Case 2: omplex potential

Whentheinterationhasanimaginarypartabovethresholdthingsgetmore

ompliated. In a rst step the potential is separated into the ontribution

fromthe ut onthe real axis and the rest:

U (s) = ¯ U (s) − 1 2 i

Z _∞

thres ²

dw π

U ⁽⁺⁾ (w) − U ⁽ ⁻ ⁾ (w)

w − s .

^(5.7)

The ontributionfrom

U ¯

^an ^be ^treated^as^before. ^It^is ^not ^yet ^lear ^how^to

treat the integral part of the equation above.

Fig. 5.1illustratesthesituationinthediagonal

K K ¯ ^∗

^hannel^with ^spin^and

isospinzeroand positiveG-parity. Theexat resultwhihontributestothe

potential

U (s)

^shows ^a^strong^energy ^dependene ^at^the^point^where^the ^ut

starts. One the integral in equ. (5.7) is subtrated the potential beomes

almostlinear.

The imaginary part of the exat potential for the physial ase of the

ex-hange of a pion between a

K

^-meson ^and ^a

K ¯ ^∗

^-meson ⁱⁿ ^the

u

^-hannel ⁱⁿ

the

(I ^G , S) = (0 ⁺ , 0)

^setor ^is ^shown ⁱⁿ ^g. ^5.2. ^In ^the ^left ^olumn, ^s-wave,

the transitionamplitude between s-and d-wave and d-wave are displayed.

A disontinuity in the imaginary part for d-wave sattering is expeted to

invalidateatreatmentof theimaginarypart. Forexampleanitejumpin

U

would lead to a logarithmi divergene in the integral (5.4). The imaginary

partisaonsequeneofthe instabilityofthe

K ^∗

^and ^its^possible^deay^into^a

kaonand apionwhihalsogivesthe

K ^∗

^it's^width. ^A^veraging^the ^imaginary

part overthe spetraldistribution of the inomingand outgoingvetor

par-tile willmakethe potential ontinuous again. A spetral distribution

ρ S (s)

1.4 1.5 1.6 1.7 1.8 1.9 2 s^1/2 [GeV]

-60 -40 -20 0 20 40 60

Amplitude [GeV²]

Figure5.1: This gureshows thereal partof theexat potential(solid blak

line), the imaginarypart ofthe exat potential(solid red line), the real part

of theintegralinequ. (5.7) (dashed line) andthe diereneof the real parts

(dottedline) in the ase

K K ¯ ^∗ → K K ¯ ^∗

^with ^a ^pion ⁱⁿ^the

u

^-hannel.

an befound in[29℄. In theproess ofaveraginganotherproblemarises: the

integral

Z _∞

(m _K +m π ) ² dm ² ₁

Z _∞

(m _K +m π ) ² dm ² ₂ ρ S (m ² ₁ )ρ S (m ² ₂ )U (s, m ² ₁ , m ² ₂ )

inludesa ombinationof

K ^∗

^masses ^for ^whih ^the ^ut ^starts ^at ^threshold:

m π + m 1 = s , c ₋ (s, m ² _π , m ² ₂ ) = s ,

where

c ₋

^is ^the ^ontour ^funtion. ^At ^this ^point ^(and ^the ^point ^with

m 1

and

m 2

interhanged) in the integration, negative powers of the enter of mass momentum willdiverge. These divergenes are integrableif the s-wave

is involved. For the d-wave it is neessary to deform the ontour of both

integrations in the averaging slightly away from the real axis. To doso the

expression for the spetral distribution has to be replaed by some analyti

funtion whih oinides with the non-analyti formula in [29℄ on the real

axis. Expliitly,the formulaused is

λ(s) = (s − (m π + m K ) ² ) · (s − (m π − m K ) ² ) , f R (s) = 1

16 π ²

q λ(s)

2 s (ln( − m ² _π − m ² _K + s − ^q λ(s) )

-25 0 25

0 2500

1.25 1.5 1.75

-50000 0 50000

1.5 1.75 2

Energy [GeV]

