A PURELY QUARK LAGRANGIAN FROM QCD

Transcription

1 Romanian Reports in Physics 69, 302 (2017) A PURELY QUARK LAGRANGIAN FROM QCD AMIR H. FARIBORZ a, RENATA JORA b a Department o Matemathics/Physics, SUNY Polytechnic Institute, Utica, NY 13502, USA b Horia Hulubei National Institute o Physics and Nuclear Engineering PO Box MG-6, Bucharest-Magurele, Romania rjora@theory.nipne.ro Received July 29, 2017 Abstract. We present a method or determining a purely quark Lagrangian by mocking up the QCD partition unction or large gauge couplings g. The resulting eective theory displays all the symmetries o low energy QCD and can be potentially used to explore hadron properties. PACS: Aw, Lg, Fe 1. INTRODUCTION Quantum chromodynamics is an SU(3) gauge theory which contains quarks and gluons that interact with each other. Whereas at high energies the coupling constant is small and one can adequately describe the reality by expanding perturbatively in the coupling constant it is still not completely clear how one can depict the lowenergy regime where the coupling constant is large and quarks orm bound states o mesons and baryons. Various methods and Lagrangians have been proposed to describe the low energy dynamics: While some o them like the Nambu Jona Lasinio model [1], [2] are still based on ermions as undamental degrees o reedom, most o them consider the mesons and baryons as the starting point. In the latter case one then constructs eective low-energy models which are widely based on the symmetries o the QCD Lagrangian like the SU(3) L SU(3) R chiral symmetry, U(1) V or U(1) A (as well as assumptions about QCD vacuum and potential) embedded in the quark lavor sector o QCD. Models o low energy QCD with notable results regarding hadrons properties and interactions include chiral perturbation theory [3],[4], linear and nonlinear sigma models [5]-[22] or other symmetry induced eective Lagrangians [23]-[32]. Thus one can conclude that chiral symmetry whether spontaneously or explicitly broken is the major ingredient or building a low energy QCD eective theory. In the present work however we shall consider a completely dierent point o view: we will not rely on symmetries but instead we shall build a purely dynamical

2 Article no. 302 Amir H. Fariborz, Renata Jora 2 Lagrangian and obtain the chiral symmetry as a derivative o the method. Our main tool is the QCD partition unction rom which one can derive all the properties o the particles and interactions. We start by considering an alternative description o the partition unction ater integrating out the quark degrees o reedom. As a consequence the original quarks will be replaced by copies (that describe the constituent quarks) with the same masses and quantum numbers but with dierent Lagrangian and thus interactions. In the end we shall obtain a Lagrangian where the gluon degrees o reedom have been eliminated and that contains only the quarks and their subsequent interactions. Note that our method implies only manipulation and calculations o the partition unction and does not involve any loop computation. We consider the eective ermion Lagrangian obtained in this paper as a purely theoretical one and as a irst step towards a more comprehensive approach. The connection with the phenomenology o the bound states o mesons and baryons will be perormed and inalized in a uture work. 2. THE QCD LAGRANGIAN AND PARTITION FUNCTION We start with the gauge ixed QCD Lagrangian with N colors and N lavors in the undamental representation: L = 1 4 (F a µν) 2 + c a ( µ µ g abc µ A b µ)c c + Ψ (iγ µ D µ m )Ψ, (1) where, and, F a µν = µ A a ν ν A a µ + g abc A b µa c ν, (2) D µ = µ iga a µt a. (3) Here as usual t a are the generators o the group SU(N) in the undamental representation. We shall ignore the ghosts in what ollows as they do not contribute essentially to our arguments. First we separate the Lagrangian in two pieces: L 1 = Ψ (iγ µ µ m + gγ µ t a A a µ)ψ L 2 = 1 4 (F a µν) 2. (4) The QCD partition unction has the orm: Z 0 = da a µ(x)d c b (x)dc d (x)d Ψ l xdψ l (x)exp[i d 4 xl], (5)

