Coupled-channel study of axial-vector mesons with realistic t- and u-channel exchanges

with realisti t- and u- hannel ex hanges

Vom Fa hberei h Physik

der Te hnis hen Universität Darmstadt

zur Erlangung des Grades

eines Doktors der Naturwissens haften

(Dr. rer. nat.)

genehmigte Dissertation von

Dipl.-Phys. Julian Hofmann

aus Frankfurt a.M.

Referent: Priv. Doz. Dr. Matthias F.M. Lutz

Korreferent: Prof. Dr. Christian Fis her

Tag der Einrei hung: 25.1.2010

Tag der Prüfung: 17.2.2010

Darmstadt 2010

1 Zusammenfassung 5

2 Introdu tion 7

3 The tree-level s attering amplitude 9

3.1 Introdu tion: QCD and Ee tive Theories . . . 9

3.1.1 Color . . . 9

3.1.2 Flavor . . . 10

3.2 The intera tion . . . 11

3.3 Partial wave proje tion . . . 14

4 Analyti stru ture of the potential 23 4.1 The potentialfromdispersion integrals . . . 26

4.1.1 Example: the ex hange of akaon in

π ρ → K ¯

K

∗

. . . . 29

4.2 Perturbative analysis of the s attering amplitude . . . 32

4.2.1 Anomalousthresholds . . . 34

4.2.2 Cuts onthe real axis:

π ρ → π ρ

. . . 36

4.2.3 Cuts onthe real axis:

π φ → ¯

K K

∗

. . . 38

4.2.4 Cuts onthe real axis:

π K

∗

_{→ K ρ}

. . . 38

5 Computation of the s attering amplitude 41 5.1 Case 1: real potential . . . 41

5.2 Case 2: omplex potential . . . 42

6 Results 47 6.1 Resultsfor Weinberg-Tomozawa Intera tion . . . 47

6.1.1

(I, S) = (

1

2

, 1)

. . . 47 6.1.2

(I

G

_{, S) = (0}

+

_{, 0)}

. . . 49 6.1.3

(I

G

_{, S) = (0}

−

_{, 0)}

. . . 51 6.1.4

(I

G

_{, S) = (1}

+

_{, 0)}

. . . 51 6.1.5

(I

G

_{, S) = (1}

−

_{, 0)}

. . . 53

6.2 Resultsfor the fullintera tion . . . 54

6.2.1

(I

G

_{, S) = (1}

+

_{, 0)}

: the

b

1

(1235)

. . . 54 6.2.2

(I, S) = (

1

2

, 1)

: the

K

1

(1270)

. . . 58

7 Summary and Outlook 61

A The invariant amplitude 63

A.1 The 4-pointverti es. . . 63

A.2 The Pseudos alars- hannel Ex hange . . . 64

A.3 The Pseudos alaru- hannel Ex hange . . . 64

A.4 The s- hannelVe tor Ex hange . . . 64

A.5 The t- hannelVe tor Ex hange . . . 66

Zusammenfassung

Aufgabe der theoretis hen Hadronenspektroskopie ist es, Eigens haften von

mesonis hen und baryonis hen Resonanzen, wie zum Beispiel Massen und

Partialbreiten, zu bere hnen. Ein mögli her Weg hierzu ist das

Quark-Modell, in dem Grundzustand und Anregungszustände von Systemen aus

Quarks und Antiquarks (und Gluonen) bere hnet werden.

Ziel dieser Arbeit ist es, einen Teil des mesonis hen Spektrums genauer zu

untersu hen. Der hier verwendete Ansatz ist ni ht das Quarkmodell. Als

Freiheitsgrade werden ni ht Quarks und Gluonen verwendet, sondern die

Mesonen selbst. Ziel ist es, die Amplitude für die Streuung der lei htesten

pseudoskalaren Mesonen, der SU(3) Goldstone Bosonen (

π, K, η

), an den Vektormesonen

ρ, K

∗

und

ω

ni ht-perturbativ zu bere hnen. Resonanzen manifestieren si hals Pole inder Streuamplitude.

Ein Formalismus, der s hon mehrfa h eingesetzt wurde, um die

Streuam-plitude ni ht-perturbativ zu bere hnen ist das Lösen einer

partialwellen-projeziertenBethe-Salpeter-Glei hung. HierzuwirddieStreuamplitudezuerst

in Störungstheorie bere hnet, ans hliessend partialwellen-projeziert und als

KernfüreineBethe-Salpeter-Glei hungverwendet. AlsModellfürdie

We h-selwirkung wurde typis herweise der hirale Lagrangian in führender

Ord-nung verwendet. Meist wurde nur s-Wellen Streuung betra htet. Die

Be-handlung der Energieabhängigkeit einer We hselwirkung, die

s−

,

t−

und

u−

Kanal Austaus hprozesse beinhaltet, ists hwierig.

IndieserArbeitwirdeinFormalismusverwendet,derni htdie

Bethe-Salpeter-Glei hungbenutzt,sondernaufeinerni ht-linearenIntegralglei hungbasiert.

Lösungen dieser Glei hung sind kausal und analytis h. Der Input in diese

Glei hung ist ein vorbehandeltes Potential. Um dieses zu erhalten sind

mehrereS hrittenötig. EineLagrange-Di hte,diemiteinerKombinationaus

hiralenundLarge-

N

c

Argumentenaufgestelltwurde,wirdauseinerfrüheren Arbeit übernommen und verwendet, um die Streuamplitude in

Störungs-theorie zu bere hnen. Enthalten sind neben der führenden hiralen

Ord-nung sowohl Gegenterme als au h pseudoskalare und vektorielle

Austaus h-prozesse. DasErgebniswirdpartialwellen-projeziertundunterAusnutzung

dieserFormkanndasPotentialnunverwendetwerden,umdievolle

Streuam-plitudezu bere hnen.

In einem ersten S hritt ges hieht das für ein Potential, in dem nur die

Weinberg-Tomozawa-We hselwirkungberü ksi htigtwird. Dieserlaubteinen

Verglei h mit früheren Ergebnissen, bei denen das glei he System mitHilfe

der Bethe-Salpeter-Glei hung untersu ht wurde. Im nä hsten S hritt wird

dievollständigeWe hselwirkungverwendet,umdas

b

1

(1235)

unddas

K

1

(1270)

genauer zu betra hten. Das Potential enthält no h zwei unbekannte

Kop-plungskonstanten, die an das experimentelle Spektrum angepasst werden

können. Die hierbei erhaltenen Werte sind zwar konsistent mit früheren

Re hnungen, jedo h unters heiden sie si h signikant für die beiden

Reso-nanzen. Dies wirdals eine Konsequenz der Abwesenheit von pseudoskalaren

Austaus hprozessen interpretiert.

