String equations for lattice gauge theories with quarks

Volume 87B, number 1,2

PHYSICS LETTERS

22 October 1979

STRING EQUATIONS FOR LATTICE GAUGE THEORIES WITH QUARKS Don WEINGARTEN

Phystcs Department, Indmna Umverstty, Bloomington, IN 47405, USA Received 10 July 1979

Exact Dyson-Schwmger equations are derived, m a U(N) lattice gauge theory, tor vacuum expectation values of products of operators for strings ending in quarks

Heuristic equations for small variations in the path of the vacuum expectation of a Wilson loop, (Tr [P exp (1 fq Aid dx~)]), in QCD have been derived in ref [1 ] These equations are similar to equations governing vacuum expectation values in the string model and perhaps provide a way of relating the string model to QCD More recently, exact equations for variations in the paths of the expectation

ITr[Pexp(ifAudxu)]Tr~ exp(,f Audxu)7), ql

qm

have been found in a theory which approximates the lattice formulation of QCD [2] and a slrmlar result has been obtained for a U(N) lattice gauge theory without approximations [3] These last two sets of equations are actually lattice versions of the Dyson-Schwmger equations of continuum field theories In the present article the equations derived in refs [2,3] will be extended, for a lattice U(N) gauge theory, to vacuum expectations of products including factors of the form ~l(Y) e exp(l fq A u dxu) t~k(X),where q is a path running from x to y In effect we reformulate the closed string equations of refs [2,3] for strings ending in quarks Similar equations can also be derived in SU(N) lattice gauge theories but will not be discussed here The full set of Dyson-Schwlnger equations for all vacuum expectation values of a U(N) or SU(N) lattice gauge theory deterrmne the vacuum expectation values uniquely and are equivalent to the more familiar path integral formula Thus these equations probably have other uses aside from providing a possible connection

between gauge theories and the string model In particular, the Dyson-Schwmger equations may suggest new methods for solving the theory or evaluating its continuum limit The derivation we will give is based on to those in refs [2,3] but uses in addition some properties of differentiation with respect to Grassmann variables [4] which we will begin by reviewing Let G be a Grassmann algebra generated by the elements gl, ,gn Then any h E G can be written in the form

h = a +gtb ,

(1)

where a and b are elements of the algebra G ' generated by the set o f g l , ,gn °mlttlnggt The derivative O/Og~ is defined to be the linear map from G to G ' given by

(OlOg,) h = b

(2)

Let the even and odd parity subspaces of G have as their members sums of homogeneous terms including, respectively, even or odd numbers of factors o f g 1, , gn" I f h and k are in G and h has parity p, then eqs (1) and (2) imply

(D/~gt)hk= [(O/Ogt)h ] k +(-1)Ph(O/~gt)k

(3)

With the usual definition o f integration over Grassmann variables [4] we have also for any h E G 0

f dgI f dgn-~lh=O

(4)

Now let us define a U(N) lattice gauge theory on a finite region A C Z 4 For each ( x , y ) in L, the set of oriented nearest-neighbor links on A, U(x, y ) is a mem97

Volume 87B, number 1,2

PHYSICS LETTERS

ber of U(N) with U(x,y) = u t ( y , x ) Let P b e the set of oriented plaquettes p on A, and for each p E P define U(p) to be the ordered product of U(x,y) around p beginning at some arbitrarily chosen starting point For any oriented path q of nearest-neighbor links on A, define U(q) to be the ordered product of U(x,y) along q Each x E A is assigned a set of Grassmann variables ~m(X), ~lb(X) with l and l spin indices and a and b U(N) Indices Define I~(x,y) for each (x, v) E L to be P(x+eu,x)=l

±3,u,

,4 =~_J~(x)~(x)-K ~

(x,y) ~L

~(x)r(x,y)U(x,y)4,(v) (5)

Q exp(-A),

(7)

U(x,v) For a set of oriented paths q l, , qm with q~ running from x~ to y~, and a set of closed oriented paths, pl, , Pn, it IS convenient to define the notation