3 3 A purely quark Lagrangian rom QCD Article no. 302 where the usual rules o the path integration apply. Here is the lavor index whereas l is the color one. Then one can integrate over the ermion variables to obtain: Z 0 = da a µ(x) det [ iγ µ µ m + gγ µ t a A a ] µ exp[ d 4 xl 2 ]. (6) 3. A SIMPLIFIED APPROACH We shall discuss in particular the real lie case o N = 3, N = 3 corresponding to QCD with three light lavors. We introduce the ermionic current that couples with the gluon ield or each ermion species: J a µ = Ψ γ µ t a Ψ. (7) For one generation o ermions there are 8 3 degrees o reedom or each ermion where 8 represents space time degrees o reedom o an o-shell ermion and 3 the number o colors. Overall we have 24N degrees o reedom or all ermions. We consider a single lavor and next extend our arguments to N lavors. We start by making a change o variables rom the elementary ermion degrees o reedom to the composite current Jµ. a We irst mention that in the case o anticommuting variables the Jacobian appear with an inverse power as compared to the case o commuting ones. Note that or this change o variables to make sense we need 24 degrees o reedom or Jµ a instead o the 32 that one might obtain rom a simple counting. This means that we need something similar to a gauge condition which we choose to be µ Jµ a = ω where ω is an arbitrary unction as in the more standard case o a gauge ield. Since there are exactly 8 constraints one obtains the desired matching o 24 degrees o reedom or both the ermion and vector boson variables. Then the change o variable is: d ΨdΨ dj a µ dj a µ dψ i, (8) where J µ a is the subset o 24 components obtained rom the variables Jµ a ater applying the constraints. The Jacobian in Eq. (8) is a determinant with dimension 24. dj a µ dψ in Since any arbitrary derivative = Ψ jm (γ µ ) ji (t a ) mn (here the irst indices are space time whereas the second ones are color) contains a ermion variable then the determinant will contain products o 24 ermion variables. Taking into account that there are exactly 24 distinct ermion variables and the anticommuting nature o these we conclude that the actual determinant will be given exactly by the product o the 24 distinct variables (since those term that contain a repeated variable will be zero)

4 Article no. 302 Amir H. Fariborz, Renata Jora 4 times an irrelevant constant actor. Thus: dj a µ dψ i = const Ψ im. (9) i,m Then one can also write quite saely: dj a µ dψ i δ( µ Jµ a ω)det[t a γ µ Jµ] a (10) Here the determinant is taken in the space γ µ t a so it has the dimension 12 which leads exactly to a product o 24 distinct ermion variables. Note that at each point we take into account the composite nature o J a µ. Finally the complete change o variables takes the orm: d ΨdΨ dj a µδ( µ J a µ ω)det[t a γ µ J a µ] djµ a det[t a γ µ Jµ]exp[ i a d 4 x ( µ Jµ) a 2 ] (11) 2ξ The parameter ξ is arbitrary and we consider it very large so that the corresponding contribution in the exponential can be neglected. Note that the let hand side contains a product o measures over an even number o Grassmann variables which means that overall the let hand side is a real commuting quantity while the right hand side contains a product o measures over real variables since Jµ a is real. Equation (11) is valid apart rom a constant proportionality actor (which does not aect in any way the unctional calculus since in computing correlators in QFT one always divides by the partition unction). We need to extend our arguments to N = 3 lavors. By applying the previous approach or simply consider a product over the number o lavors we obtain: d Ψ dψ djµ a det[γµ t a Jµ a ]δ ( µ Jµ a ω). (12) It is more convenient however to write: det[γ µ t a Jµ a ] const det[ta γ µ J a µ ]3 (13) The above relation is due to the act that the above determinant contains a product o 72 ermion variables. But this is the total number o ermion variables so that each term that contains a variable twice is zero. Thus the determinant will be a constant times the product o 72 distinct ermion variables. But this is also what one would obtain rom Eq. (12) so Eq. (13) is correct. This being settled we need to write the ull partition unction in Eq. (6) in terms o the new variables Jµ a. As we mentioned previously the change o variables

