Kaonic hydrogen versus the $K^{-}p$ low energy data

0.28 MB·

by  A. Cieply

EPJ manuscript No.
(will be inserted by the editor)
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Kaonic hydrogen versus the K p low energy data
A. Cieply´1 and J. Smejkal2
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0 1 Nuclear Physics Institute,250 68 Rˇeˇz, Czech Republic
0 2 Instituteof Experimental and Applied Physics, Czech Technical University,Horsk´a 3a/22, 128 00 Praha 2, Czech Republic
2
Received: date/ Revised version: date
n
a
J Abstract. We present an exact solution to the K−-proton bound state problem formulated in the mo-

8 mentum space. The 1s level characteristics of the kaonic hydrogen are computed simultaneously with the
−
1 available low energy K p data. In the strong interaction sector the meson-baryon interactions are de-
scribed by means of an effective (chirally motivated) separable potential and its parameters are fitted to

] the experimentaldata.
h
p
PACS. 11.80.Gw Multichannelscattering–12.39.Fe ChiralLagrangians–13.75.Jz Kaon-baryoninterac-
-
p tions – 36.10.Gv Mesonic atoms and molecules, hyperonicatoms and molecules
e
h
[ 1 Introduction [10] and whether they fit into the picture drawn by the

2 chiralmodels represents a question which is addressedby
v We developed a precise method of computing the meson- the theory [8,11] as well as by the coming SIDDHARTA
8 nuclear bound states in momentum space. The method experiment.
2
was already applied to pionic atoms and its multichannel
9
versionwasusedtocalculatethe1slevelcharacteristicsof
4
pionic hydrogen[1]. In the present work we aim at simul- 2 Formalism
.
1 taneousdescriptionofboththe1slevelkaonicboundstate
1 −
andtheavailableexperimentaldatafortheK pinitiated Our approach to solving the meson-nuclear bound state
7
processes. problem in the presence of multiple coupled channels was
0
Until recently the old Deser-Trueman formula [2] was given in Ref. [1]. Here we just remark that the method is
:
v used to determine the strong interaction energy shift and basedonthe constructionof the Jostmatrix and involves
Xi width (of the 1s level) in kaonic hydrogen from the K−p the solution of the Lippman-Schwinger equation for the
scattering length and vice versa. Recently, the Deser - transition amplitudes between various channels. Bound
r
a Trueman relation was modified to include the isospin ef- states in a specific channel then correspond to zeros of
fects and electromagnetic corrections [3]. Our exact so- the determinant of the Jost matrix at (or close to) the
−
lution of the K p bound state problem allows to check positive part of the imaginary axis in the complex mo-
the precision and limitations of those approximate ap- mentum plane. The zeros are computed iteratively and if
proaches. However, one should not forget that the strong onlythe point-like Coulombpotentialisconsideredinthe
interaction part of the scattering length is not a directly K−pchannelthemethodreproducesthewellknownBohr
measured quantity and its determination from the scat- energyof the 1s level with a precisionbetter than 0.1 eV.
tering data is always model dependent. We follow the approach of Ref. [4] when constructing
The treatment of the kaon-nucleon interaction at low thestronginteractionpartofthepotentialmatrix.Inthis
energies requires a special care. Unlike the pion-nucleon model the Λ(1405) resonance is generated dynamically
interactiontheK¯N dynamicsisstronglyinfluencedbythe bysolvingcoupledLippman-Schwingerequationswithin-
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existence of the Λ(1405) resonance, just below the K p put effective (chirally motivated) potentials. The reader
threshold. This means that the standard chiral perturba- should note that our approach differs from the recently
tion theory is not applicable in this region. Fortunately, morepopularon-shellschemebasedontheBethe-Salpeter
one can use non-perturbative coupled channel techniques equation,unitarityrelationfortheinverseoftheT-matrix
to deal with the problem and generate the Λ(1405) reso- andonthedimensionalregularizationofthescalarloopin-
nancedynamically.Suchapproachhasprovenquiteuseful tegral[12].Further,whiletheauthorsofRef.[4]restricted
andseveralauthorshavealreadyapplieditto variouslow themselves only to the first six meson-baryon channels
energymeson-baryonprocesses[4]-[8].Whethertherecent that are open at the K¯N threshold we employ all ten
experimental results on kaonic hydrogen from the DEAR coupled meson-baryonchannels: K−p, K¯0n, π0Λ, π+Σ−,
collaboration[9]areconsistentwiththeolderKEKresults π0Σ0, π−Σ+, ηΛ, ηΣ0, K+Ξ−, and K0Ξ0.
