The chiral partner of the nucleon in the mirror assignment with global symmetry

0.11 MB·English

by  Susanna Gallas

The chiral partner of the nucleon in the mirror
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assignment with global symmetry
0
0
2

n
a
J

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SusannaGallas
∗
JohannWolfgangGoethe-UniversitätFrankfurtamMain
]
h E-mail: [email protected]
p
-
FrancescoGiacosa
p
e JohannWolfgangGoethe-UniversitätFrankfurtamMain
h
E-mail: [email protected]
[

1 Dirk H.Rischke
v JohannWolfgangGoethe-UniversitätFrankfurtamMain
3
E-mail: [email protected]
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( )
1. Wecalculatethepion-nucleonscatteringlengthsa0± andthemassparameterm0,whichdescribes
0 thenucleonmassinthechirallimit,attree-levelintheframeworkofagloballysymmetriclinear
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sigma modelwith parity-doublednucleons. When recentlattice results [1] are used, we obtain
0
( ) (+)
v: m0≃300−600MeV.Whilea0− isinfairagreementwithexperimentaldata,a0 istoosmallbe-
i causeoftheemployedlargescalarmesonmass. Thisindicatestheneedtoaccountforadditional
X
scalardegreesoffreedom.
r
a
8thConferenceQuarkConfinementandtheHadronSpectrum
September1-62008
Mainz,Germany
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Thechiralpartnerofthenucleon SusannaGallas
1. Introduction
Effectivemodels which embody chiral symmetryand itsspontaneous breakdown atlow tem-
peraturesanddensitiesarewidelyusedtounderstandthepropertiesoflighthadrons. Viablecandi-
datesobeyawell-definedsetoflow-energytheorems[2,3]buttheystilldifferinsomecrucialand
interesting aspects such asthemass generation ofthe nucleon and thebehavior atnon-zero T and
m .
Here we concentrate on a linear sigma model with U(2) U(2) symmetry and parity-
R L
×
doubled nucleons. The mesonic sector involves scalar, pseudoscalar, vector, and axial-vector
mesons. Inthebaryonicsector,besidestheusualnucleondoubletfieldN,asecondbaryondoublet
N∗ with JP = 12− is included. As first discussed in Ref. [4] and extensively analyzed in Ref. [5],
inthe so-called mirror assignment thenucleon fields N and N have amassm =0inthechirally
∗ 0
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symmetricphase. Thechiralcondensatej increasesthemassesandgeneratesamasssplittingofN
andN ,butisnolongersolelyresponsible forgenerating themasses. Suchatheoreticalset-uphas
∗
been used in Ref. [6] to study the properties of cold and dense nuclear matter. The experimental
assignmentforthechiralpartnerofthenucleonisstillcontroversial: thewell-identifiedresonances
N (1535) and N (1650) are two candidates with the right quantum numbers listed in the PDG
∗ ∗
[7], but we shall also investigate the possibility of a very broad and not yet discovered resonance
centered atabout1.2GeV,whichhasbeenproposed inRef.[6].
The aim of the present work is the development of an effective model which embodies the
chiralpartnerofthenucleoninthemirrorassignmentwithinthecontextofglobalchiralsymmetry
involving alsovectorandaxial-vectormesons. Inthiswaymoretermsappearthanthoseoriginally
proposed in Ref. [4]. Moreover, we restrict our study to operators up to fourth order (thus not
including Weinberg-Tomozawa interaction terms). Theaxial coupling constant ofthenucleon can
be correctly described. Using recent information about the axial coupling of the partner [1] and
experimental knowledge about its decay width [7] we can evaluate the mass parameter m which
0
describes the nucleon mass in the chiral limit. Then, we further evaluate pion-nucleon scattering
lengths andwecomparethemwiththeexperimental values[8].
2. The model and itsimplications
ThescalarandpseudoscalarfieldsareincludedinthematrixF =(s +ih )t0+(−→a0+i−→p ) −→t
aannddLthme=(a(xwiaml-+)vefc1mt)otr0fi+e(ld−→rsmar+e−→are1pmr)e·s−→etnt(e−→dt b=y12th−→te,mwahtreicrees−→tRmar=eth(wePm a−ulif1mm )at0tr+ice(s−→ranm d−t0−→a=1m 1)21··−2→t).
