Hyperon-nucleon Scattering In A Covariant Chiral Effective Field Theory Approach

0.18 MB·

by  Kai-Wen Li

Hyperon-nucleon Scattering In A Covariant Chiral
Effective Field Theory Approach
∗
Kai-WenLi
7 SchoolofPhysicsandNuclearEnergyEngineeringandInternationalResearchCenterfor
1 NucleiandParticlesintheCosmos,BeihangUniversity,Beijing100191,China
0
E-mail: [email protected]
2

r Xiu-LeiRen
a
M SchoolofPhysicsandStateKeyLaboratoryofNuclearPhysicsandTechnology,Peking
University,Beijing100871,China

3 E-mail: [email protected]
1
Li-ShengGeng
]
h SchoolofPhysicsandNuclearEnergyEngineeringandInternationalResearchCenterfor
t
- NucleiandParticlesintheCosmos,BeihangUniversity,Beijing100191,China
l
c BeijingKeyLaboratoryofAdvancedNuclearMaterialsandPhysics,BeihangUniversity,
u
Beijing100191,China
n
E-mail: [email protected]
[

2 BingweiLong
v
CenterforTheoreticalPhysics,DepartmentofPhysics,SichuanUniversity,29Wang-Jiang
2
7 Road,Chengdu,Sichuan610064,China
2 E-mail: [email protected]
6
0
. Arecentlyproposedcovariantchiraleffectivefieldtheoryapproachisappliedtostudystrangeness
1
0 S= 1hyperon-nucleoninteractionsatleadingorder.12lowenergyconstantsareintroducedby
7 −
Lorentzinvariance,whichisdifferentfromtheheavybaryonapproach,whereonlyfiveappear.
1
: The Kadyshevskyequationis employedto iterate the chiralpotentials. A quite satisfactory de-
v
i scriptionofthe36hyperon-nucleonscatteringdataisobtainedwithχ2 17,whichiscomparable
X ≃
withthenext-to-leadingorderheavybaryonapproach.Theresultshintatamoreefficientwayto
r
a constructthechiralpotentials.
The26thInternationalNuclearPhysicsConference
11-16September,2016
Adelaide,Australia
∗
Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Hyperon-nucleonScatteringInACovariantChiralEffectiveFieldTheoryApproach Kai-WenLi
1. Introduction
Baryon-baryon interactions play a crucial role in hypernuclear physics. During the past 20
years, chiral effective field theory (χEFT), first proposed by Weinberg [1, 2], has made great
progressinderivingnucleon-nucleon(NN)[3,4,5],hyperon-nucleon(YN)[6,7],hyperon-hyperon[8,
9,10]interactions.
Nevertheless, current χEFT studies on baryon-baryon interactions, referred to as the heavy
baryon(HB)approach,arebasedonanon-relativistictheoreticalframework,whichcannotbeused
forrelativistichypernuclearfew-andmany-bodycalculations. Inaddition,theHBapproachisvery
sensitive to ultraviolet cutoffs, known as the renormalization group invariance problem [11, 12].
Recently, Epelbaum and Gegelia proposed a new χEFT approach [13, 14] (referred to as the EG
approach)fortheNN interaction. Atleadingorder(LO),thepotentialisthesameasthatoftheHB
approach but the scattering equation changes tothe Kadyshevsky equation. Itis shown that inthe
strangeness S= 1YN system [15], the description of the scattering data with the EG approach
−
remainsquantitatively similartotheHBapproach, onlythecutoffdependence ismitigated.
