Meson Properties in a renormalizable version of the NJL model

13 Pages·0.18 MB·English

by  Andre L. Mota

Meson Properties in a renormalizable version of the NJL model.
Andr´e L. Mota(a)(b) and M. Carolina Nemes(a)
, (a) Departamento de F´ısica, Instituto de Ciˆencias Exatas,
Universidade Federal de Minas Gerais,
Belo Horizonte, CEP 30.161-970, C.P. 702, MG, Brazil
9
9
9 (b) Departamento de Ciˆencias Naturais,
1
Fundac¸˜ao de Ensino Superior de S˜ao Jo˜ao del Rei,

n S˜ao Jo˜ao del Rei, MG, Brazil
a
J

9 Brigitte Hiller(c) and Hans Walliser(c)
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1 (c) Departamento de F´ısica, Universidade de Coimbra,
v P-3000 Coimbra, Portugal
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5
4 ABSTRACT
1
0 Inthepresentpaperweimplementanon-trivialandrenormalizable extension of
9 theNJLmodel. Wediscusstheadvantagesandshortcomingsofthisextendedmodel
9 compared to a usual effective Pauli-Villars regularized version. We show that both
/
h versions become equivalent in the case of a large cutoff. Various relevant mesonic
p observables are calculated and compared.
-
p
e
h
:
v
i 1. Introduction
X
r
a
The Nambu and Jona-Lasinio (NJL) model [1] and its extensions have received much attention
in low and medium energy hadronic physics [2,3]. Because of its four fermion interaction it is
non-renormalizableinthe weak-couplingexpansionford>2 dimensions,the reasonforthe model
usually being treated with a cutoff Λ introduced to regularize the appearing ultraviolet (UV)
divergencies. Althoughtheoriginaltheoryisnon-renormalizableinperturbationtheory,itbecomes
renormalizable in the mean-field expansion also for d > 2 [4,5,7]. However, in contrast to d < 4
where the NJL model represents a perfect renormalizable field theory it is supposed to collapse
for d = 4 to a trivial theory of non-interacting bosons [8–10]. Therefore in order to prevent the
collapse, a cutoff Λ has also to be retained in the ”renormalized” theory. The scale of this cutoff
may in principle be deduced from an underlying non-trivial theory, in our case presumably QCD.
IfafiniteΛcouldbechosenlargeenoughtoextendtheintegrationsofallfiniteintegralstoinfinity,
thenthecutoffappearsonlyimplicitly withinthecouplingswhicharepreventedfrombeingdriven
to zero. Formally this may be achieved by augmenting the model by bosonic kinetic terms and
quartic self-couplings, capable to absorb the cutoff dependence of the coupling constants. This
1
procedure results in a renormalizable and non-trivial field theory for d = 4 dimensions which
corresponds to a linear sigma model with quarks [11], but with fixed bosonic self-couplings. In
order to distinguish this model from the familiar cutoff or regularized NJL we will call it simply
renormalized version in the following.
This non-trivial extension of the NJL is motivated by the observation that some observables
relatedtofiniteintegralsrequireaninfinite oratleastaverylargecutoff,mostprominentexample
being the anomalous pion decay π0 →γγ [12]. Due to the underlying symmetries of the model, it
isclearthatonlyprocesseswhicharesensitivetolargemomentaaresizeablyinfluenced. Therefore
many low-energy quantities, especially of course those used to fix the parameters of the model as
e.g. the pion mass, are essentially the same as in the regularized version.
The renormalized version has all the positive features known to renormalizable theories, but
suffersfromtheoccurrenceofLandaughosts,awellknownproblemrelatedtoLagrangianswithout
asymptotic freedom [5,6].
Arecentwork[13]presentsarenormalizableextensionoftheNJL-modelbyincludingthe quark
interaction generated by one gluon exchange, which simultaneously screens out the unphysical
ghosts. The quark self-energy becomes momentum dependent with the appropriate asymptotic
behavior, in contrast to the constant value obtained in the original NJL model. We consider our
approach to be the ”minimal” non-trivial renormalization program which still keeps the simple
local structure of the original NJL Lagrangian and we think it worthwhile to analyse the results
obtained from this approach,due to its simplicity.