Thecorresponding Lagrangian describing onlymesonsreads
Lmes = Tr (Dm F )†(Dm F ) m2F †F l 2 F †F 2 l 1 Tr[F †F ] 2+c(detF †+detF )
− − −
+ Tr[hH(F †+F )] 1Tr (Lmn )2+(R(cid:0)mn )2 (cid:1)+im21Tr(cid:0)(Lm )2+((cid:1)Rm )2
−4 2
+ h2Tr(F †Lm Lm F +F R(cid:2)m Rm F †)+h3Tr(F(cid:3)Rm F †Lm (cid:2))+Lthree+Lf(cid:3)our ,
(2.1)
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Thechiralpartnerofthenucleon SusannaGallas
where Dm F =¶ m +ig (F Rm Lm F )and Lmn =¶ m Ln ¶ n Lm ,Rmn =¶ m Rn ¶ n Rm are the field
1
− − −
strengthtensorsofthe(axial-)vectorfields. L andL describe3-and4-particleinteractions
three four
of(axial-)vector fields,whichareirrelevant forthiswork,seeRef.[9].
Thebaryon sector involves thebaryon doublets Y and Y ,whereY haspositive parity and
1 2 1
Y negative parity. In the so-called mirror assignment, Y and Y transform in the opposite way
2 1 2
underchiralsymmetry,namely:
Y U Y ,Y Y U†, Y U Y ,Y Y U†, (2.2)
1R −→ R 1R 1R −→ 1R R 2R −→ L 2R 2R−→ 2R L
andsimilarly fortheleft-handed fields. Suchfieldtransformations allowtowritedownabaryonic
Lagrangian withachirallyinvariant masstermforthefermions,parametrized bym :
0
Lbar = Y 1Ligm D1mLY 1L+Y 1Rigm D1mRY 1R+Y 2Ligm D2mRY 2L+Y 2Rigm D2mLY 2R
g Y YF +Y F †Y g Y F †Y +Y YF
1 1L 1R 1R 1L 2 2L 2R 2R 2L
− −
m0(cid:0)(Y 1LY 2R Y 1RY 2L Y 2(cid:1)LY 1R+(cid:0)Y 2RY 1L), (cid:1) (2.3)
− − −
b b
where Dm = ¶ m ic Rm , Dm = ¶ m ic Lm and Dm = ¶ m ic Rm , Dm = ¶ m ic Lm are the
1R − 1 1L − 1 2R − 2 2L − 2
covariant derivatives for the nucleonic fields. The coupling constants g and g parametrize the
1 2
interaction of the baryonic fields with scalar and pseudoscalar mesons and j = 0 s 0 =Zfp is
h | | i
thechiralcondensate emerginguponspontaneous chiralsymmetrybreakbingintbhemesonicsector.
Theparameter fp =92.4MeVisthepiondecayconstantandZisthewavefunctionrenormalization
constant ofthepseudoscalar fields[11].
Thetermproportional tom generatesalsoamixingbetweenthefieldsY andY .Thephys-
0 1 2
ical fields N and N , referring to the nucleon and to its chiral partner, arise by diagonalizing the
∗
baryonic partoftheLagrangian. Asaresultwehave:
N Y 1 ed /2 g e d /2 Y
=M 1 = 5 − 1 . (2.4)
N∗! Y 2! √2coshd g5e−d /2 ed /2 ! Y 2!
−
b
Themassesofthenucleon anditspartnerareobtained upondiagonalizing themassmatrixM:
1 (g g )j
m = 4m2+(g +g )2j 2 1− 2 , b(2.5)
N,N∗ 2 0 1 2 ± 2
i.e., the nucleon mass is not only gqenerated by the chiral conbdensbate j but also by m0. The pa-
b b
rameter d in Eq. (2.4) is related to the masses and the parameter m by the expression: d =
0
Arcosh mN2+mm0N∗ .Let us consider two important limiting cases. (i) When d →¥ , corresponding
tom0 h0,nomiixingispresentandN=Y 1,N∗=Y 2.InthiscasemN =g1j /2andmN =g2j /2,
→ ∗
thus the nucleon mass is generated solely by the chiral condensate as in the linear sigma model.
