A Path-Integral Argument for the Triviality of the Thirring Non-Abelian Model in Four-Dimensions at Large Number of Colors

A Path-integral argument for the triviality of the Thirring Non-Abelian model in four-dimensions at large number of colors.

Luiz C.L. Botelho Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal Fluminense, 24220-140, Niterói, Rio de Janeiro, Brazil e-mail: [email protected]

Abstract: We argument the triviality of the chiral-vectorial SU(N) non-abelian thirring model at the t' Hooft large number of colors JV -► oo.

1 Introduction

One of the most interesting and conceptually important problem in Quantum Field Theory is to understand the triviality of quantum field theories as a "phase-transition" phenomena depending on external parameters, including the famous space-time dimen- sionality. It is even argued sometimes that does not exists non-remormalizable quantum field theories. What is really happening is the appearance of the Quantum Field Theory Triviality phenomena. However, there is some arguments on literature pointing out that through resummations-specially by means of the large N expansions - one could be able to make such non-renormalizable Field theories - like the Thirring fermion quantum field model to became non-trivial renormalizable ones ([1]). We aim in this paper to present an argument based on an approximate chiral path-integral bosonization and a non-abelian Stokes theorem for constant gauge fields to show that - unfortunatelly - such resummation

1 - renormalization phenomenon may not happens. This result is obtained through the analysis on the triviality of the attractive non-abelian thirring model at the context of a large number of colors. This study is presented on section 2. On section 3 we complement our previous studies by presenting a probabilistic - topological argument for the triviality by means of a Fermionic Loop space analysis and different on its nature from the large

Nc techniques employed on section 2.

2 The Path-integral triviality argument

We start our analysis by considering the chiral non-abelian SU(NC) thirring model Lagrangean on the euclidean space-time of finite volume Q C RA.

+ (|WT5(AAMC)2

Here (i^a,i(^ ) are the euclidean for-dimensional chiral fermion fields belonging to a fermionic fundamental representation of the SU(NC) non-abelian group with Dirichlet Boundaries condition imposed at the finite-volume región Q. In the framework of path integráis, the generating functional of the Green's functions of the quantum field theory associated with the Lagrangean eq.(l) is given by (¡zü = ij^d^)

,N2-N

1 0 ^ x exp ^* o 2 9 A b A x exp d x(^ ^(X )bc,f;cy(x n

x exp<-i d x(ifjar¡a + r¡aipa)(x) (2)

In order to procced with a bosonization analysis of the fermion field theory described by the above path-integral, it appears to be convenient to write the interaction Lagrangian in a form closely parallel to the usual fermion-vector coupling in gauge theories by making use of an auxiliary non-abelian vector field Aa (x), but with an purely imaginary coupling with the axial vectorial fermion current (at the Euclidean world).

2 2 1 „N -N „N -N 3 D A Z[Va,Va\ = ^TTTTrJ II D[M*Wa(x)] n T\ [ »\ Z^Ü)J „=1 J „=1 a=0

1 i 0 $ + igif, A Ut x exp d x(ipa,ipa [X (0 + igi5JL)* 0 \$ 1 Xexp< x A A x i"2 lQ ^ i l l){ )

x exp<-i d x(ifja r¡a + r¡a ipa) (x) (3)

At this point of our analysis we present our idea to bosonize (solve) exactly the above written fermion path integral. The maim point is to use the long time ago suggestion that at the strong coupling gbare —► 0+ and at a large number of colors (the t'Hooft limit ([2])), one should expect a greatly reduction of the (continuum) vector dynamical degrees of freedom to a manifold of constant gauge fields living or the infinite dimensional Lie algebra of SU(oo). In this approximate leading t' Hooft limit of large number of colors, we can evalúate exactly the fermion path-integral by noting that the Dirac kinetic operator in the presence of the constant SU(N) gauge fields can be written in the following suitable form

i r .,i ,, - , o v(cp) ^v(cp) exp d x(i(ja'i(ja) (4) V(

a V(cp)=exp[-gl5(A ^x,)Xa} (5) with the chiral SU(N) valued phase defined by the constant gauge field configuration

3 Note that due to the attractive coupling on the axial current - axial current interaction of our thirring model eq.(2), the axial vector coupling is made through an imaginary - complex coupling constant ig. Now we can follow exactly as in the well-known chiral path-integral bosomization scheme ([3]) to solve exactly the quark field path integral eq.(4) by means of the chiral change of variables

if)(x) = exp{-#75 ip(x)}x(x) (7)

i¡){x) =x(x)exp{-g^5ip(x)} (8)