,Pn)klll

= (~11 0'1) U(ql) ~kl (Xl)

Tr U(Pl)

Tr U(Pn))

kml m ~klmO~m)U(qm)~km(Xm) (8)

For simplicity we will suppress spin indices in the remain- t der of ttus article and adopt the convention that whenever 6xy appears for a pair o f points x , y which are locations of fermlon fields, a factor of 6kl IS also implied for the corresponding spin indices k, l The quantity (ql, , qm,Pl, ,pn ) may be thought of as the vacuum expectation of a product of field operators for closed strings and field operators for strings ending in quarks We are ready to derive Dyson-Schwlnger equations for (q,), the vacuum expectation of the operator for a single string ending In quarks The extension of these 98

(10)

the vacuum expectations of the operator for a single closed loop The action of the derivative on e x p ( - A ) yields

(qZ,)r(z,y),

(11)

where qZ is q followed by (z,y) and the sum is over z which are nearest neighbors o f T Combining eqs (9)-(11), we obtain the D y s o n Schwlnger equation for variations of (q,) with respect to the string's end point y

(6)

The Integral f d/aF represents Grassmann integration over all t~m(x ), ~lb(X), and f d/aG is Integration over one copy of U(N)-Haar measure for each Independent

X

(9)

The differentiation In eq (9) can be carried out using eq (3) and yields two terms The action of the derivative on ~O(x) gives

(q,) - r ~

Z=f duvf duc e x p ( - A )

,qm,Pl,

U(q) ~(x) e x p ( - A ) = 0

Z

and for any polynomial Q o f gauge and fermlon variables, the vacuum expectation (Q) is

(ql,

lf f Z- jd/.tFJ d/aG~

(q,)-K ~

1 + ~ Tr U(p), 2g 2 p~P

f f

equations to arbitrary (q l, , qm,Pl, ,pn ) IS straightforward and will not be given here Let us replace t~Cv) in eq (8) for (q,) by 0/O~(y) Then eq (4) implies

6xy(,q) ,

where the 7 u are euclidian "y-matrices The action A is gwen by

(Q) = Z-1

22 October 1979

(qZ,) P ( z , y ) = -6xy(,q)

(12)

Z

The left side of this equation is a euclidean lattice version oftb - g ~ - m acting on ~(y) from the right The right side of eq (12) IS a version of the 5-functions which appear In the continuum Dyson-Schwlnger equations Alternatively, the right side of eq ( 1 2 ) m a y be thought of as a source term arising from processes in which the quark and antlquark at the string's ends annihilate leaving behind a closed string By replacing ~(x) in (q,) with ~ / ~ ( x ) , we can derive an equation similar to eq (12) but governing variations of (q,) with respect to the string's starting point x The result is

(q,) - K ~

F(x, z) (qz ,) = -6xy (,q) ,

(13)

Z

where qz is q preceded by (x, z) and the sum is over z which are nearest neighbors o f x An equation for the variation of (q,) with respect to changes in the path q at some link (z 1, z2) can be obtained by replacing U(z 2 , Zl) in U(q) with b/3U(z 1, z2) and then duplicating the arguments an refs [2,3] We obtain

Volume 87B, number 1,2

PHYSICS LETTERS

N- 1__ ~ 2g2 _

G(r,,s,)

1 Z~(q;,)+2 2g 2 l l

+ 2K(t 11"(z1, z2) t 2 ,) (14) In the second term on the left side o f e q (14), the index t ranges over the s~x plaquettes contalmng the hnk (z2, z 1) If t specxfies (z2, z 1, a, b) and q is ( , z0, Zl, Z2,Z3, ) , t h e n q z l s ( , z 0 , z l , a , b , z 2 , z 3 , ) A possible q Is shown in fig la and a corresponding qt is shown m fig lb In the first term on the right of eq (14), t ranges over the six plaquettes containing (Zl, z2)