Wegen te hnis her Probleme wurden diese in der nalen Re hnung

wegge-lassen. Das ist au h der Grund, warum weitere relevante Grössen, wie zum

Beispieldas D/S Verhältnisder Zerfälleder betra hteten Resonanzen,ni ht

genauer untersu ht wurden. Die Ursa hen des te hnis hen Problemswerden

erläutert und ein mögli her Lösungsweg diskutiert. Interessant für weitere

Arbeiten istzum einen der Einuss dieser Prozesse auf das Spektrum. Zum

anderenkönnenmitdenindieserArbeitverwendetenMethodenau hV

ektor-Vektor-undPseudoskalar-PseudoskalarKanäleindasModellmiteinbezogen

werden. Dadur h würdedieBehandlung vonResonanzenmithöheren Spins

Introdu tion

The a epted theory of the strong intera tion is QCD. It des ribes the

ou-plingof the sixknown quarks, threeof whi h arelight,and the gluons. This

intera tion is based on the gauge group SU(3) that ouples to an internal

quantum number alled olor. Sin e this gauge group is non-Abelian, the

gluons an intera t with themselves. Another remarkable feature of QCD is

that therenormalized oupling onstantislarge atlowenergies. This

invali-datesstandard perturbation theory. A perturbativeexpansionismeaningful

onlyat high energies.

Although the details are not ompletely understood yet, this leads to an

interesting ee t at low energies: onnement. Color-neutral parti les are

the only hadrons observed in nature. These are divided into the bosoni

mesons and fermioni baryons. The al ulation of the properties of these

ompositeparti les, massand partialde ay widths,is thetaskof theoreti al

hadron spe tros opy. A traditional way to do so is to model for example a

meson asa bound quark-antiquarkstate [1, 2, 3℄.

The approa h followed in this thesis is based on a dierent idea: at low

energies an expansion of quark intera tions in powers of the QCD oupling

onstantisimpossible. Therelevantdegrees of freedomatthese energiesare

the baryons and mesons and not quarks and gluons. Hadroni resonan es

manifestthemselvesaspoles ins attering amplitudes. Theansatz used here

is to al ulate these s attering amplitudes with baryons and mesons as the

relevantdegrees of freedom.

Be auseofspontaneous hiralsymmetrybreakingtheintera tionofthe

light-est hadrons  the avor SU(3) multiplet ontaining the pion, also alled

pseudo-GoldstoneBosons  withany other hadron an beexpanded in

pow-ers of small momenta and the small quark masses. At leading order, this

intera tion depends ex lusively on a single parameter, the pion de ay

on-stant

f

π

. To al ulate non-perturbative ee ts like bound states or reso-nan es fromthis leadingorder intera tion, an innitesummation isneeded.

In previousworks this was provided by the Bethe-Salpeter-Equation(BSE).

S-waveboundstatesoftheGoldstoneBosons withavarietyofotherhadrons

ampli-tudeofthe GoldstoneBosonswiththeSU(3)-multipletin ludingthenu leon

has been al ulated. In [7, 8℄ the Goldstone Bosons were s attered o the

harmedsextet ontainingthe

Σ

c

andthe anti-triplet ontainingthe

Λ

c

. The works [9, 10, 11℄ onsidered the intera tion of the baryon resonan es with

spin and parity

3/2

+

, the multiplets ontaining the

∆

and the

Σ

∗

c

. The in-tera tionof the Goldstone bosons withthe lightest ve tor mesons(

ρ, K

∗

_{, φ}

and

ω

)were subje tof [15,16℄. Charmedmesonswere studiedin[12,13,14℄ and together with hidden harm in[17℄.

Insummaryone ansaythattheframeworkusedworksextremelywell

when-ever the Goldstone Bosons are intera ting with spin 0 orspin 1/2 parti les.

In the ase of spin 1 or 3/2 parti les the agreement is more s hemati . The

problemisthatforexamplein

π ∆

s atteringthe

ρ N

hannelisignoredeven soit isnot mu h heavier. A more realisti modelneeds to onsider take

ad-ditional hannels, together withamore elaborateintera tionthatin ludes

s

,

t

and

u

- hannelex hangepro esses. Su hadditionaltermsintheintera tion willlead toa more ompli ated analyti stru ture of the intera tion kernel.

A dierent s heme is ne essary to unitarize the tree-level s attering

ampli-tudes. This s heme is provided in [18℄. In that work the s attering of the

nu leon o tet o the light pseudos alar and ve tor mesons is studied.

Non-linearintegralequationsthat are basedon ausaland unitaritypropertiesof

the s attering amplitude are employed tounitarizethe s attering kernel.

In this thesis this s heme will be used toexamine resonan es in the

s atter-ing of the light pseudos alar (P) and ve tor (V) mesons. This allows for a

more realisti intera tion than in [15℄. Also the d-wave amplitude will be

al ulated,making additionalobservables liketheD/S ratioforthe

b

1

(1235)

a essible. The in lusion of additional hannels (PP or VV) is beyond the

s ope of this work, but a desirablenext step.

This work is organized as follows: in the subsequent se tion the tree-level

amplitude for the intera tion of a light pseudos alar meson with a ve tor

meson (PV

→

PV) will be al ulated and partial-wave proje ted. The inter-a tion Lagrangian is taken from [19℄ where it was onstru ted using hiral

and large

N

c

arguments.

Inthe third hapter the resultingpotentialwillbemodied su h that it an

be used as an input for the non-linear integral equations on whi h this

ap-proa h is based. Several te hni al problems an o ur in that pro ess and

will be dis ussed in detail. The methods to solve the dispersion relations

for the full s attering amplitude are summarized in the following hapter.

Finallythe resultswillbepresented. Ashortsummaryandoutlook on lude

The tree-level s attering

amplitude

Thegoalofthisthesisisto al ulatenon-perturbativeee tsintheamplitude

for s attering the lightest avor-o tet of pseudos alar (P) mesons o the

lightest avor-nonet of ve tor (V) mesons. The rst step is to al ulatethe

treelevelamplitudeforthispro ess. Forthispurposeanee tiveeldtheory

for the intera tion is needed. This will be taken from [19℄ where the same

pro ess was studiedin the presen e of photons.

3.1 Introdu tion: QCD and Ee tive Theories

In thesimplestquark model,a meson onsistsof aquark andananti-quark,

bound primarily by the strong intera tion. A ording to quantum

hromo-dynami s(QCD),the moderntheoryofthe strongintera tion,the dynami s

of quarks and gluonsfollows from the Lagrangian density

L = ¯

ψ(iD

/

− M)ψ −

1

4

G

a

µν

G

µν

a

,

(3.1)

with the gauge- ovariantderivative

D

µ

= ∂

µ

+ i g

s

λ

a

G

µ

a

,

(3.2)

where the

λ

'sare the Gell-Mann matri esand

g

s

is the strong oupling on-stant. The quark eld

ψ

impli itly arriestwoindi es, avorand olor.