The aim of this paper is twofold. In the first part, which is of formalcharacter,we show how to
implement the originalNJL Lagrangianto render it a non trivial renormalizabletheory. Secondly
we apply this Lagrangian to calculate relevant observables of the SU(2) flavor case and make a
comparativestudywiththePauli-Villarsregularizedversion. Weintendinthiswaytogetabetter
understanding of the advantages and shortcomings of the two possible descriptions of the model
in leading 1/N order.
C
2. Mean field expansion of the NJL model and triviality
In this section we briefly recapitulate the main features of the renormalization procedure for
the NJL model using the mean-field expansion. Following Eguchi [4] we isolate the UV singulari-
ties, but we consider also finite contributions which enter the renormalization scheme in order to
demonstratetheequivalencetotheprocedurepresentedbyGuralnikandTamvakis[5]. Inaddition
we allow the symmetry to be broken explicitly by a current quark mass. Finally we discuss the
issue of triviality and the introduction of a cutoff Λ preventing the model from the collapse. The
following derivation applies to SU(2) with N colors, an extension to SU(3) is straightforward.
C
Starting point is the NJL Lagrangianwith a local four-quark interaction
G
L=q¯(iγµ∂ −mˆ )q+ 0 (q¯q)2+(q¯iγ τq)2 , (1)
µ 0 5
2
(cid:2) (cid:3)
where we consider the isospin symmetric limit mˆ = mˆ = mˆ. To allow for wave function renor-
u d
malizationslateronandalsoinordertotracetheN orders,itisconvenienttoreplaceG =g2/µ2
C 0 0 0
with g2 ∼ 1/N . The subscript zero denotes bare (infinite) quantities everywhere. With boson
0 C
fields introduced in the standard way the Lagrangianbecomes
µ2
L=q¯[iγµ∂ −mˆ −g (σ +iγ τπ )]q− 0(σ2+π2)
µ 0 0 0 5 0 2 0 0
µ2 µ2
L=q¯[iγµ∂ −g (σ +iγ τπ )]q− 0(σ2+π2)+ 0mˆ σ , (2)
µ 0 0 5 0 2 0 0 g 0 0
0
2
where the latter representation is obtained by shifting the scalar field σ → σ −mˆ /g and a
0 0 0 0
term independent of the dynamical fields is omitted. Integrations over the fermion fields q and q¯
may now be performed in the path integral such that the resulting effective Lagrangian collects
the corresponding trace log contribution
µ2 µ2
L=−iTrℓn[iγµ∂ −g (σ +iγ τπ )]− 0(σ2+π2)+ 0mˆ σ . (3)
µ 0 0 5 0 2 0 0 g 0 0
0
Expecting the scalar field to possess a nonvanishing vacuum expectation value we expand σ =
0
m/g +σ′, where m represents the (finite) constituent mass
0 0
L=i ∞ 1Tr (iγµ∂ −m)−1g (σ′ +iγ τπ ) n− µ20(σ′2+π2)− µ20(m−mˆ )σ′ . (4)
n µ 0 0 5 0 2 0 0 g 0 0
nX=1 (cid:2) (cid:3) 0
To evaluate this sum is now quite straightforward. The terms for n = 1,...,4 contain UV diver-
gencies showing up as
d4q 1 d4q 1
I =i , I =i (5)
quad Z (2π)4q2−m2 log Z (2π)4(q2−m2)2
quadraticallyandlogarithmicallydivergentintegralsrespectively. TheeffectiveLagrangianisthen
given by
1 N g2
L= (−4N g2I )(∂ σ′∂µσ′ +∂ π ∂µπ )− C 0∂ σ′∂µσ′
2 C 0 log µ 0 0 µ 0 0 12π2 µ 0 0
1 1
− (µ2−8N g2I )(σ′2+π2)− 4m2(−4N g2I )σ′2+...
2 0 C 0 quad 0 0 2 C 0 log 0
+8N g3I mσ′(σ′2+π2)+2N g4I (σ′2+π2)2+...
