Infrared Behaviour of Yang-Mills Theories

Infrared Behaviour of Yang-Mills TheoriesAuthor(s): C. Nash and R. L. StullerSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 78 (1978), pp. 217-233Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20520729 .

Accessed: 12/06/2014 15:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Royal Irish Academy is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Irish Academy. Section A: Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 91.229.229.44 on Thu, 12 Jun 2014 15:30:13 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ria

http://www.jstor.org/stable/20520729?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


1 217 ]

22.

INFRARED BEHAVIOUR OF YANG-MILLS THEORIES

By

C. NASH School of Mathematics, Trinity College, Dublin

and

R. L. STULLER

Department of Physics, Imperial College, London

(Communicated by L. S. 0 Raifeartaigh, M.R.I.A.)

[Received, 24 JANUARY 1977. Read, 13 FEBRUARY 1978. Published, 22 DECEMBER 1978]

ABSTRACT

A non-perturbative method for treating the infrared structure of Yang-Mills theories is presented. The starting point is an ansatz for P the generating function for one particle irreducible gluon Green's functions. This leads to a decoupling of gluons from the three and four particle on shell S matrix elements and possibly a complete absence of gluon coupling to on shell matrix elements. In these circumstances Fer

mions can also be confined. A pair of coupled equations are obtained whose solution can derive the conditions under which the ansatz gives the desired confinement.

I. Introduction

This paper is an assault on the infrared structure of the non-abelian Yang-Mills

field. It treats, in the main, the Yang-Mills field coupled only to itself. However, we

shall refer to the theory where Fermions are also coupled to the Yang-Mills fields.

The topic of Fermions will be treated more fully in a subsequent paper by one of us

[1]. Recently there has been interest in calculating the infrared structure of Yang-Mills

theories both by perturbative [2, 3] and non-perturbative methods. The perturbative

approach of reference [2] establishes to the lowest non-trivial order in perturbation

theory the finiteness of certain transition probabilities, just as in QED. While the

non-perturbative approach of reference [3] argues that the infrared structure causes

cross sections for the production of soft gauge quanta vanish. It is to this latter

approach that our results are allied, although the methods are entirely dissimilar.

It is convenient at this point in the introduction to describe the difference in starting

point of our work from that of references (2, 3, 4]. In the latter mentioned works the

Feynman diagrams of the basic Lagrangian is the starting point, and the Green's

functions are calculated by summing finite or infinite numbers of diagrams respectively. In this paper the starting point is F(B?,(x)), where BI,,(x) is the Yang-Mills field and

r is the generator of one particle irreducible Green's functions, the effective action.

No Feynman diagrams are used, to calculate a Green's function one functionally

differentiates r the appropriate number of times with respect to B0,(x).

PROC. RI.A., VOL. 78, SECT. A [22]



218 Proceedings of the Royal Irish Academy

The structure of our calculational procedure is as follows: firstly, the target is to construct r(Br,(x)) which although not the exact F(B a(X)) contains all the infrared information about the Green's functions. This construction begins by making an infrared ansatz [5] for the Yang-Mills 2 point function <01 T(BaI(x)Bbv(O))10>. This ansatz is then incorporated into 1(Baa(x)), r is then examined for the constraints induced on it by the Ward-Slavnov [6] identities F is then parameterised so as to satisfy these constraints and this is found to imply a structure for all the higher

Green's functions <01T(Ba1 1(x1) ... Ban-,(xu)I0> n = 2, 3, . .., without having to make an ansatz for them separately. The resultant expression for F has two parameters which occur as unknown functions of the coupling constant g, these are called A(g), associated with the gluon sector, and h, associated with the ghost sector. These two functions are determined by solving two coupled Schwinger-Dyson equations: those for the gluon and the ghost self-energies. In this model we circumvent a common difficulty associated with attempts to use the Schwinger-Dyson equations to obtain properties of Green's functions. It is simply that the kernels that appear in our equations are all known since they are defined in terms of functional derivatives acting on r, which we have constructed already. Usually this is not the case, and the

kernels have to be dealt with by educated guessing since the only way of formally

determining them is by solving further Schwinger-Dyson equations with yet more unknown kernels. With this model we find it a straightforward but computationally lengthy task to calculate gluon Green's functions. In general we have calculated the

Green's functions for three and for four gluons interacting. These allow us to calculate

expressions for the decay A -B+ C (1 1)

with A, B and C gluons, and also the forward two to two amplitude

A+ B -*A + B. (1.2)

These expressions are found to be zero for on shell gluons (i.e. zero mass gluons); they are not zero, of course, when the particles are off shell. This we claim as a success

in showing the absence of such zero mass particles from the spectrum of existing particles. The burden of computation has prevented us from calculating the ampli tudes for gluon multi-particle processes in general, but in a more specialised case we

can provide information about these multiparticle processes and find these vanish also, where we have calculated them. The obvious conjecture is that all the on-shell S matrix elements for gluon scattering vanish, but we have not yet proved this, though we regard our results as optimistic, positive evidence for this attractive conjecture. The model is made more attractive by the property that when the functions 2 and h

referred to above are such that the various gluon S-matrix elements vanish, then if we couple Fermions to the gluons these Fermions will have infinite mass, this calculation is done in reference [5].