3.1.1 Color

Coloristhebasi quantumnumbertowhi hthegaugebosons,thegluons

G

µ

a

, ouple. This is analogous tothe photons whi h ouple tothe ele tri harge

in Quantum ele trodynami s. In ontrast to QED, the quantum number

U(1)but

SU(3)

. Thisisalsothereasonwhy thereisanadditional,quadrati ontributiontothe gluoni eld strength tensor,

G

a

_µν

= ∂

µ

G

a

ν

− ∂

ν

G

a

µ

+ g

s

f

abc

G

b

µ

G

c

ν

,

a = 1 , . . . , 8 ,

(3.3)

where

f

abc

are the SU(3) stru ture onstants. In equ. (3.1) this last term

gives rise togluoni self-intera tions.

The strong oupling onstant

g

s

renormalized at high energies is smalland a perturbative expansion is possible. For de reasing energy the oupling

onstant grows and perturbation theory breaks down. These two related

ee ts, the self-intera tion of the gluons and the large oupling onstant at

smallenergies lead to interesting phenomena. One of these is onnement:

only olor-neutral parti les like mesons or baryons are observed in nature.

Currently the Lagrangian (3.1) annot be used dire tly to al ulate bound

statesof quarks and gluons. Other methodshave tobe employed. One su h

methodis Latti egauge theory,whi huses extensive numeri alsimulations.

Another method is to onsider the limit when

N

c

, the number of olors approa hesinnity. Inthisspe ial asesomephysi alproblemsbe omeeasier

to solve. The hope is that these relations still hold for the physi al value

N

c

= 3

. Higher order orre tions should be suppressed by powers of 1/

N

c

. This on ept was rst introdu edto QCDby t'Hooft[20℄. An introdu tion

tolarge

N

c

QCD an be found in [21℄.

In this work, an intera tion will be used that relies on large

N

c

QCD and another symmetry of the Lagrangian (3.1), avorsymmetry.

3.1.2 Flavor

Six dierent kinds of quarks or quark avors are known. Three of them

have masses of less than 1 GeV and will be onsidered in this work: the

up, down and strange quark. The matrix

M

in the Lagrangian (3.1) on-tains the masses of these quarks. If the three quarks were massless, the

QCDLagrangian would be invariantunder a transformation that mixesthe

quarks of dierent avor. This would hold for both heli ities of the quark

separatelyandtheLagrangiandensitywouldhaveaSU(3)

×

SU(3)symmetry 1

. This symmetry is broken in two ways. On the one hand the small but

non-vanishing masses of the quarks break the symmetry expli itly. On the

other hand this approximate symmetry SU(3)

×

SU(3) is spontaneously bro-ken toSU(3),as anbeseenforexampleintheparti lespe trumwhi honly

has the latterapproximate symmetry. Thetheory of spontaneoussymmetry

breaking(asexplainedforexamplein[22℄) laimsthat foreverygeneratorof

anapproximateglobalsymmetrythatisspontaneouslybroken, thespe trum

must ontain one approximately massless s alar parti le with the quantum

numbers ofthe generator. In the aseathand,these Goldstonebosons (GB)

1

are the lightest pseudos alar mesons, the pion, the kaon and the eta. F

ur-thermore at low energies an ee tive eld theory an be used in whi h the

GBs are the relevant degrees of freedom. This orresponds toan expansion

in powers of small momentaand quark masses. In this hiral expansion, all

terms that are onsistent with the fundamental symmetries of the model,

have tobe onsidered. Power ounting rules denean orderings heme [23℄.

The zeroth orderin this expansion vanishes.

3.2 The intera tion

In[19℄,a ombinationoflarge

N

c

argumentsanda hiralexpansiontoleading orders resulted in the following intera tion density:

L = f

2

tr

n

U

µ

U

†

µ

o

−

1

4

tr

n

(D

µ

V

µα

) (D

ν

V

να

)

o

+

1

8

m

2

1

−

tr

n

V

µν

V

µν

o

+ i

m

V

h

V

4

tr

n

V

αµ

V

µν

V

α

ν

o

+ i

˜h

V

4 m

V

tr

n

(D

α

_V

αµ

) V

µν

(D

β

V

βν

)

o

+ i

h

A

8

ǫ

µναβ

_tr

n

V

µν

(D

τ

V

τ α

) + (D

τ

V

τ α

) V

µν



U

β

o

+ i

m

V

h

P

2

tr

n

U

µ

V

µν

U

ν

o

+

1

4

g

D

tr

n

V

µν

[V

µν

, U

α

]

₊

U

α

o

+

1

4

g

F

tr

n

V

µν

[V

µν

, U

α

]

₋

U

α

o

+

1

8

b

D

tr

n

V

µν

V

µν

χ

+

o

+ i

b

A

8

ǫ

µναβ

_tr

nh

V

µν

, V

αβ

i

+

χ

−

o

,

(3.4)

where the elds havethe following parti le ontent:

V

µν

=







ρ

0

µν

+ ω

µν

√

2 ρ

+

µν

√

2 K

+

µν

√

2 ρ

−

µν

−ρ

0

µν

+ ω

µν

√

2 K

0

µν

√

2 K

−

µν

√

2 ¯

K

0

µν

√

2 φ

µν







,

Φ =









π

0

₊

_√

1

3

η

√

2 π

+

√

_{2 K}

+

√

2 π

−

_−π

0

₊

_√

1

3

η

√

2 K

0

√

2 K

−

√

_{2 ¯}

_K

0

₋

_√

2

3

η









.

(3.5)

The anti-symmetri tensor

ǫ

µναβ

is given by

ǫ

µναβ

₌











+1 if µ, ν, α, β is an even permutation of 0, 1, 2, 3

−1 if µ, ν, α, β is an odd permutation of 0, 1, 2, 3 ,

0 otherwise

ǫ

µναβ

_{= −ǫ}

µναβ

.

(3.6)

Theve tormesonwasrepresentedbyananti-symmetri Lorentztensor. This

work willfollowthat onvention. The isospin-averaged masses

m

π

= 138 MeV,

m

K

= 496 MeV,

m

η

= 547 MeV,

m

ρ

= 770 MeV, m

K

∗

= 894 MeV, m

_ω

= 783 MeV, m

_φ

= 1019 MeV,

willbe used.

The other elds inthe Lagrangian (3.4) are dened by

Γ

µ

=

1

2



u

†

_∂

µ

u + u∂

µ

u

†



,

u = exp

i Φ

2 f

!