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2 A. Cieply´, J. Smejkal: Kaonic hydrogen versus the K p low energy data
The strong interaction potentials are constructed in way.First,theaxialcouplingsDandF werealreadyfixed
such a way,that in the Bornapproximationthey give the in the analysis of semileptonic hyperon decays [14], D =
same (up to O(q2)) s-wavescatteringlengths as arethose 0.80, F = 0.46 (g = F +D = 1.26). Then, we fix the
A
derivedfromtheunderlyingchirallagrangian.Hereweuse couplingsb andb tosatisfytheapproximateGell-Mann
D F
them in the separable form formulas for the baryon mass splittings,
′ 1 Mi Cij ′ 1 Mj MΞ −MN =−8bF(m2K −m2π) ,
V (k,k ) = g (k) g (k ) ,
ij r2Ei ωi i f2 j s2Ej ωj MΣ −MΛ = 136bD(m2K −m2π) , (3)
1
g (k) = , (1)
j 1+(k/α )2 which gives b =0.064 GeV−1 and b =−0.209 GeV−1.
j D F
′ Similarly, we determine the coupling b0 and the baryon
inwhichthemomentak andk refertothe meson-baryon chiral mass M0 from the relations for the pion-nucleon
c.m. system in the i and j channels, respectively, and the sigma term σ and the proton mass,
πN
kinematicalfactors M /(2E ω )guaranteeaproperrel-
j j j
ativistic flux normaplization with Ej, Mj and ωj denoting σπN = −2m2π(2b0+bD+bF) ,
tch.me.mseyssotnemenoefrgchyaannndelthj.eTbhaeryooffnsmhaesllsfaonrmd efnaecrtgoyrsing t(hke) Mp = M0−4m2K(b0+bD−bF)−2m2π(b0+2bF) .(4)
j
introduce the inverse range radii α that characterize the
j Since the value of the pion-nucleon σ-term is not well de-
radius of interactions in various channels.Finally, the pa- terminedweenforcefourdifferentoptions,σ =20−50
πN
rameter f stands for the pseudoscalar meson decay con-
MeV, which cover the interval of the values considered
stant in the chiral limit and the coupling matrix C is
ij by various authors. Finally, we reduce the number of the
determined by chiralSU(3) symmetry and includes terms
inverse ranges α to only five: α , α , α , α
j KN πΛ πΣ ηΛ/Σ
upto the secondorderinthe mesonc.m.kinetic energies.
(forboth theηΛ andηΣ channels),andα .This leaves
KΞ
For the first six channels the couplings C were listed in
ij us with 11 free parameters: the five inverse ranges, the
[4] and we intend to publish the remaining coefficients in
meson-baryonchiralcoupling f,andfivemorelowenergy
amoreelaboratepaper[13].Forillustration,weshowjust
constantsfromthesecondorderchirallagrangiandenoted
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the coupling of the elastic K p process,
by dD, dF, d0, d1, and d2.
The fitted low energy K¯N data include the three pre-
E2 −m2 D2 E2
CK−p,K−p =−EK − K2M0 K +(F2+ 3 )2MK0 + cisely measured threshold branching ratios [15]
+4m2K(bD+b0)−EK2 (2dD+2d0+d1) .(2) γ = σ(K−p→π+Σ−) =2.36±0.04,
σ(K−p→π−Σ+)
Herem andE denotethekaonmassandenergyinthe
K K σ(K−p→charged particles)
center-of-mass frame, M0 stands for the baryon mass in R = =0.664±0.011,
the chirallimit, andthe parametersF,D, andb’s andd’s c σ(K−p→all)
represent coupling constants that appear in the underly- σ(K−p→π0Λ)
ing chiral lagrangian(see [4] for more details). The origin Rn = σ(K−p→all neutral states) =0.189±0.015,(5)
and relevance of the various terms present in Eq. (2) was
discussedthoroughlyinRef.[7].Ingeneral,thecoefficients −
and K p-initiated total cross sections. For the later ones
C include contributions from the meson-baryon contact
ij we consider only the experimentaldata taken atthe kaon
interactions as well as the direct and crossedBorn terms. laboratorymomentap =110MeV(fortheK−p,K¯0n,
LAB
However,incontrastto[7]ourmodelisbasedonthestatic π+Σ−, π−Σ+ final states) and at p = 200 MeV (for
LAB
(heavy)nucleonapproximationadoptedbythe authorsof the same four channels plus π0Λ and π0Σ0). Our results
Ref.[4]inwhichtheunderlyinglagrangianisexpressedin
show that the inclusion of the cross section data taken
a fixed reference frame.
