Analytic results — Lattice and continuum

I|lllIIIIIImlil~lll[lUll

"!

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 30 (1993) 81-95 North-Holland

Analytic Results - Lattice and Continuum Manfred Salmhofer* Mathematics Department, UBC, Vancouver, Canada V6T 1Y4 A collection of results related to the continuum limit of lattice theory is reviewed.

1. I n t r o d u c t i o n There have been many analytical results in lattice theory this year, which have supplemented and stimulated numerical studies. I do not have the expertise to give an overview which is as complete and beautiful as those which Pierre van Baal and Peter Weisz have given in previous meetings. Also, there have been excellent plenary talks on subjects in which analytical work plays a central role. Therefore, I shall discuss the following topics: 1.Fermions and Hopping Expansion 2.Flow Equations and Improved Actions 3.The Horizon Function 4.Construction of YM4 with an infrared cutoff This ordering is, in seminar tradition, from simple to complicated. The sections will start with a brief reminder of well-known things to describe the setting for the new results, and will end with a brief overview over related results. Acknowledgement." I would like to thapk Pierre van Baal, Martin Liischer, Peter Weisz and in particular Erhard Seller for their guidance, advice and many discussions which helped me in preparing this talk. I also thank Dan Zwanziger for enlightening discussions. Financial support by NSERC of Canada is gratefully acknowledged.

2. Fermions and Hopping Expansion This section contains results from the study of fermion-gauge models in the simplest case, the strong coupling limit. The motivation for this is that some questions which are important at all values of the gauge couplings can be answered in this limit. *supported by the National Science a n d Engineering Research Council of C a n a d a

2.1. B a s i c F a c t s Everyone knows of course what the hopping expansion is, but for definiteness let us recall the most basic facts. The action for fermions, (¢, ( ~ - re)C) is discretized on the lattice, and the partition function on a lattice h with finitely many sites IA[ is defined by the Berezin integral Z A --

/ DCD~,DUe M ~ x ,~¢(x)-H-ZSw

(1)

where fl = 1/g 2, Sw is the usual Wilson action for gauge fields, H is shorthand notation for the nearest-neighbour hopping term which we take of one of the standard forms. Most important here is that the coefficient M of the diagonal term in the action is related to the bare fermion mass m as M = m + d for Wilson fermions [1], where d is the dimension, and M = m for staggered fermions [2]. Expansion of the exponential in powers of M leads to a corresponding expansion for Zh. On a finite lattice, this expansion terminates at a finite order L = a0[A[, a0 some fixed number, because the Grassmann variables are nilpotent. Thus ZA(M) is a polynomial in M, which can be factorized in terms of its zeros A1,..., AL E C: L

ZA(M) = CL H ( M - A,).

(2)

i---1

Changing normalization we can choose cL = 1 (it may depend on 3, but this will not cause troubles for small /3). Consequently, we get for the free energy density L

1 E

f~t(M) - -~] i=l ln(M - ,X,) and for its derivative

XA(M) =

Xh(M) = 0_~ OM

1 c 1 ~-~ "= M _ Ai

0920-5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.

(3)

-

----

(¢¢/A

X-~ an(A) n=o ]-'" Mn+l ,

(4)

82

M. Salmhofer /Analytic results - lattice and contbmum

The expansion in 1 / M is usually called the hopping expansion and x = 1 / M the hopping parameter (the usual definition differs from the one used here by a factor two). It is a classic result from the theory of cluster expansions that analyticity of the free energy density and the correlation functions in M and /3 holds for all ]M I > M0 and 1/31 < /3o, where fl0 and M0 are independent of the volume Ihl [3]. Therefore the expansion in 8 has a positive radius of convergence k in finite and infinite volume, k >_ 1 / M o , and thus the zeros must all satisfy ]A/] _ M0. In the thermodynamic limit the theory has a mass gap in lattice units, that is, a finite correlation length, for [M] > M0. Consequently, any critical point, where a continuum limit may be taken, has 8c _> 1 / M o . Needless to say, it is important to know ~c and also to get information about the location and behaviour of the zeros. In particular we will be interested in the question if ~ = ~ . This is hard to answer exactly in general; at /3 = 0 there are useful alternative representations of the system.

2.2. Staggered Ferlnions Recall that here M = rn, the bare fermion mass. For /3 = 0, the model has an exact representation as a generalized m o n o m e r - d i m e r model [4], for gauge group U(1) it is a m o n o m e r dimer model in the strict sense of [6], ZA(m) = Cnmlhl-2'~, where Cn is the number of ways of putting n dimers on the lattice without having any site touched by more than one dimer (this hard-core constraint is due to the nilpotency of the fermions). This representation is very useful for simulations [5], and one can get precise data for e.g. the chiral condensate X, which exhibit chiral s y m m e t r y breaking. For the m o n o m e r - d i m e r models associated to U(1)-theory, the rigorous knowledge is also quite complete: it has been proven [6] that all zeros Ai are purely imaginary, hence there is no phase transition for m > 0. Using methods developed in [7], one can then show that there is a mass gap p > 0 for all m > 0 [8]. Moreover, for d > 4, one can prove [8] that the continuous chiral s y m m e t r y of the model is broken, and that the pion mass

obeys (5)

c l m < m r < c2m lid.