,

U

µ

=

1

2

u

†



∂

µ

e

i

Φ

f



u

†

=

i ∂

µ

Φ

2 f

+ O(Φ

2

_{) ,}

D

µ

V

αβ

= ∂

µ

V

αβ

+ [Γ

µ

, V

αβ

] = ∂

µ

V

αβ

+

1

8 f

2

[[Φ, ∂

µ

Φ], V

αβ

] + O(Φ

3

_{) ,}

χ

+

=

1

2

uχ

0

u +

1

2

u

†

_χ

0

u

†

= χ

0

−

1

8 f

2

{{χ

0

, Φ} , Φ} + O(Φ

3

_{) ,}

χ

₋

=

1

2

uχ

0

u −

1

2

u

†

_χ

0

u

†

=

i

2 f

{Φ, χ

0

} + O(Φ

3

_{) ,}

χ

0

=







m

2

π

0

0

0

m

2

π

0

0

0

2 m

2

K

− m

2

π







,

(3.8)

whi hguaranteesanintera tioninagreementwiththe onstraintsfrom hiral

symmetry, see e.g. [24, 23℄.

Most oupling onstantsintheLagrangian(3.4)werealreadyderived in[19℄.

The mass splitting within the ve tor meson multiplet leads to an estimate

of

b

D

= 0.92 ± 0.05

.

h

P

was al ulated from the ve tor meson de ays

ρ →

ππ, φ → ¯

K K

and

K

∗

_{→ π K}

as

h

P

= 0.29 ± 0.03

. The values

h

V

and

˜h

V

were estimated from the magneti moment and the quadrupole moment of

theve tormesons. Finallythe radiativede ays

K

∗

±

→ K

±

γ, K

0

∗

→ K

0

γ

and

φ → η γ

led to the values for

h

A

and

b

A

. In this work the following values willbe used:

h

P

= 0.29 ,

h

A

= 2.10 ,

b

A

= 0.27 ,

m

V

= 776 MeV ,

h

V

= 0.45 ,

˜h

V

= 3.72 ,

b

D

= 0.92 ,

f

π

= 90 MeV .

(3.9)

Theother onstants

g

D

and

g

F

willbeused tottheresultstothemeasured spe trum.

At leading order the intera tion (3.4) results in a total of 16 diagramsthat

ontribute: four onta t intera tions, a pseudos alar

s

- and

u

- hannel ex- hange and a total of ten ve tor ex hange pro esses. A straight-forward

al ulation ofthe tree level s attering amplitude leads to

T

_tree

µ¯

¯

ν,µν

(¯

q, ¯

_{p; q, p) = −}

C

W T

4 f

2

π

g

¯

νν

_(p

µ

_{(q + ¯}

_q)

µ

¯

_{+ ¯}

_p

µ

¯

_{(q + ¯}

_q)

µ

₎

−

_{16 f}

1

2

g

¯

µµ

_g

νν

¯

n

C

D

g

D

+ C

F

g

F



(q · ¯q) + C

χ

b

D

o

−

X

x∈[8]

C

_s−ch

(x)

m

V

h

P

2 f

2

!

2

¯

p

µ

¯

q

¯

¯

ν

S

x

(p + q) p

µ

q

ν

−

X

x∈[8]

C

_u−ch

(x)

m

V

h

P

2 f

2

!

2

¯

p

µ

¯

q

¯

ν

S

x

(p − ¯q) p

µ

q

¯

ν

−

X

x∈[9]

C

_s−ch

(11,x)

h

2

A

16 f

2

Γ

¯

µ¯

ν

¯

α ¯

β

(¯

p, ¯

q) S

¯

α ¯

β,αβ

x

(p + q) Γ

µν

αβ

(p, q)

−

X

x∈[9]

h

A

b

A

8 f

2

n

C

_s−ch

(12,x)

Γ

µ¯

¯

ν

_{α ¯}

_¯

_β

(¯

p, ¯

q) S

_x

α ¯

¯

β,αβ

(p + q) ǫ

µν

_αβ

+ C

_s−ch

(21,x)

ǫ

µ¯

¯

ν

_{α ¯}

_¯

_β

S

_x

α ¯

¯

β,αβ

(p + q) Γ

µν

_αβ

(p, q)

o

−

X

x∈[9]

C

_s−ch

(22,x)

b

2

A

4 f

2

ǫ

¯

µ¯

ν

¯

α ¯

β

S

¯

α ¯

β,αβ

x

(p + q) ǫ

µν

αβ

−

X

x∈[9]

C

_u−ch

(11,x)

h

2

A

16 f

2

Γ

¯

µ¯

ν

¯

α ¯

β

(¯

p, −q) S

¯

α ¯

β,αβ

x

(p − ¯q) Γ

µν

αβ

(p, −¯q)

−

X

x∈[9]

h

A

b

A

8 f

2

n

C

_u−ch

(12,x)

Γ

µ¯

¯

ν

_{α ¯}

_¯

_β

(¯

_{p, −q) S}

_x

α ¯

¯

β,αβ

_{(p − ¯q) ǫ}

µν

_αβ

+ C

_u−ch

(21,x)

ǫ

µ¯

¯

ν

_{α ¯}

_¯

_β

S

_x

α ¯

¯

β,αβ

_{(p − ¯q) Γ}

µν

_αβ

_{(p, −¯q)}

o

−

X

x∈[9]

C

_u−ch

(22,x)

b

2

A

4 f

2

ǫ

¯

µ¯

ν

¯

α ¯

β

S

¯

α ¯

β,αβ

x

(p − ¯q) ǫ

µν

αβ

+

X

x∈[9]

C

_t−ch

(x)

h

P

2 f

2

q

¯

α

¯

q

β

¯

S

¯

α ¯

β,αβ

a

(¯

q − q)

n

3 m

2

_V

h

V

g

α

µ

¯

g

µ

β

g

νν

¯

+ ˜h

V

h

g

ν

¯

α

g

β

ν

p

¯

µ

¯

p

µ

+ g

β

¯

ν

g

µν

¯

p

µ

(p − ¯p)

α

− g

νν

¯

g

β

µ

(p − ¯p)

α

p

¯

µ

¯

io

,

Γ

µν

_αβ

(p, q) = q

γ

(p + q)

α

ε

µν

βγ

+ q

γ

p

µ

ε

α βγ

ν

.

(3.10)

Some omments about the notation are in order:

p

and

p

¯

are the in oming and outgoingve tor meson momenta while

q

and

q

¯

are the momenta of the pseudos alar mesons. The total momentum is

w = p + q = ¯

p + ¯

q

. The summation index

x

runs over the ex hanged parti les, either the o tet of Goldstone Bosons orthe nonet of ve tor mesons. Thepropagators are given

by

S

x

(p) =

1

p

2

_{− m}

2

x

+ i ǫ

,

S

_x

µν,αβ

_{(p) = −}

1

m

2

x

1

p

2

_{− m}

2

x

+ i ǫ

"

(m

2

_x

_{− p}

2

) g

µα

g

νβ

+ g

µα

p

ν

p

β

_{− g}

µβ

p

ν

p

α

_{− (µ ↔ ν)}

#

.

