⟨MΦ2

⟩+

1

3f4

⟨MΦ4

⟩+O

(Φ6

f6

). (2.14)

Since only the leading-order expression is needed nothing but the constant mass term isrelevant here. Inserting the expressions found for the meson masses in the isospin limit(1.25) the expression can be simplified to give

Aµ4,L4(Φ) = 2

8M2K + 4M2

π

f2L4

[Aµ0 +O

(Φ3

f3

)]. (2.15)

The other term of L4 that enters into the axial vector current is the one proportionalto L5:

Aµ4,L5(Φ) =

δ

δiaµ

∫d4x iL5

⟨DνU

†DνU(U †χ+ χ†U

)⟩∣∣∣∣j=j0

= 2B0 L5

∫d4x

δ

δaµ

(DνU

†DνU)αβ

(U †M+MU

)βα. (2.16)

Evaluating the functional derivative gives

δ

δaµ

(DνU

†DνU)αβ

=δ(DνU

†)αγδaµ

(∂νU)γβ + (∂νU †)αγδ(DνU)γβ

δaµ

= iδ(x− y)[δαρ(U

†∂µU)σβ + U †αρ(∂µU)σβ (2.17)

−(∂µU †)αρUσβ − δβσ(∂µU †U)αρ

].

Combining the terms yields the following expression:

Aµ4,L5(Φ) = 2iB0 L5

U † ∂µU︸ ︷︷ ︸−∂µU† U

U †M+ U † ∂µU︸ ︷︷ ︸−∂µU† U

MU + ∂µU U †M U † + ∂µU M

−M∂µU † − U MU∂µU † − U †M ∂µU †U︸ ︷︷ ︸−U†∂µU

−MU ∂µU †U︸ ︷︷ ︸−U†∂µU

= 2iB0 L5

[(∂µU − ∂µU †

)M− ∂µU † U MU + ∂µU U †M U †

+M(∂µU − ∂µU †

)− U MU∂µU † + U †M U †∂µU

]. (2.18)

Since an expansion of U that is linear in Φ is sufficient,

U †MU † ∂µU − UMU ∂µU † =M(∂µU − ∂µU †

)+O

(Φ3

f3

)(2.19)

and thus the final expression for the L5 contribution to the axial vector current:

Aµ4,L5(Φ) =

8B0

f2L5

[Aµ0M+MAµ0 +O

(MΦ3

f3

)](2.20)

13

Taking only the component that will be used for the calculation of the pion decay anddefining Aµπ = −

√2f∂µπ+ one gets[

Aµ4,L5(Φ)]12

= 24M2

π

f2L5

[Aµπ +O

(Φ3

f3

)]. (2.21)

Combining those results the complete axial vector current at O(p4) is given by

[Aµ(Φ)]12 =[Aµ2 (Φ) +Aµ4,L4

(Φ) +Aµ4,L5(Φ)]12

= [Aµ2 (Φ)]12

+

(2

8M2K + 4M2

π

f2L4 + 2

4M2π

f2L5

)[Aµπ +O

(Φ3

f3

)]. (2.22)

2.3. The Vector Current

The steps to derive the vector current in terms of the Goldstone fields are analogous tothe derivation seen above. The general expression reads

〈Ω| qγµq |Ω〉 = 〈0|V µ(Φ) |0〉 =δ

δivµZ[vµ, aµ, s, p]

∣∣∣∣j=j0

, (2.23)

which leads to

V µ(Φ) = V µ2 (Φ) + V µ

4 (Φ) + ... =

∫d4x

[δL2

δvµ+δL4

δvµ+ ...

]j=j0

. (2.24)

2.3.1. L2 Contribution

The expressions for the covariant derivative in terms of vµ instead of aµ differ only by aminus sign:

DµU(†)∣∣∣aµ=0

= ∂µU(†) + iU (†)vµ − ivµU (†) (2.25)

Taking the functional derivative of these expressions one gets

δ(DνU

(†)(x))αβ

(x)

δvµ(y)= iδνµ

(δβσU

(†)αρ − δαρU

(†)σβ

)δ(x− y). (2.26)

Thus the L2 contribution of the axial vector current becomes

V µ2 (Φ) =

f2

4

∫d4x

δ

δvµ

⟨DνU

†DνU⟩∣∣∣∣j=j0

= if2

4

[(δβσU

†αρ − δαρU

†σβ

) (∂µU

)βα

+(∂µU †

)βα

(− δαρUσβ + δβσUαρ

)]= i

f2

2

[∂µU · U † + ∂µU

† · U]

(2.27)

Using the expansion of U in the fields (A.1) this leads to the expression

V µ2 (Φ) = −i

[Φ←→∂µΦ +

1

6f2

[∂µΦ Φ3 − Φ3 ∂µΦ

]+

1

2f2Φ (Φ

←→∂µΦ) Φ

]+O(Φ6). (2.28)

14

2.3.2. L4 Contribution

The vector current stemming from L4 has to be found up to quadratic order in the mesonfields. With the same argument as in the case of the axial vector current most terms of L4

need not be considered. The only addition comes from the Lagrangian term containingL9 which leads to a contribution at order Φ2 as will be shown in the following.

The contributions of the terms of L4 proportional to the constants L4 and L5 arecalculated just as in the axial vector case. The L4 one is particularly simple and directlygives

V µ4,L4

(Φ) = L4

∫d4x

δ

δvµ

⟨DνU

†DνU⟩

︸ ︷︷ ︸4/f2 V µ2

⟨U †χ+ χ†U

⟩∣∣∣j=j0︸ ︷︷ ︸

2B0〈M(U†+U)〉=4M2K+2M2

π+O(Φ2)

= −i8L4

f2

(2M2

K +M2π

) [Φ←→∂µΦ +O

(Φ4

f4

)](2.29)

From the L5-term one gets

V µ4,L5

(Φ) = L5

∫d4x

δ

δvµ

⟨DνU

†DνU[U †χ+ χ†U

]⟩ ∣∣∣∣j=j0

= 2B0L5

∫d4x

(U †M+MU

)βα

δ

δvµ

(DνU

†DνU)αβ

∣∣∣∣j=j0

= 2iB0L5

[∂µU U †MU † + ∂µU † UMU − U †MU † ∂µU − UMU ∂µU †

+(∂µU + ∂µU †

)M−M

(∂µU + ∂µU †

)], (2.30)

which can be expanded to quadratic order in the meson fields to give

V µ4,L5

(Φ) =8iB0L5

f2([∂µΦM,Φ] + [M∂µΦ,Φ]) . (2.31)

The additional L9 contribution is given by

V µ4,L9

(Φ) = −iL9

∫d4x

δ

δvµ

⟨F ρσR DρUDσU

† + F ρσL DρU†DσU

⟩∣∣∣∣j=j0

,

where the derivative acting on the fields will not contribute due to the field strength tensorsvanishing when setting the external fields to zero:

= −iL9

∫d4x

δ

δvµ(∂ρvσ − ∂σvρ)αβ

(∂ρU∂σU † + ∂ρU †∂σU

)βα

= −iL9 ∂ν

(∂µU ∂νU † − ∂νU ∂µU † + ∂µU † ∂νU − ∂νU † ∂µU

)(2.32)

Expanding this to quadratic order in Φ leads the L9 contribution to the vector current tobe

V µ4,L9

(Φ) = −i4L9

f2

([∂ν∂µΦ, ∂νΦ] +

[∂µΦ, ∂2Φ

]+O

(Φ4

f4

)). (2.33)

15

3. Self-energy: Mass and Wave-functionRenormalization

The propagator of an interacting theory will always receive loop corrections and can there-fore only to leading order be approximated with the standard Feynman propagator. Inthe case of an effective theory like ChPT in addition to the loops, higher orders in themomentum expansion will also create additional tree-level contributions that have to beincluded. The general propagator for a scalar field φ, with the vacuum of the interactingtheory denoted by |Ω〉, can be written as an expansion in the self-energy

=

∫d4x eipx 〈Ω|Tφ(x)φ(0)|Ω〉

=

+

−iΣ +

−iΣ −iΣ + ...,

where the self-energy −Σ(p2) includes all irreducible diagrams contributing to the propaga-tor. In a quantum field theory that can be treated perturbatively the corrections throughthe self-energy should be small and so the expansion can be treated as a geometric seriesand rewritten as

=i

p2 −m20

[1 +

Σ(p2)

p2 −m20

+

(Σ(p2)

p2 −m20

)2

+ ...