at other kaon momenta is not necessary since the fit at
The potential of Eq.(1) is used not only when solving just 1 − 2 points fixes the cross section magnitude and
the bound state problem but we also implement it in the
theenergydependenceisreproducednicelybythe model.
standard Lippman-Schwinger equation and compute the
Finally, we include the DEAR results [9] on the strong
low energy K¯N cross sections and branching ratios from
interaction shift ∆E and the width Γ of the 1s level in
N
the resulting transition amplitudes.
kaonic hydrogen:
∆E (1s)=(193±43) eV, Γ(1s)=(249±150) eV . (6)
¯ N
3 KN data fits
Thus,we endup witha totalof15data points inourfits.
The parameters of the chiral lagrangian which enter the OurresultsaresummarizedinTables1-3.Thefirstta-
coefficientsC andtheinverserangeradiiα determining ble showsthe resultsofourχ2 fits comparedwiththe rel-
ij j
the off-shell behavior of the potentials are to be fitted evant experimental data. The resulting χ2 per data point
to the experimental data. Before performing the fits we indicatesatisfactoryfits.Itisworthnotingthattheirqual-
reducethenumberofthefittedparametersinthefollowing ity and the computed values do not depend much on the
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A. Cieply´, J. Smejkal: Kaonic hydrogen versus the K p low energy data 3
Table 1. The fitted K¯N threshold data
σ [MeV] χ2/N ∆E [eV] Γ [eV] γ R R
πN N c n
20 1.33 232 725 2.366 0.657 0.191
30 1.36 262 697 2.365 0.657 0.190
40 1.37 253 710 2.370 0.657 0.189
50 1.40 266 708 2.370 0.658 0.190
exp - 193(43) 249(150) 2.36(4) 0.664(11) 0.189(15)
Table 2. Chiral lagrangian parameters (b0 and d’s in 1/GeV):
σπN [MeV] b0 M0 [MeV] a+πN [m−π1] f [MeV] d0 dD dF d1 d2
20 -0.190 997 -0.016 108.6 -0.385 -0.368 -0.817 0.396 0.152
30 -0.321 864 0.001 100.0 -0.354 -0.206 -0.522 0.406 -0.211
40 -0.453 729 0.006 108.9 -0.484 -0.151 -0.459 0.448 -0.280
50 -0.584 594 0.007 108.8 -0.747 -0.092 -0.429 0.567 -0.349
27 [4] -0.279 910 -0.002 94.5 -0.40 -0.24 -0.43 0.28 -0.62
exact value of the σπN term. Tables 2 and 3 show the Table 3. Inverserange parameters αj (in MeV):
fitted parameters of the chiral lagrangianand the inverse
σ [MeV] α α α α α
rangeparametersα . The lastrowsin the tablescompare πN KN πΛ πΣ ηΛ/Σ KΞ
j 20 610 209 570 1100 530
our values with those determined in Ref. [4]. We remind
30 647 262 535 308 21
the reader that the parameter b0 and the baryon mass in 40 653 320 618 281 89
thechirallimitwerenotfittedtothedataandaregivenin
50 594 370 610 342 124
the second and third column of Table 2 only to visualize
27 [4] 760 300 450 - -
their respective values corresponding to the selected σ
πN
term. The πN isospin-even scattering length a+ shown
πN
in the fourth column of Table 2 was not included in our
fits either but we feel that its presentation is important
is not surprising that the computed πN scattering length
and deserves some comments.
