On the microscopic structure of $\pi NN$, $\pi N\Delta$ and $\pi\Delta\Delta$ vertices

0.32 MB·

by  Ju-Hyun Jung

Few Body Systems manuscript No.
(will be inserted by the editor)
Ju-Hyun Jung · Wolfgang Schweiger
On the microscopic structure of πNN, πN∆ and
π∆∆ vertices
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Received: date / Accepted: date
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Abstract We use a hybrid constituent-quark model for the microscopic description of πNN, πN∆
n
and π∆∆ vertices. In this model quarks are confined by an instantaneous potential and are allowed
a
to emit and absorb a pion, which is also treated as dynamical degree of freedom. The point form of
J
relativistic quantum mechanics is employed to achieve a relativistically invariant description of this
1
system.StartingwithanSU(6)spin-flavorsymmetricwavefunctionforN and∆ ,i.e.theeigenstates
1 0 0
ofthepureconfinementproblem,wecalculatethestrengthoftheπN0N0,πN0∆0andπ∆0∆0couplings

] and the corresponding vertex form factors. Interestingly the ratios of the resulting couplings resemble
h
strongly those needed in purely hadronic coupled-channel models, but deviate significantly from the
t
- ratios following from SU(6) spin-flavor symmetry in the non-relativistic constituent-quark model.
l
c
u Keywords Constituent-quark model · Meson-baryon form factors · Point-form quantum mechanics
n
[

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1 Motivation and Formalism
v
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OurinterestinπNN,πN∆andπ∆∆couplingsandvertexformfactorsisconnectedwithourattempt
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0 totakepion-loopeffectsintoaccountwhendescribingtheelectromagneticstructureofN and∆within
3 a constituent-quark model. As it turns out, the calculation of the loop effects boils down to a purely
0 hadronic problem, in which the quark substructure of the N and the ∆ is hidden in strong and
1. electromagneticformfactorsof“bare”baryons,i.e.eigenstatesofthepureconfinementproblem.Since
0 πNN, πN∆ and π∆∆ couplings and vertex form factors are basic building blocks of nuclear physics
7 and every hadronic model of meson-baryon dynamics, their microscopic description is also highly
1 desirable on more fundamental grounds.
:
v OurstartingpointforcalculatingthestrongπNN,πN∆andπ∆∆couplingsandformfactorsisthe
i mass-eigenvalueproblemforthreequarksthatareconfinedbyaninstantaneouspotentialandcanemit
X
andreabsorbapion.Todescribethissysteminarelativisticallyinvariantway,wemakeuseofthepoint-
r form of relativistic quantum mechanics. Employing the Bakamjian-Thomas construction, the overall
a
four-momentumoperatorPˆµ canbeseparatedintoafree4-velocityoperatorVˆµ andaninvariantmass
operator Mˆ that contains all the internal motion, i.e. Pˆµ =Mˆ Vˆµ [1]. Bakamjian-Thomas-type mass
operators are most conveniently represented by means of velocity states |V;k ,µ ;k ,µ ;...;k ,µ (cid:105),
1 1 2 2 n n
which specify an n-body system by its overall velocity V (V Vµ = 1), the CM momenta k of the
µ i
individual particles and their (canonical) spin projections µ [1]. Since the physical baryons of our
i
model contain, in addition to the 3q-component, also a 3qπ-component, the mass eigenvalue problem
J.-H. Jung · W. Schweiger
Institut fu¨r Physik, FB Theoretische Physik, Universita¨t Graz, Austria
E-mail: [email protected]
E-mail: [email protected]