]=

i

p2 −m20 − Σ(p2)

. (3.1)

The self-energy contains loop diagrams and will always be divergent. In order to calculatefinite amplitudes those divergences must be absorbed in the definition of the renormalizedphysical mass m and the field as φ =

√Zφr such that the propagator takes the shape

=iZ

p2 −m2 − Σr(p2). (3.2)

The pole of the propagator is a physical observable and must be indepent of arbitrarydefinitions so through an expansion of both (3.1) and (3.2) the value of the field-strengthrenormalization constant Z can be found. Expanding the denominator of the latter aroundthe position of the pole one gets

p2 −m2 − Σr(p2) = 0 +

(1− dΣr(p

2)

dp2

)∣∣∣∣p2=m2

(p2 −m2

)+ ..., (3.3)

where the derivative vanishes.The expansion of the bare denominator gives similarly

p2 −m20 − Σ(p2) =

(p2 −m2

0 − Σ(p2))∣∣p2=m2 +

(1− dΣ(p2)

dp2

)∣∣∣∣p2=m2

(p2 −m2

)+ ...

(3.4)

17

The first term in this expansion defines the physical mass through the condition

m2 − Σ(m2) = m20. (3.5)

In the second term the derivative of the unrenormalised self-energy does not vanish (infact it generally contains divergences), leading to the identification of the wavefunctionrenormalization as

Z =

[1− dΣ(p2)

dp2

∣∣∣∣p2=m2

]−1

. (3.6)

In the Kallen-Lehmann representation the full propagator then takes the form

=

iZ

p2 −m2 + iε+ [reg. at p2 = m2]. (3.7)

In calculating amplitudes that need renormalization this has the consequence that onemust multiply the result with a factor of

√Z for each incoming or outgoing particle to

account for the renormalization of the external legs. This can also be seen in the LSZreduction formula, which for a two-point correlation function is given by∫

d4xd4y ei(qy−px) 〈Ω|Tφ(x)φ(y)|Ω〉 =i√Z

p2 −m2 + iε

i√Z

q2 −m2 + iε〈φ(q)|S|φ(p)〉.

(3.8)

Since here on the left the fully renormalized amplitude is calculated also the right sideof the equation must be renormalized which is being achieved with the wave-functionrenormalization factor [PS95].

3.1. Calculation of the Pion Self-energy

The pion self-energy at leading order gets contributions from the contact terms of L4 aswell as through loops stemming from the interaction terms of L2:

−iΣ(p2) =

L4 +

L2 (3.9)

= −iΣ4(p2)− iΣ2(p2). (3.10)

Of all the terms of the respective Lagrangians only those can contribute that have both aπ+ and a π− field to be contracted with the incoming and outgoing pions. The additionalfields in the L2 interaction terms must be of the form φaφ

∗a to be able to contract in the

shape of a loop. The fields appearing in the interaction terms may include two derivatives.Of these none or both must act on the pion field pair. If there was only one derivativeacting on a field forming the loop, the integral over the loop momentum q would includea factor qµ making it odd and thus zero. These considerations leave terms of the form

L2 ⊃ d φaφ∗a∂µπ

+∂µπ− + d′ π+π−∂µφa∂µφ∗a + d′′ π+π−φaφ

∗a (3.11)

L4 ⊃ c ∂µπ+∂µπ− + c′ π+π− (3.12)

to be isolated from the Lagrangians.

18

3.1.1. L2 Contribution

In the general case for a term of the form d φaφ∗a∂µπ

+∂µπ− the Feynman rule for the loopvertex is id p2 resulting in a contribution to the self-energy of

p p

φa , q

π+ π+d

= id p2

∫d4q

(2π)4

i

q2 −m2φ + iε

≡ id p2 Iφ. (3.13)

Here the notation Iφ for the divergent integral is introduced since it will appear frequentlythroughout the calculations. It can be solved employing the formalism of dimensionalregularization to give an explicit expression for the contained divergence in terms of theparameter R:

Iφ ≡∫

d4q

(2π)4

i

q2 −m2φ + iε

(3.14)

= µ2ε(mφ

4π

)2[R+ ln

m2φ

µ2

]where R ≡ 1

ε+ γE − 1− ln 4π.

For a term of the shape d′ π+π−∂µφa∂µφ∗a the Feynman rule depends on the loop momen-

tum leading to

p p

φa , q

π+ π+d′

= id′∫

d4q

(2π)4

iq2

q2 −m2φ + iε

≡ id′ m2 Iφ. (3.15)

In the third type of contributing Lagrangian term d′′ π+π−φaφ∗a there is no momentum

depence in the Feynman rule so the corresponding self energy diagram just gives

p p

φa , q

π+ π+d′′

= id′∫

d4q

(2π)4

i

q2 −m2φ + iε

≡ id′′ Iφ. (3.16)

The part of the L2 Lagrangian containing only terms with four Goldstone fields is

L2,4φ =1

6f2

[⟨∂µΦ Φ ∂µΦ Φ− ∂µΦ∂µΦ Φ2

⟩+B0

⟨MΦ4

⟩], (3.17)

19

when s =M and without external fields. This includes terms generating loops of chargedand uncharged pions and kaons:

− 1

6f2

[2π+π− + 2π0π0 +K+K− + K0K0

]∂µπ

+∂µπ−

− 1

6f2

[2∂µπ

0∂µπ0 + ∂µK−∂µK+ + ∂µK

0∂µK0]π−π+ (3.18)

+1

6f2

[M2ππ−π+ +M2

ππ0π0 + (M2

π +M2K)K+K− + (M2

π +M2K)K0K0 +M2

πη28

]π−π+.

Applying the schematics developed above one needs to keep in mind that the term contain-ing four charged pion fields (π−π+)2 has four equivalent possibilities to contract, leadingto an additional factor of four and further the term proportional to π−π+∂µπ

−∂µπ+ hasto be treated both as type d and as type d′.

Finally this leads the contribution to the self-energy stemming from the L2 loops to be

Σ2(p2) =1

6f2

(4p2 −M2

π

)Iπ +

1

3f2

(p2 −M2

π

)IK −

1

6f2M2πIη. (3.19)

3.1.2. L4 Contribution

The terms of L4 proportional to L1, L2 and L3 contain a minimum of four fields since theexpansion of ∂µU starts at O(Φ). Without external fields the terms L9 and L10 vanish (ordo not couple to the Goldstone bosons in the case of H1 and H2).

The contributions from the remaining terms will be calculated in the following in thelimit of isospin symmetry and the external fields set to zero except for s =M.

L4,4 = 2B0L4

⟨∂µU

†∂µU⟩⟨M(U † + U

)⟩(3.20)

Since the first trace is already O(Φ2) only the constant term of the second trace has to betaken into account.