Thus, at m = 0, we have a Goldstone theorem: the pion is massless, and so the correlation length is infinite. On the other hand, ZA(0) is the number of close-packed dimer configurations, and thus at least one. By (2), L

(6)

1 _ ZA(0) = i=1

so

at

least

one

of the

zeros has

[Ai]

]ZA(O)I/IAI I > 1, and therefore the expansion (4) in ~ = 1 / M can converge only for ]hi < 1 in any

finite volume. In fact [9], since all ]Ait < M0, one can easily see from (6) that, for any ¢ > 0, there are O(IAI) zeros with IAI > 1 - e, thus the contribution of the residues of the corresponding poles in (4) does not vanish as IAI ---* co. Since one also knows that all zeros are on the imaginary axis, one can prove that this implies k _< 1 in infinite volume (see Proposition 2 in [10]), which differs from nc = oc, and thus the hopping parameter expansion diverges before the critical point because of the singularities away from the real axis. 2.3. W i l s o n F e r m i o n s From the above it is clear that also in this case alternative representations of the system might he useful. In the simplest case, namely d = 2 and /3 = 0, there is an exact map to an eightvertex model [11]. Labelling the vertices as in [12], the activities ai = e - t ' are al = M 2, a2 = 0, a3 = a4 = 1, as . . . . . as = 71 = 1/2. The configurations of this model can be thought of as consisting of nonintersecting loops, which get factors r / = 1/2 for every corner. This factor comes from the g a m m a matrices of the original model. The non-intersection condition a2 = 0 is due to the nilpotency of the fermions, and the factors M 2 again correspond to monomers. A generalization of this map for a parallelogram lattice has been provided in [13], but no analogues in higher dimensions seem to be known. For M > 0, this is a nonsymmetric case of the eight-vertex model which has not been solved as far as I know; however, at M = 0 it reduces to a solvable six-vertex

83

M. Salndzofer/Ana~tic results - lattice routcontinuum model which is critical and whose continuum limit has central charge c -- 1. The corresponding value gc -- oc. But one can argue as above that k _< 1: again, for a lattice with periodic boundary conditions, there are two configurations of straight lines winding around the lattice which have weight one, and all other summands in ZA are nonnegative [9]. Therefore ZA(0) > 1, which again implies that there are O(IAI) zeros with IAI > 1 - v . Thus k _< 1 holds in any finite volume. The choice of periodic boundary conditions is not essential for this, because e.g. on a square with fixed boundary conditions, ZA(0) _> 2 -4 IV~-~, and so Z A ( 0 ) 1/]AI still goes to one for large [A I. In infinite volume, k might still be larger than one. This cannot be excluded as for staggered fermions because the location of the zeros is not known as well as in the case of staggered fermions. However, one can show quite generally that either k < 1 in infinite volume or, if k > 1, the hopping expansion ceases to represent the function X(M) when one passes ~¢ = 1 [10]. A different argument for k < 1 is a resummation of the hopping expansion which can be done by expanding in powers of the corner activity ~/ defined above [14]; for 7/ -- 1/2 the O(r/s) result reads 2 (

_1-

M4 1 + 2(M 4 - 1) 2

/

(7)

for ]M I > 1 and X(M) ~ M3/(4(1 - M4) ~) for IMI < 1. The denominators which become zero at M = 1 arise from summations of geometric series. One also sees that something must go wrong at least with this resummation of the expansion already for M > 1.2, since the right side of (7) becomes negative for M < 1.2, but the actual function X(M) >_ 0 (as can bee seen from the eight-vertex representation). It was surprising that a high-precision numerical study, which is possible using the eight-vertex representation, did not show any singular behaviour in X(M) or its derivative at. or around these values of M [9]. Such a behaviour may be expected from the chiral Ward identity. The scalar susceptibility, the derivative of X, has a maximum at M ~ 1.3, but the data over a factor

four in the sidelength of the lattice are identical within error bars and indicate a maximum value of about three [9]. The data agree quantitatively with the resummation given above for large M, as they nmst, and qualitatively for small M (i.e. one sees the M 3 but the coefficient is different). Thus the data do not indicate a critical point at k. Although this does not, of course, mathematically exclude a very weak singularity, it suggests that the assumption k = gc should be checked also in the physically interesting case d = 4. The other examples given above show that if k = ~c holds for Wilson fermions in d = 4, there must be a particular reason why this is so for that type of fermions, which should be investigated. 2.4. L e e - Y a n g M e a s u r e s An interesting question arising in this context is how much information about the distribution of zeros one can get back from the hopping expansion. This is related to the problem of finding resummations of the hopping expansion which converge to the thermodynamic functions because the thermodynamic quantities can be determined from the limiting distribution of zeros. A glance at (4) reveals that

am(A) =

1

L

(8) i:1

This implies that in a finite volume, the sequence (an(A))n_>0 determines the zeros. In infinite volume, equations (4) and (7) are replaced by

-~a~M

-n-1

(9)

n:0

and

a,~ = / dZ(z)z '~.

(i0)

Here £ is the limiting distribution of zeros, which can be shown to exist as a two-dimensional measure (at least for subsequences) under quite general conditions [10]. Its support consists of those points in the plane where O([A() zeros accumulate in the thermodynamic limit [10]. The above question can then be restated as a moment problem for £, namely whether £ can be reconstructed

84

M. SalmhoferIAna~yticresults- latticeand continuum

uniquely from the moments an. The answer is no, because the an are only part of the moments, and different measures can satisfy (10) with the same an for all n. To reconstruct ~:, one needs the additional independent moments

gauge groups in [18]. Thus there is a lot of good information about the location of zeros and it will be interesting to combine these results with the hopping expansion via (10) to get results about chiral phase transitions.

b,~n = f dE(z)~mz n.