In section II we give the construction procedure for our ansatz for the effective action r(Ba,(x)). Section III is devoted to the gluon scattering amplitude and their properties. Finally, section IV discusses the coupling of Fermions to the Yang-Mills gluons and makes some general concluding remarks about the uses of weakly- and strongly-coupled Yang-Mills gluons and about the present work.



NASH AND STULLER-Infrared behaviour of Yang-Mills theories 219

II. The infrared structure of the effective action

The first thing that we wish to do is to parametrise the growth of the gluon propagator near its mass shelf. The Slavnov (6] identities allow us to write for the general propagator

G ab4V(p,) = (gv 2 + 4 d8ab (2.1)

All the dynamics is contained in the function dab(p), in 2.1 a and b are the group indices and 4 is the gauge parameter. We expect dab to be singularp -+ 0 and write for

it [51

dab j MA ij) 3ab(

In the above A(g) is the function referred to in the introduction, M2 is the mass introduced so that Gtab "(p) may be normalised at the space-like point p2 = -M2. The extraction of the factor 8ab iS our incorporation of the requirement that we maintain exact gauge invariance throughout and are dealing with a symmetric theory. This power law growth of dab is not expected to be the actual analytic behaviour of dab near p = 0 but may be modified by more slowly varying functions like logarithms, etc. We have not allowed though for essentially singular behaviour, but even if this

were present it might even strengthen our results [4]. Having made the ansatz 2.2 the next stage is to incorporate it into the effective action. To this and consider the effective action for these diagrams this is well known to be given by the spacetime integral of the Lagrangian L (7, 8, 9].

F f4dxL (2.3) where

L = 41FaUVFPv-. (8zAax)2 + 1/ DPabCb (2.4)

In 2.4 we have included* the standard gauge fixing term 1/24 (8aAMa(x))2 and Fadeev Popov ghost term ela a, DpabCb. The tensor F;a and the covariant derivative Dabs stand for the expressions

F;a, = -A9 -Aaig + igTa%cAb AC

Dab = E 3ab _ igTe0 Ae7 . (2.5)

TabC is simply the structure constantfabe of the "colour" group, it is written in this form to remind the reader of its properties as a representation matrix. Now to obtain the free particle propagator Gab v (p, q,free) from Ttree we simply use the property

[7, 8, 9] that

(27r)4 J(p + q)frab v(pq) q dx dy e2 tree f- xdePeq 3A ag(x) t5Abv,(y)

= -(2)4 3(p + q){Gab,,v (p, q,free)>-1. (2.6)

* The ghost term does not in fact contribute any thing to the tree approximation but we have included it for convenience.




A simple alteration to rtree however, will produce, instead of G ab,v (p,q, free), the

Gabt (p,q ) given in 2.1, 2. We call this rtree and write

"tree = 5 dx[ Fa'v(x)(y)) F% zv(x)-T, + Aa a DabCb (X) . (2.7)

Equation 2.7 has two drawbacks: the first is that it does not satisfy the constraints

induced by local gauge invariance; the second is that it is a polynomial of degree 4

only in A?a and hence gives zero for the irreducible 5, 6 ... point functions. It turns

out that both these drawbacks are removed by removing the first one. To this end we

begin by dealing with the gauge structure of F. Due to the appearance of the field

Aa ,(x) in the expression for Dab, in 2.5 the transformation properties of Green's

functions and, a fortiori, generating functions such as F, is quite complicated and produces in general non-linear relations between the function being transformed and its transformed partner. We wish to state in differential form the content of the

constraints induced by local gauge invariance on r. There are various ways of

treating the gauge properties of Yang-Mills theories and to refer to them all is

unfortunately impossible [6, 7, 8, 9]. A useful starting point for a discussion of these gauge properties is reference [10] and references contained therein. We wish to make use of two [11, 12] papers which modify the equations of reference 6 in a form more

readily used by us. To state the equation that we shall use we need to establish some

standard notation for generating functions. The generating functions Z and W are

defined by

Z[J] = I9GA dc dc exp S dx {4Fav(x)Fva 2 A) a Mabcb ja (x)Aa(X)

and ln Z[J] = iW[J]. (2.8)

In 2.8 @A, dc, di are suitably defined measures on the space of fields AaA(x), c(x) and

c(x); and Mab = a Dyab (2.9)