-50000 0 50000

Figure5.2: This gureshows the imaginarypart ofthe potentialinthe ase

K K ¯ ^∗ → K K ¯ ^∗

^with ^a ^pionⁱⁿ ^the

u

^-hannel. ^On ^the ^right^side ^the ^potential

wasaveragedoverthespetraldistributionoftheinomingandoutgoing

K ^∗

The rst row orresponds to

T ₁₁ ⁽¹⁺⁾

^, ^the ^seond ^row ^shows

T ₁₂ ⁽¹⁺⁾ = T ₂₁ ⁽¹⁺⁾

^and

the lastrow

T ₂₂ ⁽¹⁺⁾

− ln( − m ² _π − m ² _K + s + ^q λ(s) )) + 2 , f I (s) = i ^q λ(s)

16 π s ,

g R (s) = (s − (m π + m K ) ² ) · (f R (s) − f R (m ² _K ∗ )) , g I (s) = (s − (m π + m K ) ² ) · f I (s) ,

ρ K ^∗ = − i g ρ

N s

g I (s) ( | g I (m ² _K ∗ ) | )

· 1

(s − m ² _K ∗ + m K ^∗ g ρ (g R (s) + g I (s))/( | g I (m ² _K ∗ ) | ))

· 1

(s − m ² _K ∗ + m K ^∗ g ρ (g R (s) − g I (s))/( | g I (m ² _K ∗ ) | )) ,

^(5.8)

with

g _ρ = 0.051

^Ge^V ^and

N

^a ^onstant ^to ^normalize ^the ^integral ^over ^this

expression to unity. Compared to the expression give in [29℄, the formula

above ontains an additional fator

1/s

^whih ^makes ^the ^spetral ^funtion

normalizableand thus allows to drop the uto used in the originalwork.

The resultof the imaginarypart of the potentialwith this spetral

distribu-tion is shown inthe right olumnof g. 5.2.

Usingthespetralaveragealsomeanshangingthe

K K ¯ ^∗

^threshold^to

2 m _K + m π

^. ^In ^the^region^between^this ^new ^threshold^and^the ^nominal^threshold^the

potentialdevelops additionalimaginary ontributions, for example fromthe

exhangeofan

η

ⁱⁿ^the

u

^-hannel^but^also^from^the^exhange^of^a^rho-meson.

Thealulationsfortheexhangeofapionhavealreadyproventobetedious,

averagingtheexhange ofapartilewith widthlikethe rho-mesonisbeyond

the sope of this work. Therefore the pseudosalar exhange proesses will

not be onsidered inthe following.

Results

In the setors with isospin or strangeness larger than one no resonanes are

generated. The remaining setors will be disussed in the following. Close

toa resonane the sattering amplitude behaves like

T ab ( √

s) = − s M M ¯

g ₁ g ₂ M _R

√ s − m R + i Γ/2 .

^(6.1)

In the ase of bound states the width

Γ

^vanishes. ^T^o ^analyze ^the ^sattering

amplitudeinmoredetailstheabsolutevalueofthederivativeofthesattering

amplitude with respet toenergy,

d T ab ( √ s) d √

s

.

^(6.2)

is used in the alulations. Close to a resonane this will be tted to the

orresponding expression obtained from equ. (6.1) to evaluate the position

M R

^, ^width

Γ

^and ^oupling ^onstants

g i

^of ^a ^state. ^In ^the ^ase ^of ^a ^bound

state the inverse is used.

6.1 Results for Weinberg-Tomozawa Interation

InarststeptheresultswillbedisussedintheasewhenonlytheW

einberg-Tomozawa interation is onsidered and the xipansion is performed to rst

order. S-and d-wave willbe onsidered.