2.5. R e l a t e d R e s u l t s Series Expansions: Vohwinkel and Weisz have calculated the low-temperature expansion of the four-dimensional Ising model to 34th order to complete the study of triviality bounds [19]. An alternative method to calculate the series expansion coefficients has been used to extend the lowtemperature expansion for the free energy density of the Ising model to 50th order in d = 3 and 44th order in d = 4, and to estimate the critical temperature and the critical exponent a [20]. However, in [19] also the expansion for the second moment of the two-point function has been determined, because it is needed to define the renorrealized mass and wave function renormalization constant. This second moment is not simply a derivative of the free energy density. In the notation of the discussion about the Lee-Yang measure, this is information not contained in the an. Zeros: Kenna and Lang have derived a finitesize scaling theory for ¢4 using perturbative renormalization group methods and determined the finite-size scaling behaviour of the susceptibility and the zeros [21]. They showed that there are multiplicative logarithmic corrections to the mean-field behaviour arising from the double zero of the fl function at zero in d = 4, and were able to identify these corrections in a simulation. Exactly Solvable Models: Baxter and Bazhanov have solved a three-dimensional spin model with interactions around a cube, which is related to the sl(n) chiral Potts model. They showed that the Boltzmann weights of this three-dimensional model satisfy a restricted set of Yang-Baxter equations, and thus did not need to solve the tetrahedron equations to solve the model [22].

(11)

The b,~n determine the measure uniquely [10]. It is an interesting open problem to find an expansion by which they can be determined. T h a t the an alone do not determine E is in agreement with theorems about Pad@ convergence because these theorems require some additional assumption about the analytic structure of the function under consideration (e.g. that it should be meromorphic in a certain disk or a Herglotz function [15]) which in our case would be equivalent to some information about the form of E or its support. This additional information can be sufficient to get uniqueness. For instance, in the case of the Ising model, all zeros in the fugacity are on the unit circle by the Lee-Yang theorem [16]. This implies that the coefficients of the fugacity expansion are then just the Fourier coefficients of the limiting measure, so they determine it uniquely. This information has been used to calculate details about the limiting distribution for the Ising case [17]. The fact that the moments determine the measure means that the so performed resummation must converge to the right answer, which is good to know (even if it does not guarantee that an approximation involving a practically available number of coefficients works already well). It is true for more general curves (e.g. for conformal images of the unit circle) that once one knows that a particular curve supports the measure, the coefficients an determine the measure via (10). This may he a good way to study fermionic systems. Recently, the zeros of fermionic partition functions have been studied numerically [18], and - apparently without knowledge of the rigorous results of [6] - it has been verified that the zeros are on the imaginary axis for /3 - 0 for QED with staggered fermions. In addition to this, the zeros have been determined numerically also for/3 > 0 and more complicated

3. F l o w E q u a t i o n s and I m p r o v e d A c t i o n s The topic of this section are recent rigorous developments in Symanzik's improvement programme and the flow equation renormalization

M. Salmhofer IAna~ytic results - lattice and continuum group technique which was used to derive them in a rather simple way. 3.1. S y m a n z i k I m p r o v e m e n t Recall that for nonzero lattice spacing a > 0, the right side of the Callan-Symanzik-equation is not zero, but that for a physical quantity P which depends on a and the bare coupling gB

(°0 -

~

+fl(gB)

P(gB,a) = O(a21oga)(12)

and even only O(aloga) for fermions (here and in the following, this is always supposed to mean a(loga) z with some /~ > 0). Symanzik's improvement programme [23] aims at reducingthis to higher powers of a, O(a n log a), by modifying the action by local, irrelevant terms [23]. Improvement was first shown to work for ¢4, the O(N)-sigma model [23], Yang-Mills theory [24]. That it is possible at all is a nontrivial statement, which depends also on the observables which are to be improved. The important concept of onshell improvement was introduced in [25]: if only on-shell quantities such as masses are improved, the conditions are less restrictive. This facilitates consistency checks and allows to keep certain parameters free. On-shell improvement for fermions [26] leads to the clover action which is also used in simulations aiming at weak interaction matrix elements now [27]. Improvement may spoil Osterwalder-Schrader positivity: for onedimensional spin systems, it has been proven that actions with more than next-to-nearest neighbour terms are not OS-positive unless they have infinite range [28]; these problems are discussed

in [29]. One important, and still open, question is whether improvement can be carried out to all orders in perturbation theory. There is a similar problem in continuum theory: in a renormalizable theory with momentum cutoff A0 which plays the role of the inverse lattice spacing a - 1 the renormalized Greens functions converge like A0-1 log A0 as A0 --+ oo, in every order in perturbation theory. Improvement means in this context speeding up the convergence to get an additional factor Ao-N which corresponds to higher powers of a. This was shown to be possible to all

85

orders in perturbation theory and for all Greens functions in ¢4-theory and QED. The proofs were given by Wieczerkowski (for ¢4) [30], and more recently independently by Keller, who has further simplified the proof, for ¢4 and QED [31]. Although similar, the continuum problem is technically simpler than the one on the lattice because one can use the full euclidean invariance, but it is nevertheless good to have a proof that improvement works to all orders in perturbation theory in this case. 3.2. F l o w equation Renormalization Invented by Wilson [32] for non-perturbative purposes, this method has by now also become a very elegant tool in perturbative renormalization. This development started with Polchinski's drastically simplified proof of perturbative renormalizability of ¢4 [33]. The method was adapted to the O(N)-nonlinear sigma model in two dimensions [34], further simplified and extended to physical renormalization conditions [35], and shown to work for QED with a photon mass [36], furthermore applied to composite operator renormalization and the short-distance expansion [37]. The simplifications over all previous methods are substantial: combinatorics of graphs is replaced by the Leibniz rule for derivatives of products, the only integrals needed are of the form f x~(ln x)~dx. It is fair to remark, however, that massless particles have not yet been treated and also no proof of local Borel summability of the perturbative series has been given so far by this method. These proofs have been given using the Gallavotti-Nicol5 method [38, 39] which uses a discrete scale decomposition of the propagator and therefore is still more powerful, although not quite as simple. For simplicity, consider a scalar field theory with ultraviolet cutoff A0, and introduce a flow parameter h < A0. The effective action at scale A, L h'A°, is obtained by integrating out the fluctuation fields which have momentum between A and A0,

e--LA'Ao(¢) = const,f

(¢ -

The constant is chosen such that LA'A°(0) : 0,

M. Salmhofer/Analytic results- latticeand continuum

86

and Pc]o is the Gaussian measure with propagator c , *°(p) -

+1

I~(~-~-7o2) - K(~-7)