5 5 A purely quark Lagrangian rom QCD Article no. 302 makes sense only with the additional constraint on the ield J a µ. We shall consider a particular case o it with ω(x) = 0. On the other side this constraint is equivalent to constraining the ree equation o motion o the ermion ield. Thus it can be ulilled only i, iγ µ µ Ψ mψ = a(x)2 Ψ(x). (14) M We use the unction a2 (x) M (where M is an arbitrary scale) instead o simply a(x) because we do not want to urther constrain the ermion ields Ψ. Then the kinetic term or the ermion ield will be replaced in the Lagrangian by: Ψ(iγ µ µ m)ψ = a2 (x) M ΨΨ. (15) Furthermore since the unction a(x) is arbitrary one can use the equation o motion to eliminate the kinetic term altogether rom the Lagrangian. The partition unction in Eq. (6) will thus become: Z 1 = da a µ djρ b det[ta γ µ Jµ a ]3 exp i d 4 x[g Jµ a Aaµ + L 2 ] (16) The same partition unction is obtained i one introduces three lavors o ermion copies χ, χ o the original quarks such that: Z 1 = da a µ djρ b d χ dχ exp i d 4 x 1 M 2 χ γ µ t a [ Jµ a ]χ exp i d 4 x[ Jµ a gaaµ + L 2 ]. (17) We make the change o variable Kµ1 a = J µ a, Ka µ2 = J µ2 a, Ka µ3 = J µ3 a to determine: Z 1 = da d σdkµ1k a ρ2dk b ν3 c d χ dχ exp i d 4 x 1 M 2 χ γ µ t a Kµ1χ a [ ] exp i d 4 x[gkµ1a a aµ + L 2 ] = da a µdkρ2dk b ν3δ c A aµ + 1 gm 2 χ γ µ t a χ exp[i d 4 xl 2 ], (18) where the integral over Kµ1 a is a delta unction. Here one can drop the unwanted

6 Article no. 302 Amir H. Fariborz, Renata Jora 6 integrals over Kρ2 b and Kc ν3 since they do not contribute in any process and apply the delta unction to obtain that the eective Lagrangian is just: L e = L 2 A aµ = 1 gm 2 χ γ µ t a χ (19) Although this procedure seems oversimpliying it gives a correct glimpse o what kind o Lagrangian we should expect in terms o the current J a µ i the ermion kinetic term is neglected. The next section will contain a more general and comprehensive approach. 4. A COMPREHENSIVE APPROACH We start with Eq. (6) which we rewrite here or completeness: Z 0 = da a µ(x) det [ [ iγ µ µ m + gγ µ t a A a ] µ exp i d 4 xl 2 ]. (20) We shall now try to reproduce the above partition unction using a dierent set o variables. We irst need an identity that we shall prove in what ollows. Consider the integral: Z x = exp[i d w i dw i dȳ j dy j dj k ds m d 4 x[ 1 M wak wj k + ȳb k ys k + M 2 k S k J k ]]. (21) Here w i, y i and their conjugates are each a set o n ermion variables (we include in n the space time components) and J k, S k bare a set o regular scalars with mass dimension 1. The index k goes also rom 1 to m and the matrices A k, B k are each a set o m (n n) matrices where A k have mass dimension 1 and B k have mass

7 7 A purely quark Lagrangian rom QCD Article no. 302 dimension 0. We will solve irst the integral over the ermion ields to get: Z x = dj n ds n det[i 1 A k J k ]det[i B m S m ] M n k m exp[i d 4 xm 2 J r S r ] r dj n ds n det[ 1 A k B m J k S m ]exp[i d 4 x M 2 J r S r ] M n k,m r dj n ds n d z i dz i exp[i d 4 1 x[ z[ M 2 A k B m J k S m ]z + M 2 J r S r ]] = n k,m r ds k d z i dz i δ(m 2 S k + z 1 M 2 A k B m S m z) k,i m ds k d z i dz i δ(s k (M 2 + z 1 M 2 Ak B k z) + z 1 M 2 A k B m S m z) k,i m k 1 d z i dz i M (22) za M k B k z 2 i k Here in the third line we expressed the determinant as a path integral over an extra pair o n ermion variables z i and z i and we dropped all unimportant constant actors. In order to urther process the result in Eq. (22) we consider a lattice in the coordinate Euclidean space such that: d 4 x = 1 Λ 4 x En, (23) where Λ is the cut-o corresponding to the lattice and x En goes over all coordi-