6.1.1

(I, S) = ( ¹ ₂ , 1)

Fig. (6.1)shows the real andthe imaginarypart ofthe sattering amplitude

for

(I ^G , S) = ( ¹ ₂ , 1)

^. ^The ^dominant ^eet ^is ^a ^narrow ^resonane ^that ^has ^a

mass of1.233 GeV anda widthof 4MeV. The ouplingonstantsasdened

-100 0 100

0 100

-100 0 100

Amplitude [GeV]

-100 0

1 1.2 1.4 1.6 1.8

-100 0 100

1 1.2 1.4 1.6 1.8 2

Energy [GeV]

0 100

Figure6.1: This gure shows the amplitudes for

(I ^G , S) = ( ¹ ₂ , 1)

^when ^only

theWeinberg-Tomozawainterationisonsidered. The leftolumnshowthe

realpart, theseondolumnthe imaginarypartof thesattering amplitude.

The rst row orresponds to the s-wave,the seondrow to tothe transition

amplitudebetweens-andd-waveandthelastrowtod-wave. Thesolidblak

lineorrespondstothe

π K

^hannel,^the ^dashed^blak^line^to

ρ K

^,^the ^dotted

blak line to

ω K

^, ^the ^solid ^green ^line^to

η K ^∗

^and ^the ^dashed ^green ^line ^to

φ K

1 1.2 1.4 1.6 1.8

Energy [GeV]

-500 0 500

Amplitude [GeV]

1.2 1.4 1.6 1.8 2

-150 0 150

Figure 6.2: This gure shows the amplitude for

(I ^G , S) = (0 ⁺ , 0)

^, ^the

K K ¯ ^∗

hannel, when only the Weinberg-Tomozawa interation is onsidered. The

left gure shows the real part, the right gure the imaginary part of the

sattering amplitude. The solid line orresponds to the s-wave, the dashed

linetotothetransitionamplitudebetweens-andd-waveand thedottedline

tod-wave.

inequ. (6.1) are

wave π K ^∗ ρ K ω K η K ^∗ φ K mass width s 0.35 3.1 0.41 2.1 0.58 1.233 GeV 4 MeV s − d 0.22 2.4 0.34 1.4 0.37 d 0.14 1.8 0.27 0.88 0.24

.

^(6.3)

This resonane orresponds to the

K 1 (1270)

^whih ^has ^a ^width ^of ⁹⁰ ^MeV.

Asin[15℄ the

ρ K

^and^the

η K ^∗

^hannels^are ^dominant,^but^here ^the^binding

energy is higher. This alsoexplains the smallwidth: the resonane position

is near the

ρ K

^and ^the

ω K

^threshold. ^With ^those ^to ^hannels ^losed, ^the

width gets tosmall.

When only the s-wave is onsidered as in [15℄ the position and width turn

out to be1.216 GeV and 5MeV respetively.

In the amplitudes that are not dominatedby the

K 1 (1270)

^one ^an ^also^see

aweak signalofthe

K 1 (1400)

^. ^Aording ^to^the ^PDG ^this^state ^has ^a^width

of 174 MeV. A quantitative analysis willbe avoided.

6.1.2

(I ^G , S ) = (0 ⁺ , 0)

In this one hannel problem there is a bound state with a mass of 1348

MeV. The oupling onstants are 3.6, 2.5 and 1.7 for s-wave, the transition

between s- and d-wave and for d-wave respetively. The state orresponds

to the

f 1 (1285)

^whih ^ame ^out ^at ^a ^higher ^mass ⁱⁿ ^[15℄. ^When ^the ^d-wave

-1000 -500 0

500 1000 1500

0 500

Amplitude [GeV]

-1000 -500 0

0.8 1 1.2 1.4 1.6

-250 0 250

1 1.2 1.4 1.6

Energy [GeV]

250 500 750

Figure 6.3: This gure shows the amplitudes for

(I ^G , S) = (0 ⁻ , 0)

^when

only the Weinberg-Tomozawa interation is onsidered. The left olumn

shows the real part, the seond olumnthe imaginary part of the sattering

amplitude.The rst roworresponds to the s-wave,the seond rowto tothe

transitionamplitudebetween s-and d-wave andthe lastrowtod-wave. The

solid line orresponds to the

π ρ

^hannel, ^the ^dashed ^line^to

η ω

^, ^the ^dotted

lineto

K K ¯ ^∗

^and ^the dash-dotted lineto

η φ

6.1.3

(I ^G , S ) = (0 ⁻ , 0)

In this setorthere are two lear signals. The one at higherenergies has the

properties

wave π ρ η ω K K ¯ ^∗ η φ mass width s 1.4 2.5 6.4 3.6 1.225 GeV 85 MeV s − d 1.1 2.0 4.8 2.3 d 0.94 1.6 3.6 1.5

.