To define F, the generator of one particle irreducible graphs, we introduce the field Ba(x) given by,

- W Ba (x) = 6Ja(x) (2.10)

and F is then defined by the equation

rI[] = W[J] + I dx J"a(x)B#a(x) (2.11)

so that

6Ba(X) = Ja(X). (2.12)

To derive the desired equation for F one first works with W[J], and examines the

behaviour of equation 2.8 when a local gauge transformation is performed:

Aax(x) AaI(X) + DabIob(x) (2.13)




where wb(x) is an infinitesimal gauge parameter, noting that since 2.13 is simply a change of variable in 2.8 then

a6W[JJ , -- = 0. (2.14)

After judicious work with equations 2.10, 11, 14 one finially derives the required equation for r In reference 8 it takes the form

L B] o

-=0. (2.15)

Where we have suppressed group and tensor indices and Fo is related to F by

dx uaX2

ro[B] = r[B] + f d (O,BMa(x))2. (2.16)

The differential operator L[B] has the form

L[B] = Dab [B] + yab [B] (2.17)

where yab2 is a functional containing in its definition a generating function for generating proper vertices with two ghost lines and an arbitrary number of B'a? lines.

The form of our required equation for r in reference [9] has a similar form to that

of equation 2.15. However, the proper vertices for the ghosts are more explicitly dealt with by introducing sources Ka and K' for the ghosts and conijugate fields, in

analogy to B?A, C&' and co. Equation 2.15 is replaced by

Dab [B] -F _j &aiyal a2b coba2i = 0 (2.18)

where it is easy to see that yabcA is a generating function for proper vertices with two

ghost lines and an arbitrary number of B", lines. We have now reached the point where

we may return to our original problem. A few remarks are now in order: the quantity

f dXFAh,(X)F,,,a(X) is gauge invariant and satisfies

Dab 6Bb(X-) I dx F,. FAVC

e 0. (2.19)

However, this is not true of the expression 0 2 gAg)

S dxPFv ? mi Fva

this results eventually in Ftree of equation (2.7) not satisfying the requirements of

gauge invariance, i.e. equation (2.18). But if we alter the expression to

I dxFa;V(X) ( ; FG;)

where D2 is the square of the covariant derivative, then this expression is invariant

under infinitesimal gauge transformations. Indeed we can go further and use the more

general expression f dx Fag d-lab (-D2) FPVb, where dab(D2) is obtained by Fourier transforming the dab(p) defined in equation (2.1), and replacing the space-time




argument x2 by the covariant differential operator -D2. We now write for r the equation

F =JS dx[Fnv(x) d-lab(-D2)Fnvb- g (a,B%a)2 + zja a Dgabh-l(_D2)WbI (2.20)

where nothing is assumed about the properties of the function h-' in the ghost term.

It is now straightforward to check that (2.18) is satisfied provided h and yab, are identified by the equation

yabe = 6Bb c (x) aA DAUbh-/(-D2) (2.21)

this indeed we do. To return to a concrete form for d'ab i.e.

7D2, A(g)

d-laob - i '5mab (2.22)

we see that because D2 is a quadratic in B then d'ab may be expressed as an infinite

power series in B with each coefficient giving a one particle irreducible of Green's

function. Thus we have a full structure of irreducible vertices and the second drawback

mentioned above is removed as promised. Furthermore, we see that the extremely

powerful non-linear structure of the gauge transformations of the Yang-Mills theory are such that, to make a consistent gauge covariant ansatz for the two point function,

we are automatically committed to a predetermined structure for the higher n point

functions. Thus the burden of gauge invariance is seen to be a virtue in this case. This

leads us to our next important remark which is that (2.20) can never be a non-trivial

F for an abelian gauge theory. For in an abelian theory the covariant derivative is

simply the partial derivative 8,n and because it does not depend on the field Bn; then

(2.20) would only yield an expression for the vector two point function, all the higher vector proper vertices would be zero. This property we also regard as a virtue, because the intention is to use the theory to implement the infrared slavery program

[13] of quark confinement, and this gives us a firm reason for having to use a non

abelian gauge theory rather than some other sort of theory. Finally, we wish to draw

attention to the fact that (2.20) may well be of use to study problems other than those

of the infrared structure. The reason is that one computes all Green's function in

terms of two unknown functions dab and h. Further, dab and h may be found by

realising that the Schwinger-Dyson self-energy equations for the vector and the ghost self-energies form a coupled system of two equations for the two unknown functions

dab and h. The kernels in these self-energy equations which are usually not known but

are treated by educated approximations, are simply determined in terms of dab and h

and their derivatives. This is, of course, because the kernels are certain proper vertices



