THE MOMENTUM–SPACE BOSONIZATION OF THE NAMBU–JONA-LASINIO MODEL WITH VECTOR AND AXIAL-VECTOR MESONS

V?ronique Bernard, Ulf-G. Mei?ner e

arXiv:hep-ph/9312203v1 1 Dec 1993

Centre de Recherches Nucl?aires et Universit? Louis Pasteur de Strasbourg e e Physique Th?orique e BP 20Cr, 67037 Strasbourg Cedex 2, France and A.A. Osipov Joint Institute for Nuclear Research, Laboratory of Nuclear Problems 141980 Dubna, Moscow Region, Russia

ABSTRACT The momentum-space bosonization method is extended to the case of a Nambu–Jona-Lasinio type model with vector and axial-vector mesons. The method presented gives the possibility of deriving any meson vertex function to all orders in momenta and to the leading order in 1/Nc . Two-point functions, which describe one-particle transitions to the hadronic vacuum, and meson selfenergies are considered. We ?nd new relations which generalize the well-known KSFR relation and both the ?rst and the second Weinberg sum rules. These result from a consistent treatment of higher order terms in the momentum expansion.

CRN 93–57 0

December 1993

1. Introduction The Nambu–Jona-Lasinio (NJL) model in a version incorporating all essential QCD symmetries may be a reasonable low-energy approximation for QCD. In our previous paper [1] we suggested a systematic method for evaluating any mesonic N-point function in the bosonized NJL model. The main idea consists in the construction of special bosonic variables to be used for the description of the observable mesonic states. As a result, it is possible to extend the usual treatment of bosonized NJL models, which was formulated in papers [2] and developed in Refs. [3]-[6]. The standard approach is essentially oriented to the derivative expansion of the e?ective meson Lagrangian. Our method does not use this approximation. Therefore, one can expect it to be a powerful tool for exploration of the extended NJL model, which includes not only scalar and pseudoscalar ?elds but also heavier vector and axial-vector meson states. One of the principal di?culties we come across on going over to higher energies that are typical of vector particles is the con?nement property of QCD. The NJL model enables one to see dynamic symmetry breaking mechanism at work, which leads to meson formation from quark-antiquark pairs, but it does not forbid emission of constituent quarks (with mass m) into the continuum. The model has to deal with this phenomenon already at energies p2 ? 4m2 ? m2 . ρ In its formal part, our method does not require an a priori solution to the con?nement problem. It only covers the general bosonization scheme for the model and mainly applies to the separation procedure of the collective degrees of freedom in theories with four-fermion interaction. For that aspect the con?nement mechanism is not of relevance. Our goal here is to construct a formal scheme which involves the bosonization procedure and has all the advantages of the pure fermionic approach (Hartree-Fock plus Bethe-Salpeter approximation). The advantages of this approach are the explicit use of boson variables for describing the dynamics of collective excitations and the possibility of gaining full information on the momentum dependence of vertex functions. The NJL model belongs to the set of nonrenormalizable theories. Hence, to de?ne it completely as an e?ective model, a regularization scheme must be speci?ed to deal with the quark-loop integrals in harmony with general symmetry requirements. As a result, an additional parameter Λ appears, which characterizes the scale of the quark-antiquark forces responsible for the dynamic chiral symmetry breaking. From the meson mass spectrum it is known that Λ ? 1 GeV. Here, we will make use of the Pauli–Villars [7] regularization, which preserves gauge invariance. In this form it was used in Refs.[8]. 1

For simplicity we consider the linear bosonized version of the extended NJL model with U (2)?U (2) symmetry which can be explicitly violated by the current quark masses. The extension to the case of U (3) ? U (3) symmetry will be done elsewhere. 2. Momentum-space bosonization of the extended NJL model Consider the extended U (2) ? U (2) NJL Lagrangian with a local four-quark interaction GS (qτa q)2 + (qiγ5 τa q)2 L(q) = q(i? ? m)q + 2 GV ? (qγ ? τa q)2 + (qγ5 γ ? τa q)2 , (2.1) 2 where q = (u, d) are coloured current quark ?elds with current mass m = diag(mu , md ), τa = (τ0 , τi ), τ0 = I, τi (i = 1, 2, 3) are the Pauli matrices of the ?avour group SU (2)f . The constants of the four-quark interactions are GS for the scalar and pseudoscalar cases, GV for the vector and the axial-vector cases. The current mass term explicitly violates the U (2) ? U (2) chiral symmetry of the Lagrangian (2.1). In what follows, we shall only consider the isospin symmetric case mu = md = m. Introducing boson ?elds in the standard way, the Lagrangian takes the form L(q, σ, π , v, a) = q (i? ? m + σ + iγ5 π + γ ? v? + γ5 γ ? a? ) q ? ? ? ? ? ? v?a + a2 ?2 ??a σ 2 + πa ?2 ? a + . 2GS 2GV

