Results for the full interation

Insummarythepreviousresults[15℄anbereproduedinthenewformalism.

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

tobe justied, asexpeted fromphysialreasons.

-800 -400 0

1 1.2 1.4 1.6

Energy [GeV]

0 500 1000

Amplitude [GeV]

Figure 6.6: This gureshows the amplitudes for

(I ^G , S) = (1 ⁺ , 0)

^{when the}

fulltreelevelsatteringamplitudeinequ. (3.10)(exludingthepseudosalar

u

^{-hannel) is onsidered to zeroth order in the xipansion. The upper graph}

orresponds to the real part of the amplitude while the lower one shows the

imaginary part. The solid line orresponds to the

π ω

^{hannel, the dashed}

lineto

π φ

^{,the dotted lineto}

η ρ

^{and the} dash-dotted lineto

K K ¯ ^∗

^.

-600 0 600

600 1200

0 1000 2000

Amplitude [GeV]

-2000 -1000 0

1 1.2 1.4 1.6

-2000 0 2000

1.2 1.4 1.6

Energy [GeV]

2000 4000

Figure6.7: This gure shows the amplitudes for

(I ^G , S) = (1 ⁺ , 0)

^{when the}

fulltreelevelsatteringamplitudeinequ. (3.10)(exludingthepseudosalar

u

^{-hannel) is onsidered to rst order in the xipansion. The upper graph}

orresponds tothe real part of the amplitude while the lower one shows the

imaginary part. The solid line orresponds to the

π ω

^{hannel, the dashed}

lineto

π φ

^{, the dotted lineto}

η ρ

^{and the} dash-dotted lineto

K K ¯ ^∗

^.

The situationhangeswhen the rst orderof the xipansionisinluded. The

result for Weinberg-Tomozawa was already presented, with a mass of 1.304

GeVitwasalmost70MeVabovethe physialvalue. Toreproduethe

physi-almass andwidth,the parameters

g D = 0.7

^and

g F = 2.8

^{were neessary in}

[19℄. Thesevaluesareagoodstartingpointforthetinthepresentsenario.

Herethe

b 1

^{anbereproduedwiththe values}

g D = 0.8

^and

g F = − 3.56

^,the

result isshown in g. 6.7.

The ontributions to the most important proess

K K ¯ ^∗ → K K ¯ ^∗

^are

illus-tratedin g. 6.8.

In summarythe results are

Scenario g D g F mass[GeV] width[MeV] π ω π φ ρ η K K ¯ ^∗ xiorder 0 4.15 − 5.75 1.230 135 1.8 0.73 4.4 6.9

xiorder 1 1.9 1.8 4.3 6.9

s + d wave 0.8 − 3.56 1.231 138 3.2 2.1 6.1 8.7 6.0 2.4 8.6 11.2 [

¹⁹

] 0.7 − 2.8 1.230 142 2.1 1.0 2.3 4.2

.

(6.8)

100 200

20 40

200 400

200 400

-400 -200 0

1.4 1.6

-40 -20 0

1.4 1.6 1.8

0 20

Figure6.8: This gure shows the amplitudes for the proess

K K ¯ ^∗ → K K ¯ ^∗

with

(I ^G , S) = (1 ⁺ , 0)

^{using the ounter terms speied in 6.8. The rst}

row shows the Weinberg-Tomozawa interation (left) and the term whih

is multiplied by

b _D

^{, the seond line shows the} interations multiplied by

g D

^and

g F

^{(with the tted value of equ. 6.8), the third line shows the}

s

-and

t

^-hannel ontributions to the potential and the lastline illustratesthe

u

^{-hannel. Solid lines orrespond to the xipanded potential atually used}

while the dashed lines indiate the exat potential without xipansion. The

xipansionpointisthe

K K ¯ ^∗

^{thresholdat1.39GeV,thepotentialisnotneeded}

below that point.

-200 0 200 400

1 1.2 1.4 1.6

Energy [GeV]

0 200 400 600

Amplitude [GeV]

Figure6.9: This gure shows the amplitudes for

(I ^G , S) = (1 ⁺ , 0)

^{when the}

fulltreelevelsatteringamplitudeinequ. (3.10)(exludingthepseudosalar

u

^{-hannel) is onsidered to rst order in the xipansion. The upper graph}

orresponds tothe real part of the amplitude while the lower one shows the

imaginary part. The solid line orresponds to the

π ω

^{hannel, the dashed}

lineto

π φ

^{, the dotted lineto}

η ρ

^{and the} dash-dotted lineto

K K ¯ ^∗

^.

Thesize of the ounter terms isonlyreasonable (andinlinewith [19℄)when

the rst order of the xipansion is onsidered. The open

π φ

^{hannel plays a}

moreimportantrolewhenallheliitiesandtherstorderofthexipansionare

onsideredwhilethe dominantdeayhannel

π ω

^{remainsalmostunhanged}

ompared tothe previous work [19℄.