=4L4

f2(2M2

K +M2π) 〈∂µΦ∂µΦ〉+O

(Φ4)

⊃ 8L4

f2(2M2

K +M2π) ∂µπ

−∂µπ+ (3.21)

In the last line all terms without pion fields were neglected. The next contribution stemsfrom the L5 part of the Lagrangian:

L4,5 = 2B0L5

⟨∂µU

†∂µU[U †M+MU

]⟩=

8B0L5

f2〈∂µΦ∂µΦ · M〉+O

(Φ4)

⊃ 8L5

f2M2π ∂µπ

−∂µπ+ (3.22)

Another contribution comes from the L6 part of L4:

L4,6 = 4L6B20

⟨M(U † + U

)⟩2

= const.− 16L6

f2

(2M2

K +M2π

)B0〈MΦ2〉+O(Φ4)

⊃ −16L6

f2

(2M2

K +M2π

)M2π π−π+ (3.23)

20

Potentially there could also be a contribution from the L7 term given by

L4,7 = 2B0L7

⟨M(U † − U

)⟩2

=16B0L7

f2

⟨MΦ +O(Φ3)

⟩2

=16B0L7

f2

(1√3

(m−ms)η8 +√

2mπ0

)2

+O(Φ4), (3.24)

which turns out to be indepent of the charged pion fields at O(Φ2). The last source ofcontact terms is the Lagrangian containing L8:

L4,8 = 4L8B20

⟨M(UMU + U †MU †

)⟩= const.− 16L8B

20

f2

⟨Φ2M2 + ΦMΦM

⟩+O(Φ4)

⊃ −16L8

f2M4ππ−π+ (3.25)

Combining these results the terms of L4 that contribute to the pion self-energy are

L4 ⊃8

f2

[L4(2M2

K +M2π) + L5M

2π

]∂µπ

−∂µπ+

− 16

f2

[L6

(2M2

K +M2π

)M2π + L8M

4π

]π−π+. (3.26)

The Feynman rule for the charged pion contact term can be read off this expression leadingto the self-energy contribution

Σ4(p2) =8

f2

[(2L6M

2π − L4p

2) (

2M2K +M2

π

)+ 2L8M

4π − L5p

2M2π

]. (3.27)

Combining the previous results one gets for the full self-energy of the charged pion theexpression

Σ(p2) = − 1

f2

[1

6Iπ +

1

3IK +

1

6Iη − 16L6

(2M2

K +M2π

)− 16L8M

2π

]M2π

+1

f2

[2

3Iπ +

1

3IK − 8L4

(2M2

K +M2π

)− 8L5M

2π

]p2.

(3.28)

3.2. Field-strength Renormalization

The field-strength renormalization factor Z needed to renormalize the external legs of theamplitudes to be calculated in the following chapters can be determined directly fromequation (3.6) to be

Z =

[1− 1

f2

(2

3Iπ +

1

3IK − 8L4

(2M2

K +M2π

)− 8L5M

2π

)]−1

. (3.29)

Keeping in mind that the self-energy is a correction, Z−1 can be expanded around 1 togive

= 1 +1

f2

(2

3Iπ +

1

3IK − 8L4

(2M2

K +M2π

)− 8L5M

2π

)+O(p4) (3.30)

21

3.3. Mass

The equation to determine the mass at O(p4) was found in (3.5). In an effective fieldtheory the divergences appearing at a certain order will always be renormalized by thetree-level amplitudes at that order, so the “bare” pion mass in this context is the pionmass determined at O(p2), which has already been substituted into the results until now.Taking now the self-energy at p2 = m2 means substituting the pion mass at O(p4), whichwill be called Mπ,4, into the equation.

Separating the self-energy into a constant and a momentum dependent part one has

Σ(p2) = C1 + C2p2, (3.31)

which inserted into the equation for the mass gives

M2π,4 =

M2π + C1

1− C2. (3.32)

Since both the “bare” pion mass and the coefficient C2 are of O(p2) while C1 is of O(p4)the denominator can be expanded such that

M2π,4

M2π

= (1 + C2) +C1

M2π

= 1 +1

f2

[1

2Iπ −

1

6Iη + 8(2M2

K +M2π)(2L6 − L4) + 8M2

π(2L8 − L5)

], (3.33)

where the infinities contained in the integrals Iπ and Iη can be explicitly expressed using(B.1), inserting the Gell-Mann Okubo mass relation (1.26) for the η mass, to give

= 1 +1

f2

[µ2ε

(4π)2

(5

9M2π −

2

9M2K

)R+

1

2

M2π

(4π)2lnM2π

µ2− 1

6

M2η8

(4π)2lnM2η

µ2

+ 8(2M2K +M2

π)(2L6 − L4) + 8M2π(2L8 − L5)

]. (3.34)

Although these divergences can be perfectly absorbed in the four low-energy couplingsappearing here, their exact definitions have to be determined elsewhere. Using the generalexpression

Li = Lri (µ) + Γiµ2ε

2(4π)2R (3.35)

the corrective divergence produced by the couplings is given by[8M2

K(2Γ6 − Γ4) + 4M2π(2Γ8 − Γ5 + 2Γ6 − Γ4)

] µ2ε

(4π)2f2R. (3.36)

For the pion mass to take a finite value this leads to the conditions

2Γ8 − Γ5 + 2Γ6 − Γ4 = − 5

36and 2Γ6 − Γ4 =

2

72, (3.37)

which are consistent with Γ4 = 18 , Γ5 = 3

8 , Γ6 = 11144 and Γ8 = 5

48 given in [GL85].The renormalized pion mass at O(p4) is then given by

M2π,4 = M2

π

1 +

M2π

32π2f2lnM2π

µ2−

M2η8

96π2f2lnM2η

µ2

+8

f2

( (2M2

K +M2π

)[2Lr6(µ)− Lr4(µ)] +M2

π [2Lr8(µ)− Lr5(µ)])

. (3.38)

22

4. The Pion Decay Constant

4.1. Parametrization of Charged Pion Decay

Charged pions decay almost exclusively into a muon and anti-muon neutrino (or anti-muon and muon neutrino). On the level of the Standard Model this corresponds to theproduction of a W boson by the pion’s constituent quarks and it’s subsequent decay intothe lepton-neutrino pair. In the case of a π+ the leading order process is given by

W+

u

d

µ+

νµ

and can in principle be calculated through the Standard Model Lagrangian in leadingorder as

iM(π+ → µ+νµ) =⟨µ+νµ

∣∣T exp

(i

∫d4x LSM

) ∣∣π+⟩

(4.1)

= −Vudg2

16

∫d4x

∫d4y

⟨µ+νµ

∣∣W †µ(x)Wν(y)ν(x)γµ(1− γ5)µ(x) |0〉×

× 〈Ω| d(y)γν(1− γ5)u(y)∣∣π+

⟩, (4.2)

where the matrix element has been pulled apart into the electroweak part that can be an-alytically determined and the hadronic part involving the non-perturbative QCD vacuum.Because of the parity of the pion only the axial vector component can contribute leavingthe matrix element of the pion coupling to the axial vector quark current, that can beparametrized while defining the pion decay constant:

〈Ω| uγµγ5d∣∣π+

⟩= 〈Ω|Aµ(0)

∣∣π+⟩≡ i√

2fπpµ. (4.3)

In the QCD Lagrangian used here the coupling to the W is introduced through the externalfield

aµ =e

2√

2 sin θW

W †µ0 Vud Vus

0 0 00 0 0

+ h.c.

. (4.4)

Operators of Goldstone fields that correspond to the ud-annihilation into a W+ can thusbe deduced from the effective Lagrangian through the functional derivative with respectto a12

µ , which makes it clear that the relevant component of the matrix-valued axial vectorcurrent needed for the calculation of the pion decay is [Aµ]12.