(2.2)

? ? ? ? a ? Here σ = σ a τa , π = πa τa , v? = v?a τa , ? ? = a?a τa . The vacuum expectation value of the scalar ?eld σ 0 turns out to be di?erent from zero (< σ 0 > =0). To obtain the physical ?eld σ0 with < σ0 >= 0 one performs a ?eld shift leading to ? ? a new quark mass m to be identi?ed with the mass of the constituent quarks σ 0 ? m = σ0 ? m, ? σ i = σi , ? (2.3)

where m is determined from the gap equation (see Eq.(2.6) below). Let us integrate out the quark ?elds in the generating functional associated with the Lagrangian (2.2). Evaluating the resulting quark determinant by a loop expansion one obtains L(? , π , v, ?) = ?iTr ln 1 + (i? ? m)?1 (? + iγ5 π + γ ? v? + γ5 γ ? a? ) σ ? ? a σ ? ? ? ? σ2 a + 2GS πa ?2 + v?a ?2 + 2GV a2 ??a 2 .

Λ

(2.4)

The index Λ indicates that a regularization of the divergent loop integrals is introduced. We apply here the Pauli–Villars regularization [7], which preserves vector gauge invariance and at the same time might allow to reproduce the quark condensate for physical values of the current quark mass [6]. The Pauli–Villars cut-o? Λ is introduced by the following replacements e?im m e

2

z

→ R(z) = e?im

′ 2

2

z

1 ? (1 + izΛ2 )e?izΛ

2

,

2 ?im2 z

→ iR (z) = m R(z) ? izΛ e

4 ?iz(Λ2 +m2 )

(2.5)

,

where the minimal number of Pauli–Villars regulator has been introduced. In this case the expressions for some loop integrals Ii (see formulae (2.7), (2.11), and (2.24)) coincide with those obtained by the usual covariant cut-o? scheme. Consider the ?rst terms of the expansion (2.4). From the requirement for the terms linear in σ to vanish we get a modi?ed gap equation ? m ? m = 8mGP I1 . The integral I1 is equal to

Λ

(2.6)

I1 = iNc

3 Λ2 d4 q = Λ2 ? m2 ln 1 + 2 (2π)4 (q 2 ? m2 ) (4π)2 m

,

(2.7)

where Nc = 3 is the number of colours. The terms quadratic in the boson ?elds lead to the amplitudes

?? Πππ (p2 ) = 8I1 ? G?1 + p2 g ?2 (p2 ) ?+a ??a , π ? π ? S

(2.8a) (2.8b) (2.8c) (2.8d) (2.8e) (2.8f )

Π

σσ ??

(p )

2

?? Πvv (p2 ) ?? Πaa (p2 )

?? Ππa (p2 ) ?? Πaπ (p2 )

a a

= 8I1 ? G?1 + (p2 ? 4m2 )g ?2 (p2 ) ?+a ??a , S σ ? σ ? ?a ??a v ? ν ?ν 2 ?2 2 ?ν ?1 = g GV + 4(p p ? g p )gV (p ) ε? (p)εv (p), ν = g ?ν G?1 + 4m2 g ?2 (p2 ) V ?2 ?a ??a ? ν + 4(p p ? g ?ν p2 )gV (p2 ) ε?a (p)εa (p), ν ??a = 2img ?2 (p2 )p? ε?a (p)??a , π ? ?2 2 ? aa ? = ?2img (p )p ε? (p)?+a . π ?

? ? Here εv (p), εa (p) are the polarization vectors of the vector and axial-vector ? ? ?elds. We have introduced the symbols ??a = 1 and ??a = 1 to explicitely σ ? π ?