,

(14)

where K is a monotonically decreasing smooth function with K ( x ) = 1 for x < 1 and K ( x ) = 0 for x _> 4. Thus, C~°(p) = 0 if Ipl < A or [p[ > A0, which restricts the m o m e n t a of the integrated fields to be between A and A0 (the precise definition of C A° contains a p-independent factor [35] which helps to show certain smoothness properties; it has been dropped here because it plays a more technical role; for a thorough discussion of this factor see [37]). The action at cutoff scale

LA°=6m2f £--6Z/¢A¢+ABf ¢

(15)

contains the bare parameters 5m 2, 6Z and the bare coupling AB, that is, interaction and counterterms. Later on, the improvement term will be introduced as an extra term in L A°. The map L A° ~ L A'A° is a continuous version of a blockspin transformation, and variation of A generates the renormalization group flow. At A = 0, all fluctuations are integrated out, and L °,A° generates the connected a m p u t a t e d correlation functions with UV cutoff A0. We are thus mainly interested in l i m A o ~ L 0'A°. The functions are connected because the logarithm is taken to get the effective action. Taking b-X, 0 one gets Wilson's renormalization group equation which is a non-linear functional differential equation for L A'h°. In perturbation theory, this can be rewritten as an infinite system of equations which can be controlled by simple integration. Perturbation theory starts by expanding L in a formal power series O<3

r=l

X¢(pl) .. . ¢(pn)

n>_l

A,A0 (pl, • . . , p ~ - : ) t:r,,

(16)

in a parameter A which will soon be fixed to be the renormalized coupling. Note that 1. the integral kernels ~QA,Ao are the connected a m p u t a t e d Greens functions of order r >_ 1 in A, with n external m o m e n t a and propagator CAA°(p),

in other words they contain the effects of the interaction terms and nothing else, i.e. the zeroth order terms are already in the propagator. 2. If A < A1 < A0, C A° = C A I + C A° , so the flow A ~ L A'h° has the semigroup property. This justifies the name effective action for nh'A°: for any A < A1, the Greens functions £a,ho can be calculated using C hi as propagator and £hl,ao r,7~ as vertices. This point is also important technically because it allows to stop the flow at any A1 > 0 and then continue to A = 0 by calculating loop diagrams of the effective theory. Thus limh0-,~ £0,Ao ~7",Vt exists if the same can be shown for because all loop integrals are finite. Note rh,,Ao ,n that here the fact that the propagator is massive, m > 0, is used. In terms of the £, the flow equation reads

69 A Ao ~A Ao~ O'i~r,n (Pl,..,Pn-i)~-r'n (.Pl,..,Pn-1)

(17)

where, abbreviating p = ( P l , . . . , P , ~ - I ) , dropping unimportant constants,

.T~A,A° (p] = rjrt \ /

+ nt.Jl-nllE =n-~2

and

f dq(OAChAo(q))Lr,g+e(q, A Ao --q, P) {t~ ~Ao~t,A,AorA,Ao "~

S i~uAwA )J-'~,,,~,~'r,,,,~,,) (p).(18)

rtq-rtt~r

The first contribution to the right side is a loop integral, with the integration over q restricted to h < Iql -< 2A because 0AC~°(q) = 0 elsewhere. The second term generates tree diagrams, the mom e n t a in the two £ and COACA° are permutations and linear combinations of P l , . . - , pn. The symbol S stands for symmetrization in Pl, • •., P,~ and will not play any role in the following. For details and the derivation, see [33, 35]. The physical renormalization conditions are imposed on the Greens functions and thus enter as boundary conditions at A = 0. The renormalized coupling can be defined as the value of the connected a m p u t a t e d four-point function at zero external m o m e n t u m , which in terms of the

means •0'A°(0, 0, 0) : ~r,1r,4

(19)

There are two more conditions, for the twopoint function, which can for instance be taken as

87

M. Salrnhofer /Analytic results - lattice and continuum ~ ~ 0 Ao ~r,2/'0'A°= 0 and OuO~L~I 2 = 0. With this choice,

the mass m appearing in the propagator in (14) is the physical one. The conditions described here are only one possible example, the general ones are given in [35]. There is another boundary condition at A = A0. Here the action must take the form (15). The use of these mixed boundary conditions requires some care about consistency. It can, however, be shown to all orders in r that the three physical conditions at A = 0 uniquely - and consistently determine the bare parameters ~m 2, 6Z and AB. Moreover, all three are of order at least one in A, as it must be. Using the semigroup property mentioned under 2. above, it is possible to transport the boundary conditions from A = 0 up to a scale A1 > 0. This is technically convenient and gives rise to a condit.ion/,~,, ~,A14' Ao/r~ (u 0 , 0 ) : ~ , l + O ( r - 1 ) w h e r e O ( r - 1 ) stands for a finite sum over Feynman diagrams with vertices I"~'pIl~71 fA~,Ao with n' > 4 or r' < r. The boundary conditions imposed in the early work [33, 34, 30] for the effective action at some "physical mass scale" are similar to this in spirit, but the physical ones are those given above because the only a priori given mass scale is m here. Note also that h i is an auxiliary device and does not correspond to any physical scale. The bounds given below for the Greens functions l:0,Ao may depend on A1, but the functions themselves do not. It is quite instructive to verify that the form of the boundary conditions at A1 induced by those at A = 0 is such that the proof of renormalizability goes through. The main result about renormalizability, originally proven by Polchinski, but in its present stronger form due to Keller, is L e m m a : For any M > O, all A between A1 and Ao, for all orders r and n u m b e r s of external legs n, and all m o m e n t a [Pi[ <_ M , A a o (Pl, . . . , Pn-1)[ < A4-~P(log A___) [l:r,'n