(3.11)

A list with all oupled hannels onsidered in this work is given in table

(0, 2)

(1, 2)

(

1

2

,

1)

(

i

√

2

K

t

µν

σ2

K)

(

√

i

2

K

t

µν

σ2

~

σ K)

(

1

√

3

π

· σ K

µν

)

(

1

√

3

ρµν

· σ K)

(ωµν

K)

(η Kµν)

(φµν

K)

(

3

2

,

1)

(0

+

,

0)

(0

−

_,

₀₎

(π · T Kµν

)

(ρµν

· T K)

1

2

( ¯

K Kµν

− ¯

Kµν

K)

(

1

√

3

ρµν

· π)

(ωµν

η)

1

2

( ¯

K Kµν

+ ¯

KµνK)

(φµν

η)

(1

+

_,

₀₎

₍₁

₋

_,

₀₎

_{(2, 0)}

(~π ωµν)

(~π φµν

)

(~

ρµν

η)

1

2

( ¯

K ~

σ Kµν

+ ¯

Kµν

~

σ K)

−

i

√

2

(~

ρµν

× ~π)

1

2

( ¯

K ~

σ Kµν

− ¯

Kµν

~

σ K)

1

2

(π

i

_ρ

j

µν

+ π

j

ρ

i

µν

) −

1

3

δ

ij

_π

_{· ρµν}

Table3.1: Coupled- hannelstates

(I

G

_{, S)}

,withisospin(I),G-parity(G)and

strangeness (S).

hannels; they are given in the tables (3.2-3.6). The

s

- hannel oe ients are listed ina fa torizedform:

h

C

_s−ch

(x)

i

ab

= G

(x)

a

G

(x)

b

,

h

C

_s−ch

(ij,x)

i

ab

= G

(i,x)

a

G

(j,x)

b

.

(3.12)

3.3 Partial wave proje tion

The next step is a partial wave expansion of the tree-levelamplitude (3.10)

asdemonstratedintheappendixof[15℄. Tosummarizethis appendixbriey:

on e the amplitudes

G

i

fullling

h¯λ|T |λi ≡ ǫ

†

¯

µ¯

ν

(¯

p, ¯

λ) T

¯

µ¯

ν, µν

tree

ǫ

µν

(p, λ)

= ǫ

†

¯

µ

(¯

p, ¯

λ)

n

G

1

g

µµ

¯

+ G

2

w

µ

¯

w

µ

+ G

3

w

µ

¯

p

¯

µ

+ G

4

p

µ

¯

w

µ

+ G

5

p

µ

¯

p

¯

µ

o

ǫ

µ

(p, λ) ,

(3.13)

(a) [G

(1,ρ)

₍₁

+

_,0)

]

_a

[G

(2,ρ)

(1

+

_,0)

]

_a

(1)

1

2 m

2

π

(2)

0

0

(3)

√

1

3

2

√

3

m

2

π

(4)

1

2 m

2

K

[G

(1,ω)

₍₀

−

_,0)

]

_a

[G

(2,ω)

(0

−

_,0)

]

_a

√

3

√

3 2 m

2

π

1

√

3

2

√

3

m

2

π

1

2 m

2

K

0

0

[G

(1,φ)

₍₀

−

_,0)

]

_a

[G

(2,φ)

(0

−

_,0)

]

_a

0

0

0

0

√

2

√

8 m

2

K

−

√

2

3

4

√

3

m

2

π

−

√

8

₃

m

2

K

(a) [G

(π)

₍₁

−

_,0)

]

_a

[G

(η)

(0

+

_,0)

]

_a

(1)

√

2

√

3

(2)

1

₋

(3)

₋

₋

(4)

₋

₋

(5)

₋

₋

[G

(K)

₍

1

2

,1)

]

a

√

3

2

−

√

2

3

−

_√

1

2

3

2

1

√

2

[G

(1,K

₍

1

∗

)

2

,1)

]

_a

[G

(2,K

₍

1

∗

)

2

,1)

]

_a

√

3

2

√

3 m

2

π

√

3

2

√

3 m

2

K

1

2

m

2

K

−

₂

√

1

3

√

3 m

2

π

−

√

4

₃

m

2

K

1

√

2

√

2 m

2

K

Table 3.2: Coupling onstantsspe ifyingthe s- hannelmeson ex hange

on-tributions (see (3.12)).

are known, the partial-waveproje tion an be writtendown. 2

AdditionalLorentz stru tures ontaining

p

µ

or

p

¯

¯

µ

do not ontribute in equ.

(3.13) sin e they vanish when ontra tedwith the polarizationve tors. The

latter an be transformed between ve tor and tensorrepresentation by

ǫ

µν

(p, λ) =

i

√

p

2

n

p

µ

ǫ

ν

(p, λ) − p

ν

ǫ

µ

(p, λ)

o

.

(3.14)

The onvention used for the polarizationve tors is

ǫ

µ

(p) =













0

±1

√

2

−i

√

2

0













,











p

cm

M

0

0

ω

M











,

ǫ

µ

(¯

p) =













0

∓ cos Θ

_√

2

−i

√

2

± sin Θ

_√

2













,











p

cm

M

ω

M

sin Θ

0

ω

M

cos Θ











,

(3.15)

with

ǫ

µ

(p) = ǫ

µ

(p, ±1), ǫ(¯p, 0)

,

ǫ

µ

(¯

p) = ǫ

µ

(¯

p, ±1), ǫ(¯p, 0)

and the further def-initions

ω

2

_{= M}

2

_{+ p}

2

cm

and

ω

¯

2

_{= ¯}

_M

2

_{+ ¯}

_p

2

cm

.

(Usingequ. (3.14)inequ. (3.13)determineshowtotransformthe s attering

amplitude between ve tor and tensor representation:

T

tree

µ,µ

¯

=

¯

p

ν

¯

¯

M



T

tree

µ¯

¯

ν,µν

− T

tree

ν ¯

¯

µ,µν

− T

tree

µ¯

¯

ν,νµ

+ T

tree

ν ¯

¯

µ,νµ



p

_ν

M

.

(3.16)

The transformations of the s attering amplitude in the equation above and

in the polarizationve tors in equ. (3.14) inprin ipleredu e the problemto

the situation in the appendix of [15℄ where only the ve tor representation

was used.)