is becoming negative for too low σ terms. Anyway, it
Thegoalofthepresentworkwastocheckthecompat- is interesting that our fits aimed atπNthe K¯N interactions
ibility of the DEAR kaonic hydrogen data with the low
allowforsogoodreproductionoftheπN quantity.Specif-
−
energy K p cross sections and branching ratios. There-
ically, the parameter set obtained in the fit for σ = 30
fore, we have not included in our fits the Λ(1405) mass MeVgivesa+ inaniceagreementwithexperimeπnNtwhile
spectrum and other processes considered e.g. in Ref. [8]. the χ2/N isπNonly slightly inferior to our best fit. Many
Infact,thelowenergyconstantsinvolvedinthefitsshould otherauthors(e.g.[4]or[8])includethea+ valuedirectly
bealsoconstrainedbyotherobservablescalculatedwithin πN
in their fits. The d couplings (of the second order chiral
theframeworkofChPTinvolvingthesamemeson-baryon
lagrangian)contributetothecontactmeson-baryoninter-
lagrangian. The spectrum of baryon masses and the πN
actions in the second order of meson momenta. Although
isospin-even scattering length may come to one’s mind in
our fits confirm their mostly negative signs it is difficult
this respect. The later quantity to order q3 is given by
to come to any conclusions concerning their values. The
[16]:
factthat eventhe signof d2 is not welldetermined in our
1 analysis speaks for itself.
a+ = ×
πN 4π(1+m /M ) We have also tried to perform fits with the b param-
π N
eters taken from the analysis of the baryon mass spec-
m2
× f2π(−2bD−2bF −4b0+dD+dF +2d0)− tTroummoz[1a8w]aa)ntdermwitchonotnrilbyutthinegctuorrtehnetCalgecboreaffi(cWieenitnsb(etrhge-
(cid:20) ij
m2 g2 3g2m3 approachadoptedinRef.[5]).Unfortunately,wewerenot
− π A + A π . (7)
f2 4M 64πf4 able to achieve satisfactory results in those cases. Thus,
N (cid:21) we conclude that the low energy constants derived in the
Since the experimental value of a+ is practically consis- analysisofbaryonmassesarenot suitable in the sector of
πN
tent with zero, a+ = −(0.25±0.49)·10−2m−1 [17], it meson-baryon interactions and that the inclusion of the
is encouraging to0n+ote the mostly negative signπs of the d- q2 terms is necessary for a good description of the K¯N
parameters that cancel the positive contributions due to data. The later point is in agreementwith the analysis of
the b terms and the q3 correction represented by the last Ref. [7].
term in Eq. (7). As a smaller σ term means a smaller The inverse range parameters given in Table 3 are in
πN
absolute value of the negative parameter b0 (and hence a line with our expectations. The values corresponding to
smallerpositivecontributionduetotheb0 termina+πN)it the open channels K¯N, πΛ and πΣ seem to be well de-
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4 A. Cieply´, J. Smejkal: Kaonic hydrogen versus the K p low energy data
termined and show only a moderate dependence on the Table 4. Precision of the Deser-Trueman formula. The com-
adoptedvalue ofthe σ term.Ingeneral,the rangesob- plexenergies∆E −(i/2)Γ aregivenfortheapproximateDT
πN N
tained for the open channels correspond to the t-channel andMDTformulasandcompared withourcomputed“exact”
exchanges that are believed to dominate the interactions. values.
Ochnantnheelsotishenrothawnedlltdheefinraedngienothfeinfittesraacntdiotnhseifinttthede vcaloluseeds aK−p [fm] ∆EN −(i/2)Γ [eV]
DT 207−(i/2)832
α and α exhibit relatively large statistical errors.
ηΛ/Σ KΞ −0.50+i1.01 MDT 251−(i/2)714
This feature also justifies our use of only one range pa- exact 232−(i/2)725
rameter for both η channels. DT 247−(i/2)830
In Figure 1 we present the low energy K−p initiated −0.60+i1.01 MDT 285−(i/2)689
cross sections calculated using our best fit with σ =20 exact 266−(i/2)708
πN
MeV. The results obtained for the other adopted values
of σ are quite similar, therefor we decided to not in-
πN
clude them in the figure. Though we declined from using
all experimental data in our fits and took only the data fact, the correction due to Coulomb interaction is taken
pointsavailablefortheselectedkaonlaboratorymomenta only in its leading order in the MDT formula. The inclu-
p = 110 MeV and p = 200 MeV, the description sion of more terms of the relevant geometric series would
LAB LAB
of the data is quite good. Specifically, we do not observe bring the MDT value into a better agreement with our
the lowering of the calculated cross sections in the elas- exact solution [19].