NASH AND STnLLER-Inf rared behaviour of Yang-Mills theories 223

which may be calculated by functionally differentiating r. To show the structure of these coupled equations and their kernels we write them below :*

-1 -.1

2! 31 W %

2.23 (a) -

/Y\Avc: V/7__ wwv_

- - ---- -----

2 23 (b)

The graphical notation in equations (2.23) is as follows: the wavy line denotes the vector particle and the dotted line the ghost. Insertion of an unshaded circle in a line denotes an exact propagator, while a shaded circle with various external lines means that the Green's function is one particle irreducible, i.e. a proper vertex. A black dot

denotes a bare vertex. Equation (2.23a) contains d 1ab' its first two derivatives and h-1, while (2.23b) contains d -'ab and h ', the solution of this coupled system would determine d -ab and h as functions and their dependence on g and the representation

structure. It is not our purpose here to pursue this general programme but rather to

investigate the infrared behaviour, therefore we choose for d- l.b the specific form given

in equation (2.22). Here the unknown is the function A(g), hence the Schwinger-Dyson equations determine 2 and h-' as functions of g. With reference to the prospect of

confining quarks, conditions are described in reference [5], under which use of the propagator of equations (2.1), 2 in the Schwinger-Dyson equation for the Fermion self-energy, gives an infinitely massive Fermion. The main condition, apart from various technical considerations, is that A(g) > i (it is for these values of A that the static potential calculated by Fourier transforming Gab,v (p, -p) into coordinate space give a rising potential at spatial infinity). Thus we expect there to be a value of g

which when reached by, say, increasing g from 0 causes A to exceed 1. This value of g

should be in the strong coupling region appropriate for the strong interactions. However, since A(g) is ultimately to be determined by the Schwinger-Dyson equations we can find the values of g for which A exceeds j by this method. The use of the

Schwinger-Dyson equations to do this has not yet been done. However, it can be shown that the equations can probably be uncoupled and replaced by one equation [141. A successful solution, or approximation method for obtaining the solution, should contain ultraviolet approximations for the propagators and kernels since the

1 * In general there is also a tadpole-like term:

but because we assume that we are dealing a symmetric solution to the theory this term is zero.




loop integrals in the Schwinger-Dyson equations are over all momentum space. For example an expression for dab containing a sensible ultraviolet approximation is

dab (p) = (P j() aab (2.24)

In a more sophisticated attempt one can exploit the asymptotic freedom of the theory to derive better ultraviolet approximations and use these. This has not been done yet.

We now proceed to the next section where we present the calculations of the proper

vertices.

IMI. The on-shell gluon scattering amplitudes

To begin with we must establish our notation. The general one particle irreducible Green's function is written as

F a,

n

.

l . an

D Xn) !t'.I In

and its definition is

r1 a, ( - -n) ti(xl) x.. . ........... bBnzBXj .. (3.1)

With B,, and wa set equal to zero. Its Fourier transform

fal.af (P... *PnI)

is defined by

IJdx1 ..dx eiPIX1. eX a. ...a

a (x . .. x,) = (2) 6(p1 ... Pn)

'll1...jL-n

(3.2)

To obtain S matrix elements from the

Fy a,..u (Pi *Pn)

we require the connected Green's functions in momentum space from which the S matrix elements are obtained. The way to do this, as is well known [15], is suc cessively to differentiate equation (2.11) with respect to Ba,,(x). This analyses the one particle irreducible structure and relates the

r atthcnet aG (Pien' fupn

to the connected Green's functions.




We can now begin the calculation of S matrix elements starting with the three particle decay shown in Fig. 1.

p +q

Fig. 1.

P q

2 2 2 (p+q)

_ p = q =0

a33

3Bcv(x3) 6Bbi(x2) 6BCA(x3)

the quantity we have to calculate, has three sorts of term contributing to it. To

identify these terms we define

d -tab; a( x . * x = n d-lab(_Dz(x)) pi .. P. X; XI... .)

=Bal ,(xl) .. 6B.. . d Xn) (3(3)

evaluated with Bay = 0. We can then classify the terms which contribute to

3rabc Av (X1, x2, X3)

into three kinds. These are:

EcbaV (X3, X2, XI) = dx bIJCx) d-' aa2(X) b ( (x) (3.3a)

f 62FaIA2() ala2 6ra122a2 (X) F )ba dx FAIA2(X) d- 2 FAAX) (X (3.3b) FVIA (X3, X2, X1) - d BCv(x3) 3Bb(x2) ) 6Ba(xI)

cba 6'P'1110 - _a_a2; 6FZIlAZa2(X) G VpA (X3, X2, XI) = - dx 3;%j) d 1aja2,b(x,x2) 6Ba (X,) (3.3b)

where Ba (x) is set = 0. The expression for the vertex is now readily obtained.

rabc Agv (x, X2, X3) - Fac%vO (X3, X2, XI) + FbcVA (X3, X1, X2) + Fcba AYv (x2, XI, x3)

+ G bcavA (X3, X2, X1) + Gacb%AP', (X3, X1, X2) + EcbaVG,A (X3, X2, X1)

? GCbaMY,A (X2, X3, X1) + Gabc MAV (X2, X1, X3) + EbCaP,VA (X2X X3, X1)

+ GCAVbA (X1, X3, X2) + GbaCAMV (X1, X2, X3) + EaCbP (X1, X3, X2).