2

(I, S)

h

C

ρ

t−ch

C

11,ρ

u−ch

C

12,ρ

u−ch

C

21,ρ

u−ch

C

22,ρ

u−ch

C

u−ch

π

(0, 2)

11

3

4

3

4

3

2

m

2

K

3

2

m

2

K

3 m

4

K

3

4

(1, 2)

11

−

1

4

1

4

1

2

m

2

K

1

2

m

2

K

m

4

K

1

4

(

1

2

, 1)

11

1

0

0

0

0

0

12

0

0

0

0

0

−1

13

0

√

3

2

√

3 m

2

π

√

3 m

2

K

2

√

3 m

2

π

m

2

K

0

22

1

0

0

0

0

0

24

0

1

2

m

2

K

m

2

π

2 m

2

π

m

2

K

0

(

3

₂

, 1)

11

−

1

2

0

0

0

0

0

12

0

0

0

0

0

1

2

22

−

1

2

0

0

0

0

0

(0

+

_{, 0)}

11

3

4

−

3

4

−

3

2

m

2

K

−

3

2

m

2

K

−3 m

4

K

3

4

(0

−

_{, 0)}

11

2

0

0

0

0

−2

12

0

1

2 m

2

π

2 m

2

π

4 m

4

π

0

33

3

4

3

4

3

2

m

2

K

3

2

m

2

K

3 m

4

K

−

3

4

(1

+

_{, 0)}

11

0

1

2 m

2

π

2 m

2

π

4 m

4

π

0

33

0

1

3

2

3

m

2

π

2

3

m

2

π

4

3

m

4

π

0

44

−

1

4

−

1

4

−

1

2

m

2

K

−

1

2

m

2

K

−m

4

K

1

4

(1

−

_{, 0)}

11

1

0

0

0

0

1

22

−

1

4

1

4

1

2

m

2

K

1

2

m

2

K

m

4

K

−

1

4

(2, 0)

11

−1

0

0

0

0

1

Table 3.3: Coupling onstantsspe ifyingthe intera tion strength as dened

(I, S)

h

C

ω

t−ch

C

u−ch

11,ω

C

12,ω

u−ch

C

21,ω

u−ch

C

22,ω

u−ch

C

η

u−ch

(0, 2)

11

−

1

4

−

1

4

−

1

2

m

2

K

−

1

2

m

2

K

−m

4

K

−

3

4

(1, 2)

11

−

1

4

1

4

1

2

m

2

K

1

2

m

2

K

m

4

K

3

4

(

1

2

, 1)

12

0

1

2

m

2

π

m

2

K

2 m

4

K

m

4

π

0

34

0

1

2

√

3

1

√

3

m

2

K

√

1

₃

m

2

π

√

2

₃

m

2

K

m

2

π

0

(

3

₂

, 1)

12

0

1

2

m

2

π

m

2

K

2 m

2

K

m

2

π

0

(0

+

_{, 0)}

11

1

4

−

1

4

−

1

2

m

2

K

−

1

2

m

2

K

−m

4

K

3

4

(0

−

_{, 0)}

11

0

1

2 m

2

π

2 m

2

π

4 m

4

π

0

22

0

1

3

2

3

m

2

π

2

3

m

2

π

4

3

m

4

π

0

33

1

4

1

4

1

2

m

2

K

1

2

m

2

K

m

4

K

−

3

4

(1

+

_{, 0)}

13

0

1

√

3

2

√

3

m

2

π

√

2

₃

m

2

π

√

4

₃

m

4

π

0

44

1

4

1

4

1

2

m

2

K

1

2

m

2

K

m

4

K

−

3

4

(1

−

_{, 0)}

11

0

−1

−2 m

2

π

−2 m

2

π

−4 m

4

π

0

22

1

4

−

1

4

−

1

2

m

2

K

−

1

2

m

2

K

−m

4

K

3

4

(2, 0)

11

0

1

2 m

2

π

2 m

2

π

4 m

4

π

0

Table 3.4: Coupling onstants spe ifyingthe intera tion strength as dened

(I, S)

h

C

K

∗

t−ch

C

11,K

∗

u−ch

C

12,K

∗

u−ch

C

21,K

∗

u−ch

C

u−ch

K

(

1

2

, 1)

11

0

−

1

4

−

1

2

m

2

π

−

1

2

m

2

π

−

1

4

12

1

4

0

0

0

0

13

−

√

3

4

0

0

0

0

14

0

−

1

4

−

1

2

m

2

π

3

2

m

2

π

− 2 m

2

K

3

4

15

q

3

8

0

0

0

0

22

0

−

1

4

−

1

2

m

2

K

−

1

2

m

2

K

−

1

4

23

0

√

3

4

√

3

2

m

2

K

√

3

2

m

2

K

√

3

4

24

−

3

4

0

0

0

0

25

0

q

3

8

q

3

2

m

2

K

q

3

2

m

2

K

−

q

3

8

33

0

1

4

1

2

m

2

K

1

2

m

2

K

1

4

34

−

√

3

4

0

0

0

0

35

0

1

√

8

1

√

2

m

2

K

√

1

₂

m

2

K

−

√

1

₈

44

0

1

12

2

3

m

2

K

−

1

2

m

2

π

2

3

m

2

K

−

1

2

m

2

π

3

4

45

q

3

8

0

0

0

0

55

0

1

2

m

2

K

m

2

K

1

2

(

3

₂

, 1)

11

0

1

2

m

2

π

m

2

π

1

2

12

−

1

2

0

0

0

0

22

0

1

2

m

2

K

m

2

K

1

2

(0

−

_{, 0)}

13

√

3

2

√

3

2

√

3 m

2

π

√

3 m

2

K

−

√

3

2

23

√

3

2

−

1

2

√

3

√

3 m

2

π

−

√

4

₃

m

2

K

−

√

1

₃

m

2

K

−

√

3

2

34

−

q

3

2

−

1

√

6

−

q

2

3

m

2

K

√

6 m

2

π

−

4

√

6

3

m

2

K

q

3

2

(1

+

_{, 0)}

14

1

2

1

2

m

2

π

m

2

K

−

1

2

24

−

1

√

2

1

√

2

√

2 m

2

π

√

2 m

2

K

√

1

₂

34

√

3

2

−

1

2

√

3

√

3 m

2

π

−

√

4

₃

m

2

K

−

√

1

₃

m

2

K

−

√

3

2

(1

−

_{, 0)}

12

1

√

2

−

1

√

2

−

√

2 m

2

π

−

√

2 m

2

K

√

1

₂

Table 3.5: Coupling onstantsspe ifyingthe intera tion strength as dened

in (3.10) ontinued. The oe ients

C

22,K

∗

u−ch

are not displayed. They obey the relation

C

22,K

∗

u−ch

= C

12,K

∗

u−ch

· C

21,K

∗

u−ch

/C

11,K

∗

u−ch

(I

G

_{, S)}

h

C

φ

t−ch

C

11,φ

u−ch

C

12,φ

u−ch

C

21,φ

u−ch

(0, 2)