(ii) Inthe chirally restored phase where j 0, one has mass degeneracybm =m =m .bWhen
N N 0
→ ∗
chiralsymmetryisbroken, j =0,asplitting isgenerated. Bychoosing 0<g <g theinequality
1 2
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m <m isfulfilled.
N N
∗
Notealsothat,wheng =c =c andh =h =h =0,thechiralsymmebtrybebcomeslocal[10,
1 1 2 1 2 3
12]. ThecorrespondingmodelhasbeenstudiedinRef. [13]. Itwasnotpossibletomakeaclear-cut
prediction astowhetherthemassofthenucleon isdominantly generated bythechiralcondensate
or bymixing withits chiral partner. Inaddition tothis, the description of themesonic decays was
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Thechiralpartnerofthenucleon SusannaGallas
not correct in a locally symmetric framework as shown in Ref. [14]. Also, the expression for the
axial charge reads gN = tanhd . Since tanhd <1 for all d and Z >1, we obtain gN <1, at odds
A Z2 | | A
with the experimental value gN =1.267 0.004. Thus, in the context of local symmetry one is
A ±
obligedtointroducetermsofhigherordersuchastheWeinberg-Tomozawaone. Asafinalremark,
notethatbysettingZ=1(whichinturnmeansg =0)thevectormesonsandaxial-vectormesons
1
drop out and only the (pseudo-)scalar and nucleonic terms survive in the Lagrangian. Then, for
gN∗avaluemuchlargerthan0.5ispredicted, whichisindisagreement withlatticedata[1].
A
In Ref. [6] a large value of the parameter m ( 800 MeV) is claimed to be needed for a
0
∼
correct description of nuclear matter properties, thus pointing to a small contribution of the chiral
condensate to the nucleon mass. Validating this claim through the evaluation of pion-nucleon
scattering atzerotemperature anddensityisthesubjectofthepresentpaper.
Theexpressions fortheaxialcoupling constants ofthenucleonandthepartneraregivenby:
d d d d
e e e e
gN = g(1)+ − g(2), gN∗ = − g(1)+ g(2), (2.6)
A 2coshd A 2coshd A A −2coshd A 2coshd A
where
c 1 c 1
(1) 1 (2) 2
g =1 1 , g = 1+ 1 (2.7)
A −g −Z2 A − g −Z2
1 1
(cid:18) (cid:19) (cid:18) (cid:19)
refertotheaxialcouplingconstantsofthebare,unmixedfieldsY andY .Notethatwhend ¥
1 2
→
one has gNA = g(A1) and gNA∗ = g(A2). Also, when c1 = c2 = 0 (or Z = 1) we obtain the results of
Ref. [4]: gN =tanhd and gN∗ = tanhd ,which in the limit d ¥ reduces to gN =1 and gN∗ =
A A − → A A
1. However, in our model the interaction with the (axial-)vector mesons generates additional
−
contributions togN andgN∗,whicharefixedviatheexperimentalresultforgN andthelatticeresult
A A A
forgNA∗,seethenextsection. FromtheLagrangian (2.3)onecancomputethedecayN∗→Np and
( )
thescattering amplitudes a ± [15].
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3. Results anddiscussion
We consider three possible assignments for the partner of the nucleon. (1) The resonance
N∗(1535), with mass MN (1535) = 1535 MeV and G N (1535) Np = (67.5 23.6) MeV, which –
∗ ∗ → ±
beingthelightestbaryonicresonancewiththecorrectquantumnumbers–surelyrepresents oneof
the viable and highly discussed candidates for the nucleon partner. (2) The resonance N (1650),
∗
with a mass lying just above, MN (1650) =1650 MeV and G N (1650) Np =(92.5 37.5) MeV. (3)
∗ ∗ → ±
A speculative candidate N (1200) with a mass M 1200 MeV and a very broad width
∗ N (1200)
∗ ∼
G N (1650) Np &800MeV,suchtohaveavoided experimental detection uptonow[6].