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can be formulated as a 2-channel problem of the form
(cid:18)Mˆconf Kˆ (cid:19)(cid:18) |ψ (cid:105) (cid:19) (cid:18) |ψ (cid:105) (cid:19)
3q π 3q =m 3q , (1)
Kˆπ† Mˆ3cqoπnf |ψ3qπ(cid:105) |ψ3qπ(cid:105)
with |ψ (cid:105) and |ψ (cid:105) denoting the two Fock-components of the physical baryon states |B(cid:105). The mass
3q 3qπ
operators on the diagonal contain, in addition to the relativistic particle energies, an instantaneous
confinement potential between the quarks. The vertex operator Kˆ(†) connects the two channels and
π
describes the absorption (emission) of the π by one of the quarks. Its velocity-state representation
can be directly connected to a corresponding field-theoretical interaction Lagrangean [1]. We use a
pseudovector interaction Lagrangean for the πqq-coupling
L (x)=−fπqq (cid:0)ψ¯ (x)γ γ τψ (x)(cid:1)·∂µφ (x). (2)
πqq m q µ 5 q π
π
After elimination of the 3qπ-channel the mass-eigenvalue equation takes on the form
(cid:2)Mˆconf +Kˆ (m−Mˆconf)−1Kˆ†(cid:3)|ψ (cid:105)=m |ψ (cid:105) , (3)
3q π 3qπ π 3q 3q
(cid:124) (cid:123)(cid:122) (cid:125)
Vˆπopt(m)
where Vˆopt(m) is an optical potential that describes the emission and reabsorption of the pion by the
π
quarks. One can now solve Eq. (3) by expanding the (3q-components of the) eigenstates in terms of
(cid:80)
eigenstates of the pure confinement problem, i.e. |ψ (cid:105) = α |B (cid:105), and determining the open
3q B0 B0 0
coefficients α . Since the particles that propagate within the pion loop are also bare baryons (rather
B0
thanquarks),theproblemofsolvingthemasseigenvalueequation(3)reducesthentoapurehadronic
problem,inwhichthedressingandmixingofbarebaryonsbymeansofpionloopsproducesfinallythe
physical baryons. The quark substructure determines just the coupling strengths at the pion-baryon
verticesandleadstovertexformfactors.Tosetupthemass-eigenvalueequationonthehadroniclevel
one needs matrix elements (cid:104)B(cid:48)|Vˆopt(m)|B (cid:105) of the optical potential between bare baryon states. The
0 π 0
general structure of these matrix elements is (B and B(cid:48) are at rest)
0 0
(cid:88)(cid:90) d3k(cid:48)(cid:48) 1
(cid:104)B(cid:48)|Vˆopt(m)|B (cid:105)∝ π J5 (k(cid:48)(cid:48)) J5 (k(cid:48)(cid:48)) , (4)
0 π 0 (cid:112)m2 +k(cid:48)(cid:48)2(cid:113)m2 +k(cid:48)(cid:48)2 πB0(cid:48)B0(cid:48)(cid:48) π m−mB(cid:48)(cid:48)π πB0(cid:48)(cid:48)B0 π
B0(cid:48)(cid:48) π π B0(cid:48)(cid:48) π 0
where m is the invariant mass of the B(cid:48)(cid:48)π system in the intermediate state and spin- as well as
B(cid:48)(cid:48)π 0
0
isospin dependencies have been suppressed.
For the cases we are interested in, i.e. the N and the ∆, the currents occurring in Eq. (4) can be
cast into the form1:
f
J5 (k )=i πN0N0F (k2)u¯(−k )γ γ u(0)kµ,
πN0N0 π m πN0N0 π π µ 5 π
π
f
J5 (k )= π∆0∆0 F (k2)(cid:15)µνρσu¯ (−k )u (0)k k ,
π∆0∆0 π m m π∆0∆0 π µ π ν ∆0,ρ π,σ
π ∆0
f
J5 (k )=−i πN0∆0 F (k2)(cid:15)µνρσu¯(−k )γ γ u (0)k k ,
πN0∆0 π m m πN0∆0 π π σ 5 ν ∆0,µ π,ρ
π ∆0
f
J5 (k )=i πN0∆0 F (k2)(cid:15)µνρσu¯ (−k )γ γ u(0)k k , (5)
π∆0N0 π m m π∆0N0 π ν π 5 σ ∆0,µ π,ρ
π ∆0
where u(.) is the Dirac spinor of the nucleon and u (.) the Rarita-Schwinger spinor of the ∆. Here we
µ
have again suppressed the isospin dependence and also omitted the spin labels. From Eqs. (4) and (5)
one can then infer the analytical expression for the combination f F (k2) in terms of quark
πB0(cid:48)B0 πB0(cid:48)B0 π
degrees of freedom [3].