11

−

1

2

−

1

2

−m

2

K

−m

2

K

(1, 2)

11

−

1

2

1

2

m

2

K

m

2

K

(

1

₂

, 1)

45

0

−

q

2

3

−

4

√

6

3

m

2

K

+

2

√

6

3

m

2

π

−

q

8

3

m

2

K

(0

+

_{, 0)}

11

1

2

−

1

2

−m

2

K

−m

2

K

(0

−

_{, 0)}

33

1

2

1

2

m

2

K

m

2

K

44

0

4

3

16

3

m

2

K

−

8

3

m

2

π

16

3

m

2

K

−

8

3

m

2

π

(1

+

_{, 0)}

44

1

2

1

2

m

2

K

m

2

K

(1

−

_{, 0)}

22

1

2

−

1

2

−m

2

K

−m

2

K

Table 3.6: Coupling onstants spe ifyingthe intera tion strength as dened

in(3.10) ontinued. The oe ients

C

22,φ

u−ch

are not displayed. They obeythe relation

C

22,φ

u−ch

= C

12,φ

u−ch

· C

21,φ

u−ch

/C

11,φ

u−ch

Inthenext steppartialwaveamplitudesare introdu edinthe enterof mass

frame,

ǫ

†

µ¯

¯

ν

(¯

p, ¯

λ) T

¯

µ¯

ν, µν

tree

ǫ

µν

(p, λ) =

X

J

(2 J + 1)h¯λ|T

(J)

|λid

(J)

λ¯

λ

(Θ) ,

(3.17) where

d

(J)

λ¯

λ

(Θ)

denoteWigner'srotationfun tionsand

Θ

isthe anglespanned bythein omingandoutgoingve tormesonthree-momentum

3

. Inthe enter

of massframethe lefthand side ofthis equation anbe al ulated expli itly

in terms of the invariant amplitudes

G

i

using the right hand side of equ. (3.13) and the expressions inequ. (3.15)for the polarizationtensors.

If the

G

i

are known, the obje ts

h¯λ|T

(J)

|λi =

Z

1

−1

d cos Θ

2

h¯λ|T |λi d

(J)

λ¯

λ

,

(3.18)

are now a essible. These are the desired obje ts, but in a dierent basis.

The transformation of basis onsistsof several steps.

In the rst step, the previous equation isrewritten in terms of parity

eigen-states whi h are dened by:

h1

±

| = (±h+1| + h−1|)/

√

2 ,

h2

+

| = h0| .

(3.19)

The se ond step is almosttrivial for negative parity. There is justone state

in equ. (3.18). The orresponding one-dimensional matrix will be res aled

3

In [15℄ there is an ambiguity in the denition of that angle. If

p

· ¯

p

= ω ¯

ω

−

by a fa tor

1/(p

cm

p

¯

cm

)

J

. For positive parity the two by two matrix in equ.

(3.18)is multiplied fromthe right by

U

p

J

cm

=

1

p

J

cm





1

p

−1

cm

−

q

J

J+1

ω

M p

cm

0

_{M p}

1

_cm





,

(3.20)

and fromthe leftby

¯

U

T

¯

p

J

cm

,

U = U(p

¯

cm

↔ ¯p

cm

, ω ↔ ¯ω, M ↔ ¯

M ) .

(3.21)

Those steps above will lead tothe invariant amplitudes

M

listed in the ap-pendix of[15℄. Thereisanadditionalfreedomthatwillbeusedinthis work.

Theresultinginvariantamplitudes anberes aledwithapowerof

s = (p+q)

2

ifthe phasespa e (tobeintrodu edinthe next hapter)isres aled withthe

inverse expression. For positive parity the following exponents willbeused:

T

_i,j

(1+)

= s

d

i,j

_M

(1+)

i,j

,

d

1,1

= 1 ,

d

1,2

= d

2,1

=

3

2

,

d

2,2

= 2 .

(3.22) In this work only positive parity and angular momentum one,

J = 1

is onsidered. The nal expression is

T

₁₁

(1+)

=

Z

1

−1

dx

2

n

−

1 + x

2

2

s G

1

+

x

3

_{− x}

2

p

¯

cm

p

cm

s G

5

o

,

T

12

(1+)

=

Z

1

−1

dx

2

n

3 x

2

− 1

√

8

ω s

3

2

p

2

cm

G

1

+s

2

x

2

_{− 1}

√

2

G

4

+

s

3/2

_{(1 − x}

2

₎

√

8 p

cm

(3 ω ¯

p

cm

x − 2 ¯ω p

cm

) G

5

o

,

T

21

(1+)

=

Z

1

−1

dx

2

n

3 x

2

− 1

√

8

¯

ω s

3

2

¯

p

2

cm

G

1

+s

2

x

2

_√

− 1

2

G

3

+

s

3/2

_{(1 − x}

2

₎

√

8 ¯

p

cm

(3 ¯

ω p

cm

x − 2 ω ¯p

cm

) G

5

o

,

T

22

(1+)

=

Z

1

−1

dx

2

n

s

2

4 p

2

cm

p

¯

2

cm



4 x p

cm

p

¯

cm

+ ω ¯

ω(3 − 9 x

2

)



G

1

+

s

3

_x

p

cm

p

¯

cm

G

2

+s

5/2

2 ¯

ω x p

cm

+ ω ¯

p

cm

(1 − 3 x

2

₎

2 p

2

cm

p

¯

cm

G

3

+s

5/2

2 ω x ¯

p

cm

+ ¯

ω p

cm

(1 − 3 x

2

₎

2 p

cm

p

¯

2

cm

G

4

+

s

2

4 p

2

cm

p

¯

2

cm

h

2 (1 − 3 x

2

) · (ω

2

p

¯

2

_cm

+ ¯

ω

2

p

2

_cm

)

+x p

cm

p

¯

cm

ω ¯

ω (9 x

2

− 1)

i

G

5

o

,

(3.23)

with

x = cos(Θ)

and the usualMandelstam variables

s = (p + q)

2

,

(3.24)

t = (p − ¯p)

2

,

(3.25)

u = (p − ¯q)

2

.