23

4.2. The Pion Decay Constant at O(p4)

The matrix element for the pion decay at O(p4) will include the tree-level contributionsof L2 and L4 as well as 1-loop diagrams of L2. Since the divergences of the loops makerenormalization necessary also corrections to the external pion field stemming from thepion self-energy need to be included. Thus the full amplitude will involve the followingcontributions:

〈0| [Aµ2 (Φ)]12 ∣∣π+(p)

⟩=

Aµ2 +

Aµ2 +

Aµ4

+

Aµ2 L2 +

Aµ2 L4

Here the last two diagrams represent the contribution from the renormalization of theexternal pion field.

Of the axial vector current derived in chapter 2 only the terms in the first line ofequation 2.12 can contribute to the pion decay in 4.3, since the contraction of a field andit’s derivative vanish (second line) or there is no possibility of contraction for the incomingpion (third line). The remaining terms of the axial vector current are

[Aµ2 (Φ)]12 ⊃ −

√2f

[1− 1

3f2

(2π0π0 + 2π+π− +K+K− +K0K0

)]∂µπ+. (4.5)

Thus the evaluation of equation 4.3 at O(p4) involves matrix elements of the shape

〈0|φ(x)φ∗(x)∂µπ+(x)∣∣π+(p)

⟩= 〈0|φ(x)φ∗(x) |0〉 〈0| ∂µπ+(x)

∣∣π+(p)⟩︸ ︷︷ ︸

−ipµe−ipx

= −ipµe−ipx Iφ, (4.6)

where the definition of the integral B.1 is used. In the isospin limit (mu = md = m) theaxial vector current stemming from L2 acts in the pion decay amplitude 4.3 as

〈0| [Aµ2 ]12 ∣∣π+

⟩bare

= −√

2f 〈0|[1− 1

3f2

(2π0π0 + 2π+π− +K+K− +K0K0

)]∂µπ+

∣∣π+⟩

= i√

2f

(1− 4

3f2Iπ −

2

3f2IK

)pµ. (4.7)

The axial vector current stemming from L4 differs from the tree-level L2 result only byfactors (see eq. 2.22) so the full, bare amplitude reads

〈0| [Aµ]12∣∣π+

⟩bare

= i√

2f

(1− 4

3f2Iπ −

2

3f2IK

+28M2

K + 4M2π

f2L4 + 2

4M2π

f2L5

)pµ. (4.8)

24

The renormalized amplitude is calculated by multiplying the bare one with a factor of√Z for the external pion field as elaborated in section 3.2. The necessary term for the

pion decay is taken directly from 3.30:

√Z =

(1− dΣ(p2)

dp2

∣∣∣∣p2=m2

)−1/2

= 1 +1

2

dΣ(p2)

dp2

∣∣∣∣p2=m2

+O

((dΣ(p2)

dp2

)2)

= 1 +1

3f2Iπ +

1

6f2IK −

8M2K + 4M2

π

f2L4 −

4M2π

f2L5. (4.9)

All terms differing from one in this expression are of O(p2). Since the contributions to theaxial vector current from loops and the ones from L4 are already suppressed at O(p2) withrespect to the tree-level L2 result, multiplying those terms with the non-trivial terms of√Z gives results that are suppressed at O(p4) and have to be neglected in this calculation.

Thus the final, renormalized expression for the pion decay matrix element becomes

〈0| [Aµ(Φ)]12∣∣π+(p)

⟩= i√

2fpµ[√

Z − 4

3f2Iπ −

2

3f2IK (4.10)

+28M2

K + 4M2π

f2L4 + 2

4M2π

f2L5

]= i√

2fpµ[1− Iπ

f2− IK

2f2+

8M2K + 4M2

π

f2L4 +

4M2π

f2L5

]= i√

2fπpµ,

where a comparison with the last line immediately gives the full expression for the piondecay constant at O(p4).

fπf

= 1− Iπf2− IK

2f2+

8M2K + 4M2

π

f2L4 +

4M2π

f2L5 (4.11)

= 1− 2µπ − µK −µ2ε

32π2f2

(2M2

π +M2K

)R+

8M2K + 4M2

π

f2L4 +

4M2π

f2L5

The divergence parametrized in R can now be absorbed in a redefinition of the couplingconstants L4 and L5 to give the final and finite result.

fπf

= 1− 2µπ − µK +8M2

K + 4M2π

f2Lr4(µ) +

4M2π

f2Lr5(µ) (4.12)

L4 ≡ Lr4(µ) +1

8

µ2ε

32π2R L5 ≡ Lr5(µ) +

3

8

µ2ε

32π2R (4.13)

µφ ≡m2φ

32π2f2lnm2φ

µ2(4.14)

4.3. Determination of Lr5(µ)

Analogously to the shown derivation the kaon decay constant at O(p4) is [Pic98]

fKf

= 1− 3

4µπ −

3

2µK −

3

4µη8 +

8M2K + 4M2

π

f2Lr4(µ) +

4M2K

f2Lr5(µ). (4.15)

25

Having in mind that fπ = f(1 +O(p2)) and using the expansion 1/(1− ε) = 1 + ε+O(ε2)one can simplify the ratio of the two decay constants to be

fKfπ

=1 + ∆fK +O(p4)

1 + ∆fπ +O(p4)= 1 + ∆fK −∆fπ +O(p4) (4.16)

= 1 +1

4(5µπ − 2µK − 3µη8) +

4

f2

(M2K −M2

π

)Lr5(µ) +O(p4),

which may be used to determine the value of the renormalized coupling Lr5(µ) to be

Lr5(µ) =1

4(M2K −M2

π)

[f2

(fKfπ− 1

)− 1

128π2

(5M2

π lnM2π

µ2− 2M2

K lnM2K

µ2− 3M2

η8 lnM2η8

µ2

)].

(4.17)

Already in the calculation of the currents the terms in the final results involving the quarkmasses were substituted by the expressions for the meson masses found at O(p2) since theerror produced thereof would appear only at O(p6). Applying the same concept here letsone exchange f for fπ.

The error being made by identifying the measured ratio fK/fπ with the O(p4) resultcan be estimated by introducing a theoretical uncertainty of

∆th =p4

Λ4χ

, (4.18)

where Λχ ∼ 4πfπ ∼ 1.2 GeV is the typical breakdown scale of ChPT and p2 = M2K may

be fixed since the kaon mass is the highest energy scale in the contributing decays.Using the experimental values [O+14]

fKfπ

= 1.198± 0.005, (4.19)

fπ = 92.21± 0.14 MeV (4.20)

and the pseudoscalar masses as given by the Particle Data Group, the value of the low-energy coupling constant Lr5 can be determined. At the mass of the ρ meson its value isgiven by

Lr5(Mρ) = (1.2± 0.3) · 10−3. (4.21)

The uncertainty of Lr5 is mainly due to the neglected contributions of the next orders inthe chiral expansion. Ignoring the experimental uncertainties of the pion decay constantand fK/fπ as well as the error in the meson masses due to isospin breaking effects, theuncertainty of Lr5 changes only insignificantly:

∆Lr5 = 0.277 · 10−3 −→ 0.272 · 10−3. (4.22)

Although the low-energy constants of ChPT are most commonly determined at the ρmass this is pure convention. The scale-dependence of Lr5(µ), including all mentionederror sources, is shown in figure (4.1) for energies up to Λχ.

26

0 Mπ MK Mη Mρ

μ / MeV

2

4

6

8L5(μ)103

Figure 4.1.: The scale-dependence of the renormalized low-energy constant Lr5(µ) with theuncertainty given by the shaded region.

27

5. Electromagnetic Form Factor

The coupling of the charged pions to the electromagnetic current, and as such to photons,is parametrized in the electromagnetic form factor F (q2) as⟨

π+(p′)∣∣ Jµ(0)

∣∣π+(p)⟩

= (p+ p′)µ · F (q2) , where q = p′ − p. (5.1)

Due to Lorentz and gauge invariance other contributions are not possible and the formfactor at zero momentum transfer is fixed at F (0) = 1 [DGH92].