Iflspeclfies(zl,Z2,b,a) andqls( ,ZO,Zl,Z2,Z 3, ), ¢

q~ls( ,Zo,Zl,Z2,b,a, zl,z2,z3, ) For the q m fig t la, a possible q~ xs shown m fig lc In the second term on the right side of eq (14), the sum Is over any occurrences of the hnk (z2, Zl) at some location m q The path with ends, rt, and the closed path, st, are obtained by removing (Zl, z2) and the lth occurrence o f (z2, Zl) and rejoining the open string ends A possible q with only one occurrence of (z2, Zl) is shown m fig le and the corresponding r and s are shown m fig 1f In the last term on the right side o f eq (14), t 1 and t 2 are obtained by removang the hnk (Zl, z2) from q The path t 1 Is the remaining segment running from x to z 1 and t 2 is the segment from z 2 to y The notahon t 1F(Zl, Z2) t 2 indicates that the spin index of tk(Zl) at the end of t 1 is contracted with the first spin index of l-'(z 1, z2) and the spin index o f @(z2) at the beginning of t 2 is contracted with the second

(o)

(b)

(c)

(d)

22 October 1979

mdex of l-'(Zl, z2) For the q shown m fig la, possible t 1 and t 2 are shown m fig ld The left side of eq (14) Is something hke a lattice string version o f D + m 2 acting on q at the hnk ( Z l , Z 2 ) The second term on the rtght side o f e q (14) is a string version of the f-functions which appear m continuum Dyson-Schwmger equations This term may also be thought of as arising from processes in which the string 4 crosses itself along the hnk (Zl, z2) and a closed strmg loop breaks off at the crossing The first term on the rtght side of eq (14) arises from string self-interactions which occur m the U(N) gauge theories but are absent in the simplest N a m b u - G o t o field theory considered m ref [2] The third term arises from the interaction of strings with quarks and is absent from equahons for variations of string paths m the pure gauge theory In ref [3] Ttus term corresponds to processes m which the string breaks somewhere on its interior and a q u a r k antlquark pair appears at the open ends This completes the denvahon of Dyson-Schwlnger equations A few addmonal comments might be useful The procedure used here can easdy be apphed to the N a m b u - G o t o field theory m ref [2] with quarks lntro. duced by an adaptahon of eq (5) One obtains equatmn of the same form as eqs (12)-(14) except that the first term on the right m eq (14) is absent and the second ar third terms appear without factors of 2 Does the introduction of quarks stablhze the vacuum of the N a m b u Goto field theory 9 By carrying out the integral over fer. mlon fields exphcltly and obtaining a Matthews-Salam formula, one can show the vacuum is not stabdlzed Fo~ a fimte lattice A the Matthews-Salam determinant Is a polynomial in the U(x, y) and cannot control the exp, nentmlly dwergent factor e x p ( - A ) Smce the U(N) theory does have a satisfactory hamfltoman and vacuum [5], however, it appears that the first term on the right side of eq (14), whxch is missing m the N a m b u - G o t o theory, must be the origin of this stabdlty I would hke to thank Tohru Eguchl for sending me a copy of ref [3] prior to pubhcatlon and Don Llchten berg for a helpful discussion Tlus work was supported in part by the U S Department of Energy

(e)

(f)

References Fig 1 Terms appearing in eq (14) Points corresponding to a ~O(x) are represented by circles and those corresponding to a @(y) are represented by dots

[1] Y Nambu, Phys Lett 80B (1979) 372, J L Gervals and A Neveu, Phys Lett 80B (1979) 255,

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Volume 87B, number 1,2

PHYSICS LETTERS

E Corngan and B Hasslacher, Phys Lett 81B (1979) 181, A M Polyakov, Phys Lett 82B (1979) 247, T Eguchl and S Wadla, Umv of Chicago preprmt EFI 78/65 (1978), L Durand and E Mendel, Umv of Wisconsin preprmt COO-881-88 (1979)

100

22 October 1979

[2] D Wemgarten, Indiana Unlv preprmt IUHET-45 (1979) [3] T Eguchl, Plays Lett 87B (1979) 91 [4] F A Berezm, The method of second quantlzatlon (Acadern Press, New York, 1966) [5] M Luscher, Commun Math Phys 54 (1977)283