A new covariant power counting is recently proposed to study the NN [16] andYN [17] sys-
temsinχEFT,whichispartlymotivatedbythesuccessesofcovariantchiralperturbation theoryin
heavy-light andonebaryonsystems[18,19,20,21]. Atthepotentiallevel,thebaryonsaretreated
inacovariant waytomaintain allthesymmetries andanalyticities. Inthepresent work, wereport
onthestudyofthestrangeness S= 1ΛN ΣN systeminthiscovariantχEFTapproach atLO.
− −
2. Formalism
Thedetails canbefound inRefs. [16,17]. Inthefollowing webriefly explain theprocedures
in obtaining the chiral potentials, calculating the scattering equation, and performing the fits. We
startfromthecompleteDiracspinorsothatLorentzinvariance isretained,
1 ε
B
uB(p,s)=Np σp χs, Np= , (2.1)
ε·B ! r2MB
whereε =E +M andE = p2+M2,whilethesmallcomponentofu isomittedbyemploy-
B B B B B B
ing anon-relativistic reductionqinthe HBapproach. Here naivedimensional analysis isapplied in
obtaining the Feynman diagrams atLO,which consist of nonderivative four-baryon contact terms
and one-pseudoscalar-meson exchanges. Specifically, following Refs. [6, 15], there are 3 for the
formerand7forthelatterinthestrangeness S= 1ΛN ΣN system,asshowninFigure.1.
− −
Λ N Σ N Σ N Λ N N Λ Σ N N Σ Σ N N Σ
η K π K π,η K
Λ N Λ N Σ N Λ N Λ N Λ N Λ N Σ N Σ N
Figure1:Nonderivativefour-baryoncontacttermsandone-pseudoscalar-meson-exchangediagramsforthe
ΛN ΣN system. Thesolid linesdenoteincomingandoutgoingbaryons,andthe dashedlinesdenotethe
−
exchangedpseudoscalarmesons.
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Hyperon-nucleonScatteringInACovariantChiralEffectiveFieldTheoryApproach Kai-WenLi
The relevant chiral Lagrangians in this framework consist of the nonderivative four-baryon
contact termsandthemeson-baryon interactions
C1 C2
L1 = i tr(B¯ B¯ (ΓB) (ΓB) ), L2 = i tr(B¯ (ΓB) B¯ (ΓB) ),
CT 2 a b i b i a CT 2 a i a b i b
C3
L3 = i tr(B¯ (ΓB) )tr(B¯ (ΓB) ), (2.2)
CT 2 a i a b i b
D F
L(1) =tr B¯ iγ Dµ M B B¯γµγ u ,B B¯γ γ [u ,B] , (2.3)
MB µ − B − 2 5{ µ }− 2 µ 5 µ
!
(cid:0) (cid:1)
where tr indicates trace in flavor space; Γ are the elements of the Clifford algebra, andCm (m=
i i
1,2,3) are the low energy constants (LECs) corresponding to independent four-baryon operators.
Inaddition, DµB=∂ B+[Γ ,B],Γ = 1 u†∂ u+u∂ u† ,u =i(u†∂ u u∂ u†)andu2=U =
µ µ µ 2 µ µ µ µ − µ
exp i√2φ , where f is the pseudoscalar meson decay constant in the chiral limit. The matrices
f0 0 (cid:2) (cid:3)
B an(cid:16)d φ c(cid:17)ollect the octet baryons and pseudoscalar mesons. In numerical calculations, we use
D+F =1.26, F/(F+D)=0.4 and f =93 MeV [7]. Potentials for the Feynman diagrams in
0
Figure.1arederivedfromtheaboveLagrangians. StrictSU(3)symmetryisimposedonthecontact
terms thus 12 independent LECsremain: CΛΛ,CˆΛΛ,CΣΣ,CˆΣΣ,CΛΛ,CˆΛΛ,CΣΣ,CˆΣΣ,CΛΣ,CˆΛΣ,
1S0 1S0 1S0 1S0 3S1 3S1 3S1 3S1 3S1 3S1
CΛΛ, CΣΣ . SU(3) symmetry is only broken by the mass difference of the mesons in the one-
3P1 3P1
pseudoscalar-meson exchanges. Thepotentialscorresponding totheFeynmanndiagramsshownin