A1

-

0 ,,AAo, -O~O l..'r 'n ( P l . . , Pn - 1 )

< A h - n p q o A0, "-~0 2

(

g "~l ~

(20)

(21)

where P and P are polynomials whose coefficients depend on M, A1, m but not on A or Ao.

These bounds imply that limao--.~ £A1,A0 ex--rlr~ ists: (20) means that, as a function of A0, l:A,io is r,n bounded everywhere, and (21) that the derivative vanishes like A0-2(log A0)Z as A0 ~ oo. Thus the l:h,Ao have a finite limit as A0 --~ 0% and the rate r~n of convergence is A0- I log A0. One can avoid transporting the boundary conditions to A1 by a modified induction hypothesis [37]. In Polchinski's version [Pi[ _< A was required, whereas here M is independent of A. Also, Polchinski had only log Ah--~ in the first bound, which by itself was not sufficient for boundedness. The proof [33, 35] is simple enough to be sketched here. The functions ~£A,Ao are labelled r~n by a pair of indices (r,n), r the order in A, n the number of external momenta. At a given r, l:A,Ao .=-- 0 if n is too large, n > no(r), because the l: are given by connected diagrams and one needs a minimum nmnber of vertices to join a given number of legs. For example, n0(1) = 4, because one cannot even make a tree diagram in ¢4 theory with six external legs and only one vertex. Similarly, no(2) = 6, and so on. The flow equation (17). is an infinite system of differential equations labelled by (r, n). The structure of this system can be visualized as (r, n) ~- (r, n + 2) + ~

( / , n')(r", n").

(22)

rt~r rll~r

The restriction to r' < r, r" < r in the sum comes from the fact that all l:A,Ao are of order r > 1 in r,n A. The main point is that all terms on the right side have either lower order r' < r or a larger number of legs n' > n. Thus one can do an inductive proof, ordering the indices (r, n) upwards in r, and at fixed r downwards in n, starting above no(r). That is, one proceeds with (r, n) = (1, 6), (1,4), (1,2), (2, 8), (2, 6), ( 2 , 4 ) , . . . , using eq. (20) as induction hypotheses. Because of (22), in every step of this procedure the right side of the equation c0AI: = .T is then already known to obey (20) by the induction hypothesis, and one can simply integrate to get t" A H

£a'.Ao _ ,:a",Ao = l a.~r,,ao -r,n ~,n J a I - - - r,n .

(2a)

At this point the boundary conditions enter: one chooses A~ = A1, A" = A for the 'relevant' terms

88

M. Salmhofer /Analytic results - lattice and continuum

(n _< 4), and A' = A, A" = A0 for the 'irrelevant' terms (n > 5). After that, elementary integration and, for the relevant terms, a Taylor expansion in the momenta suffice to establish (20) by induction. Knowing (20), one then does a similar induction for (21). 3.3. I m p r o v e m e n t The goal is to modify the £ by changing the boundary condition at A0 slightly, so that the modified functions still have the same limit as the ones above, but convergence is sped up by a factor A0N. To motivate the choice of the improvement term, it is instructive to look back at the unimproved situation: the rate of convergence as A0 --* oc is determined by eq. (21), which is derived similarly as sketched above for eq. (20), by integrating the derivative w.r.t A0 of the flow equation. One ends up with an equation similar to (23), but containing 0ho.~, which suggests that the improvement may be achieved by subtracting a suitable part of •. Keller's improvement term [31] is given by oo

Ao I

fA°--/dAot~-~--oI/dAIr4+N-nT' ~'A"A°'r,n Ao

r,n

(24)

A1

where r k means Taylor expansion up to order k if k > 0, and zero otherwise. Note that 1. once the cutoff action is changed to L A° -tIho, all Greens functions are changed, call them ~A,ho The ~- on the right side of (24) is given r~n " by (18), with £ replaced by/~ everywhere. (23) is a definition of 2" in the same recursive sense as above, i.e. on the right side only terms of lower order or with more legs appear. 2. The Taylor expansion makes the improvement terms polynomial in the momenta p, hence local in position space. 3. Because r4+N-'LT" = 0 for n > 4 + N, only a finite number of terms is added in every order and for every n. The improvement term has the effect that (21) is replaced by A 5-n+N OAo£r'~ (Pl,..,Pn-1) < A02+ N ~A A0

_ - - P ( ] o g

A0 )

(25)

This inequality implies the improved convergence.

Improvement T h e o r e m : Given N >_ 1, adding (24) to the cutoff action, for all r and n, the improved Greens functions /~0,ho have the same limit as the unimproved ones, and the convergence to the limit is as A0-1-N(log A0) ~ , for some ~>_0.