(3.4)



226 Pro,ceedings of the Royal Irish Academy

The actual scattering amplitude is given by the Fourier transform of equation 3.2 multiplied by i, i.e. i rabc (p, q, r). The Fourier transforms of E, Fand G are defined

n an analogous way:

dx1 dx2 dx3 eiPxl eiqx eirXs abeApv (x1, X2, X3) = (2m)4 6(p + q + r)E Abc (p, q, r)

I dx1 dx2 dx3 eiPXI ei2 eirx3 Fab1,LV (xI, x2, x3)=(2n)4 3(p + q + r) F lbcv (p, q, r)

5 dx1 dx2 dx3 eiPxl e iqx eXrX3 GabcAfiV (xl, X2, x3) (2n)4 b(p + q + r) GabcA,V (p, q, r).

(3.5)

We find that with d- lala2 given by equation (2.22):

E abc (p, q, r) = 2T bc (Ptgxv -PvgA) ( j)

F Abz (p, q, r) = T b, (rAg,v -

r,gAv)(_.2)

G abc ApV (p, q, r)= g bg Tac{(q-2r),(p.r)gAv-pvqOrA-2pvr r (3.6)

Finally, we obtain the expression for the scattering amplitude i Fr Apv (p, q, r):

i rabcAv =ga [Pgvx-pVgf(;$2) + (qvq,g - qAq,v) (12)A

t2 AL bA

+ (rkg,1-r g -) (y) - {(q - 2p)1(r.p)gvA - rAq,pv - 2rApupv

(p - 2q)A(r.q)gv, + rypAqv + 2ruqAqv} (_)M2)k

2 -2 1-(r -2p),(q.p)g,4 +r qAPvPy + 2qAPpPV + (P - )(w)

(q2) A-I A qvpArp - 2q,,rAr,} (-M2) -

I

{(r - 2q),(p.q)g, -prvq,q - 2PqqA

_M2) 2~ (2) 1

- (q - 2r),f(p.r)g,A + P/9I/rA + 2PvrurA} (_M2)A J . (3.7)

The scattering amplitude i fabc A;V (p, q, r) has an S matrix element formed by cal

culating ieA, ,,v C FabcVUy (p, q, r) where s6, qU and 'v are the polarisation vectors of the

external particles. Hence the on-shell S matrix element is calculated by using transverse polarisation vectors where the following kinematics must be noted,

E.p = .q =r.r= 0 2 2 2 = _1

p2 2 q 2-r -2 (3.8)

p +q +=r 0



NASH AND SmLLER-Infrared behaviour of Yang-Mills theories 227

The kinematics mean that the decay is a colinear process. With this information it is

quite straightforward to calculate i sntpe rbc Apv and the result is

Mabc(p, q, - p - q) = iekqpzrabC (p, q -p -q) = 0. (3.9)

This we regard as a success of programme.

The next amplitude to calculate is the gluon-gluon scattering amplitude. It may be calculated by using the following equation relating the connected matrix element

and the proper vertices:

a d a a d

+

D c b c

b

ad a d

b c b ; c+ (3.10) b~~~~~~~~~~~~~~~ The unshaded circle on the LHS of the equation denotes the connected matrix element.

The only unknown object on the RHS is the irreducible 4 vertex. The 4 vertex is now

to be calculated. There are four types of term contributing to the 4 vertex

rabcdA V(X1, X2, X3, x4); we write these as E1234, F1234, G1234 and H1234 where

JJ A(X;2)p, IA2(X) d C a(X) E1234 d I 3Baz(xi) 3BbM(x2) X l1; (x3) 65Bd0(X4)

Gt234 = |dx Ba ( ) d lXX2)a,AXX,3 aDB C sFalA AA(X 62FxiA2a2(X)

F1234 = dx 6B(xr) 6Bdbt(X2) 1(X) 3BC,(X3) 6Bd (x4)

G13 =- (dx 3F a121122(x) d-l'2b XXx3Fk,1a22(X) 1234 -41 X6B aA(xl) da Az;c xx,x 3) 3B d0(X4)

H1234 = -4 dx ,2(X)d-laba2;b.\X; 62

FA1A2a2(X) (.1 H1.2 3 J aBA(XJ) 2(xx Bcv(X3) 6Bd a(X4) (.1

with Baf set equal to zero.