Fig.1canbeexpressed inthefollowingway:
VCBT1B2→B3B4 =Ci(u¯3Γiu1)(u¯4Γiu2), (2.4)
(u¯ γµγ q u )(u¯ γνγ q u )
VOBM1BE2→B3B4 =−NB1B3φNB2B4φ 3 5q2µ 1q2 4m25 ν 2 IB1B2→B3B4, (2.5)
0− −
wherethesubscriptsCTandOMEdenotethecontacttermandone-pseudoscalar-meson exchange,
respectively. N istheSU(3)coefficientandI theisospinfactor. Ddetailscanbefound
BBφ B1B2 B3B4
→
inRefs.[6,16,17].
TheKadyshevsky equation isusedtoiteratethepotentials duetotheinfrared enhancement in
two-baryon propagations followingRef.[15]:
Tνν′,J(p,p;√s)=Vνν′,J(p,p)+ ∑ ∞dp′′p′′2 2µν2′′ Vρνρν′′′′,J(p′,p′′)Tρν′′′ρ′ν′′,J(p′′,p;√s) ,
ρρ′ ′ ρρ′ ′ ρ ,ν Z0 (2π)3 p 2+4µ2 q2 +4µ2 p 2+4µ2 +iε
′′ ′′ ′′ ν′′ ν′′ ν′′− ′′ ν′′
(cid:0) (cid:1)(cid:16)q q (2.6) (cid:17)
where √s is the total energy of the baryon-baryon system in the center-of-mass frame, q is
ν
′′
the relativistic on-shell momentum. The labels ρ and ν denote the partial waves and particle
channels, respectively. Relativistic kinematics is used for the relation of the laboratory momenta
and the center-of-mass momenta. Wesolve theKadyshevsky equation in particle basis inorder to
accountforthephysicalthresholdsproperly. TheCoulombforceistreatedwiththeVincent-Phatak
method [22]. Toregularize the integration in the high-momentum region, the chiral potentials are
multipliedwithanexponential formfactor f (p,p)=exp (p/Λ )4 (p/Λ )4 .Oneshould
ΛF ′ − F − ′ F
h i
2
Hyperon-nucleonScatteringInACovariantChiralEffectiveFieldTheoryApproach Kai-WenLi
Table1: Low-energyconstants(inunitsof104GeV 2)atΛ =600MeVinthecovariantχEFTapproach.
− F
LECs CΛΛ CΣΣ CΛΛ CΣΣ CΛΣ CˆΛΛ
1S0 1S0 3S1 3S1 3S1 1S0
0.0223 0.0528 0.0032 0.0842 0.0232 3.9185
− −
LECs CˆΣΣ CˆΛΛ CˆΣΣ CˆΛΣ CΛΛ CΣΣ
1S0 3S1 3S1 3S1 3P1 3P1
4.0681 0.4190 0.4132 0.7326 0.2044 0.2616
−
note that although the potentials are fully covariant, the scattering amplitude isnot because of the
useoftheKadyshevsky equationandthecutoffregularization.
The LECscan be pinned down by fitting to the 36YN scattering data, as it has been done in
Ref. [7]. In actual calculations we introduce the following constraints, i.e. the Λp 1S , 3S and
0 1
Σ+p3S scattering lengths. TheΛhypertriton isnottakenintoaccount because wearenotableto
1
perform 3-bodycalculations atpresent.
3. Results anddiscussion
ThefitwasdonebyvaryingthecutoffvalueΛ aroundtheρmesonmass. Cutoffdependence
F
of the χ2 in describing the 36 YN data is shown in Figure. 2, in comparison with other χEFT
approaches. It can be seen that the LO covariant χEFT approach yields a smaller χ2 value and
a milder cutoff dependence of the χ2, compared with the LO HB and EG approaches. They are
similar to the next-to-leading (NLO) HB approach, which has 23 LECs, however. Nevertheless,
renormalization group invariance is still not achieved, as exhibited by the variation of χ2 with
the cutoff (more clearly shown in the insert). The minimum value of the χ2 is 16.7, located at