The proof and the extension to QED are contained in [31] (see also [30]). 3.4. N o n p e r t u r b a t i v e R e s u l t s Outside of formal perturbation theory, one really has to deal with the non-linearity of the functional differential equation. Nevertheless, the use of flow equations has led to simplifications of proofs and the clarification of concepts there as well. Existence of solutions for small changes of A ('short flow times') bounded uniformly in the volume for lattice systems has been shown using Hamilton-Jacobi theory [40, 41]. The block-spin analysis of the hierarchical sine-Gordon model was treated very elegantly in [42]. In [43] the results of [40] were combined with a novel definition of an exponential as a function on sets which automatizes the cluster expansion and directly rewrites the effective action in terms of the polymer activities. This allowed for a simplified proof of analyticity in the fugacity of the Coulomb and Dipole Gases without a splitting into smallfield and large-field regions [43]. In [44], the Kosterlitz-Thouless phase of the full sine-Gordon model has been analyzed using the techniques of [43]. Infrared asymptotic freedom has been shown in noncompact lattice QED for small hopping parameter ~¢ [45]. 4. T h e H o r i z o n Function This section will focus on the geometry of non-abelian gauge theory, and in particular on Zwanziger's recent work on the effects of the boundary of the fundamental domain for the continuum limit of lattice gauge theory. 4.1. Continuum Recall that in a gauge theory, the physical configuration space is given by the space of gauge orbits, which is obtained from the space of gauge

M. Salmhofer/Analytic results- lattice and continuum connections by factoring out the action of the gauge group. For practical purposes it is more convenient to deal with the gauge fields A~,(x) instead of the gauge orbits {Ag, I g(x) E G}, where

Ag~,(x) = g-l(x)(O~, + Ai,(x))g(z);

(26)

for this one has to fix the gauge, that is, choose one point on every orbit to represent this orbit. From such a gauge fixing procedure one then gets a definition of a fundamental domain .T which is in one-to-one correspondence with the space of gauge orbits. Gribov showed that in non-abelian theory, the gauge cannot be fixed by imposing O~,A~, = 0 because the hyperplane F defined by this condition is intersected more than once by the gauge orbits [46]. In particular, the naive Faddeev-Popov trick does not work because the determinant of the change of variables has zeros, and 9r ¢ F. In 1982, Zwanziger obtained further information about 9r [47] in the following way. Consider the functional

I(A,g) =

IIAgll2 = f

(A)~, (x) 2

(27)

aJ.z

(the sum over a is over the components with respect to generators t a of the Lie algebra). Obviously, I(A, g) > O, and for given A, one can look for its minima on the gauge orbit, i.e. the minima of g ~-~ I(A, g) at fixed A. Such minima exist on every gauge orbit, as proven by Dell'Antonio and Zwanziger [48]. Selecting one of these minima, one would then get a representative on the gauge orbit. Doing the derivatives, one finds that stationary points are at those g for which O,A~(x) = O, i.e. those g for which Ag E F. The second derivative is M(Ag) = -O,D~,(Ag), the usual FaddeevPopov operator whose determinant appears in the Faddeev-Popov formula (D~,(A) denotes the covariant derivative, Dz(A)f = O~,f + [A,, f]). Defining the first Gribov region f~ as the set of all the minima (for different A), one thus has the theorem [47] that .T C f~ C r. However, f~ is still larger than 5r because there may be more than one minimum on every gauge orbit [52]. Taking

89

only absolute minima does not help because there may be degenerate ones at the boundary. To get )r one must either select one of them or identify them; the space obtained by the latter procedure is then topologically nontrivial, as is the space of gauge orbits [50]. But the inclusion .T C f~ is very useful because f~ can be studied more easily than .T itself. Zwanziger showed [47] that ~ is convex, A = 0 is in f~, and f~ is contained in an ellipsoid, thus is bounded in all directions. That A E f~ restricts the size of A can be seen from the fact that in f~, the Faddeev-Popov operator M(A) is nonnegative for A E f~ because there A is a minimum of g ~ I(A,g), but that M(A) is also linear in A. The presence of the boundary of f~ suppresses low Fourier modes, and the gluon propagator vanishes at zero momentum [49]. 4.2. L a t t i c e For a gauge-invariant observable O(U), consider

1/ v A u e - s ( v ) ° ( u )

(28)

(olc = -2

where S is some gauge-invariant action, and

~DAU = IL dUl, with dU the invariant measure on the group, and the product running over all links of A, Z fixed by (1)c = 1. If this compact formulation (integration over the group variables) is chosen, then gauge fixing is not necessary a priori. It is, however, needed in perturbation theory, and it can be convenient for various reasons. Taking gauge fields U,(z) E SU(n), one can define [51] a gauge fixing term by

IL(U,g)= ~ links l

where U / =

(1-1Re

tr U f )

(29)

n

+ e.)-i

for l =

Note that the sum in /L is over all links of the lattice. In the classical continuum limit, 1L(U) ½1lAgll~, and (at least for SU(2)) the statements about Y" and f2 made above carry over to the lattice. Moreover, on a finite lattice, the boundaries 8f2 and g ~ are of measure zero in the space of all configurations [51]. One can do the usual Faddeev-Popov trick, inserting a factor 1 = A(r, U) f r L : dg(x) e-r~L(u,g)

90

M. Salmhofer/Analytic results- lattice and continuum

into (26). The limit r -+ c~ then selects the minima of IL and thus (O)c is exactly the same as =

1 f:DAUe-S(U)O(U) x6 (cOuAu) det M(U)xy(U)