Assuming a scalar isoscalar confinement potential, the masses of the bare nucleon and the bare
Delta are degenerate, the momentum part of the wave function will be the same and the spin-flavor
part of the wave function is SU(6) symmetric. Rather than solving the confinement problem for a
1 This form exhibits the correct chiral properties and avoids problems with superfluous spin degrees of
freedom when treating spin-3/2 fields covariantly by means of Rarita-Schwinger spinors [2].
3
2.0 1.0
πN0N0 πN0N0
π∆0N0 πN0N0(RCQM)
πN0∆0 0.8 πN0N0(SL)
1.5 π∆0∆0 πN0N0(PR)
)
2Q )0.6
F( 1.0 2Q
fπBB′00fπqq F(0.4
0.5
0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5
Q2 [GeV2] Q2 [GeV2]
Fig. 1 Theleftplotshowsthe(unnormalized)πN N ,π∆ ∆ ,πN ∆ ,andπ∆ N formfactorsasfunctions
0 0 0 0 0 0 0 0
ofQ2 =−2M (M −(M2+k2)1/2).IntherightplottheQ2 behaviorofF (normalizedto1atQ2 =0)
0 0 0 π πN0N0
is compared to the outcome of another relativistic constituent-quark model [6] and to phenomenological fits
obtained within two purely hadronic dynamical coupled-channel models [7; 8] (SL and PR).
particular potential, we parameterize the momentum part of the 3q wave function of N and ∆ by
0 0
means of a Gaussian
ψN0,∆0(k ,k ,k )∝exp(cid:0)−α2(k2 +k2 +k2 )(cid:1) , k +k +k =0, (6)
3q q1 q2 q3 q1 q2 q3 q1 q2 q3
and consider the mass of N and ∆ , i.e. M = M =: M as free parameter. The parameters of
0 0 N0 ∆0 0
our model are therefore the oscillator parameter α, the mass M , the constituent-quark mass m :=
0 q
m =m and f , the πqq coupling strength. For fixed m =263 MeV (taken from Ref. [4]) we have
u d πqq q
adapted the remaining parameters such that the physical N and ∆ masses, resulting from the mass
renormalizationduetopionloops(withN and∆ intermediatestates),agreewiththeirexperimental
0 0
values. In order to tune these parameters we started with a fixed M and α and calculated the masses
0
ofthephysicalnucleonandDeltaasfunctionoff bysolvingthemasseigenvalueequation(3).Then
πqq
wehavevariedM andαsuchthatthephysicalnucleonandDeltamasses(i.e.m =0.9385GeVand
0 N
m =1.233 GeV) are obtained for a reasonable value of f (which also leads to a reasonable value
∆ πqq
for f ). This gives us M = 1.350 GeV, α = 2.915 GeV−1 and f = 0.6602. Note that in our
πN0N0 0 πqq
simple model the physical ∆ is still a stable particle, since the threshold of the only possible decay
channel πN is larger than the mass of the physical ∆. In order to get a ∆ with a finite decay width
0
within our model one would need in Eq. (1) an additional 3qππ channel. For a purely hadronic model
which provides a finite decay width for the ∆ and makes use of the same relativistic coupled-channel
framework as employed here, see Ref. [5].
2 Results and Discussion
The left plot of Fig. 1 shows (unnormalized) πN N , π∆ ∆ , πN ∆ , and π∆ N form factors as
0 0 0 0 0 0 0 0
function of the (negative) four-momentum transfer squared (analytically continued to small time-like
momentumtransfers).ItisworthnotingthatF andF donotcoincide.Thisis,ofcourse,no
π∆0N0 πN0∆0
surprise,sinceinthefirstcasetheN isrealandthe∆ virtual,whereasitisjusttheotherwayround
0 0
inthesecondcase.Theformfactorsdescribethuscompletelydifferentkinematicalsituations,butthey
coincideataparticularnegative(i.e.unphysical)valueofQ2.Sincethereisonlyonecouplingstrength
at the πN ∆ -vertex (i.e. f = f , see Eq. (5)), this is the natural point to normalize the
0 0 π∆0N0 πN0∆0
form factors and extract the coupling constants. Its value Q2 =−0.079 GeV2 is close to the standard
0
normalization point, namely the pion pole Q2 = −m2. The resulting coupling strengths are given in
0 π
Tab. 1 and compared with those from other models.