8 Article no. 302 Amir H. Fariborz, Renata Jora 8 nates and lattice points. Then: 1 M za M k B k z = 2 k,n exp[ k,n ln[m M 2 zak B k z]] = exp[ ln[m 2 ] ln[1 + 1 M 4 zak B k z]] = k,n k,n exp[ ln[m 2 ] 1 M 4 [ zak B k z] [ 2 M 4 zak B k z] ] k,n k,n k,n const exp[ Λ4 d 4 x E za k B k z] = M 4 const exp[i d 4 x k k za k B k z]. (24) In the last line o Eq. (24) we considered Λ = M and we neglected terms o order 1 and higher. We also expressed everything in the Minkowski space through the M 4 substitution x 0 = ix E0. Then rom Eqs. (22) and (24) we obtain: Z x = const d z i dz i exp[i d 4 x za k B k z] = const det[ia k B k ]. (25) Here we implicitly assume that repeating indices are summed over, convention which will be respected also in what ollows. Now consider again Z x and this time we integrate irst over J k and S k : Z x = d w i dw i dȳ j dy j dj n ds n exp[i d w i dw i dȳ j dy j const p n d 4 x[ 1 M wak wj k + ȳb k ys k + k dj p 1 M 2 δ(j p + 1 d w i dw i dȳ j dy j exp[ i M 2 S k J k ]] M 2 ȳbp y)exp[i 1 M wak wj k ] d 4 x 1 M 3 wak wȳb k y] const det[ia m B m ]. (26) The result in the last line o Eq. (26) will be o most importance in what ollows. There is an important extension to the results in Eqs. (22) and (26) which we will use here but state without proo because the proo is just a simple generalization

9 9 A purely quark Lagrangian rom QCD Article no. 302 o the arguments above: Z z = d w i dw i dȳ j dy j dj n ds n exp[i = n d 4 x[ 1 M wak wj k + ȳb k ys k + M 2 k d w i dw i dȳ j dy j exp[ i S k J k + T ( w,w)]] = d 4 x 1 M 3 [ wak wȳb k y]] const det[ia m B m ]. (27) Here T is any polynomial o the type k ( wc kw) k. This is true because both integrals over w and y must be satisied simultaneously and the number o variables w that are integrated in the ermion path integral must match the number o variables y. We thus start rom: Z x = d w i dw i dȳ j dy j dj n ds n exp[i = n d 4 x[ 1 M wak wj k ȳb k ys k + M 2 S k J k + T (w, w)]] = k d w i dw i dȳ j dy j exp[i d 4 1 x[ M 3 wak wȳb k y + T (w, w)]] const det[a k B k ], (28) and consider the set w i as being N N 2 ermions Ψ i, where N is the number o colors and N 2 is the number o lavors. Similarly the set y i is a similar set o N N 2 ermions χ i. In total we consider to have N = 6 lavors o ermions corresponding to the 6 lavors o quarks o the standard. We urther deine the matrices A k, B k as: A k = [i µ t a mt a γ µ + g A a µ] B k = γ µ t a (29) T = 1 m 2 ( 0 Ψ Ψ v 3 ) 2, (30) where v is a constant with mass dimension 1 and m 0 is an arbitrary scale. Then: A k B k == [iγ µ µ t a t a 4mt a t a + g γ µ t a A a µ] = [iγ µ N 2 1 µ 2N 1 4mN2 2N + g γ µ t a A a µ] = N 2 1 2N [iγµ µ 4m + gγ µ t a A a µ] (31)