C 0 log 0 0 0 C 0 log 0 0
µ2
− 0(m−mˆ )−8N g mI σ′ . (6)
(cid:20)g 0 C 0 quad(cid:21) 0
0
Thedotsafterthefirsttwolinesdenotefinitehigherderivativetermsofg2σ2andg2π2proportional
0 0 0 0
to N0 not explicitely shown, but a finite kinetic term for the scalars is kept and finally leads to
C
different wave-function renormalizations σ′ = Z−1/2σ′ and π = Z−1/2π . The dots in the third
σ 0 π 0
line indicate finite higher order terms also not shown. The renormalized parameters may now be
introduced as follows
g2 N g2
Z−1 = 0 =−4N g2I − C 0 (7)
σ g2 C 0 log 6π2
σ
g2
Z−1 = 0 =−4N g2I (8)
π g2 C 0 log
π
µ2
σ =µ2−8N g2(I +2m2I ) (9)
Z 0 C 0 quad log
σ
µ2
π =µ2−8N g2I . (10)
Z 0 C 0 quad
π
Together with the gap-equation
µ2
0(m−mˆ )=8N mI , (11)
g2 0 C quad
0
3
which makes the linear term in σ′ vanish, we obtain the renormalizedLagrangianin its final form
0
1 1
L= (∂ σ′∂µσ′+∂ π∂µπ)− (µ2σ′2+µ2π2)
2 µ µ 2 σ π
2mg 1
− σσ′(g2σ′2+g2π2)− (g2σ′2+g2π2)2 +... (12)
g2 σ π 2g2 σ π
π π
Thecurrentquarkmasshasdisappearedanditisnoticedthattheremainingparametersarerelated
according to eqs. (7-10)
1 1 N µ2 4m2+µ2
= − C , σ = π (13)
g2 g2 6π2 g2 g2
σ π σ π
suchthattherenormalizedmodelischaracterizedbythreeparameters(m,g ,µ )astheregularized
π π
one(G,m,mˆ). Fromtheseequationsusingthegap-equation(11)wemayalsodefinearenormalized
four-fermion coupling G
µ2 µ2 mˆ mˆ
m π =mˆ 0 = 0 = (14)
g2 0g2 G G
π 0 0
by reintroduction of the renormalized (physical) current quark mass. Although this relation is
beyond the scope of the renormalized model it will nevertheless prove useful for the evaluation of
the quark condensate.
The N orders are now carriedby the couplings g2 ∼g2 ∼1/N . For practicalcalculations we
C π σ C
consider only the leading order N for each process. From the quadratic terms in the Lagrangian
C
we read off the renormalized meson propagatorsof order N0
C
p2
∆−1(p2)=p2−µ2 −4N g2 (p2−4m2)Z (p2)−
σ σ C σ(cid:20) 0 24π2(cid:21)
∆−1(p2)=p2−µ2 −4N g2p2Z (p2), (15)
π π C π 0
where the momentum dependent terms due to the finite higher derivative terms not explicitely
shown in eq.(12) are contained in the finite function
1 1 p2
Z (p2)= dzℓn 1− z(1−z)
0 16π2 Z (cid:20) m2 (cid:21)
0
(16)
1+ 4m2arsinh |p2| −1 p2 ≤0
|p2| 4m2
 q q
= 8π12  q4pm22 −1arcsinq4pm22 −1 0<p2 ≤4m2
1− 4m2 arcosh p2 −iπ −1 4m2 <p2
 q p2 (cid:18) q4m2 2(cid:19)
which possesses different branches. The physical meson masses are defined as usual via the poles
of the propagators
m2 =4m2+µ2 +4N g2(m2 −4m2)Z (m2)
σ π C π σ 0 σ
m2 =µ2 +4N g2m2Z (m2). (17)
π π C π π 0 π
In the chiral limit (µ2 =0) we obtain m =0 and m =2m.
π π σ
4
Similarly,alsointhechirallimittheleadingN 3-and4-bosonvertexfunctionsatzeromomenta
C
are obtained from the cubic and quartic self-couplings in accordance with Guralnik and Tamvakis
[5], who derive these results from the corresponding Ward identities.