The couplings quoted for the hadronic model [8] are the physical (“dressed”) couplings. The cou-
plings given in Ref. [6] may be interpreted as “bare” couplings, since meson-cloud effects are not
included explicitly, but physical N and ∆ masses are used for their extraction. The values for f
πN∆
and f taken from Ref. [7] are bare couplings. Dressing of the nucleon, however, is not consid-
π∆∆
ered by the authors. Our results are also bare couplings, since they have been extracted from vertex
matrix elements (cid:104)B(cid:48)π|K†|B (cid:105) involving only bare baryons. For the ratio of the coupling strengths
0 π 0
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Table 1 Our prediction for πN N , πN ∆ and π∆ ∆ coupling constants in comparison with results from
0 0 0 0 0 0
anotherrelativisticconstituent-quarkmodel[6].Shownarealsovaluesforthesecouplingconstantsusedintwo
popularhadroniccoupled-channelmodels[7;8].Theπqq couplingsemployedinthequarkmodelsaregivenin
the second column.
f f f f
πqq πN0N0 πN0∆0 π∆0∆0
Our model 0.6602 1.0027 1.2123 0.6097
Ref. [6] 0.5889 0.9318 1.537
Ref. [7] 1.0027 1.256 0.415
Ref. [8] 0.9708 2.451
we get f : f : f = 1.209 : 1 : 0.608. This may be compared with the predic-
πN0∆0 πN0N0 π∆0∆0
tion from the non-relativistic constituent-quark model assuming SU(6) spin-flavor symmetry, i.e.
√
f : f : f = 4 2/5 : 1 : 9/5 = 1.13 : 1 : 1.8 [9]. The differences can solely be ascribed to
πN∆ πNN π∆∆
relativistic effects and are obviously significant, in particular for the π∆ ∆ -vertex. Remarkably, our
0 0
fractions resemble very much those needed in the dynamical coupled-channel model of Ref. [7], i.e.
f :f :f =1.26:1:0.42.
πN∆ πNN π∆∆
IntherightplotofFig.1ourresultforF iscomparedwiththeoutcomeofanotherrelativistic
πN0N0
constituent-quark model [6] and with two parameterizations of this form factor that have been used
in the hadronic models [7; 8]. Up to Q2 ≈ 1 GeV2 our prediction is comparable with the form factor
parametrization of Ref. [8], but for higher Q2 it falls off slower2. The form factors of Refs. [6; 7] fall off
much faster already at small Q2. Deviations of our result from the one of Ref. [6] have their origin in
different 3q wave functions of the nucleon, but also in different kinematical and spin-rotation factors
entering the microscopic expression for the pseudovector current of the nucleon.
Having determined the πN N , π∆ ∆ and πN ∆ vertices from a microscopic model, we are
0 0 0 0 0 0
now in the position to calculate the electromagnetic form factors of physical nucleons and Deltas
and determine the effect of pions on their electromagnetic structure. Such calculations exist in the
literature (see, e.g., Ref. [4]), but in all the work known to us the strong pion-baryon vertices were
parameterized.Werathertrytotreatstrongandelectromagneticverticesonthesamefooting,i.e.start
with the same three-quark wave function for the (bare) baryons, calculate strong and electromagnetic
form factors of bare baryons and with these form factors as input the effect of pion loops (for physical
baryons). First exploratory calculations for the nucleon (with another three-quark wave function and
withoutDeltas)gavereasonableresultsandshowthatsignificantpion-loopeffectscanbeobservedfor
Q2 (cid:46)0.5GeV2 [3].Itwill,ofcourse,bemoreinterestingtoinvestigateelectromagnetic∆andN →∆
transition form factors, where pionic effect are expected to play a more significant role.
Acknowledgements J.-H.JungacknowledgesthesupportoftheFondszurFo¨rderungderwissenschaftlichen
Forschung in O¨sterreich (Grant No. FWF DK W1203-N16).
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2 It is, of course, an extreme point of view to attribute the N-∆ mass difference solely to the pions. As a
consequence, the 3q wave function in Eq. (6) and thus also strong and electromagnetic form factors may fall
off too slowly. If this will turn out in our calculations of the electromagnetic form factors one has to think of
including other mechanisms (like one-gluon exchange) that can also contribute to the N-∆ mass difference.