10 Article no. 302 Amir H. Fariborz, Renata Jora 10 where we redeined g = N 2 1 2N g and 4m = m. We observe that the operator A k B k is the operator that appears in the standard model between two ermion states. Note that we can include in each A k and B k a diagonal lavor matrix with the same inal result. Then the counterpart o the Eq. (28) in terms o the above deinition will be: Z 0 = da a µ(x) det [ [ iγ µ µ m + gγ µ A a ] µ exp i[ d 4 xl 2 + ] d 4 xt ] = = exp 3 da a µ [ i[ d 4 xt ] 1 M 3 ] [ d Ψ dψd χ dχ exp i d 4 x d 4 xl 2 ] [ Ψ (it a µ mt a γ µ + g A a µ)ψ ] [ χ γ µ t a χ ] + =1. (32) Here M is an arbitrary scale that reestablishes the correct dimensionality. All constants m 0, v and M with mass dimension 1 are arbitrary and should be determined subsequently rom phenomenological arguments. We denote: Z 1 = exp [ i[ d 4 xt ] 3 d Ψ dψ d χ dχ 1 M 3 ] d 4 x [ Ψ (it a µ mt a γ µ + g A a µ)ψ ] [ χ γ µ t a χ ] + =1, (33) and work only with it. First we will make the change o variable Jµ a = 1 χ M 2 γ µ t a χ (see section II or details). Note that this implies a subsequent gauge condition or Jµ a which we

11 11 A purely quark Lagrangian rom QCD Article no. 302 shall discuss later. The partition unction Z 1 will become: 3 Z 1 = d Ψ dψ djµ a det[j µ a γµ t a ] exp [ i[ 1 M ] d 4 xt ] d 4 x Ψ (it a µ mt a γ µ + g A a µ)ψ J aµ +. (34) Noting that in the change o variable we can use det[ J µ a γµ t a ] 3 instead o det[j µ a γµ t a ] 3 (see Eq. (13)) we can urther write: 3 Z 1 = d Ψ dψ dψ djµ a det[ Jµ a γµ t a ] 3 [ exp i[ 1 d 4 x [ M Ψ (it a µ mt a γ µ + g A a µ)ψ ] J aµ + d 4 xt ] ]. (35) We can urther make a change o variables Yµ a = 3 J µ a, Za µ = Jµ2 a, U µ3 a = J µ3 a and drop the unwanted integrals over Zµ a and Uµ a as they would not contribute to any process. This yields: 3 Z 1 d Ψ dψ det[yµ a γ µ t a ] 3 dy a µ exp i[ 1 d 4 x M dy a µ [ Ψ (it a µ mt a γ µ + g A a µ)ψ ]Y aµ + d 4 xt ] = 3 d Ψ dψ d ξ dξ [ exp i[ 1 d 4 x [ M Ψ (it a µ mt a γ µ + g A a µ)ψ ]Y aµ + ] a ξ γ µ t a ξ Y aµ + d 4 xt ]. (36)

12 Article no. 302 Amir H. Fariborz, Renata Jora 12 Here we introduced another set ξ and ξ o 3 ermions to account or the determinant in the irst line o Eq. (36). Here a is an arbitrary dimensionless coupling constant. As we mentioned previously in order to be able to make a change o variable rom the ermions χ to the currents Jµ a one would need something similar a to gauge condition to cut o the number o degrees o reedom rom 32 to 24. We shall consider this constraint as µ Jµ a = 0. Upon the change o variables to Yµ a, Zµ a and Uµ a this constraint will become µ Yµ a = 0 or the only variable o interest. We shall introduce this in the partition unction Z 1 as: δ( µ Yµ a ) = ds a exp[i d 4 xm 1 S a µ Yµ a ] = ds a exp[ i d 4 xm 1 µ S a Jµ], a (37) where M 1 is an arbitrary constant with mass dimension 1. With the addition o the gauge condition the partition unction in Eq. (36) will become: 3 Z 1 dyµ a ds a d Ψ dψ det[yµ a γ µ t a ] 3 exp i[ 1 d 4 x M dy a µ [ Ψ (it a µ mt a γ µ + g A a µ)ψ ]Y aµ + d 4 xt ] = 3 d Ψ dψ d ξ dξ [ exp i d 4 x[ 1 [ M Ψ (it a µ mt a γ µ + g A a µ)ψ ]Y aµ + ] a ξ γ µ t a ξ Y aµ M 1 µ S a Y aµ + T ]. (38) We shall integrate Y aµ by observing that it couples only linearly that the Lagrangian is Hermitian and thus it leads to a product o delta unctions. This yields: 3 Z 1 = ds a d Ψ dψ d ξ dξ δ(g A a µ Ψ Ψ + +Ma [ Ψ (it a µ mt a γ µ )Ψ ] ξ γ µ t a ξ M 1 M µ S a ) exp[i d 4 xt ] (39) We shall regard the delta unction in the space o variables A a µ.