^(6.4)

Theothersignalliesontopofthe

π ρ

^-threshold^at⁹⁰⁸^MeV^and^is^dominated

by the

π ρ

^and ^the

K K ¯ ^∗

^hannel. ^The ^PDG ^[30℄ ^lists ^the

h 1 (1170)

^with ^a

width of 360 MeV and the

h ₁ (1380)

^with ^a ^width ^of ⁹⁰ ^Me^V. ^These ^two

resonanes already were overbound in the previous alulations [15℄, this

problemhas even worsened.

6.1.4

(I ^G , S ) = (1 ⁺ , 0)

Inthe

(I ^G , S) = (1 ⁺ , 0)

^setor^the^PDG^lists^the

b 1 (1235)

^with^a^width^of¹⁴²

MeV. In the alulated spetrum a resonane with the following properties

are found:

wave π ω π φ η ρ K K ¯ ^∗ mass width s 1.0 1.6 2.7 5.0 1.304 GeV 92 MeV s − d 0.81 0.97 2.2 3.4 d 0.64 0.72 1.7 2.3

.

^(6.5)

This is a good plae to pause and take a look at the inuene of the

pa-rameters of the model. The following table lists the results when a single

parameter ishangedto the value indiated in the rst olumn.

Scenario mass[ GeV] width[MeV] wave π ω π φ η ρ K K ¯ ^∗ only s − wave 1.265 114 s 1.1 1.8 3.0 5.7

s 1.4 1.5 2.4 5.2 b D = 0.92 1.338 90 s − d 1.1 0.84 1.9 3.5 s 0.83 0.52 1.5 2.4 s 1.7 1.6 3.5 6.4 g _D = 0.5 1.259 147 s − d 1.3 0.96 3.0 4.6 s 1.0 0.61 2.5 3.4 s 1.2 1.8 3.2 5.9 g F = − 0.5 1.259 112 s − d 0.94 1.1 2.7 4.2 s 0.74 0.72 2.3 3.1 s 1.2 1.8 3.1 5.6 Λ S = 1.8 GeV 1.265 113 s − d 0.93 1.1 2.5 3.8 s 0.74 0.71 2.0 2.6

(6.6)

-400 -200 0 200

200 400 600 800

-200 0 200 400

Amplitude [GeV] ^-600

-400 -200 0

1.2 1.4 1.6

-100 0 100

1.2 1.4 1.6

Energy [GeV]

100 200 300

Figure6.4: This gureshows the amplitudesfor

(I ^G , S) = (1 ⁺ , 0)

^when ^only

theWeinberg-Tomozawainterationisonsidered. Theleftolumnshowsthe

realpart, theseondolumnthe imaginarypartof thesattering amplitude.

The rst row orresponds to the s-wave,the seondrow to tothe transition

amplitude between s- and d-wave and the lastrow tod-wave. The solid line

orresponds tothe

π ω

^hannel, ^the ^dashed^line^to

π φ

^,^the ^dotted^line^to

η ρ

and the dash-dotted lineto

K K ¯ ^∗

-50 0

50 100

-20 0 20 40

Amplitude [GeV] ^-80

-60 -40 -20

0.8 1 1.2 1.4 1.6

0 10

1 1.2 1.4 1.6

Energy [GeV]

20 40

Figure6.5: Thisgure shows the amplitudes for

(I ^G , S) = (1 ⁻ , 0)

^when^only

theWeinberg-Tomozawainterationisonsidered. Theleftolumnshowthe

realpart, the seondolumnthe imaginarypartof thesattering amplitude.

