In-medium \eta' mass and \eta'N interaction in vacuum in the linear sigma model

0.15 MB·

by  Shuntaro Sakai

h h
In-medium mass and N interaction in vacuum in
′ ′
the linear sigma model
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n
a
J
0 ShuntaroSakai†
1 DepartmentofPhysics,KyotoUniversity,Kitashirakawa-Oiwakecho,Kyoto606-8502,Japan
E-mail: [email protected]
]
h
DaisukeJido
t
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l DepartmentofPhysics,TokyoMetropolitanUniversity,Hachioji,Tokyo192-0397,Japan
c
E-mail: [email protected]
u
n
[
We investigatethe h N two-body interaction in the contextof the h meson mass modification
′ ′
1
inthenuclearmedium. Ithasbeenarguedinseveralarticlesthatthemassesofh andtheother
v ′
5 pseudoscalarmesons(p , K, h ) shoulddegeneratein the chiral-symmetricphase. Itis expected
3
thatthereductionofthemassdifferencebetweenh andh wouldtakeplaceinthenuclearmatter
2 ′
2 ifoneassumesthatthedecreaseofthequarkcondensateatthenormalnucleardensityoccurswith
.
1 partialrestorationofchiralsymmetry.Atlowdensity,thein-mediumself-energygivingthemass
0 modificationbythemediumeffectcanbeobtainedbytheh Ntwo-bodyT matrix.Thus,wealso
4 ′
1 estimatetheh N interactionstrengthinvacuumwiththelinearsigmamodelwhichinvolvesthe
′
:
v effect of partial restoration of chiral symmetry. In the view of the linear sigma model, we find
Xi thattheh ′Ninteractionisattractiveandgeneratedthroughthesigmamesonexchange.Weexpect
r thattheinteractionisenoughstrongandfortheexistenceofaboundstateoftheh ′N system.
a
XVInternationalConferenceonHadronSpectroscopy-Hadron2013
4-8November2013
Nara,Japan
ReportNo.:KUNS-2473
∗
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
In-mediumh massandh N two-bodyinteraction ShuntaroSakai
′ ′
1. Introduction
Themassspectrumofhadronsreflectsthesymmetryofitsfundamentaltheory,QuantumChro-
modynamics (QCD). Ifone considers simply, nine Nambu–Goldstone (NG)bosons would appear
according to the NGtheorem if one assumes the axial part of U(3) U(3) isspontaneously bro-
L R
×
ken downtoU(3) . Nevertheless, whenoneseesthepseudoscalar mesonmasses, onecannot find
V
the singlet pseudoscalar meson around the pion mass [1]. Taking account of the U (1) anomaly,
A
we can regard the U (1) symmetry as broken explicitly and the h mass can be expained with the
A ′
U (1)anomaly[2].
A
Other than the U (1) anomaly, chiral symmetry breaking also plays an important role for
A
thegeneration oftheh mass. Asdiscussed inRefs. [3],thepseudoscalar singlet mesonandoctet
′
mesondegeneratewhenchiralsymmetryisrestored,e.g. athightemperatureorhighdensity,inthe
chirallimiteveniftheU (1)symmetryisbrokenexplicitly. Takingaccountofthedegeneracyofthe
A
pseudoscalar flavor-singlet and octet meson in the chiral restored phase and the partial restoration
of chiral symmetry, weexpect the reduction of the h mass in the nuclear matter. Concerning the
′
chiralrestorationinthenuclearmatter,the35%reductionofthequarkcondensateissuggestedfrom
the analysis of the experimental data [4]. Recent experiments have suggested that the h -nucleus
′
optical potential could be attractive with certain strength and could hava a smaller imaginary part
[5]. We can interpret the h mass reduction in the nuclear matter as the attractive potential of
′
h in the nuclear matter. The existence of h -mesic nuclei is suggested theoretically [6] and the
′ ′
experimental attempt to observe the h -mesic nuclei is discussed [7]. However, it is not known
′
wellwhethertheinteraction betweenh andnucleonisattractiveorrepulsivedespitetheexistence
′
of someexperimental data [8]. Suchapoor knowledge ofthe h N interaction makesitdifficult to
′
analyze theh properties inthenuclearmatter.