5.1. Electromagnetic Current

The photon coupling is introduced in the Lagrangian through the external vector currentvµ = eAµQ. In the QCD Lagrangian this creates the interaction term

LQCD ⊃ eAµqγµQq = eAµ(

2

3uγµu− 1

3dγµd− 1

3sγµs

), (5.2)

allowing to read off the classical contribution to the electromagnetic current in terms of thequark fields as Jµ = (2uγµu− dγµd− sγµs)/3, which can also be calculated by taking thetrace of the quark charge matrix and the vector-current. In this sense the electromagneticcurrent as given by the meson fields will take the form

LChPT ⊃ 〈vµV µ〉 = eAµ 〈QV µ〉 = eAµJµ (5.3)

By determining this trace with all the contributions to the vector current calculated insection 2.3 the electromagnetic current stemming from L2 reads

Jµ2 = 〈QV µ2 〉 =

[1− 1

6f2

(2π0π0 + 4π+π− +K0K0 + 4K+K−

)]i(π−∂µπ+ − π+∂µπ−

),

(5.4)

where the leading order contribution may be defined as Jµ0 ≡ i (π−∂µπ+ − π+∂µπ−), andthe contribution from L4 is given by

Jµ4 =⟨Q(V µ

4,L4+ V µ

4,L5+ V µ

4,L9

)⟩=

1

f2

[8L4

(2M2

K +M2π

)+ 8L5M

2π

]Jµ0

− i4L9

f2

[∂νπ

−∂ν∂µπ+ − ∂νπ+∂ν∂µπ− + ∂2π−∂µπ+ − ∂2π+∂µπ−]. (5.5)

29

5.2. Calculation of the Form Factor

The leading order contribution to the electromagnetic form factor of the charged pion isgiven by the matrix element

π+ π+Jµ0 = i⟨π+(p′)

∣∣π−∂µπ+ − π+∂µπ−∣∣π+(p)

⟩= (p+ p′)µ, (5.6)

showing that the form factor will be of the form

F (q2) = 1 + ∆F (q2). (5.7)

This means that at very low energies the interaction of a charged pion with a photon canbe reasonably well simplified by treating the pion as a point charge.

5.2.1. Loop Contribution from L2

For the form factor calculation at O(p4) two different types of loop diagrams will be rel-evant. On the one hand there are the loops produced by calculating the current matrixelement with the NLO expression for the electromagnetic current and on the other handthere are also diagrams that have a four meson interaction vertex producing a chargedmeson loop, which then interacts at leading order with the photon:

p p′

φ

π+ π+Jµ2

+

p p′π+ π+L2

Jµ2

Inserting the electromagnetic current from L2 (5.4) at O(p4) into expression 5.1 oneneeds to contract matrix elements of the form⟨

π+(p′)∣∣φφ∗(π−∂µπ+ − π+∂µπ−)

∣∣π+(p)⟩. (5.8)

If φ does not stand for the charged pion field the solution will be the loop integral Iφ (seeappendix B) multiplied to the LO solution. In the case of the charged pion field there aretwo possibilities to contract the expression leading to an additional factor of 2. Summingup and going into the isospin limit again the contribution to the form factor is given by

∆F2,i(q2) = − 5

6f2[2Iπ + IK ] . (5.9)

To identify the contribution stemming from the second type of loop diagram, first therelevant four-meson vertices need to be identified. Only terms of the Lagrangian that haveboth a π+ and π− field can contribute, as well as another charged particle-antiparticle pairto allow for the photon coupling. The relevant terms for the interaction of four charged

30

pions are given by

L2 ⊃1

6f2

[M2ππ

+π−π+π− + π−π−∂µπ+∂µπ+ + π+π+∂µπ

−∂µπ− − 2π+π−∂µπ+∂µπ−

+(M2π +M2

K

)K+K−π+π− −K+K−∂µπ

+∂µπ− − π+π−∂µK+∂µK− (5.10)

+ 2π−K−∂µπ+∂µK+ + 2π+K+∂µπ

−∂µK− − π−K+∂µπ+∂µK− − π+K−∂µπ

−∂µK+].

Keeping in mind that there are two possibilities each to contract the π+ and π− fields,which yealds a factor of four in each term, the matrix element for the interaction of fourcharged pions can be directly calculated.∫

d4x⟨π+(p′)π+(k′)

∣∣iL2

∣∣π+(p)π+(k)⟩

=i

6f2

∫d4x ei(p+k−p

′−k′)x ×

× 2[2M2

π − 2pk − 2p′k′ − pk′ − pp′ − kk′ − kp′]

= (2π)4δ(p+ k − p′ − k′) i

3f2

(t+ u− 2s+ 2M2

π

)(5.11)

Reading off the Feynman rule gives the result of

p

k

p′

k′

π+

π+

π+

π+

i

3f2

(t+ u− 2s+ 2M2

π

), (5.12)

where the arrows denote the direction of the momenta flowing and s = (p+k)2, t = (p−p′)2

and u = (p − k′)2 are the usual Mandelstam variables. Crossing symmetry immediatelygives the Feynman rule for the π+π− → π+π− vertex to be

p

k

p′

k′

π+

π−

π+

π−

i

3f2

(t+ s− 2u+ 2M2

π

). (5.13)

The Feynman rules for the corresponding diagrams involving kaons are calculated in thesame way. With the momenta defined equivalently to the pion case the result reads

iM(π+K+ −→ π+K+

)=

i

6f2

(t+ u− 2s+M2

π +M2K

)and (5.14)

iM(π+K− −→ π+K−

)=

i

6f2

(t+ s− 2u+M2

π +M2K

). (5.15)

31

Noticing that the coupling of the leading order electromagnetic current to positively andnegatively charged pions has exactly the opposite sign, one can simplify the calculationslightly by summing both over the π+ as well as over the π− loop. These are in princi-pal equivalent, since, due to crossing symmetry, an amplitude depending on a incomingparticle with momentum k is always equivalent to the same amplitude with the respectiveantiparticle going out with momentum −k [PS95].

The full pion loop contribution Iππ is thus equal to the sum of the π+ and π− loopsdivided by two:

iMloop =1

2

p p′

π+, k π+, k′

π+ π+L2

Jµ2

+

p p′

π−, k π−, k′

π+ π+L2

Jµ2

=

1

2

∫d4k

(2π)4

i

k2 −M2π + iε

i

(k − q)2 −M2π + iε

i

3f2(k + k′)µ ×

×[(t+ u− 2s+ 2M2

π)− (t+ s− 2u+ 2M2π)]

= − i

6f2

∫d4k

(2π)4

(k + k′)µ

[k2 −M2π + iε] [(k − q)2 −M2

π + iε]3(u− s) (5.16)

= i(p+ p′)ν

f2

∫d4k

(2π)4

(2k − q)µkν

[k2 −M2π + iε] [(k − q)2 −M2

π + iε](5.17)

The divergences appearing in this integral need to be brought into a form that will allow thesummation and cancellation of divergences appearing in the different O(p4) contributionsto the amplitude of the electromagnetic current. The rather lengthy calculation can befound in the appendix B and finally gives

iMloop = −(p+ p′)µ[

1

6f2

q2

M2π

Iπ −1

f2Iπ +

1

6f2

q2

(4π)2Aπ(q2)

], (5.18)

where the function

Aπ(q2) ≡ 8M2π

q2− 5

3+ σ3

π lnσπ + 1

σπ − 1(5.19)

has been defined and σπ =√

1− 4M2π

q2is the pion phase space factor. Adding also the

contribution from the kaon loop, which is the same expression divided by two and withthe proper substitution of masses, one can read off the correction to the form factor fromthe second type of loop diagrams as

∆F2,ii(q2) = − q2

6f2

(IπM2π

+IK

2M2K

)+

1

f2

(Iπ +

1

2IK

)− q2

96π2f2

(Aπ(q2) +

1

2AK(q2)

).