∗ →
For all these scenarios, we want to determine the values of the parameters c , c , and m .
1 2 0
Beyond the width, which isdifferent inthethree cases mentioned above, wealso usegN∗ =0.2
A ±
0.3, aspredicted by lattice QCD[1], and gN =1.26. Werepeat the evaluation fordifferent values
A
ofZ.Remarkably, m doesnotdependonZ.
0
Figure 1 shows the mass parameter m as a function of the axial coupling constant of N for
0 ∗
differentmassesofN∗. FortherangeofgNA∗ givenbyRef.[1],m0=300−600MeVforN∗(1535)
andN (1650), meaningthathalfofthenucleonic masssurvives inthechirally restored phase. On
∗
the contrary, for N (1200) the value for m lies above 1000 MeV. This result suggests that the
∗ 0
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Thechiralpartnerofthenucleon SusannaGallas
contribution of the chiral condensate to the nucleonic mass should be negative, which is rather
unnatural. Wecan then discard the possibility that a hypothetical, not yet discovered N (1200) is
∗
thechiralpartnerofthenucleon.
According to Ref. [16], when the fields N and N belong to a parity doublet, gN 1 and
∗ A ∼
gNA∗ ∼−1. Then,inRef. [16]thelattice resultof[1]isusedagainsttheidentification ofN∗(1535)
asthepartner ofthenucleon. However,within ourmodelwecanstill accommodate N (1535) (or
∗
alsoN∗(1650))asthepartnerofthenucleon. ThesmallvalueofgNA∗ arisesbecause ofinteractions
ofthepartnerwiththeaxial-vector mesons.
Figure1: m0asafunctionofgNA∗.
In Figure 2 we plot the isospin-odd and isospin-even scattering lengths as a function of the
axialchargegN∗forZ=1.5. Acomparisonofourresultstotheexperimentaldataonp N scattering
A
lengths, as measured in Ref. [8] by precision X-ray experiments on pionic hydrogen and pionic
( )
deuterium, yields the following: (a) the isospin-odd scattering length a − is close to the experi-
0
mentalrange, butweexpectanevenbetterresultwiththeintroduction oftheD resonance. (b)The
(+)
isospin-even scattering length a is an order of magnitude smaller than the experimental band
0
a(+) =( 8.85783 7.16)10 6 MeV.Thereason forthis isthestrong dependence ofa(+) onthe
0,exp − ± − 0
scalarmesons. Hereweusems =1370MeV.Asmallermassofthesigmamesonmaybefavored,
howeverthisresultseemstobeexcluded [9].
Figure2: Thescatteringlengths(a)a(0−)and(b)a(0+)asafunctionofgNA∗.
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Thechiralpartnerofthenucleon SusannaGallas
4. Summary andoutlook
We have computed the pion-nucleon scattering lengths at tree-level in the framework of a
globally symmetric linear sigma model with parity-doubled nucleons. Within the mirror assign-
ment the mass of the nucleon originates only partially from the chiral condensate, but also from
the mass parameter m . Using the lattice results of Ref. [1] we find that m 300 600 MeV.
0 0
≃ −
Approximately half of the nucleon mass survives in the chirally restored phase. The isospin-odd
scattering lengthliesclosetotheexperimental band,butcouldbeimprovedinfurtherstudies. The
isospin-even scattering length istoo small: future inclusion of alight tetraquark state givesrise to
alargecontribution, andthusisexpectedtoimprovetheresults. Wealsoplantoextendourmodel
by including the D resonance, necessary to correctly reproduce the p-wave scattering lengths [17]
andtoevaluatetheradiativeh -photoproduction.
Acknowledgements
TheauthorsthankT.Kunihiro, T.TakahashiandD.Parganlijaforusefuldiscussions.
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