′
In the below, we study the h N two-body interaction with the linear sigma model as a chiral
′
effective model. In the construction of the model, we assume the 35% reduction of the quark
condensate. In addition to the h N interaction, we calculate the in-medium h mass which is
′ ′
expected toreduceinthenuclearmatter. ThedetailofthisworkisshowninRef. [9]
2. Method
Forthecalculation, weusetheSU(3)linear sigmamodel asachiral effective model[10,11].
The linear sigma model can describe both the chiral restored phase and the spontaneously broken
phase. To describe the h N interaction, we introduce the nucleon degree of freedom explicitly
′
basedonthechiralsymmetry. TheLagrangian isgivenas
L = 1tr¶ m M¶ m M† m 2trMM† l tr(MM†)2 l ′ trMM† 2
2 − 2 − 4 − 4
(cid:0) (cid:1)
+Atr c M†+Mc † +√3B detM+detM†
(cid:0) s (cid:1) s (cid:0) ~t ~p h (cid:1) h
+N¯i¶/N gN¯ 0 + 8 +ig · +ig 0 +ig 8 N. (2.1)
5 5 5
− (cid:18)√3 √6 √2 √3 √6(cid:19)
Here,themesonfield,thenucleon field,andthequarkmassaregiven,respectively, by
8 s l 8 p l
M= (cid:229) a a +i(cid:229) a a, N =(p,n)t, c =diag(m ,m ,m ). (2.2)
q q s
√2 √2
a=0 a=0
2
In-mediumh massandh N two-bodyinteraction ShuntaroSakai
′ ′
V] 280 1000
1/3） [Me 226700 (〈- ss〉)1/3 ]V 890000 mη’
densate 224500 eM[ ssaM 567000000 m
Quark Con 222123000 0 0.04 (〈- q 0q.0〉8)1/3 0.12 0.16 noseM 123400000000 0 0.04 η 0m.08π 0.12 0.16
（-
Nuclear Density [fm-3] Nuclear Density [fm-3]
Figure 1: In-medium quark condensate. The Figure 2: In-medium meson mass. The solid,
dashed and solid lines represent the in-medium dotted and dashed lines representthe h , h and
′
u,dandsquarkcondensates,respectively. p masses,respectively.
TheLagrangianisconstructedtopossessthesameglobalsymmetryasQCD.Thetermproportional
toAexpresses theeffectofthecurrentquarkmass. Here, c corresponds tothecurrentquarkmass
and we assume the isospin symmetry, m =m =m , and introduce the SU(3) flavor symmetry
q u d
breaking withm =m . Thetermproportional toBrepresents theeffectoftheU (1)anomalyand
q s A
6
thistermisnotinvariant undertheU (1)transformation.
A
TheLagrangian contains 6freeparameters whichcannot befixedfrom thesymmetry. Wefix
these parameters using the observed meson masses, the meson decay constants and the fact that
the 35% reduction of the quark condensate at the normal nuclear density. For the calculation of
thein-medium quantities, i.e. themeson masses, weintroduce theeffectofthesymmetric nuclear
matterwiththemeanfieldapproximation ofnucleon.
In the linear sigma model, the vacuum expectation value of the sigma field s is an order
0
h i
parameter ofthechiral symmetry breaking. Now,wehavenon-zero s duetotheexplicit flavor
8
h i
symmetrybreaking. Wedetermine s and s tominimizetheeffectivepotential.
0 8
h i h i
3. Results
3.1 In-mediummesonmass
First, we show the in-medium quark condensate in Fig. 1. As mentioned above, we assume
the35%reduction ofthequarkcondensate, sothevalueoftheu,d quarkcondensate atthenormal
nuclear density istheinput value. Here,wehaveassumed theisospin symmetry, sotheu,d quark
condensates coinside. Next,wediscussthein-mediummesonmass. Thein-mediumself-energyof
themesonscomesfromthediagramsshowninFig. 3. Diagram(a)inFig.3comesfromthenucleon
meanfield,while(b)and(c)comefromtheparticle-hole excitation andthecrossedchannel ofthe
particle-hole excitation, respectively. Inthechirallimit,theh masscanbewrittenas
′
mh2 =6B s 0 . (3.1)
′ h i
B is the coefficient of the determinant term and represents the effect of the U (1) anomaly. From
A
thisexpression, onecanfindthenecessity oftheU (1)anomalyandthechiralsymmetrybreaking
A
for the generation of the h mass, and the restoration of chiral symmetry, the reduction of s ,
′ 0
h i
leads to the reduction of the h mass. Thecalculated in-medium meson masses are shown in Fig.