(5.20)

32

5.2.2. Contributions from L4

The contributions to the form factor stemming from L4 are found by evaluating the matrixelement of the electromagnetic current found as given in 5.5⟨

π+(p′)∣∣ Jµ4 (0)

∣∣π+(p)⟩. (5.21)

The first part of the current is directly proportional to the LO current and therefore thecorrection of the form factor is simply

∆F4,i(q2) =

1

f2

[8L4

(2M2

K +M2π

)+ 8L5M

2π

]. (5.22)

For the second part the matrix element is given by

−i4L9

f2

⟨π+(p′)

∣∣ ∂νπ−∂ν∂µπ+ − ∂νπ+∂ν∂µπ− + ∂2π−∂µπ+ − ∂2π+∂µπ−∣∣π+(p)

⟩= −i4L9

f2

[−ipp′(p+ p′)µ + ip′2pµ + ip2p′µ

]=

2

f2L9

[q2(p+ p′)µ − 1

2(p′2 − p2)(p′ − p)µ

]. (5.23)

The contribution to the form factor therefore becomes

∆F4,ii(q2) =

2L9

f2q2, (5.24)

as the incoming and outgoing pion are on-shell.

5.2.3. Renormalization of the Amplitude

As already seen for the case of the pion decay constant, the full renormalised amplitudeis calculated by multiplying the bare one with a factor of

√Z

2= 1 +

dΣ(p2)

dp2

∣∣∣∣p2=m2

+O

((dΣ(p2)

dp2

)2)

(5.25)

= 1 +2

3f2I(M2

π) +1

3f2I(M2

K)− 28M2

K + 4M2π

f2L4 −

8M2π

f2L5, (5.26)

which leads to a correction of the form factor of

∆FZ(q2) =2

3f2Iπ +

1

3f2IK −∆F4,i(q

2), (5.27)

where the last two terms have already been identified with the L4 contribution found inequation (5.22).

The full correction to the form factor is found by adding up the changes caused throughthe O(Φ4) terms of the electromagnetic current (∆F2,i, 5.9), the L2 loop diagram contain-ing the O(p2) vertices (∆F2,ii, 5.20), the tree-level L4 contributions (∆F4,i + ∆F4,ii, 5.22and 5.24) and the correction from the wave-function renormalization (∆FZ). The loopintegrals Iπ and IK that are not proportional to q2 as well as the terms containing L4 orL5 cancel directly, such that the remaining correction reads

∆F =2L9

f2q2 − q2

6f2

(IπM2π

+IK

2M2K

)− q2

96π2f2

(Aπ(q2) +

1

2AK(q2)

). (5.28)

33

Inserting the solutions for the integrals one can determine the necessary redefinition of L9

to make the amplitude finite. Thus the final expression is found to be

= 2L9q2

f2− µ2εR

2 · 32π2

q2

f2− q2

96π2f2

(lnM2π

µ2+

1

2lnM2K

µ2+Aπ(q2) +

1

2AK(q2)

),

leading to

F (q2) = 1 +2Lr9(µ)

f2q2 − q2

96π2f2

(lnM2π

µ2+

1

2lnM2K

µ2+Aπ(q2) +

1

2AK(q2)

), (5.29)

where the renormalised coupling is defined as

L9 = Lr9(µ) +1

4

µ2ε

32π2R. (5.30)

5.3. The Electromagnetic Radius

The electromagnetic form factor F (q2) for on-shell pions is completely determined bythe square of the transfered three-momentum and thus equal to the Fourier transformof the electric charge density ρ(x) and can therefore be calculated for certain exemplarydistributions. One sees that in the limit of a point-like particle the form factor is constantand (if properly normalised) equal to one. More extended charge distributions will leadto a decrease with q2. Since for wavelengths much smaller than the extension of thescattering potential the phases of the contributions will vary rapidly and cancel the formfactor must be assumed to go to zero for very big momentum transfers, justifying a Talylorexpansion around q2 = 0 [PRSZ95].

Further assuming ρ to be spherically symmetric the form factor is given by

F (q2) =

∫d3x eiqxρ(x)

= 4π

∫d|x| ρ(|x|) sin |q||x|

|q||x|︸ ︷︷ ︸1− q2

6|x|2+O(q4)

x2

= 4π

∫d|x| ρ(|x|)x2 − q2

64π

∫d|x| ρ(|x|)x4 + ... ,

which can be written as

= 1 +q2

6〈r2〉+ ... (5.31)

when choosing the proper normalisation and defining the mean square charge radius 〈r2〉.In order to identify the charge radius the formula for the form factor 5.29, specifically theterm containing the logarithm of the phase space factor, needs to be expanded in termsof the square of the momentum transfer q2. The relevant expansion reads

σ3π ln

σπ + 1

σπ − 1= −8M2

π

q2+

8

3− q2

10M2π

+O(q4), (5.32)

34

leading to the expression

Aπ(q2) = 1− q2

10M2π

+O(q4). (5.33)

Inserting this in the form factor formula and neglecting all terms of higher order than q2

one finds

F (q2) = 1 +1

6

[12Lr9(µ)

f2− 1

32π2f2

(2 ln

M2π

µ2+ ln

M2K

µ2+ 3

)]q2 +O(q4), (5.34)

which directly identifies the pion electromagnetic charge radius at O(p4) of ChPT as

〈r2〉 =12Lr9(µ)

f2− 1

32π2f2

(2 ln

M2π

µ2+ ln

M2K

µ2+ 3

). (5.35)

Mainly through pion scattering from electrons the radius has been experimentally deter-mined as [O+14] √

〈r2〉exp = (0.672± 0.008) fm, (5.36)

leading to the value for the low-energy coupling of

Lr9(Mρ) = (6.9± 0.6) · 10−3, (5.37)

where the error in the charge radius due to the truncation of the chiral expansion has beenestimated as

∆th〈r2〉 =p4

Λ4χ

fm2. (5.38)

The relevant energy scales in the pion electron scattering used to determine the electro-magnetic form factor are the pion mass and the momentum transfer q2. In [A+86], oneof the more precise sources for the charge radius cited by the PDG, the form factor ismeasured for q2 up to 0.26 GeV2 ≈M2

K , thus justifying to set p2 = M2K for the theoretical

error of identifying the measured charge radius with the O(p4) result.As already seen for Lr5, the uncertainty is almost entirely due to the neglected O(p6) con-

tributions to the form factor. Ignoring isospin breaking and the experimental uncertaintyof the pion decay constant ∆Lr9 changes only in the third significant figure.

The value of Lr5(µ) for energies up to Λχ is shown in figure (5.1).

35

Mπ MK Mη Mρ

μ / MeV

6

8

10

12

L9(μ)103

Figure 5.1.: The scale-dependence of the renormalized low-energy constant Lr9(µ) with theuncertainty given by the shaded region.

36

6. Conclusion

The strong interaction is described in the Standard Model through quantum chromody-namics (QCD), which is a gauge theory in terms of quarks and gluons. Although its pre-dictions in the high-energy regime are all consistent with experimental data, the dynamicsin the low-energy regime cannot be calculated analytically due to its running couplingconstant, which increases dramatically when lowering the energy scale. In the limit ofmassless quarks the QCD Lagrangian is symmetric under the chiral group, which consistsof separate SU(3) transformations for the left- and the right-handed quarks. This sym-metry is spontaneously broken leading to eight Goldstone bosons, which can be identifiedwith the pseudoscalar meson octet.