13 13 A purely quark Lagrangian rom QCD Article no. 302 Beore going urther we need to make an important amendment. When we introduce the change o variable rom χ, χ to Jµ a we replaced (we consider here or simplicity only one ermion species): d χ dχ djµ a det[γµ t a Jµ a ] (40) The reason stems orm the act that when we transorm rom one set o variables one gets always products o 24 dierent ermion components. However one can extend the above transormation to include some unction in the determinant. Consider that we have: dj a µ det[iγ µ µ m + kt a γ µ J a µ] (41) and we apply the inverse transormation to ermion variables χ : dj a µ det[iγ µ µ m + kt a γ µ J a µ ] = d χ dχ dj a µ dχ 1 det[iγ µ µ m + kt a γ µ χ γ µ t a χ ] (42) Moreover we also need to consider a unction (or one lavor) to be integrated which is o the type det[xµj a µ]. a Assume we express this in terms o ermion components. Those variables that repeat themselves will get canceled. In the end det[xµj a µ] a = χ 1 χ 2..χ 1 χ 2 where is an arbitrary unction and the product is over all 24 ermion components or one lavor (note that the product might contain derivative which we omit or simplicity). Then Eq. (42) will become: djµ a det[iγ µ µ m + kt a γ µ Jµ a ]det[xa µjµ] a = 1 d χdχ χ 1 χ 2...χ 1 χ 2.. [a 0 + a 1 χ i χ j +...a 12 χ 1 χ 2...χ 1 χ 2..] [ χ 1 χ 2...χ 1 χ 2..] = d χdχ[a 0 + a 1 χ i χ j +...a 12 [ χ 1 χ 2...χ 1 χ 2..]] = d χdχ[a 12 χ 1 χ 2...χ 1 χ 2..](43) But the end result corresponds to that part o det[iγ µ µ m + kt a γ µ Jµ a ] that contains only Jµ a so the addition we make to det[t a γ µ Jµ] a is irrelevant. The main point o the above discussion is that even when we integrate over Jµ a the intrinsic ermion nature o the variables should be considered. For most o the purposes we shall consider Jµ a however as a regular commuting variable. However our discussion has a counterpart by discussing purely the Jµ s. a This being settled one can add in Eq. (39)

14 Article no. 302 Amir H. Fariborz, Renata Jora 14 a kinetic term or the ermions ξ, ξ as in: 3 Z 1 = d Ψ dψ d ξ dξ ds a δ(g A a µ Ψ Ψ + [ Ψ (it a µ mt a γ µ )Ψ ] + Ma M 1 µ S a )) exp[i[ d 4 x[ [ ξ (iγ µ µ M )ξ ] + ξ γ µ t a ξ d 4 xt ]]. (44) The expression in Eq. (44) is the inal partition unction we will work with. It appears that the delta unction in Eq. (44) is badly deined. In order to ix this we go back to the ull Lagrangian o interest (including the pure gluon one) which is: L = [ ξ (iγ µ µ M )ξ ] + T + Jµ[g a A a µ Ψ Ψ + [ Ψ (it a µ mt a γ µ )Ψ ] + Ma ξ γ µ t a ξ M 1 M µ S a )] + L 2 (A a µ,g). (45) We irst recall that g = N 2 1 2N g and make the change o variables gaa µ A a µ. Note that we can do this at any time without aecting in any way the dynamics. This yields (we use or simplicity the same notation): L = [ ξ (iγ µ µ M )ξ + T + Jµ[ a N 2 1 2N Aa µ Ψ Ψ + [ Ψ (it a µ mt a γ µ )Ψ ] + Ma ξ γ µ t a ξ M 1 M µ S a )] + 1 g 2 L 2(A a µ), (46) where L 2 does not contain any more any trace o g. We are interested in the regime where g is large so we can consider 1 a small parameter. Next we take into account g 2 that one can solve rom the delta unction A a µ as unction o the other variables. With this substitution the Lagrangian becomes: L = L 0 = [ ξ (iγ µ µ M )ξ ] + T + 1 g 2 L 2( ξ,ξ, Ψ,Ψ,S a ) [ ξ (iγ µ µ M )ξ ] + T, (47)