After implementing the variable change eq.(7)-eq.(8), the fermion sector of the Gen- erating functional takes the a form where the independent euclidean fermion fields are decoupled from the interacting - intermediating non-abelian constant vector field Aa, namely ,N2-N

Z[VaJla] = Z((by / II D[Xa(x)]D[Xa(x)]

p+oo+oo N2N-N2-N , x / II d[Al]xexp\-- v>

1 xdet+ [(^ +^75 4)(^ + ^75/)* 0 0 A x exp<¡ -- / d x(xa,Xa 0* 0

A Í9 x exp | -i J d x(xae- ^^Va + V^-^^Xa) (x) J (9)

Let us now evalúate exactly the fermionic functional determinant on eq.(9) which is exactly the Jacobian functional associated to the chiral fermion field reparametrizations eq.(7)-eq.(8). In order to compute this fermionic determinant, £ndetj1[(^ + ig fi)(ft -\- ig $)*], we use the well-known theorem os Schwarz-Romanov ([4]) by introducing a cr-parameter (0 < a < 1) dependent family of interpolating Dirac operators

pW = (@ + tgA'^)=exp{-g(Tl5

4 Since we have the straightforward relationship £ p& = (-^^ pw+ pWi-g-yw) (ii) and the usual proper-time definition for the involved functionals determinants on our study, namely:

logdetí;1^^^*)

lim / ' liTrJe-'VW^) (12) e^0+ Je S one obtains the following differential equation for the Fermionic functional determinant

^{logdet+1(^)^)*)}

4 = 4 lim i / d xTrF ig^5(p x exp(-£ pW pW) 1 (13) where Trp denotes the trace over the color, Dirac and space-time índices. At this point we note that the diagonal part of exp(—e p^ p^*) has a well-known asymptotic expansión in four-dimensions ([4]) (where oiW = ^(7^7^ — 7^7^))

exp(-£ 0W pW) = ¿ {¿ + -£(F^(aA)a^Xb)

(14)

After substituting the Seeley-Hadamard expansión on eq.(14), taking into account eq.(6), the fact that Tr Dirac (75) = 0 and Tr Dirac (75 a^) = 0, one obtains that the only possible non-zero term in our evaluations would be given by the expression '^16{(//^B>"

x (

By supposing explicitly space-time symmetry of the finite-volume región Q, one has that the "symmetry integral" vanishing holds true

d4x -x^ = 0 (16) n As a consequence, we get the somewhat expected result that the Fermion functional determinant on the presence of constant gauge external fields coincides with the free one, namely:

det; (0 + ig4)(0 + ig4y det/ mw i ;i7) since VF[A"] = 0 on basis of eq (16). Let us return to our "Bosonized" Generating functional eq.(9) (after substituting the above obtained results on it)

. N2-N Z[Va a] = ^ ^0) I II D[Xa(x)]D[xa(x)]

2 +OQN -N , . d ex vo1 x y J] K\ P { + 2 (V)Trsu{N)(A,y j

0 0 x exp ^ x\Xai X, 0* 0

4 A x 1 x exp <^ -i / d x(xae fl75(A-A0)x^o + r¡ae-™( 1 *>' Xa){x) (18)

Let us argument in favor of the theory's triviality by analyzing the long-distance behavior associated to the SU(N) gauge-invariant fermionic composite operator B(x) = ipa{x) ij)a(x). It is straightforward to obtain its exact expression from the bosonized path- integral eq.(18)

B{x)B{y)¡ (0) X G X 19 (Xa(x)Xa(x))(Xa(y)Xa(y))y ' (( ~ V)) ( ) here the reduced model's Gluonic factor is given exactly in its structural-analytical form by the path-integral (without bothering us with the 75-Dirac indexes)