The rst row orresponds to the s-wave, the seond rowto tothe transition

amplitude between s- and d-wave and the last rowtod-wave. The solidline

orresponds to the

π ρ

^hannel^and ^the ^dashed^line^to

K K ¯ ^∗

Again, omitting the d-wave lowers the mass of the resonane. With about

40MeV the eet is largerthan the 15MeVfound in the hannelsdisussed

previously. Even so the mass is lowered, the width is inreased.

Setting

b D = 0.92

^shifts^the^mass^away^from^its^physial^value^without

hang-ingthe width. The values 0.5and -0.5 for

g D

^and

g F

^both ^have ^an^idential

eetof themass, butthe widthisinreasedfurtherby thehange in

g D

re-etingastronger ouplingtothe lightest hannel. Aninrease of theupper

bound of the xipansion by 100 MeV has an eet very similar tothe hange

g F

^listed ^above.

6.1.5

(I ^G , S ) = (1 ⁻ , 0)

In the

(I ^G , S) = (1 ⁻ , 0)

^setor ^the ^PDG ^lists ^the

a 1 (1260)

^with ^a ^width

of 250-600 MeV. Between the two thresholds at 908 MeV and 1390 MeV a

struture an be seen. A quantitative analysis is not possible and will be

avoided. A detailed disussion on the nature of the

a 1

ⁱⁿ ^the ^framework ^of

Insummarythepreviousresults[15℄anbereproduedinthenewformalism.

The approximation to omit the kinematially suppressed d-wave turns out

tobe justied, asexpeted fromphysialreasons.

Im Dokument Coupled-channel study of axial-vector mesons with realistic t- and u-channel exchanges (Seite 42-54)

U (s) = ¯ U (s) − 1 2 i

Z ∞

thres 2

dw π

U (+) (w) − U ( − ) (w)

w − s .

U ¯

K K ¯ ∗

U (s)

K

K ¯ ∗

u

(I G , S) = (0 + , 0)

U

K ∗

K ∗

ρ S (s)

1.4 1.5 1.6 1.7 1.8 1.9 2 s^1/2 [GeV]

-60 -40 -20 0 20 40 60

Amplitude [GeV²]

K K ¯ ∗ → K K ¯ ∗

u

Z ∞

(m K +m π ) 2 dm 2 1

Z ∞

(m K +m π ) 2 dm 2 2 ρ S (m 2 1 )ρ S (m 2 2 )U (s, m 2 1 , m 2 2 )

K ∗

m π + m 1 = s , c − (s, m 2 π , m 2 2 ) = s ,

c −

m 1

m 2

λ(s) = (s − (m π + m K ) 2 ) · (s − (m π − m K ) 2 ) , f R (s) = 1

16 π 2

q λ(s)

2 s (ln( − m 2 π − m 2 K + s − q λ(s) )

-25 0 25

-25 0 25

0 2500

0 2500

1.25 1.5 1.75

-50000 0 50000

1.5 1.75 2

Energy [GeV]

-50000 0 50000

K K ¯ ∗ → K K ¯ ∗

u

K ∗

T 11 (1+)

T 12 (1+) = T 21 (1+)

T 22 (1+)

− ln( − m 2 π − m 2 K + s + q λ(s) )) + 2 , f I (s) = i q λ(s)

16 π s ,

g R (s) = (s − (m π + m K ) 2 ) · (f R (s) − f R (m 2 K ∗ )) , g I (s) = (s − (m π + m K ) 2 ) · f I (s) ,

ρ K ∗ = − i g ρ

N s

g I (s) ( | g I (m 2 K ∗ ) | )

· 1

(s − m 2 K ∗ + m K ∗ g ρ (g R (s) + g I (s))/( | g I (m 2 K ∗ ) | ))

· 1

(s − m 2 K ∗ + m K ∗ g ρ (g R (s) − g I (s))/( | g I (m 2 K ∗ ) | )) ,

g ρ = 0.051

N

1/s

K K ¯ ∗

2 m K + m π

η

u

T ab ( √

s) = − s M M ¯

g 1 g 2 M R

√ s − m R + i Γ/2 .