The Fourier transforms, Epqr,, Fpr Gpqr and Hpqrs are defined as in equation (3.5).




The resulant vertex is related to the above quantities by the equation

rabcdgva (p, q, r, s)=Epqrs + Epqsr + Fpqrs + Eprqs + EprSq + Fprqs

+ Epsqr + Epsrq + Fpsqr + Gpqrs + Gpqsr + Hpqrs

+ Gprsq + Hprqs + Hpsqr + Eqrps + Eqrsp + Fqrps

+ Eqspr + Esrp +F qspr + Gqprs + Gqpsr + Hqprs

+ Gqrsp + Hqrps + Hqspr + Erspq + Ersqp + Frspq

+ G,pqs + Grpsq + Hrpqs + Grqsp + Hrqps + Hrspq

+ Gspqr + Gsprq + Hspqr + Gsqrp + Hsqpr + Hsrpq (3.12)

where

-Ag2 vaT"06T6S)}14 Epqrj = 2(_aM2)c

{sAg 5-syg20}(2s + r)v{(r + s)2}

g2 Te abTecd

Fpqrs = (-M2A (gAvg,0 _ g,XgA0){(P + q)2}A

g2A(A _ 1) e V 2 Gpqrs 2(-M2) Teab_T cd{pSA -(p.s)g2,}(2p+q)p(2s+r)y(p2)AZ

t g2As {T1, T }ad ggv{sAps - (pS)ga}(p2)

-A,g2 T60bT6Ca Hpqrs = - m (p2 g4 -pVg2)(2p + q)/p2)'* (3.13)

Finally, the somewhat lengthy expression for the vertex Fabcd Asvu (p, q, s) is rabcd AfvG (p, q, r, s)

=

[ zTebTed TeacTeb g ( 2) rc(gd g - gv ){& ? q)2}A + g2 (acM bd (g,Ag, - _g &){(p + q)2}A

+ r (_Ma)A (g,gav_gagXV){(p + 3)2}A A g2 r_M2)C

Ag2 ( Mz)A (p7gl6 - pg,A)(2p + q)A{(p2)Atl + ((p + q)2)A1 }

+2 (rMhAc (pgae -cpgAA)(2p + r) {(p2)Al- + ((p +)2

Ag2 T ad TCbC + 2 ( M2)A &Plc v

-pPv"IgAX2p +

S),{(p2)1 + ((P Qy S)2)A ^1 }

Ag 2 + ~7 Tba,T? (q,g.,,

- qvg0,,)(2q + p),A{(q 2)'-' + ((q + p)2)A1- }



NASH AND STULEMR-Infrared behaviour of Yang-Mills theories 229

+ A2

T ebdTea

(qA1g,, -

q,g,,A)(2q + s)e,{(q 2)X

-

I- ((q .

s) 2) `- 1 g2

TC_dTCaC

+2 (-c2)e (qzgil - qaga,)(2q + r)s{(q 2)A -

I + ((q + r)2)A - 1I Ag2

TC

M2Tad

+ 2 (_f2)A (qcgd -sqg1p)(2s + r 2{(q)

1 + ((q + r)2)1 }

Ag2 TabT%d +2 (-M 2)T (s7gA10-s1g )(2s + r),{(s2) + ((s r)2)1}

Ag2 TCabTCd +2 Q M2) (r, gA -rAgg,)(2r + S)0{(r2)- + ((r + S)2)'

g2 TaTa2d +2 ( M2)1 (scg26 - sagA0)(2s + q),{(s2)11l + (s + q)2)-1}

Ag2 TbebTed +2 (-M2)1 (svgAcr ssgV)2s + p)2{(s2) + (p + 5) 1 }

+ 2 (_M2)b (rAg,jy - rrg0,f)(2r + pq){(r2)A- I + ((r + p)2)A-1 -}

+12 ( M2a (2p + q)2(2s + r),,(p2)

+g2 {Tb TC}ad ) + 1 } SS2p p S

* 2 i(_M2) *11,g 2)1 2 )#(2S + r p), I(S 2)

+2 g (M2/ gya(p2)1' J{S 2p-2gA 1-P.S}

+{ 2 ( M2)2 (2, + r)V(2r + p XJp2)1 r

* g2A(,Z _-

1) Te abTecd (2q + q),,(2s + r) v(p2,I 2

g2A {Tb, Td}bad g {r1pV - g1p.{} * 2 (_M2)1- gp0(p2/ g _pS}

+ 2A_-1) TeabGTadc (2p + r),(2q + s)0(p2)1I-2

~2 (_M2)1l

g2AZ {TC, Td}a 21i

2 (_M2)1L

?TtA(A _ 1) TeabaTed (q+P(S+ )q2)A12 + t 2 (_M2 )A 2 )(q+s,(

g2A {Tc, TC}abd 2 -l (_M2)A g(q2){ {psjqA-gsq

- A( 1) TabaTade (2q + p),(2r + S)0~(q2)1k-2 t2 (M




g%4 2 {-Ta Tc}bC )