100
20
The LO HB approach
19 The NLO HB approach
80 18 The LO EG approach
Covariant EFT approach
17
60 16
15
500550600650700750800850
F (M eV)
40
20
0
500 550 600 650 700 750 800 850
F (MeV)
Figure2:χ2asafunctionofthecutoffintheLO(bluedottedline)[6],NLO(orangedashed-dottedline)[7]
HBapproach,theLOEGapproach(reddashedline)[15]andtheLOcovariantχEFTapproach(greensolid
line).
Λ =600 MeV, and the corresponding LECs are listed in Table 1. Note that the LECsin the Λp
F
1S partial wave cannot be precisely determined for the Λ hypertriton binding energy is not used
0
3
Hyperon-nucleonScatteringInACovariantChiralEffectiveFieldTheoryApproach Kai-WenLi
as a constraint. We show only a typical case here. Moreover, the description of the experimental
dataareshowninFig.3. TheLOHBapproach[6]andtwophenomenological models[23,24]are
p - p p - p p - n p - n p - p
300 250 300 500 300
Sechi-Zorn et al. Eisele et al. Engelmann et al. Engelmann et al. Eisele et al.
250 Alexander et al. 200 250 400 250
Hauptman et al.
b) 200 Kadyk et al. 200
m 150 200 300
( 150 150
100 150 200
100 100
50 50 100 100 50
0 0 50 0 0
100 300 500 700 900 140 150 160 170 180 100 120 140 160 100 120 140 160 140 150 160 170
Plab (MeV/c) Plab (MeV/c) Plab (MeV/c) Plab (MeV/c) Plab (MeV/c)
Figure 3: Cross sectionsin the HB approach(blue dottedlines) [6] and covariantχEFT approach(green
solidlines)atLOasfunctionsofthelaboratorymomentumatΛ =600MeV.Forreference,theNSC97f[23]
F
(red dash-dottedlines) and Jülich 04 [24] (orangedashed lines) results are also shown. The experimental
dataarethesameasRef.[15].
taken asreferences. Clearlythecovariant χEFTapproach canbetter reproduce thedata compared
withtheLOHBapproach, especially fortheΛp ΛpandΣ+p Σ+preactions.
→ →
The improvements in our new scheme mainly come from the contact terms. In the LO HB
approach, contact terms only contribute to central and spin-spin potentials, and they only appear
in S-waves. In our new scheme, tensor, spin-orbit, asymmetric spin-orbit and quadratic spin-orbit
terms also appear in the contact interactions. These terms are momentum dependent and improve
thedescription. TheseareparalleltotheNN case. Detaileddiscussions canbefoundinRef.[16].
4. Summary
Wehave applied anewcovariant chiral effective fieldtheory approach tothestrangeness S=
1 ΛN ΣN system at leading order. Starting from the covariant chiral Lagrangians, the full
− −
baryon spinor isretained tocalculate thepotentials. Assuch, Lorentz invariance ispreserved. We
usestrictSU(3)symmetryinthecontacttermsthusonly12lowenergyconstants areindependent.
SU(3) symmetry is broken in the one-pseudoscalar-meson exchanges due to the mass difference
of the exchanged mesons. The Kadyshevsky equation is employed to iterate the chiral potentials.
A very satisfactory description of the 36 hyperon-nucleon scattering data is obtained, as well as
a mitigated cutoff dependence. Our approach yield results similar to the next-to-leading order
heavy baryon approach but with11fewerlowenergy constants. Ithintsatamoreefficient wayto
construct thebaryon-baryon potentials.
Acknowledgements
This work is partly supported by the National Natural Science Foundation of China under
Grants No. 11375024, No. 11522539, and No. 11375120, the China Postdoctoral Science Foun-
dation under GrantNo. 2016M600845, andtheFundamental Research FundsfortheCentral Uni-
versities.
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Hyperon-nucleonScatteringInACovariantChiralEffectiveFieldTheoryApproach Kai-WenLi
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