N2-N 1 +00 ( G((x - y)) JI d[A;} exp +- vol (n)Trsu{N)(A, G(0) a=l

x Trsu{Nc)F I exp-g ¿ Aadx0 (20) with Cxy a planar closed contour containing the points x and y and possesing an área S given roughly by the factor S = (x — y)2. The notation ( )(°) means that the Fermionic average is defined solely by the fermion free action as given in the decoupled form eq (18). Let us pass to the important step of evaluating the Wilson phase factor average eq (20) on the coupling regime of gbare —► 0+ and formally at the limit of t'Hooft of large number of colors N —► oo. As the first step to implement such evaluation, let us consider our loop Cxy as a closed contour lying on the plañe /i = 0, v = 1 bounding the planar región S. We now observe that the ordered phase-factor for constant gauge fields can be exactly evaluated by means of a triangularization of the planar región S, i.e M s = \J A2 (21)

1=1

Here, each counter-clock oriented triangle A¡¡¿ is adjacent to next one A¡Zl fl A¡¿, ' = conimon side with the opposite orientations. At this point we note that 9¡ Aa dXa 1} 2) 3) f{e~ 4l ' j g* e-gAa-£Í . e-s^-¿ . e-gAa-eí ^

are where {¿a } 1=1,2,3 the triangle sides satisfying the (vector) identity va +£a +£a = 0. Since we have that n , ¥{e-9§{x)AadXa} = Um TT p{e-^A«^^} (23)

í=l and by using the Campbel Hausdorff formulae to sum up the product limit eq.(23) at g2 —► 0 limit with X and Y denoting general elements of the SU(N) - Lie algebra:

ex.ey = eX+Y+\[X,Y]+^g2-) (24) one arrives at the non-Abelian stokes theorem for constant Gauge Fields

AadXa F 1(3 f>le-9lcxy \ = p|e-9//s- 01^° h

2 = p{e+(s) [^o,A1]-S| (25)

7 As a consequence, we have the effective approximate result (formally exact at N —► oo) to be used in our analysis that follows

+ 0(i) (26)

Let us now substitute eq.(26) into eq.(20) and taking into account the natural two- dimensional degrees of freedom reduction on the average eq.(20)

2 1 r+00N -N . Gttx-y» = 7^ II d^ d^ exP +9 vo1 (fi) Trm*)(A¡ + A\) L CT^UJ j-oo a=1 L ¿ 2 x exp UÜlt(Trsu(N)[A0, A,]) ) (27) where G(0) is the normalization factor given explicitly by

a a 2 2 G(0) = JU d[A MA 0] exp |-- vol (fi)[(A«) + (A?) ] j (28)

By looking closely at eq (27)- eq(28), one can see that the behavior at large N of the Wilson phase factor average is asymptotic to the valué of the usual integral

(TrSU(N)(\ah = Sab)-

G((x-y))N>>1 < da exp < — vol (Tí)a >

(g2s)2 x exp <^ — ———a

N2-N x ( í da exp | -- vol (Vl)a21 ) l (29)

By using the well-known result (see ref [5] - pag 307, eq(3). 323 -3)

J™ exp(-/3V ~ 2j2x2)dx = 2-1 (l} eW Ki í^\ (30) we obtain the closed result at finite volume

G((x - y))N»i ~

x e N

(31)

Let us now give a (somewhat formal) argument for the theory's triviality at infinite volume vol(íl) —► oo. Let us firstly define the infinite-volume theory's limit by means of the following limit

vol (Q) = S2 (32) and consider the asymptotic limit of the correlation function at \x — y\ —► oo (S —► oo). By using the standard asymptotic limit of the Bessel function

lim Ki (z) ~ e~z J — (33) one obtains the result ( lim g2 N = g2^ < oo) in four dimensions

N2-N N 16vr 2 2 ^^te^i-i„,, ^ ,. )1' ^4- ^/ 6 1g6^ l (33) \x _ y|4(W2-W)

So, we can see that for N big, there is a fast decay of eq.(19) without any bound on the power decay. However in the usual L.S.Z. framework for Quantum Fields, it would be expected a non decay of such factor as in the two-dimensional case (see eq (33) for vol(íl) = S), meaning physically that one can observe fermionic scattering free states at large separation. Naively at N —► oo where we expect the full validity of our analysis, one could expect on basis of the behavior of eq.(33) the vanishing of the above analyzed fermionic correlation function eq.(19) as faster as any power of \x — y\ for large |rc — j/|. This