Jr 2 M2lgA(qZ)Al>. {rZqv-ggvq.r} 2 (_M2)A 1

+ {g%A( - 1) T,cTeda (2q + r) (2p + s)0(q 2)-2

t - ) 2 (_M t2)A

}{YAgPN g221 {TC, Td}ba } + -

(T?M2)A grA(q2)-l

-

Ip; ;-garp.q}

+ {g2A(A - 1) ?Tt7 d (2r + p)A(2s + q)f(r2)AZ2

g2A {,Tb) a 2 -I + {2 (M

I

gA,4(r2)A . {r0s,- g6s.r}

+Ci {Ta, Td}Cbg(2>l}{ pr

2 (..M2)A

+12 (-M2)" (2r + p)&(2p

+ S)0fi(r 2)A- 2

2 (-22)A )

{g2(l ) Td4T?b(2s + p)S(2r + q)v(32)A

g 22 {T?a Tb}dcb

+2 (_M2 gAV(s2)- j{rusv - qergv.s}

+ 2 (_M2)A (2r ~2 )(2A

+2 (_M2)A g,s {0~-g,qs

+ 2 -(A _ 1) (M2 T;eA (2s + q),(2p + r),(s2)A 2

2 {Tb, TC}da -g6ps} c (3.14)

2 (2 _M2)A 1 (3) {PS

We now wish to calculate the gluon-gluon scattering amplitude using equation 3.10. Because of the length of such a calculation we content ourselves, for the present,

with calculating the forwardj = amplitude rather than the general case. With this simplification, the t-channel exchange graph on the RHS of equation 3.10 is zero because of the matrix structure of the three gluon vertex. Also the s and u



NASH AND STULLER-Inf rared behaviour of Yang-Mills theories 231

channel exchange graphs represent practically the same calculation. Extending our previous notation we call the connected S matrix element Mabcd (p. q, r, s). We may

now rewrite equation 3.10, bearing in mind the absence of the t-channel graph, without diagrams as:

Mabba (-p, q, q, p) = rabe (-p -q, p + q) GteIK (p + q) '1t' (p, -p, -q, An + Jt)zCK (p, q, -p - q) GK$ (-p + q) (-p, p - q, q)AA

+ F-tg, (-p, -q, q, p)8AaAs1'q4'. (3.15)

Where s is the polarisation vector of the gluon of momentum p and t1 is the polarisation

vector of the gluon with momentum q, E valuating the RIHS of 3.15, for convenience in the centre of mass and in the Coulomb gauge say, we find that it is zero, the last

term vanishing separately from the first two. Thus, in summary we have the vanishing the on shell S matrix elements for three gluons and as described above for forward

gluon-gluon scattering. We would like to be able to prove that all gluon S matrix elements vanish on shell but we have not yet done so. However, there is a special

case about which more may be said, i.e. when A(g) = 1. Thus we assume that the

coupling constant g is increased so that A(g) > j and that the increase continues until A = 1. There are some general reasons for this being a desirable situation. These are

the following: A 1 corresponds to a linear confining potential for quarks and this is suggested by a general cluster decomposition argument for infinitely massive quarks by Wilson [16]; also the string models for hadrons have a basic one dimensional

character giving rise to a force which is proportional to the separation of quarks [17]; further, the density of bound states as a function of increasing energy is com patible with that expected from Regge theory. Finally, on the experimental side, there is good agreement with experiment for a linear potential [18]. There is also an in

triguing possibility for A = 1 in this article. To begin with, there is an apparent

drawback because F(B) is only a polynomial of degree six in B0,(x) rather than a

power series as it is for non integral A. This means that

F a, . . .an

is zero unless n $ 6; however, this may be intepreted as a virtue since the connected S

matrix elements for n > 6 are all made up of I particle reducible components from

the

r a, ... an

PI . .. 1-Un

for which n ? 6; these we may call generalised tree amplitudes. Thus, in this case we

have a much simpler structure to study, and have the opportunity to derive more

general results about the gluon scattering amplitudes which may have a more general application. We now turn to the next section to refer to our remaining topics.