(30)

where A~(U) = tr (t~(Uu(x) - U;(x))), cOuAu denotes the lattice divergence of A(U) and - M ( U ) , the second derivative of IL, is the lattice Fadeev-Popov operator (the generators t a are normalized such that tr(tat b) = - - 216ab) The characteristic function X7 restricts the U • F, for which OuAu = 0, to 9t', and thus incorporates the effects of the Gribov horizon. Clearly, the function X~- is the object which is hardest to treat in (30). Zwanziger has given an argument [51] that X5 can be replaced by a Boltzmann factor which can then be treated by known techniques. In contrast to the preceding ones, this derivation is not rigorous; however, the implications of the result are strong and it contains very interesting ideas. 4.3. T h e H o r i z o n F u n c t i o n One can order the eigenvalues of the operator M(U) so that they form an increasing sequence, and define the n'th Gribov horizon as the set where the n'th eigenvalue of M(U) vanishes. For U = 1, M(1) = - A , and the eigenvalues A(1,p) to every value of the momentum p are degenerate. This degeneracy will be lifted for U ¢ 1, and the eigenvalues are then labelled as Ai(U, p). Writing M(U) = - A + A T / ( U ) , the Ai(U,p) can be calculated using degenerate perturbation theory in ~/(U). Zwanziger has resummed the perturbative series for large volumes under the assumption that no level-crossings between different p occur [51]. For the "center of gravity" of the eigenvalues

m(U,p) = g-1 -~ A,(U,p) i=1

(31)

A(1,p)

where g is the degeneracy of A(1, p), the result is 1

m(U,p) = 1 - (n 2 _ 1)dvH(U)

Z<~ =

/ dpT(U)e-~n(v)

(34)

where a > 0 is a dimensional parameter which has to be fixed similarly to the determination of the temperature in statistical mechanics. In our case, a is fixed by the horizon condition [51]

(H), = d(n 2 - 1)V.

(35)

which is due to the concentration of the measure at the boundary. One can argue that leaving out )~- is justified in perturbation theory once the new Boltzmann factor is included to take into account the effects of the boundary [51]. The new term in the action H(U) modifies already the zeroth order propagators; in particular, the gluon propagator now becomes

(32)

where d is the dimension, V the volume, and the horizon function is

H(U) = (n 2 - 1)IL(U) + (B, M(U)-IB).

B is a field given explicitly in terms of U [51], the brackets denote an inner product. By definition of.T, m(U,p) >_ 0 for all U • 5 , so 0 < H(U) < (n 2 - 1)dV for all U • ~'. Starting from U • .T and moving towards the boundary of .T, the eigenvalues vanish when one arrives at the boundary OF. Note that the right side of (32) does not depend on p, so there are points on the intersection of all horizons. In classical statistical mechanics, the phase volume f dnpdnq concentrates in a thin layer under the surface of every sphere if the number of degrees of freedom n becomes very large. An analogous phenomenon may also happen [51] in the case of the gauge measure d#y(U) = DA(U)e-S(U)xj:(U ) det M(U)6(OuAu). One can rewrite Z y = f dEe ~(E) where the entropy a(E) is defined as (r(E) = ln P ( E ) = In f dpy(U)6(H(U) - E), thus, extending the above analogy to include the horizon function H as a Hamiltonian, Z~- is a microcanonical partition function. By equivalence of ensembles, it can then be replaced by a canonical one, Z7 = Z~,

(33)

b,,(k)

=

k', + nT0

(36)

where 70 is a constant of dimension (mass) 4 proportional to o~, so D(k) vanishes like k 2 at k = 0.

M. Salmhofer /Analytic results - lattice and continuum

The theory obtained by taking the continuum limit of the action is still perturbatively renormalizable, and it has a BRS symmetry [53] (because of the second term in H(U), there are more ghost fields than in the ordinary theory). The horizon condition, which in the formal continuum limit contains a logarithmically divergent integral can also be renormalized, and the quantity Ah = (nTren) 1/4 is a renormalization group invariant. Bound-state masses and string tension remain non-perturbative issues also in the horizon-theory [51], but the conclusions listed here, in particular the fact that the perturbative expansion can be renormalized, substantiate the analysis. Last but not least, the assumptions made can, and should, be further checked by simulations (see Appendix D of [51]. 4.4. Further R e s u l t s A different approach to finding a useful parametrization of the space of gauge orbits is pursued by Loll [54, 55]. In this parametrization one works with the holonomy variables (path ordered products over loops). This has the advantage of keeping gauge-invariance. The problem to solve in this approach is that the traces of these loop variables, the Wilson loops, are not all independent, but subject to constraints due to the identities between the traces of products of matrices (Mandelstam constraints). Loll has solved these constraints for the SU(2)-lattice gauge theory in d = 2, 3, 4 by writing down a cotnplete set of independent loop variables [54]. This can be considered as a characterization of the configuration space of SU(2) gauge theory. To extend this to the quantized theory given by the integral (28), one has to rewrite the gauge measure in terms of the loop variables. Results in that direction have been obtained in [55].

5. C o n s t r u c t i o n o f YM4 w i t h a n i n f r a r e d cutoff The title of this section is the same as that of a preprint by J.Magnen, V.Rivasseau, R.Sdndor (to whom I shall refer to as MRS in the following) which appeared in February [56, 57] and on the

91

results of which I am to report. This is again a continuum topic, but since Yang-Mills theory is the subject that essentially started lattice gauge theory, it is certainly of interest to lattice theorists. I will try to give an account of it as well as I can as a non-expert. The presentation I give here is extremely simplified and probably does not do [57] justice, so the critical reader who wants to know how things are really done in detail should read [57]. 5.1. Overview The result is that the continuum euclidean correlation functions of SU(2)-Yang Mills theory without matter fields have been constructed in a finite volume (this volume is the infrared cutoff) and in the trivial topological sector, for small gauge coupling. "Constructed" means that they have been proven to converge as the ultraviolet cutoff is sent to infinity. MRS have also shown that in the limit, the Slavnov-Taylor identities, which express the gauge structure, hold. This is necessary in their approach because the cutoffs they use are not gauge- invariant. This may be a good place to remember the work of T.Ba~aban [58], who, in a series of about ten papers, has shown that the effective actions obtained by his blockspin technique in the framework of lattice gauge theory converge in the UV limit. Since Bataban uses the lattice approach, gauge invariance is no problem in his work. MRS state as an advantage of their method that they also construct correlations, not only the effective action, and that their method is simpler. "Simpler" does not necessarily mean simple, as one can see from the starting point (II.78) [57] of their paper. It is fair to remark, however, that "starting point" here means the formula where all preparations have been made for a mathematical analysis. The analysis of MRS is nonperturbative, but restricted to small couplings, they do not prove any statement about confinement or a mass gap (remember that this is all in finite volume). The perturbative results are not needed to all orders but only to two-loop order. This construction has been done using the multiscale expansions which were already sufficient