′
3
In-mediumh massandh N two-bodyinteraction ShuntaroSakai
′ ′
(a) (b) (c) (a) (b) (c)
Figure 3: The diagrams contributing to the in- Figure4: Thediagramswhichcontributetothe
h N interaction
mediummesonself-energy. ′
2Fromthecalculation, theh massreducesabout80MeVandtheh massenhances about50MeV
′
at the normal nuclear density. The mass difference between the h and h mass reduces about
′
130MeV,goingtowardthedegeneracy ofh andh ,aswehaveexpected.
′
3.2 h N 2-bodyinteraction
′
Here,weshowtheh N 2-bodyinteraction evaluatedwiththesamelinearsigmamodel. Inthe
′
linear sigma model, the diagrams shown in Fig. 4 contribute to the h N interaction. The diagram
′
(a)intheFig. 4showsthecontribution fromthesigmamesonexchange,whilediagram(b)and(c)
in Fig. 4 are the contributions from the Born term which contain the nucleon intermediate state.
From these diagrams, we have obtained the low-energy h ′N 2-body interaction Vh N in the chiral
′
limitas
6gB
Vh N = . (3.2)
′ −√3m2
s
0
Substitutingthevalueofthefixedparameters,wefindthattheh Ninteractioniscomparablystrong
′
totheK¯Nsystem. IntheK¯Nsystem,aboundstate,L (1405),existsduetothestrongK¯Nattraction.
WiththeanalogytoL (1405),weexpecttheexistenceoftheh N boundstate. Fortheinvestigation
′
ofthepossibilityoftheboundstate,weanalyzedtheT-matrixoftheh Nsystembecausethebound
′
state appears as the pole of the T-matrix. We obtain the T-matrix with solving the single-channel
Lippman-Schwinger equation,
T = V+VGT. (3.3)
Here,Gistheloopintegral ofh andnucleon,
′
d4q 1 1
G(W) = 2m , (3.4)
NZ (2p )4(P q)2 m2 +ie q2 m2 +ie
− − N − N
P=(W,0)is the 4-momentum ofthe h N in the center ofmass system. As the interaction kernel
′
V, we use the h N interaction obtained with linear sigma model shown in Eq. (3.2). Now, the
′
interaction kernel is momentum-independent, so the equation can be solved with the algebraic
way,
1
T(W)= . (3.5)
V 1 G(W)
−
−
The obtained T matrix contains a divergence in the loop integral G. Here, we regulate the diver-
gence with dimensional regularization and we fix the subtraction constant with the natural renor-
malization scheme, which exclude the other dynamics than h and N [12]. We have found a h N
′ ′
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In-mediumh massandh N two-bodyinteraction ShuntaroSakai
′ ′
boundstateasapoleoftheobtainedTmatrix. Thebindingenergyis6.2MeV,thescatteringlength
is-2.7fmandtheeffectiverangeis0.25fm. Thescatteringlengthistherepulsivesigninournota-
tion. Here, we note that the obtained h N scattering length is somewhat larger value compared to
′
thevaluesuggested inRef. [8].
4. Conclusion
In this paper, we have calculated the in-medium meson mass and the h N 2-body interaction
′
with the SU(3) linear sigma model. The medium effect is introduced as one nucleon loop for the
calculation of the in-medium meson mass. We have obtained about 80 MeV reduction of the h
′
mass and 130 MeV decrease of the mass difference between h and h . Concerning the h N two-
′ ′
body interaction, we have found the strong attraction of h N comparable to the K¯N system. The
′
h N interactionobtainedfromthelinearsigmamodelisprovidedfromthesigmamesonexchange.
′
ThisisadifferentcharacterfromthatoftheordinaryNGboson,theWeinberg–Tomozawa interac-
tionwhichisenergydependent. WiththeanalogyofL (1405), wehaveinvestigated thepossibility
oftheh N bound state. Asaresult, wefound ah N bound statewiththebinding energy 6.2MeV
′ ′
and thescattering length -2.7fm. Thecoupling between s andh isnecessary forthegeneration
0 ′
oftheh massandtheh -s coupling leadstotheattraction throughthesigmamesonexchange.
′ ′ 0
Acknowledgments
S.S.isaJSPSFellowandappreciates thesupportbyaJSPSGrant-in-Aid(No. 25-1879).
Thisworkwaspartiallysupported byGrants-in-AidforScientificResearchfromMEXTandJSPS
(No. 25400254 andNo. 24540274).
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