(3.26)

Itremainsto ndsome pra ti alway to al ulatethe

G

i

inequ. (3.13)from the tree levelamplitude (3.10). Progressis made by the denitions

v

µ

_p,¯

_p,w

= ε

µνρσ

p

ν

p

¯

ρ

w

σ

,

(3.27)

p

µ

_⊥

= p

µ

₋

(p· ¯p)

¯

p

2

p

¯

µ

−

p·(w −

(w·¯

p)

¯

p

2

p)

¯

(w −

(w·¯

p

¯

2

p)

p)

¯

2

(w −

(w· ¯p)

_p

_¯

2

p)

¯

µ

_,

(3.28)

¯

p

µ

_⊥

= ¯

p

µ

₋

(¯

p·p)

p

2

p

µ

−

p·(w −

¯

(w·p)

p

2

p)

(w −

(w·p)

p

2

p)

2

(w −

(w·p)

_p

2

p)

µ

_,

(3.29)

w

_⊥

µ

= w

µ

₋

(w· ¯p)

¯

p

2

p

¯

µ

−

w·(p −

(p·¯

p)

¯

p

2

p)

¯

(p −

(p·¯

p

¯

2

p)

p)

¯

2

(p −

(p· ¯p)

_p

_¯

2

p)

¯

µ

_.

(3.30)

Theseare onstru ted insu haway that

p

µ

⊥

is orthogonalto

p

¯

µ

and

w

µ

,

w

µ

⊥

is orthogonal to

p

µ

and

p

¯

µ

and nally

p

¯

µ

⊥

is orthogonal to

p

µ

and

w

µ

.

v

µ

p,¯

p,w

is perpendi ular to all three ve tors. With these denitions it is

straight-forward to read o the

G

i

fromequ. (3.13),

G

1

=

v

p,¯

µ

p,w

T

µν

v

p,¯

ν

p,w

v

2

p,¯

p,w

,

(3.31)

G

2

= (T

µν

− G

1

g

µν

)

w

_⊥

µ

w

ν

⊥

(w

_⊥

)

4

,

(3.32)

G

3

= (T

µν

− G

1

g

µν

)

w

µ

_⊥

p

¯

ν

⊥

(w

_⊥

)

2

_(¯

_p

⊥

)

2

,

(3.33)

G

4

= (T

µν

− G

1

g

µν

)

p

µ

_⊥

w

ν

⊥

(p

_⊥

)

2

_(w

⊥

)

2

,

(3.34)

G

5

= (T

µν

− G

1

g

µν

)

p

µ

_⊥

p

¯

ν

⊥

(p

⊥

)

2

(¯

p

⊥

)

2

.

(3.35)

Analyti stru ture of the

potential

In previousworks only the leadingkinemati ontribution

T

(1+)

11

was onsid-ered. This was used in a (partial wave proje ted) Bethe-Salpeter equation

to al ulatethe full s attering amplitude. Inthis work the formalism of[18℄

willbe used. It rests ona dispersion relationfor the omplete amplitude

T

_ab

(J P )

(s) = U

_ab

(J P )

(s) +

Z

∞

thres

2

dw

π

s − µ

2

M

w − µ

2

M

∆T

_ab

(J P )

(w)

w − s − i ε

,

(4.1)

wherethepotential

U

ontains allthe left-hand utsandthe right-hand uts are summed up in the integral.

(J, P )

denotes the angular momentum and parity. In ontrast to [18℄ where baryons were onsidered, the variable of

integration doesn'thavethe dimensionofanenergy butof asquaredenergy.

Sin etheamplitudesaredened withrespe t tothe transformedbasis(3.20)

there are no additional kinemati onstraints as would be the ase if for

example heli ity eigenstates were used.

The mat hing point

µ

M

is hosen

µ

2

_M

= m

2

_min

+ M

_min

2

,

(4.2)

wherethemassesontherightside orrespondtothepseudos alarandve tor

massesof the lightest hannel. This willmakethe fullamplitude

T

identi al to the tree-level result at a point where the s attering amplitude an be

al ulated perturbatively.

When the potential

U

has no utsabove threshold,

∆T

is given by all right-hand uts [25℄:

∆T

_ab

(J P )

(s) =

1

2 i

[T

(J P )

ab

(s + iε) − T

(J P )

ab

(s − iε)]

=

X

c,d

h

T

_ac

(J P )

(s)

i

†

ρ

(J P )

_cd

(s) T

_db

(J P )

(s) .

Fora single- hannel problem,the phase spa e isgiven by

ρ(s) =





1

s

(

3

2

+

p

2

cm

2 M

2

)

_s

3/2

1

(

p

2

cm

ω

√

2 M

2

)

1

s

3/2

(

p

2

cm

ω

√

2 M

2

)

1

s

2

p

4

cm

M

2





.

(4.3)

It is not diagonal be ause of the transformation (3.20). The res aling of

equ. (3.22) was ompensated with orresponding

s

−d

ij

. For large energies

the res aled phase spa e will approa h a nite onstant, a property that is

ru ialfor the integralthe master-equ. (4.1)to be well-dened.

It turns out that to solve equ. (4.1), the potential is needed at all energies

abovethreshold, in the ase of o-diagonalmatrix elements abovethe lower

threshold. As it stands, the expression (3.10) annot be used as input for

(4.1). Considering the asymptoti behaviorthis isalready obvious: for large

energies the potentialwill behave like a polynomial. Su h a behavior is not

onlyphysi ally unreasonable, it alsoinvalidates the integration.

Progress is made by looking at the analyti stru ture of the potential, for

example for

K ρ

s attering. The

u

- hannel ut starts at

√

s = m

ρ

− m

K

. A Taylor expansion at the rho meson mass will therefore only onverge up

to threshold,

√

s = m

ρ

+ m

K

, while we need the potential above threshold. Todeal with this problema tri k is used. The idea isthe following: instead

of expanding the fun tion in

s

, the argument is substituted with a fun tion

s(ξ)

. When the potential as fun tion of

ξ

is Taylor expanded the radius of onvergen e willnowbea ir lein

ξ

,notin

s

. Ifthatfun tion

s(ξ)

is hosen properly the expansion in

ξ

(or

ξ

pansion) will onverge for

(m

ρ

− m

K

)

2

_<

s < Λ

2

s

where

Λ

s

an be mu h larger than threshold. To be more pre ise a Taylor expansionleads to

f (s) = f (s

0

) + f

′

(s

0

) · (s − s

0

) + f

′′

(s

0

) ·

(s − s

0

)

2

2 !

+ . . . .

(4.4) When the variable is substitutedthis be omes

f (s(ξ)) = f (s(ξ

0

)) +

df (s(ξ))

d ξ











ξ=ξ

0

· (ξ − ξ

0

)

+

d

2

_{f (s(ξ))}

d ξ

2











ξ=ξ

0

·

(ξ − ξ

0

)

2

2 !

+ . . . ,

(4.5)

and substituting ba k usingthe hain rule the result is

f (s) = f (s

0

) + f

′

(s

0

) · s

′

(ξ

0

) · (ξ(s) − ξ

0

)

+



f

′′

_(s

0

) (s

′

(ξ

0

))

2

+ f

′

(s

0

) s

′′

(ξ

0

)



·

(ξ(s) − ξ

0

)

2

2 !

+ . . . ,

(4.6) where

s

0

= s(ξ

0

) .

(4.7)