In terms of these mesons chiral perturbation theory (ChPT) is constructed as a low-energy effective field theory for the strong interaction. The explicit chiral symmetry break-ing through quark masses and charges, as well as the coupling to the non-hadronic sectorof the Standard Model, is implemented through the introduction of external classical fieldsinto both the QCD and the ChPT Lagrangians.

Using the path integral formalism, operators involving quarks can be related to thosecontaining meson fields. This is being done through functional derivatives of the generat-ing functional with respect to the external fields. In this way both the axial vector currentand the vector current have been determined up to an order in the meson fields sufficientfor the O(p4) calculations performed afterwards. In the case of the axial vector current,the contribution from L2, the leading-order term of the effective Lagrangian, has been cal-culated up to cubic order in the meson fields, as needed to calculate loop corrections to thepion decay constant later on. The contribution stemming from the next-to-leading orderterm L4 was only needed in tree-level amplitudes and has therefore only been determinedup to linear order in the meson fields.

As a further preliminary step, the pion self-energy has been calculated including loopcontributions from L2 and tree-level contributions stemming from L4. From this resultthe field-strength renormalization Z has been determined and its necessity for the renor-malization of physical amplitudes explained. As a byproduct the O(p4) expression for thepion mass has been found.

It was shown that the amplitude for the decay of a charged pion into a lepton-antineutrinopair can be factorized into a non-hadronic part and the hadronic part, where the latterinvolves the axial vector current and the decaying pion. This hadronic part cannot be cal-culated analytically within QCD and is parametrized in terms of the pion decay constant.In ChPT though, the amplitude involves the axial vector current in terms of pseudoscalarmeson fields, as found before. Using this current, the amplitude has been calculated whichinvolved solving loop diagrams and the inclusion of the field-strength renormalization. Theappearing divergences have been absorbed in a redefinition of the low-energy constants L4

and L5 which fixed the infinite parts of these constants. Thus the next-to-leading orderresult for the pion decay constant was found. By taking the ratio of the expressions for thekaon and pion decay constants, the renormalized low-energy constant Lr5 was determinedto be Lr5(Mρ) = (1.2± 0.3) · 10−3.

Finally, the electromagnetic form factor of the pion was calculated using the vector

37

current determined before. The part of the vector current stemming from L2 led to twodifferent types of loop diagrams: On the one hand it involved terms quadratic in the fieldsand on the other hand its leading order contribution had to be combined with leading orderinteraction vertices. Through the renormalization of this amplitude the infinite part ofL9 has been fixed and the renormalized Lr9 has been determined from the electromagneticcharge radius to be Lr9(Mρ) = (6.9± 0.6) · 10−3.

38

A. Expansions in Terms of the Meson Fields

A.1. Expansion of U

In the calculations an expansion of the unitary matrix U and its derivative in terms of themeson fields was needed, which are given here:

U(Φ) = exp

i

√2

fΦ

= 1 + i

√2

fΦ− 1

f2Φ2 − i

√2

3f3Φ3 +

1

6f4Φ4 +O

(Φ

f

)5

(A.1)

U †(Φ) = exp

−i√

2

fΦ

= 1− i

√2

fΦ− 1

f2Φ2 + i

√2

3f3Φ3 +

1

6f4Φ4 −O

(Φ

f

)5

∂µU(Φ) = i

√2

f∂µΦ− 1

f2[Φ ∂µΦ + ∂µΦ Φ]− i

√2

3f3

[∂µΦ Φ2 + Φ ∂µΦ Φ + Φ2 ∂µΦ

]= +

1

6f4

[∂µΦ Φ3 + Φ ∂µΦ Φ2 + Φ2 ∂µΦ Φ + ∂µΦ Φ3

]+O

(Φ

f

)5

∂µU†(Φ) = −i

√2

f∂µΦ− 1

f2[Φ ∂µΦ + ∂µΦ Φ] + i

√2

3f3

[∂µΦ Φ2 + Φ ∂µΦ Φ + Φ2 ∂µΦ

]= +

1

6f4

[∂µΦ Φ3 + Φ ∂µΦ Φ2 + Φ2 ∂µΦ Φ + ∂µΦ Φ3

]−O

(Φ

f

)5

39

A.2. Expansion of L2

The interaction terms of L2 involve a big quantity of terms, of which only few were neededfor the calculations of the thesis. The mass-independent interaction term depends on thefollowing trace, as seen in (1.22).

⟨(Φ↔∂µ Φ

)(Φ↔∂µ Φ

)⟩= (A.2)

+ 2 ∂µπ+∂µπ+π−π− + 4 ∂µπ

+∂µπ0π−π0 − 4 ∂µπ−∂µπ+π0π0 − 4 ∂µπ

0∂µπ0π−π+

+ 2 ∂µπ−∂µπ−π+π+ + 4 ∂µπ

−∂µπ0π+π0 − 4 ∂µπ−∂µπ+π−π+

− 2 ∂µπ−∂µπ+K−K+ − 2 ∂µK

−∂µπ+K+π− − ∂µπ0∂µπ0K−K+ − 3√

2 ∂µK0∂µπ−K+π0

− 2 ∂µπ−∂µπ+K0K0 + 4 ∂µK

0∂µπ+K0π− − ∂µπ0∂µπ0K0K0 − 3√

2 ∂µK+∂µπ0K0π−

− 2 ∂µK0∂µK0π−π+ + 4 ∂µK

0∂µπ−K0π+ + ∂µK−∂µπ0K+π0 − 3

√2 ∂µK

−∂µπ0K0π+

− 2 ∂µK−∂µK+π−π+ + 4 ∂µK

−∂µπ−K+π+ + ∂µK0∂µπ0K0π0 + 3

√2 ∂µK

0∂µπ0K−π+

− 2 ∂µK0∂µπ−K0π+ + 4 ∂K+∂π+K−π− + ∂µK

+∂µπ0K−π0 + 3√

2 ∂µK0∂µπ0K+π−

− 2 ∂µK0∂µπ+K0π− − ∂µK−∂µK+π0π0 + ∂µK

0∂µπ0K0π0 + 3√

2 ∂µK+∂µπ−K0π0

− 2 ∂µπ−∂µK+K−π+ − ∂µK0∂µK0π0π0 − 3

√2 ∂µK

0∂µπ+K−π0 + 3√

2 ∂µK−∂µπ+K0π0

− 2 ∂µK−∂µK+K0K0 − 2 ∂µK

0∂µK0K−K+ − 2 ∂µK+∂µK0K−K0 − 2 ∂µK

−∂µK0K+K0

+ 2 K−K−∂µK+∂µK+ + 2 K+K+∂µK

−∂µK− + 2 ∂µK0∂µK0K0K0 + 2 ∂µK

0∂µK0K0K0

− 4 ∂µK0∂µK0K0K0 − 4 K−K+∂µK

−∂µK+ + 4 ∂µK−∂µK0K+K0 + 4 ∂µK

+∂µK0K−K0

−√

3 ∂µη∂µK0K0π0 −

√3 ∂µK

0∂µπ0ηK0 +√

3 ∂µK−∂µπ0ηK+ +

√3 ∂µη∂

µK+K−π0

−√

3 ∂µη∂µK0K0π0 −

√3 ∂µK

0∂µπ0ηK0 +√

3 ∂µK+∂µπ0ηK− +

√3 ∂µη∂

µK−K+π0

+√

6 ∂µη∂µK0K+π− +

√6 ∂µη∂

µK+K0π− +√

6 ∂µK−∂µπ+ηK0 +

√6 ∂µK

0∂µπ+ηK−

+√

6 ∂µη∂µK0K−π+ +

√6 ∂µη∂

µK−K0π+ +√

6 ∂µK+∂µπ−ηK0 +

√6 ∂µK

0∂µπ−ηK+

+ 2√

3 ∂µη∂µπ0K0K0 + 2

√3 ∂µK

0∂µK0ηπ0 − 2√

3 ∂µη∂µπ0K−K+ − 2

√3 ∂µK

−∂µK+ηπ0

− 2√

6 ∂µK−∂µK0ηπ+ − 2

√6 ∂µη∂

µπ+K−K0 − 2√

6 ∂µη∂µπ−K+K0 − 2

√6 ∂µη∂

µπ−K+K0

− 3 ηη∂µK−∂µK+ + 3 ηK−∂µη∂

µK+ − 3 K−K+∂µη∂µη + 3 ∂µη∂

µK0ηK0

+ 3 ηK+∂µη∂µK− − 3 ∂µK

0∂µK0ηη − 3 ∂µη∂µηK0K0 + 3 ∂µη∂

µK0ηK0

40

The mass-dependent interaction terms in L2 are proportional to the following trace:

⟨MΦ4

⟩= (A.3)

+ (mu +md) π+π−π+π− + (mu +md) π

0π0π+π−

+1

4(mu +md) π

0π0π0π0

+2√3

(mu −md) π+π−π0η +

1√3

(mu −md) π0π0π0η

+ (2mu +md +ms) π+π−K+K− + (2md +mu +ms) π

+π−K0K0

+1

2(3mu +ms) π

0π0K+K− +1

2(3md +ms) π

0π0K0K0

+1√2

(mu −md) π+π0K−K0 +

1√2

(mu −md) π−π0K+K0

+ (mu +md) π+π−ηη +

1

2(mu +md) π

0π0ηη

+1√3

(mu −ms) π0ηK+K− − 1√

3(md −ms) π

0ηK0K0

+1√6

(mu +md − 2ms) π+ηK−K0 +

1√6

(mu +md − 2ms) π−ηK+K0

+ (mu −md) π0ηηη

+1

2(mu + 3ms) ηηK

+K− +1

2(md + 3ms) ηηK

0K0

+ (mu +ms) K+K−K+K− + (md +ms) K

0K0K0K0

+ (mu +md + 2ms) K+K−K0K0 1

36(mu +md + 16ms) ηηηη

41

B. Loop Integrals

The basic integrals used are given by

Iφ ≡∫

dDk

(2π)Di

k2 −m2 + iε(B.1)

= µ2ε(m

4π

)2[R+ ln

m2

µ2

]where R ≡ 1

ε+ γE − 1− ln 4π

and [PT84]

Iφφ =

∫dDk

(2π)Di

[k2 −m2 + iε] [(k − q)2 −m2 + ε](B.2)

=µ2ε

(4π)2

1

ε− ln 4π + γE + ln

m2

µ2+ σφ ln

σφ + 1

σφ − 1− 2

where σφ =

√1− 4m2

q2

=µ2ε

(4π)2

R+ ln

m2

µ2+ σφ ln

σφ + 1

σφ − 1− 1

=

Iφm2

+µ2ε

(4π)2

σφ ln

σφ + 1

σφ − 1− 1

The divergent integral found in the calculation of the pion form factor has to be expressed

through the functions Iφ and Iφφ. Further it will need to be separated into a part thatis proportional to q2 and will therefore be renormalized using the coupling L9 and into apart that will be independent of q. The expression found was

f2 iMloop = i(p+ p′)ν

∫d4k

(2π)4

(2k − q)µkν

[k2 −m2 + iε] [(k − q)2 −m2 + iε]. (B.3)

The second summand in this expression can be simplified by substituting the integrationvariable and dropping all uneven functions that give zero when the integration over allmomentum space takes place. Omitting the iε for shortness now and in the following theintegral reads

−qµ∫

d4k

(2π)4

kν

[k2 −m2] [(k − q)2 −m2]= −qµ

∫d4k

(2π)4

(k + q2)ν[

(k + q2)2 −m2

] [(k − q

2)2 −m2]

= −qµqν

2

∫d4k

(2π)4

1

[k2 −m2] [(k − q)2 −m2]

=i

2qµqν Iφφ. (B.4)

The remaining integral has the most general form

2

∫d4k

(2π)4

kµkν

[k2 −m2] [(k − q)2 −m2]= 2

[qµqνA(q2) + gµνq2B(q2)

]. (B.5)

43

Therefore the whole loop integral becomes

f2 iMloop = i(p+ p′)ν

[qµqν

(2A(q2) +

i

2Iφφ

)+ 2gµνq2B(q2)

]. (B.6)

Since q is just the difference between the momenta of the incoming and outgoing pionsand thus

(p′ + p)νqν = (p′ + p)(p′ − p) = p′2 − p2 (B.7)

the first term only contributes for off-shell pions. Neglecting that case the whole integralonly depends on the function B as

f2 iMloop = 2i(p+ p′)µq2B(q2). (B.8)

B can be determined by contracting B.5 with gµν to give

q2[A(q2) +DB(q2)

]=

∫d4k

(2π)4

k2

[k2 −m2] [(k − q)2 −m2](B.9)

=

∫d4k

(2π)4

1

(k − q)2 −m2+m2

∫d4k

(2π)4

1

[k2 −m2] [(k − q)2 −m2]

= −iIφ − im2Iφφ

and with qµqν as

q4[A(q2) +B(q2)

]=

∫d4k

(2π)4

(kq)2

[k2 −m2] [(k − q)2 −m2], (B.10)

where the numerator can be rewritten as

(kq)2 =1

4

[(k − q)2 − k2 − q2

]2=

1

4

[(k − q)2 −m2

]2+

1

4

[k2 −m2

]2+q4

4− 1

2

((k − q)2 −m2

) (k2 −m2

)− 1

2

((k − q)2 −m2

)q2 +

1

2

(k2 −m2

)q2. (B.11)

The last two terms give the same conribution but with the opposite sign canceling eachother while the first two terms both result in the integral

1

4

∫d4k

(2π)4

[(k − q)2 −m2

]2[k2 −m2] [(k − q)2 −m2]

=1

4

∫d4k

(2π)4

(k − q)2 −m2

k2 −m2(B.12)

=1

4

∫d4k

(2π)4

k2 −m2 + q2 − 2kq

k2 −m2

=1

4

∫d4k

(2π)4+q2

4

∫d4k

(2π)4

1

k2 −m2,

where the first integral is canceled by the fourth term in B.11. Summing up one receives

q2[A(q2) +B(q2)

]=

1

2

∫d4k

(2π)4

1

k2 −m2+q2

4

∫d4k

(2π)4

1

[k2 −m2] [(k − q)2 −m2]

= − i2Iφ − i

q2

4Iφφ. (B.13)

44

Having now two equations for the unknown functions A and B these can be solved for Bto give

B(q2) =1

D − 1

[−i 1

2q2Iφ − i

(m2

q2− 1

4

)Iφφ

]. (B.14)

The prefactor must be expanded around four dimensions by setting D = 4 + 2ε such that

1

D − 1=

1

3

1

1 + 2/3ε=

1

3

(1− 2

3ε+O(ε2)

), (B.15)

which leads to the final result

iMloop = − 2

3f2(p+ p′)µ

[1

2Iφ +

(m2 − q2

4

)Iφφ −

1

(4π)2

(m2 − q2

6

)](B.16)

= − 2

3f2(p+ p′)µ

[(3

2− q2

4m2

)Iφ −

1

(4π)2

(2m2 − 5q2

12

)− 1

(4π)2

q2

4σ3φ ln

σφ + 1

σφ − 1

]

45

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47