15 15 A purely quark Lagrangian rom QCD Article no. 302 Assume we consider separately shits in the variables ξ, Ψ and their subsequent conjugates: ξ = ξ + 1 g 2 ξ ξ = ξ + 1 g 2 ξ Ψ = Ψ + 1 g 2 Ψ Ψ = Ψ + 1 g 2 Ψ. (48) Then one can still obtain the interaction Lagrangian o order 1 provided that the g 2 ields Ψ and ξ satisy the equation o motion or the ree Lagrangian L 0. Recalling the deinition o T rom Eq. (30) we get: T = 2 Ψ g m 3 Ψ g ( 0 Ψ Ψ v 3 ) = 0 (49) whose only non trivial solution is Ψ Ψ = v 3. Thus the unwanted denominator in the δ unction in Eq. (44) is conveniently ixed. We redeine 1 M = y 1 and 1 MM v 3 v 2 v 3 1 = z 1 v where y and z are two adimensional coeicients. From Eqs. (44) and (47) we obtain the eective ermion Lagrangian in terms o the two sets o ermions Ψ (the light quarks) and ξ (the heavy quarks): L = [ ξ (iγ µ µ M )ξ ] + 1 m 2 ( 0 Ψ Ψ v 3 ) g 2 L 2(A a µ(ψ,ξ,s b )), (50) where, A a µ(ψ,ξ,s b ) = 2N N 2 1 [ 1 v 3 Ψ (it a µ mt a γ µ )Ψ +y 1 ξ v 2 γ µ t a ξ z 1 v µs a ]. (51) We can urther reine Eq. (51) by taking A a µ equal to the real part o the right hand side which leads to the additional constraint on the ield Ψ : µ ( Ψ t a Ψ ) = 0. (52) In Eq. (50) the term L 2 is the pure gluon Lagrangian independent o the gauge

16 Article no. 302 Amir H. Fariborz, Renata Jora 16 coupling constant: L 2 = 1 4 F aµν F a µν. (53) Note that in Eq. (51) the presence o the scalars S a (a = 1...8) is purely optional as there are many ways in which one can implement this constraint. Also i M are considered large one can integrate out the ξ ermions this leading to an equivalent Lagrangian expressed only in terms o the light degrees o reedom Ψ. 5. DISCUSSION When one constructs an eective theory or the low energy degrees o reedom o QCD, be these ermions or hadrons, one uses oten as a guiding principle the approximate symmetries already established experimentally such as the chiral SU(3) L SU(3) R symmetry or the U(1) A axial one. In the present work we considered a dierent approach; thus instead o dwelling on symmetries we ocused on the intrinsic dynamics encapsulated in the partition unction o the QCD Lagrangian. Using alternative sets o variables we were able to reproduce the exact partition unction obtained ater integrating out the quark degrees o reedom. It turns out that a partition unction depending on a set o ermion variables can be described alternatively only by new sets o ermion variables, consider them copies o the irst ones, that however have a completely dierent Lagrangian. The inal result (see Eq. (50)) is an eective Lagrangian with unusual terms depending on two groups o ermions (where Ψ are the three light quarks and ξ are the three heavy ones) and with couplings which go up to 8 ermion interaction terms. This Lagrangian gives an adequate partition unction corresponding to that o the original QCD Lagrangian or the case when the coupling constant g is large. Moreover it has an intrinsic SU(3) L SU(3) R chiral symmetry in the two sets o ermion sectors or the case when the quark masses m and M are set to zero and also an SU(3) V symmetry or the case when the two sets o masses are equal within one set. The Lagrangian in Eq. (50) is o the same type as the Nambu Jona-Lasinio model [33], [34] which in its various versions has achieved remarkable success in describing not only the mass spectrum o the low lying scalars, pseudoscalars, vectors and axial vectors but also their main decaying modes. Both our model and the NJL one are nonrenormalizable models which enclose multiple ermion interactions but there are also important dierences. Our Lagrangian incorporates our, six and eight quark interaction terms whereas the NJL model stops at our quark terms. The our ermion interaction term contains derivatives that however can be reduced by using adequate techniques. Moreover the Lagrangian in Eq. (50) includes heavy