IV. Fernlions, weak versus strong coupling, and conclusion

The actual object of using a non-abelian theory in the first place is to confine

quarks, but since there is no experimental evidence for strongly coupled massless vector particles then the non-abelian gluons must be confined also. This we have




found evidence for in the preceding section. Further, as referred to in Section II, in reference [5] the Schwinger-Dyson equation for the Fermion coupled to the Yang

Mills gluon yields an infinitely massive Fermion provided g is large enough to make A(g) > j. In reference 1 Ferm-ions are treated somewhat differently with the treatment more tailored to the physics of the present article. The central result that A(g) > i provides confinement remains unaltered however. A "mass shell" projection operator for the quarks can be developed in this approach. The nature of the Fermionic bound with "colour zero" are discussed in reference 4 and the mesons which are bound turn out to be only vector and pseudo-scalar. This of course is in agreement with the naive quark model, and may be thought of as due in part to the infinite quark

mass which forces some non-relativistic dynamics on the system, a set of circumstances for which SU(6) should be a symmetry. The question of whether there is "gluonic

matter" in the form of bound states is, for the moment, open. The next question which naturally arises is if g is decreased so that A(g) < i what happens. In the strong

coupling limit we have maintained gauge invariance and a symmetric solution to the tlhcory; the infrared singularities have then caused the gluons to decouple from the on shell S matrix thus avoiding macroscopic transmission of charge via exchange of

massless charged particles. For the weak coupling limit (2 < -) we expect it to be no

longer possible for this to happen and the system gets rid of its infrared singularities by spontaneously breaking its own symmetry and generating massive gluons so that only abelian massless components remain. No Fermions will be confined in this case. This then is a model appropriate for a gauge theory of the weak and electromagnetic

interactions. Thus we expect there to be two phases of non-abelian gauge theories, 2 <c ~, 2 > -1, both are infrared divergent, one gets rid of its divergence by decoupling the gauge particles from the S matrix and maintaining exact gauge invariance, the other gets rid of its divergence by spontaneous breakdown of the symmetry and conse

quent acquiring of mass by the gauge particles. The first of these situations is needed

to describe the strong interactions and the second is appropriate for the weak and electromagnetic interactions [19].

In conclusion we have described conditions under which a non-abelian Yang-Mills theory may describe strong interactions with no strongly interacting massless particles and no fundamental Fermion, only bound states. Finally, the conditions for which the programme is successful can in principle be derived from two coupled Schwinger Dyson equations for only two unknown functions. The results of this paper are we feel hopeful for further study.

References

[1] Nash,C. Yang-Mills theories. To appear. s

[2] Appelquist, T., Carazzone, J., Kluberg-Stern, H. and Roth, M. 1976 Infrared finiteness in Yang-Mills theories. Phys. Rev. Lett. 36,768.

[3] Yao, Y. 1976 Infrared problem in non-abelian gauge theory. Phys. Rev. Lett. 36,653.

[4] Cornwall, J. M. and Tiktopoulos, G. 1975 On-shell asymptotics of non-abelian gauge theories. Phys. Rev. Lett. 35,338.

[5] Stuller,R.L. 1976 Quark self energy. Phys. Rev. D13,513.

[6] Slavnov,A. 1972 Gauge theories. Theor. math. Phys. 10,99,153.

[7] Taylor, X C. 1971 Yang-Mills gauge identities. Nucl. Phys. B 33,436,

[8] Lee, B. W. and Zinn Justin, J. 1972 Spontaneously-broken gauge theories. Phys. Rev.

D5,3121.



NASH AND STULLER-Intfrared behaviour of Yatng-Mills theories 233

[9] 't HooFT, G. and Veltman, M. 1973 Diagrammar. CERN Rep. 73.

[10] Lee, B. W. and Zinn Justin, J. 1972 Spontaneously-broken gauge theories. Phys. Rev. D 5,3121.

[11] Lee, B. W. 1973 Transformation properties of proper vertices in gauge theories. Phys. Z<?tt.46B,214.

[[12] LohPingYu 1974 Gauge problem of non-abclian theories. Rutgers University Rep.

?13] Cf., for example, Weinberg, S. 1973 Yang-Mills theories, J. Phys., Parisl CoJloq.34C, 1.

14] Nash, C. Yang-Mills theories with Fermions. To appear.

[15] Jona-Lasinio, G. 1964 Relativistic field theories with symmetry breaking solations. Nuovo Cimento 34,1790.

[16] Wilson, K. G. 1974 Confinement of quarks. Phys. Rev. D10,2445.

[17] Cf., for example, Nielsen, H. B. and Olesen, P. 1973 Vortex line models for dual strings, Nucl.Phys.B6\945.

[18] Tyron, E. P. 1976 Unified relativistic quark model for hadrons. Phys. Rev. Lett. 36,455.

[19] For further arguments connecting the weak coupling limit of Yang-Mills theories with spon taneous breakdown, cf. r?f. 16.



Documents

Infrared Behaviour of Yang-Mills Theories