In the followingwe wantto discuss the triviality of the NJL model whichis suggestedby lattice
calculations [10]. In the mean-field expansion it follows immediately from (8)
g2 =−(4N I )−1, (18)
π C log
namely in the continuum limit the couplings g and hence also g are driven to zero rendering
π σ
the Lagrangian (12) a theory of non-interacting mesons. This is caused by the mesonic kinetic
terms and quartic self couplings in (12) being created purely by radiative corrections: they were
not present in the original Lagrangian. To avoid the collapse of the model, g must be kept
π
fixed at some finite value. For that purpose a cutoff Λ has to be introduced in order to keep the
logarithmically divergent integral in (18) finite (note that the quadratic divergence has already
disappeared in the renormalized parameters). In the continuum limit g2 ∼ (4π/N )/ℓn(Λ/m)
π C
tends to zero logarithmically, and a finite coupling g requires also a finite Λ, in fact, in order to
π
reproduce a reasonable coupling strength a rather low cutoff of the order of 1GeV is needed in
contrast to the situation in QED where the collaps is prevented by a cutoff located way above all
physical energies of interest. Nevertheless, if it were possible to choose Λ large enough such that
all finite integrals may be evaluated in the continuum limit then the cutoff would disappear from
the theory being only implicitely contained in the coupling g which is kept finite [14]. Exactly
π
this is achievedby adding mesonic kinetic terms andquarticself-couplingsto the modelwhich are
capable to absorb the troublesome radiative terms in (12) leading to a non-trivial renormalizable
extension of the NJL discussed in the following section. Mesonic properties calculated in the two
versions of the model are then presented in section 4.
3. Non-trivial extension of the NJL model
We have seen in the previous section that the triviality of the NJL model is connected with the
fact that the mesonic kinetic and interaction terms are created purely by radiative corrections.
In fact triviality may be avoided by adding these contributions to the Lagrangian (3) from the
beginning [15]
f2
L=−iTrℓn[iγµ∂ −g (σ +iγ τπ )]+ 0(∂ σ ∂µσ +∂ π ∂µπ )
µ 0 0 5 0 µ 0 0 µ 0 0
2
µ2 λ µ2
− 0(σ2+π2)− 0(σ2+π2)2+ 0mˆ σ +... (19)
2 0 0 2 0 0 g 0 0
0
Of course this may lead beyond the NJL model, we will comment on this later. Repeating the
steps which lead to eq. (6) we find the renormalized parameters as
g2 N g2
Z−1 = 0 =f2−4N g2I − C 0 (20)
σ g2 0 C 0 log 6π2
σ
g2
Z−1 = 0 =f2−4N g2I (21)
π g2 0 C 0 log
π
µ2 6λ
σ =µ2−8N g2(I +2m2I )+ 0m2 (22)
Z 0 C 0 quad log g2
σ 0
µ2 2λ
π =µ2−8N g2I + 0m2 (23)
Z 0 C 0 quad g2
π 0
5
λ λ
= 0 −4N I (24)
g2 g4 C log
π 0
together with the gap-equation
µ2 2λ
0(m−mˆ )−8N mI + 0m3 =0. (25)
g2 0 C quad g4
0 0
The renormalized Lagrangianin the shifted scalar fields becomes
1 1
L= (∂ σ′∂µσ′+∂ π∂µπ)− (µ2σ′2+µ2π2)
2 µ µ 2 σ π
2mλ λ
− g σ′(g2σ′2+g2π2)− (g2σ′2+g2π2)2 +... (26)
g2 σ σ π 2g2 σ π
π π
formally identical to the bosonized NJL Lagrangian(12) if we choose λ=1. In generalthe model
parameters are now related
1 1 N µ2 4λm2+µ2
= − C , σ = π (27)
g2 g2 6π2 g2 g2
σ π σ π
quitesimilarto(13)andalsotherelation(14)remainsunchanged. Wewanttoemphasizehowever,
that this Lagrangian does not represent the trivial NJL model and constitutes instead a different
non-trivial theory. Asaconsequencethecouplingsg andg donolongervanishinthecontinuum
π σ
limit. Thenumericalresultsofthismodelwithλ=1areofcourseidenticaltothoseobtainedfrom
the NJL model (12) with g and g kept fixed at some finite values as discussed in the preceeding
π σ
section.
Concluding, there seems to be two options to treat the problem of triviality which appears in
the NJL model:
(i) AcutoffΛisretainedtopreventthemodelfromthecollapse. Becausenumericallythecutoff
is of the order of 1 GeV only, it has to be kept also in all finite integrals.