3

show the pseudoscalar and scalar ?eld contents of the pertinent two-point functions. The functions g(p2 ) and gV (p2 ) are determined by the following integrals g ?2 (p2 ) = 4I2 (p2 ),

?2 gV (p2 ) = 1 ∞

(2.9) (2.10)

2 J2 (p2 ), 3

3 I2 (p2 ) = 16π 2

0

dy

0 1

p2 2 dz R(z)eiz 4 (1?y ) , z

(2.11)

∞

9 J2 (p ) = 32π 2

2 0

dy(1 ? y )

0

2

p2 2 dz R(z)eiz 4 (1?y ) . z

(2.12)

Let us diagonalize the quadratic form (2.8a) + (2.8d) + (2.8e) + (2.8f ) rede?ning the axial ?elds

?? ?? ε?a (p) → ε?a (p) + iβ(p2 )p? ?+a . π ?

a a

a ? a ? ε? (p) → ε? (p) ? iβ(p2 )p? ??a , π ?

a a

(2.13)

This determines the function β(p2 ), β(p2 ) = 8mI2 (p2 ) . G?1 + 16m2 I2 (p2 ) V (2.14)

Consequently, one has no more mixing between pseudoscalar and axial-vector ?elds. The self-energy of the pseudoscalar ?eld takes the form

?? Πππ (p2 ) = 8I1 ? G?1 + p2 g ?2 (p2 ) 1 ? 2mβ(p2 ) S

?+a ??a . π ? π ?

(2.15)

Now we can construct special boson variables that will describe the observed mesons. These ?eld functions (πa , σa , va , aa ) correspond to bound quarkantiquark states and are derived via the following transformations

?1/2 π a (p) = Zπ gπ (p2 )π a (p), ? ?1/2 σ a (p) = Zσ g(p2 )σ a (p), ? 1 ?1/2 v a (p) = Zv ? gV (p2 )v a (p), 2 1 ?1/2 a a (p) = Za ? gV (p2 )aa (p), 2

(2.16)

4

where gπ (p2 ) =

g(p2 ) 1? 2mβ(p2 )

= g(p2 )

1 + 16m2 GV I2 (p2 ).

(2.17)

The new bosonic ?elds have the self-energies

?1 Ππ,σ (p2 ) = δab Zπ,σ p2 ? m2 (p2 ) , π,σ ab

?1 Πv,a (p2 ) = δab Zv,a p? pν ? g?ν p2 ? m2 (p2 ) v,a ?ν,ab

.

(2.18)

The p2 -dependent masses are equal to

2 m2 (p2 ) = (G?1 ? 8I1 )gπ (p2 ), π S m2 (p2 ) = 1 ? 2mβ(p2 ) m2 (p2 ) + 4m2 , σ π 2 gV (p2 ) 3 = = , 4GV 8GV J2 (p2 ) I2 (p2 ) m2 (p2 ) = m2 (p2 ) + 6m2 . a v J2 (p2 )

(2.19a) (2.19b) (2.19c) (2.19d)

m2 (p2 ) v

The constants Zπ,σ,v,a are determined by the requirement that the inverse meson ?eld propagators Ππ,σ,v,a(p2 ) satisfy the normalization conditions Ππ,σ (p2 ) = p2 ? m2 + O (p2 ? m2 )2 , π,σ π,σ (2.20)

Πv,a (p2 ) = ?g?ν p2 ? m2 + O (p2 ? m2 )2 ?ν v,a v,a

,

around the physical mass points p2 = m2 π,σ,v,a , respectively. The conditions (2.20) lead to the values Zπ = 1 + Zσ = 1 + Zv = 1 + Za = 1 + m2 [1 ? 2mβ(m2 )] ?I2 (p2 ) π π I2 (m2 ) ?p2 π m2 ? 4m2 ?I2 (p2 ) σ I2 (m2 ) ?p2 σ m2 ?J2 (p2 ) v J2 (m2 ) ?p2 v m2 ?J2 (p2 ) a J2 (m2 ) ?p2 a ,

p2 =m2 σ

,

p2 =m2 π

(2.21a) (2.21b) (2.21c)

,

p2 =m2 v

p2 =m2 a

?

6m2 ?I2 (p2 ) J2 (m2 ) ?p2 a

.

p2 =m2 a

(2.21d)

5

Using the expressions (2.18), one can obtain the two-point Green functions ?(p). For example, in the scalar and vector ?eld case the relations Πσ (p2 )?σ (p2 ) = δac , ab bc

v,νσ 2 σ Πv ?ν,ab (p )?bc (p) = δac δ? .