17 17 A purely quark Lagrangian rom QCD Article no. 302 quark states that can be integrated out leading to a more comprehensive eective Lagrangian in terms o the light quark states. The purely quark Lagrangian we obtained can be urther processed by using QCD rules to extract the bound states o mesons or baryons and their interactions. However the overall method goes ar beyond this. One could at any intermediate stage introduce directly instead o currents the meson states. In this case one must take into account that initially there are 72 degrees o reedom or the light quarks corresponding to a product o 3 colors, 3 lavors and 8 space time coordinates o an o-shell ermion. This means that the partition unction must accommodate only that number o mesons, be they scalars, pseudoscalar, vectors etc. whose total number o degrees o reedom sum up 72. All alternatives that conveniently express the initial Lagrangian should be considered and all Lagrangians that can be constructed in this way are possible outcomes. This is also true or the Lagrangian in Eq. (50) as it is only one o the many possible choices compatible with the initial QCD partition unction. In this paper we propose a new theoretical eective ermion Lagrangian obtained rom QCD not by integrating out the gluons but by eliminating these as a result o mocking up the exact partition unction o QCD. We shall leave the discussion o phenomenological implications o our model or urther work. Acknowledgements. The work o R. J. was supported by a grant o the Ministry o National Education, CNCS-UEFISCDI, project number PN-II-ID-PCE REFERENCES 1. Y. Nambu and G. Jona-Lasinio, Physical Review 122, (1961). 2. Y. Nambu and G. Jona-Lasinio, Physical Review 124, (1961). 3. H. Leutwyler, Annals o Physics, 235, 165 (1994). 4. H. Leutwyler, Phys. Lett. B 378, 313 (1996). 5. J. Schechter and Y. Ueda, Phys. Rev. D 4, 733 (1971). 6. H. Gomm, P. Jain, R. Johnson and J. Schechter, Phys. Rev. D 33, 801 (1986). 7. H. Gomm, P. Jain, R. Johnson and J. Schechter, Phys. Rev. D 33, 801 (1986). 8. F. Sannino and J. Schechter, Phys. Rev. D 52, 96 (1995). 137 (1997). 9. D. Black, A.H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D 58, (1998). 10. J.A. Oller, E. Oset and J.R. Pelaez, Phys. Rev. Lett. 80, 3452 (1998). 11. V. Elias, A.H. Fariborz, Fang Shi and T.G. Steele, Nucl. Phys. A 633, 279 (1998). 12. K. Igi and K. Hikasa, Phys. Rev. D 59, (1999). 13. D. Black, A.H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D 59, (1999). 14. D. Black, A.H. Fariborz, S. Moussa, S. Nasri and J. Schechter, Phys. Rev. D 64, (2001). 15. D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D (2000). 16. A.H. Fariborz, R. Jora and J. Schechter, Phys. Rev. D 72, (2005). 17. A.H. Fariborz, R. Jora and J. Schechter, Phys. Rev. D 76, (2007).

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