(ii) The NJL model is augmented by kinetic terms and mesonic self-interactions. This results in
the linear sigma model coupled to quarks and constitutes a perfect non-trivial field theory.
The latter theorycontainsoneadditionalparameterλ. Forλ=1 (ii)givesthe same resultsasthe
conventionalNJL in case of a large cutoff. In principle the assumption λ=1 may be tested in ππ
scattering,of coursenot in the leading chiralorderwhich is fixed by a famous low energytheorem
[16], but in the next to leading orders (subsection 4.2). Many other meson properties are quite
independent of this parameter. In the following section we compare some mesonic observables
calculated in the two versions of the NJL model.
4. Meson properties
In this section the parameters of both versions of the model are fixed. We show that in the
renormalized version the chiral expansion becomes quite simple and we discuss the issue of the
additional parameter λ appearing in the linear sigma model with quarks. Finally we are going to
calculate several mesonic observables.
6
1. Determination of the model parameters
Forbothversionsofthemodelweusedtwosetsofmodelparameters: onewithalowconstituent
quarkmassfixedatm=210MeV,andtheotherwiththeconstituentquarkmassfixedatm=350
MeV,valueswhichareusedwidelyintheliterature. Theremainingmodelparametersareadjusted
to reproduce the pion decay constant f = 93.3 MeV and the pion mass m = 139 MeV. As a
π π
result, one obtains for the Pauli-Villars regularized version a large dimensionless ratio Λ/m ≃ 6
in the small constituent quark mass case, and a small one for the large mass case, Λ/m ≃ 2, see
Table I. As will be discussed in section 4.2, chiral expansions of the renormalized and regularized
versionscoincide in the Λ/m→∞ limit, showing that this ratio is a measure for the deviations in
the two models. On the other hand, in the renormalized version the pion decay constant is
m µ2
f =g π , (28)
π πqqg2 m2
π π
and the pion mass is given by (17). Here and in the following we use the pion quark and sigma
quark couplings
g−2 =g−2−4N Z (m2)+m2Z′(m2)
πqq π C 0 π π 0 π
g−2 =g−2−4N (cid:2)Z (m2)+(m2 −4m2(cid:3))Z′(m2) , (29)
σqq π C 0 σ σ 0 σ
(cid:2) (cid:3)
forabbreviation. Thus,f andm fixtheparametersg andµ listedalsoinTableI.Furthermore,
π π π π
for the evaluation of the quark condensate according to (14)
mˆ µ2
mˆ <q¯q >=−(m−mˆ) =−m(m−mˆ) π (30)
G g2
π
a value for the current quark mass has to be adopted which strictly speaking is not a parameter
of the renormalized version of the model. The standard value of mˆ =7.5 MeV leads to the result
quoted in Table II.
TABLEI. Parameters of the regularized and renormalized versions of the NJL model.
regularized model renormalized model
model m=350 (210) MeV m=350 (210) MeV
parameters G=17.6 (5.11) GeV−2 g =3.752 (2.250)
π
mˆ =8.5 (4.1) MeV µ =141 (141) MeV
π
related Λ=769 (1190) MeV g =7.006 (2.610)
σ
parameters µ =1333 (513.8) MeV
σ
7
2. Chiral expansion and ππ scattering
In the renormalized version of the NJL model the chiral expansion for constituent quark mass,
pion decay constant, pion mass and quark condensate become quite simple:
◦2
m=m◦ 1+ mπ +... (31)
◦2
4m
 
◦2 m◦2 N m◦2
f2 =f 1+ π − C π +... (32)
π π ◦2 ◦2
2m 8π2f
 π 
◦2 ◦2
m2 =m◦2 1− mπ + NCmπ +... (33)
π π ◦2 ◦2
4m 12π2f
 π 
mˆ <q¯q >=−f◦2m◦2 1− mˆ + m◦2π +... . (34)
π π m◦ 4m◦2
 
These formulas agree with those obtained in the Pauli-Villars regularized model [17] for large
cutoff Λ ≫ m. It is noticed that the current quark mass appears only in the expression for the
condensate.