(2.22)

give δab Zσ , ? m2 (p2 ) σ δab Zv p? pν ? g ?ν m2 (p2 ) v ?v,?ν (p) = 2 2 . ab mv (p ) p2 ? m2 (p2 ) v ?σ (p2 ) = ab p2 (2.23a) (2.23b)

The formal scheme developed here gives the possibility of evaluating any mesonic N-point Green function through the parameters of the model: Λ, m, GS , GV , and the one-loop integrals I1 , I2 , J2 , I3 , . . . IN . Two examples of these are

Λ

I2 (p2 ) =

Λ

(?iNc ) d4 q , 4 (q 2 ? m2 )[(q + p)2 ? m2 ] (2π)

I3 (p1 , p2 ) =

d4 q (?iNc ) . 4 (q 2 ? m2 )[(q + p )2 ? m2 ][(q + p )2 ? m2 ] (2π) 1 2

(2.24)

This picture corresponds to the calculations in the framework of the pure fermionic NJL model where the Bethe-Salpeter equation sums an in?nite class of fermion bubble diagrams. 3. Scale-invariant relations and matching conditions The purpose of this section is to consider some general consequences of our approach and to compare them with the well-known current algebra results. We start with the form factor fπ (p2 ), which describes the week pion decay π → lνl and can be expressed in the following form fπ (p2 ) = √ m . Zπ gπ (p2 ) (3.1)

From here on, when omitting an argument of a running coupling constant or a running mass, we always assume that its value is taken on the mass-shell of the corresponding particle. The symbol of this particle will be used for that. For 6

example, on the pion mass-shell fπ (m2 ) = fπ = 93.3 MeV. Combining Eq.(2.19a) π for the pion mass and the gap equation (2.6) with (3.1) one ?nds

2 m2 (p2 )fπ (p2 ) = π

mm 2m < qq > =? . Zπ GS Z 1? m

π m

(3.2)

The right-hand side of this equality does not depend on p2 . This is an example of a scale-invariant relation that can be found in the model under consideration. 2 It extends the well-known current-algebra result m2 fπ = ?2m < qq > derived π by Gell-Mann, Oakes and Renner [9] that is exact at the lowest order of chiral expansion (in powers of external momenta p2 and quark masses). Another relation can be found in the sector of vector mesons. The form factor of the electromagnetic ρ → γ transition is equal to 2gρ (p2 ) 1 J2 (p2 ). = 2) fρ (p 3 Zρ (3.3)

Using this formula and Eq.(2.19c) one can obtain the following scale-invariant result m2 (p2 ) 1 ρ (3.4) . = 2 (p2 ) fρ 4Zρ GV It is interesting to note here that the constant GV is related to the pion decay form factor too. This is a direct consequence of the π?a1 mixing,

2 fπ (p2 ) 1 = . 2) mβ(p 2Zπ GV

(3.5)

The comparison of the last two equations leads to a matching condition which relates physical quantities from di?erent sectors of the model. In the particular case we deal with, properties of the ρ meson and of the pion are related via

2 2 m2 = afρ fπ . ρ

(3.6)

The constant a in our case is equal to a= Zπ . 2mZρ βπ 7 (3.7)

For a = 2 this is known as the KSFR relation [10]. The constant of the ρ meson decay into two pions fρππ is usually included in the KSFR relation or eliminated from it as required by the universality condition fρππ = fρ . Let us see to what extent the model behaviour of form factors agrees with this hypothesis. For that, we calculate the form factor fρππ (p2 ), fρππ (p2 ) = gρ (p2 ) F (p2 ). Zρ (3.8)

The function F (p2 ) represents the contribution of the ρ → ππ triangular diagrams (including π?a1 mixing e?ects) and has the form F (p2 ) = 1 mβπ p2 1 ? 2mβπ (p2 ? 2m2 ) 1? 2 2 + 2 π Zπ mρ (p ) p ? 4m2 π + 2m4 π I2 (p2 ) ?1 I2 (m2 ) π I3 (?p1 , p2 ) I2 (m2 ) π . (3.9)

Here p = p1 + p2 , with p1 , p2 the pion momenta. From this one easily deduces the following relation which holds o? the ρ mass shell F (p2 ) fρππ (p2 ) . = fρ (p2 ) Zρ In particular, at p2 = 0 the equality I2 (0) ? I2 (m2 ) ? m2 I3 (?p1 , p2 ) π π

p2 =0

(3.10)