Γ

d T ab ( √ s) d √

s

.

M R

Γ

g i

(I, S) = ( 1 2 , 1)

Z _∞

thres ²

U ⁽⁺⁾ (w) − U ⁽ ⁻ ⁾ (w)

K K ¯ ^∗

K ¯ ^∗

(I ^G , S) = (0 ⁺ , 0)

K ^∗

K ^∗

K K ¯ ^∗ → K K ¯ ^∗

Z _∞

(m _K +m π ) ² dm ² ₁

Z _∞

(m _K +m π ) ² dm ² ₂ ρ S (m ² ₁ )ρ S (m ² ₂ )U (s, m ² ₁ , m ² ₂ )

K ^∗

m π + m 1 = s , c ₋ (s, m ² _π , m ² ₂ ) = s ,

c ₋

λ(s) = (s − (m π + m K ) ² ) · (s − (m π − m K ) ² ) , f R (s) = 1

16 π ²

2 s (ln( − m ² _π − m ² _K + s − ^q λ(s) )

K K ¯ ^∗ → K K ¯ ^∗

K ^∗

T ₁₁ ⁽¹⁺⁾

T ₁₂ ⁽¹⁺⁾ = T ₂₁ ⁽¹⁺⁾

T ₂₂ ⁽¹⁺⁾

− ln( − m ² _π − m ² _K + s + ^q λ(s) )) + 2 , f I (s) = i ^q λ(s)

g R (s) = (s − (m π + m K ) ² ) · (f R (s) − f R (m ² _K ∗ )) , g I (s) = (s − (m π + m K ) ² ) · f I (s) ,

ρ K ^∗ = − i g ρ

g I (s) ( | g I (m ² _K ∗ ) | )

(s − m ² _K ∗ + m K ^∗ g ρ (g R (s) + g I (s))/( | g I (m ² _K ∗ ) | ))

(s − m ² _K ∗ + m K ^∗ g ρ (g R (s) − g I (s))/( | g I (m ² _K ∗ ) | )) ,

g _ρ = 0.051

K K ¯ ^∗

2 m _K + m π

g ₁ g ₂ M _R

(I, S) = ( ¹ ₂ , 1)

(I ^G , S) = ( ¹ ₂ , 1)

(I ^G , S) = ( ¹ ₂ , 1)

η K ^∗

(I ^G , S) = (0 ⁺ , 0)

K K ¯ ^∗

wave π K ^∗ ρ K ω K η K ^∗ φ K mass width s 0.35 3.1 0.41 2.1 0.58 1.233 GeV 4 MeV s − d 0.22 2.4 0.34 1.4 0.37 d 0.14 1.8 0.27 0.88 0.24

η K ^∗

(I ^G , S ) = (0 ⁺ , 0)

(I ^G , S) = (0 ⁻ , 0)

K K ¯ ^∗

(I ^G , S ) = (0 ⁻ , 0)

wave π ρ η ω K K ¯ ^∗ η φ mass width s 1.4 2.5 6.4 3.6 1.225 GeV 85 MeV s − d 1.1 2.0 4.8 2.3 d 0.94 1.6 3.6 1.5

K K ¯ ^∗

h ₁ (1380)

(I ^G , S ) = (1 ⁺ , 0)

(I ^G , S) = (1 ⁺ , 0)

wave π ω π φ η ρ K K ¯ ^∗ mass width s 1.0 1.6 2.7 5.0 1.304 GeV 92 MeV s − d 0.81 0.97 2.2 3.4 d 0.64 0.72 1.7 2.3

Scenario mass[ GeV] width[MeV] wave π ω π φ η ρ K K ¯ ^∗ only s − wave 1.265 114 s 1.1 1.8 3.0 5.7