Similarly, the ππ scattering amplitude (box-diagram + sigma exchange) is obtained as
2
◦2
s−m◦2 N m◦2 N2m◦4 (cid:18)s−2mπ(cid:19)
A(s,t,u)= π +1− C + C 
◦2 ◦2 ◦4 ◦2◦2
f 3π2f 16π4f 4m f
π  π π π
◦2 2
+(1 −1) 1 s−m◦2 − NCm s−2m◦2 
λ ◦2◦2 π ◦2 (cid:18) π(cid:19)
4m f 4π2f
π  π 
+ NC s(u+t)−ut−2m◦4 +... (35)
◦4 (cid:20) π(cid:21)
24π2f
π
As mentioned already, the leading term is fixed by a low energy theorem [16] and is therefore
independentofthestrengthλofthe quarticself-interaction. However,inthe nexttoleadingorder
there appears a term in addition to those obtained in the regularizedmodel [17] with large cutoff,
which is effective for λ6=1.
In refs. [17,3] it was shown that compatibility with chiral perturbation theory (ChPT) requires
a small constituent quark mass of the order of m ≃ 250 MeV. For such a small quark mass the
cutoff is of minor importance and the terms in (35) which survive for λ = 1 fit the ChPT ππ
scattering threshold parameters [18] in the renormalized version as well. We may conclude that
the comparison with ChPT requires a value of λ close to 1. Of course, this conclusion is correct
only if we accept a small constituent quark mass.
3. Sigma meson properties
The sigma meson mass can be evaluated using eq.(17). In order to obtain this mass, we made
use of the real part of the propagatoronly, in both, the regularized and the renormalized version.
8
Theimaginarypartrelatedtothedecaysintoq¯q pairsisverysmallandcanbeneglected[19]. The
decay width of σ →ππ is :
3 m2 −4m2
Γ = σ πf2 . (36)
σππ p8πm2 σππ
σ
All the expressionsfor the Pauli-VillarsregularizedNJL model usedhere andin the followingsec-
tionsmaybefoundin[20]andarenotrepeatedhere. Theamplitudef readsintherenormalized
σππ
version
1 2m2 −m2
f =16mN g g2 −Z (m2)+ π σI (m2,m2,m2) ,
σππ C σqq πqq(cid:20)4N g2 0 σ 2 3 σ π π (cid:21)
C π
d4q 1
I (p2,p2,p2)=i (37)
3 1 2 Z (2π)4(q2−m2)[(q−p )2−m2][(q−p )2−m2]
1 2
with p2 =(p −p )2.
1 2
As is noticed from Table II, the scalar decay width into two pions depends most sensitively on
thetwoversionsofthemodel,speciallyofcourseinthelargemass(lowcutoff)case. Thisdifference
persists also in the chiral limit.
4. Pion charge formfactor
In the renormalized version, the pion electromagnetic formfactor is given by
1
F (p2)=2N g2 +m2 Z′(m2)−I (p2,m2,m2) , (38)
π C πqq(cid:20)4N g2 π 0 π 3 π π (cid:21)
C π (cid:0) (cid:1)
andthecorrespondingpionelectromagneticradiusisdefinedasusual. InFig. 1theelectromagnetic
pionformfactorisshowninthespace-likeregiononly,sinceinthetime-likeregiontheroleofvector
mesons, which are not considered here, is very important. For the same reason the charge radius
turns out too small compared to its experimental value, see Table II. We see however that the
renormalizable version yields a value close to the chiral limit result
3N
<r2 > = C ≃(0.59fm)2, (39)
π 4π2f2
π
whereas the regularized version tends to decrease this value further with decreasing ratio Λ/m.
From Fig.1 we see that the formfactors in the two versions of the model are very similar for the
lower mass case, but for the larger mass case the renormalized version improves on the results,
whereas the regularizedversion does not yield a satisfactory fit as noticed already in [12].
9
FIG.1. Pionformfactorinthespace-likeregion. Theresultsoftherenormalized(solid)andregularized
(dashed) models for small and large constituent quark masses are compared to experimental data [21].
5. Anomalous π0→γγ decay
The formfactorassociatedwith the anomalousprocessπ →γ∗γ, withone ofthe photons being
0
off shell is
8N e2
Fπγ∗γ(p2)=− 3C gπqqmI3(0,p2,m2π). (40)
10