6.2.2

(I, S ) = ( ¹ ₂ , 1)

^{: the}

K 1 (1270)

In this setor the previous work [15℄ found two resonanes, the

K 1 (1270)

whihhas awidth of90MeV and the

K 1 (1400)

^{withawidthof 174 MeV.In}

the more detailed analysis [19℄ the oupling onstants of the ounter terms

had to be set to

g D = 0.2

^and

g F = − 0.1

^{. Fig. 6.10 shows the result when}

these two ouplings are set to zero. A lear signal an be seen lose to the

nominal

ρ K

^{thresholdat1.266GeVandthe nominal}

ω K

^{threshold at1.278}

GeV.Thewidthofthe

ρ

^{mesonwillhavea}onsiderableeetonthewidthof

0 150 300

100 200 300

-200 0 200 400

Amplitude [GeV] ^-200

0 200 400

1.2 1.4 1.6

0 1000 2000 3000

1.2 1.4 1.6

Energy [GeV]

1000 2000

Figure 6.10: This gure shows the amplitudes for

(I ^G , S) = ( ¹ ₂ , 1)

^{when all}

interationsexeptthepseudosalarexhangesareonsidered. Theouplings

of the ounter-terms,

g _D

^and

g _F

^{are put to zero. The left olumn show the}

realpart, the seondolumnthe imaginarypartof thesattering amplitude.

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

amplitudebetweens-andd-waveandthelastrowtod-wave. Thesolidblak

lineorresponds tothe

π K

^{hannel,the dashedblak lineto}

ρ K

^,thedotted

blak line to

ω K

^{, the solid green lineto}

η K ^∗

^{and the dashed green line to}

φ K

^.

the resonane, therefore we willnot try to t the resonane alulated with

sharp masses to the data more aurately. Nevertheless it is enouraging

that the result with all interations and no ounter terms losely resembles

the result of [15℄ whih agreed well with experimental data one the width

of the vetor mesons wastaken into aount (see g. 2 in[15℄).

Summary and Outlook

In this work we studied the sattering of the lightest otet of pseudosalar

mesons, the pion, the K-meson and the eta, o the nonet of vetor mesons

whihinludesthe

ρ

^{meson. Aformalismbasedonanon-linearintegral}

equa-tion is used to ompute sattering amplitudes. Solutions of the non-linear

integral equations omply with onstraints set by ausality and unitarity.

The main input into this equation are pretreated tree-level sattering

am-plitudes that were derived in the followingsteps: aninteration Lagrangian

basedonhiralandlarge

N c

^{argumentswasutilizedtoalulatethetreelevel}

amplitudes. It inluded the Weinberg-Tomozawa interation, three ounter

terms and the exhange of vetor mesons in the

s

^-,

t

^{- and}

u

^{-hannel. The}

exhange of pseudosalar mesons wasnot onsidered in the nal alulation

beause of tehnial hallenges. It was illustrated how these issues an be

masteredinpriniplebut theatualalulationwasbeyondthe sopeofthis

work.

Thetree levelamplitudeswere thenpartial-waveprojeted andextrapolated

tohigherenergies basedonthe knowledge of theiranalyti struture. These

modied potentials denethe inputof the non-linear integralequations.

In a rst step, the spetrum was alulated for the Weinberg-Tomozawa

interation onlyand ompared to the previous results. The inuene of the

ounter termsand the orderof the analytialextrapolationof the sattering

amplitude were addressed. Then the spetrum for the full interation was

disussed in setors of partiular interest. Counter-terms were adjusted to

reprodue the mass and width of the

b 1 (1235)

^{exatly. In addition, the}

K 1 (1270)

^wasinvestigatedinmoredetail. Itsmassisreoveredintheabsene ofanyounterterm. Thisisinonitwiththe signiantsizeoftheounter

terms obtained from the study of the

b 1 (1235)

^{. We take this asymmetry as}

a onsequene of the absene of the pseudosalar

u

^-hannel.

Themostobviousextensionofthisworkistheinlusionofthesepseudosalar

exhange proesses. One possible way to ahieve this has already been

dis-ussed, but improvements are desirable, espeially to keep the neessary

CPU-time in reasonable limits. Inluding these proesses we expet that a

reliableomputation of the D/S ratios inthe deays of the axial resonanes

isfeasible.

Anotherinterestingpossibilityistheinlusionofvetor-vetorand

pseudosalar-pseudosalar hannels. From a formal point of view these hannels an be

treatedanalogously. The partialwave projetion willbeome more tedious.

Withallthesehannelsinludeditisinteresting toalulateresonaneswith

otherspin and parity thanjust

1 ⁺

^{with relativelylittleadditionaleort.}

In summarythe formalismemployed inthis thesis reprodues the measured

spetrumand previous resultswithin reasonablelimits. It alsoallows touse

amorerealistiinteration andtoinludemore hannelsinaonsistent way

sothatin futureworks morephysialsystems beome aessibleaswell asa

widervariety of observables.

The invariant amplitude

In this appendix we give expliit expressions for the G's whih result from

thepartial-waveprojetionofequ. (3.10). Onlynon-vanishingG'sareshown

withoneexeption: if

G 3

^{doesnotvanish,then}

G 4

^{isthehermitianonjugate}

(inomingand outgoingmasses interhanged) unless displayed expliitly.

Im Dokument Coupled-channel study of axial-vector mesons with realistic t- and u-channel exchanges (Seite 54-63)