9 + result would imply that g\aTe ~ 0 may be zero from the very beginning and making a sound support to the fact that the chiral thirring model - at g2 ~ 0+ and for large number of colors - may still remains a trivial theory, a result not expected at all in view of previous claims on the subject that resummation may turn non-renormalizable field theories in non-trivial renormalizable useful ones ([1]). Finally as a last remark on our formulae eq.(31)-eq.(33), let us point out that a math- ematical rigorous sense to consider them is by taking as the Eguchi-Kwaia continuum space-time Q, a set formed of n hyper-four-dimensional cubes of a side a - the expected size of the non-perturbative vaccum domain of our theory - and the surface S being formed, for instance, by n squares on the Q plañe section contained on the plañe /i = 0 , v = 1. As a consequence of the construction above exposed, we can see that the large behavior is given exactly by (N = Nc)

(N2n2a8 JV 32 (al,na2^ n—>oo G(na)N>>1 "~^ t —2 ^ ^

T\T2Í 2 8\ ^ N2-N xKi 4 \16(5f^)2n2a4 N2-N (34) na4

3 The Loop Space Argument for the Thirring Model Triviality

In order to argument the triviality phenomenon of the SU(N) non-abelian thirring model of section 1 for finite N, let us consider the generating functional eq.(3) for vanished fermionic sources Tja = r¡a = 0, the so-called vaccum energy theory's content

2 f.N -N 3 z(o,o)= / II UDK(x)]e~ -ynd*x(A*A*)(x) a=l fj,=0

x detF[(^ + ¿075 4){0 + iglb /)*] (35)

At this point of our analysis, let us write the functional determinant on eq (35) as a

10 functional on the space of closed bosonic paths {XM(

= E 1 Fsu(N) ' P Dirac exP ~ 9 f A^Xp^dX^a) Cxy { cxy

l a íi + -b n ]§FaP(Xp(a))ds] l (36)

The sum over the closed loops Cxy with fixed end-point x^ is given by the proper-time bosonic path integral below £ = f % Id^ I DFÍX^

Cxy JO -L J JXM(0)=XM=XM(T) xexpj-^ X\a)da)\ (37)

Note the symbols of the path ordenation P of the both, Dirac and color indexes on the loop space phase factors on the expression eq.(36).

By using the Mandelstam área derivative operator 5 /5 alp(X(a)) ([6]), one can re- write eq.(36) into the suitable form as an operation in the loop space-with Dirac matrizes bordering the loop Cxx, i.e.:

tg detF [(0 + ¿075 4) (0 + ¿075 /) *]

FSU{N) [exp(-0 / ^(Jfy^dX»)] | (38)

In order to show the triviality of functional fermionic determinant when averaging over the (white-noise!) auxiliary non-abelian fields as in eq.(35), we can use a cummulant expansión, which in a generic form reads off as

f (e )A, = exp |(/)^ + \{{P)A^ - (f)^) + • • • } (39)

11 So let us evalúate explicitly the first order cummulant gP"-{ld"5^^lOT))

x (FSu(N)[exp(-g l All{Xp{a))dXiJ{a)\) (40) x A Jcxx ' ^ with the average ( )A defined by the path-integral eq.(35). By using the Grassmanian zero-dimensional representation to write explicitly the SU(N) path-order as a Grassmanian path integral ([6])

FSU{N)[exp(-g í A^{Xp{a))dX^a))} J ^xx N2-N N2-N F F I] D [6a(a)]D [e:(a)}(Y^ ^(O)C(^)) a=l a=í T í • r N2~N ( ñ ñ xexpUjf da ^ U(a)-9:(a)+9:(a)-9a(a)

xexpíffjí da(A;(XP(a))(6b(\a)bc6*c)(a)dX»(a))) (41) one can easily see that the average over the A^(x) fields is straightforward and producing result the following self-avoiding loop action

Fsu(N)[exp(-g

ewí¡l da J^ \ea(a)j-6:(a)+6:(a)±6a(a)

exp{|-y da j da>[(6b(\a)bc6*c)(a)(6b(\a)bc6*c)(a>)

x ¿(D)(^» " X^dX^dX^)} (42)

At this point one can use the famous probabilistic - topological Parisi argument ([7]) well-used to understand the A

12 due to the fact that Hausdorff dimensión of our Browmian loops {XM(

(¥su{N)[exp(-g (f A^Xp(a)dX,(a))\) =1 (43) x A Jcxx ' v which by its turn, leads to the thirring model's triviality for space- time RD with D > 4.

Acknowledgments: We are thankfull to CNPq for a grant (until February of 2004 and not renewed from then on).

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