= 2m2 π

?I2 (p2 ) ?p2

(3.11)

p2 =m2 π

leads to F (0) = 1 and as a consequence one has fρππ (0) = fρ (0)/Zρ . Let us consider now the properties of the axial-vector meson (a1 ). For this purpose we have to calculate the axial-vector form factor fa (p2 ) which describes the weak a1 → lνl matrix element. The relevant calculations give 1 2gρ (p2 ) J2 (p2 )[1 ? 2mβ(p2 )] = = √ fa (p2 ) 3 Za which leads to the result m2 (p2 ) 1 ? 2mβ(p2 ) a = , 2 fa (p2 ) 4Za GV 8 (3.13) Zρ 1 ? 2mβ(p2 ) , Za fρ (p2 ) (3.12)

where we used m2 (p2 ) a

m2 (p2 ) ρ = . 1 ? 2mβ(p2 )

(3.14)

The relations (3.5) and (3.13) lead us to the another matching condition which relates the properties of the a1 meson to pionic ones,

2 2 m2 = bfa fπ . a

(3.15)

The constant b is equal to b= Zπ (1 ? 2mβa ) . 2mZa βπ (3.16)

From the relations (3.6) and (3.15) one can deduce a set of equivalent formulae. In particular, one of the direct consequences are the well-known Weinberg relations [11]. These are known to play a crucial role in the calculation of the electromagnetic π + –π 0 mass di?erence in the chiral limit. An analogue of Weinberg’s ?rst sum rule is the equality m2 m2 βa 2 ρ a Za 2 + Zπ fπ = Zρ 2 . fa βπ fρ The second sum rule is described by the relation m2 (m2 ) ? m2 m4 m4 ρ a ρ ρ a . Za 2 ? Zρ 2 = fa fρ 4GV (3.18) (3.17)

The coe?cients Z, β and a non-zero right-hand side in (3.18) result from determining physical properties of particles at di?erent values of p2 corresponding to real values of their masses. In any approach where all physical characteristics of the mesons are ?xed at a certain scale these relations take the standard form (in this case Z = 1, βa = βπ , m2 (m2 ) = m2 ). O? the mass shell (3.18) is replaced by ρ a ρ the equality m4 (p2 ) m4 (p2 ) ρ a (3.19) Za 2 2 = Zρ 2 2 . fa (p ) fρ (p ) 9

4. Summary Let us brie?y summarize our results. We have considered an extended NJL lagrangian with four quark interactions. It includes some of the salient features of QCD in the low–energy (long distance) regime. We have generalized the bosonization procedure of Ref.[1] to the case with vector and axial-vector mesons. This allows to calculate any mesonic N–point function to all orders in momenta. In particular, we have considered some important relations connecting pion properties to ones of vector and axial-vector mesons. As speci?c examples we have shown that a generalized KSFR relation and generalized Weinberg sum rules are obtained from matching conditions relating the various sectors. These modi?ed relations stem from a consistent treatment of the higher order terms in momenta. To lowest order one recovers the celebrated results [10,11]. Finally, we note that similar issues were studied in ref. [12]. There, however, all physical properties of mesons were ?xed at p2 = 0, which leads to substantial di?erences in the ?nal results. REFERENCES [1] V.Bernard, A.A.Osipov and Ulf-G.Mei?ner, Phys.Lett. B285 (1992) 119. [2] T.Eguchi, Phys.Rev. D14 (1976) 2755; K.Kikkawa, Progr.Theor.Phys. 56 (1976) 947. [3] D.Ebert and M.K.Volkov, Yad.Fiz.36 (1982) 1265; Z.Phys. C16 (1983) 205. [4] M.K.Volkov, Ann.Phys. 157 (1984) 282. [5] A.Dhar and S.R.Wadia, Phys.Rev.Lett. 52 (1984) 959; A.Dhar, R.Shankar and S.R.Wadia, Phys.Rev. D31 (1985) 3256. [6] D.Ebert and H.Reinhardt, Nucl.Phys. B271 (1986) 188; Phys.Lett. B173 (1986) 453. [7] W.Pauli and F.Villars, Rev.Mod.Phys. 21 (1949) 434. [8] V.Bernard and D.Vautherin, Phys.Rev. D40 (1989) 1615; C.Sch¨ren, E.Ruiz Arriola and K.Goeke, Nucl.Phys. A547 (1992) 612. u [9] M.Gell-Mann, R.Oakes and B.Renner, Phys.Rev. 175 (1968) 2195. [10] K.Kawarabayashi and M.Suzuki, Phys.Rev.Lett. 16 (1966) 255; Riazuddin and Fayyazuddin, Phys.Rev. 147 (1966) 1071. [11] S. Weinberg, Phys.Rev.Lett. 18 (1967) 507. [12] J.Bijnens, E.de Rafael and H.Zheng, Preprint CERN-TH.6924/93 (1993). 10

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