−
tic K p channel reported by Borasoy et al. [7] for their
fitsincludingthekaonichydrogencharacteristics.Though
−
our K p cross sections are also slightly below the exper- 4 Conclusions
imental data the difference is not significant. In addition,
the inclusion of electromagnetic corrections discussed in
We have computed the characteristics of kaonic hydro-
Ref.[7]shouldpartlyimprovethe descriptionforthe low-
gen exactly and compared the results (the 1s level energy
est kaon momenta. −
shiftandwidth)withthevaluesdeterminedfromtheK p
Finally,letusturnourattentiontothecalculatedchar- scattering length by means of using the standard Deser-
acteristics of the 1s level in kaonic hydrogen. The strong Trueman formula and its modified versionwhich includes
interaction energy shift of the 1s level in kaonic hydrogen the corrections due to electromagnetic effects. It looks
is reproduced well but we were not able to get a satis- thattheapproximateDTformulagivesthe1senergylevel
factory fit of the 1s level energy width as our results are stronginteractionshiftandwidth about10%and15%off
significantly larger than the experimental value. This re- the exactly computed values, respectively. Although the
sult is in line with the conclusions reached by Borasoy, modified DT formula does much better job on account of
MeissnerandNissler[11]onthebasisoftheircomprehen- the width the energy level shift remains about 10% off
−
siveanalysisofthe K pscatteringlengthfromscattering the exact value that lies approximately in the middle be-
experiments. However, when considering the interval of tween the DT and MDT values. In view of the current
three standard deviations and also the older KEK results levelofthe experimentalprecisionthe use ofthe modified
[10] (which give less precise but larger width) we cannot DT formula is sufficient. Nevertheless, the situation may
conclude that kaonic hydrogen measurements contradict change after the coming SIDDHARTA experiment being
the other low energy K¯N data. prepared in Frascati.
In Table 4 we compare our results (for σ = 20 Aneffectivechirallymotivatedseparablepotentialwas
πN
−
and 50 MeV) for the 1s level characteristics in kaonic hy- usedinsimultaneousfitsofthelowenergyK pcrosssec-
drogen with the approximate values determined from the tions, the threshold branching ratios and the character-
K−p scattering lengths aK−p. The later quantity is ob- istics of kaonic hydrogen. The fits are quite satisfactory
tainedfromthemultiplechannelcalculationthatusesthe except the 1s level energy width being much larger than
same parametrization of the strong interaction potential, theexperimentalvalue.Inviewofthefactthattheexperi-
Eq. (1). The 1s level complex energies are shown for: the mentalprecisionofthekaonichydrogendataisstillrather
standard Deser-Trueman formula (DT) [2], the modified low one cannot say that the data contradict the chirally
Desert-Trueman formula (MDT) [3] (see also Ref. [7] for motivated model used to describe the low energy meson-
therelationsusedtoobtaintheDTandMDTvalues)and baryon interactions. However, as the opposite statement
our “exact” solution of the bound state problem. The re- cannot be made either we should wait for the new exper-
sults obtainedforσ =30and40MeVarequite similar iment to clarify the situation.
πN
−
(with almostidentical K p scattering lengths), so we did
not include them in the table in orderto keepthe presen- Acknowledgement: A. C. acknowledges the finan-
tation more transparent. Since the numerical precision of cial support from the GA AVCR grant A100480617. The
determiningtheboundstateenergybyourmethodisbet- work of J. S. was supported by the Research Program
terthan0.1eV,thediscrepancybetweenthe“MDT”and Fundamentalexperimentsin thephysics ofthemicroworld
the “exact” values should be attributed to higher order No. 6840770029of the Ministry of Education, Youth and
corrections not considered in the derivation of MDT. In Sports of the Czech Republic.
−
A. Cieply´, J. Smejkal: Kaonic hydrogen versus the K p low energy data 5
Fig. 1. Total cross sections for K−p scattering and reactions to the meson-baryon channels open at low kaon laboratory
momenta p . The experimental data are thesame as those compiled in Fig. 1 of Ref. [4].
LAB
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