92

M. Salmhofer/Analytic results- latticeand continuum

to show infrared asymptotic freedom of (I)~. The reader who is interested in this technique should consult [60] which also contains a good discussion of the problems in Yang Mills theory in the last chapter. To proceed, we need some basic definitions and notations. The finite volume is taken to be the torus IR4/Z 4 in Euclidean space, so the momenta are discrete. The zero modes are deleted. The gauge group is SU(2), Au(x ) = t~A~(x), t ~ = icr~/2, and the topologically trivial sector means that At,(x), as well as the gauge transformations g(x), are periodic funtions of x. Calling the coupling A, the gauge transformed field is A (g) = gAug -1 + ~(Ot, g)g -1, and F ~ = Ot,A ~ O,A i, - A [mu,A,]. Write g(x) = exp(AT(x)), where 7(x) is an element of the Lie algebra. Cutoffs are imposed as A0 = MP where M > 1 is some fixed scale and p a positive integer. The technique used is the phase space expansion, which is in some respects similar to the flow equation method described in Section 3, only that here a discrete scale decomposition is used, where one analyses slices M i <<_IPl < Mi+l in each step. Among other things, MRS show that the flow of the coupling at scale MJ, ,~j is close to the perturbative result Aj 2 =

1 ( - ~ 2 ) j log M + ~ logj + C

(37)

if C is very large, i.e. the starting coupling is very small (here ~2 < 0 and /~3 are the usual perturbative coefficients of the/? function). 5.2. P r o b l e m s The following is a list of problems [57, 60] peculiar to gauge theory. 1.Positlvity Problem: By definition, the action F 2 = ¼f F ~ F ~ >_ 0, and so one may hope to define a decent Boltzmanu factor by e -r2. But the formal expression I L dA~(x) does not exist as a measure in the continuum, and so the quadratic part of the action is taken out and put into a Gaussian measure. Thus, as in perturbation theory, writing F 2 = F2 + AF3 + A2F4, F3 and F4 denoting the cubic and quartic terms in the action, one will define a propagator from F2+ gauge fixing term, and regard AF3 + A2F4 as interac-

tions. But in contrast to the ¢4-case, this interaction term is not bounded below in a useful way: although F4 = - tr ([Au, A~] 2) > 0, it is not strictly positive, and the decay of e x p ( - F 4 ) is too weak in some directions. This problem is already present in the cutoff theory. 2. Gauge non-invariance of the scale decomposition: Any momentum cutoff A0 which restricts the allowed values of the momenta to IPl < A0 is non-gauge invariant, and the same is true for scale decompositions by a partition of unity by momentum slices M i g IPl < Mi+l. Because of this, non-gauge invaxiant counterterms are necessary to achieve finiteness of the UV limit and also to restore gauge invariance in the UV limit in perturbation theory (one has a similar problem also in the flow equation method described in Section 3). For QED, Hurd has shown that renormalization and restoration of the Ward identities is possible using such non-invariant counterterms [59]. The analogous problem in the flow equation approach has been solved for QED by Keller and Kopper [36]. Besides terms like A S, A A A and others, one in particular needs a counterterm of type A4 = E a

E ( A ~ ,)4

(38)

tt

to obtain finite limits for the correlations and to restore the Slavnov-Taylor identities. 3. Gribov problem 4. Domination problems: these concern terms in the effective action which couple to lowmomentum large fields. These are difficult technical problems which cannot be touched here. The positivity problem is absent if the action contains an A 4 term of the type in (38) which has a positive coefficient ~ > 0 so that the Boltzmann factor is e x p ( - a f A4). It turns out that the gauge-restoring counterterm (38) has a positive coefficient, so that the solution to the second problem, namely restoration of gauge invariance, also provides a solution for the first one. The Gribov problem is circumvented by first choosing axial gauge where there is no Faddeev-Popov determinant and performing a change of gauge only in regions where the fields are small.

M. Salndwfer /Analytic results- lattice and continuum 5.3. S o l u t i o n s The analysis starts with a decomposition into large field and small field regions. This decomposition may be motivated through the well-known (see, e.g. [61]) example of the one-dimensional integral oo

I(A) =

dee

a ___ 0.

(a9)

93

nical complication because the axial propagator is anisotropic. Axial gauge does not completely fix the gauge, and to do perturbation theory, t h e gauge is changed in the small field region by an analogue of the usual Faddeev-Popov trick, i.e. insertion of 1 = det(Ot, D ~ (A))

/ r l dT(x)e-¢/2(°"a(/))~ (40) d

The perturbative expansion I(A) .v ~a,~,V ~ diverges because a,~ = f ~ d¢(-¢4)~e -e2 ..~ n! although I(A) is well-defined and continuous for all ~ _> 0. Nevertheless, after rescaling to y = All4¢, a standard application of Watson's lemma [62] shows that the expansion in A is asymptotic to I(A). The idea behind this is that the contribution of the region Y _ c is exponentially small in A for any c > 0, and that for the integral over [0, s], Taylor expansion works. In fact, if one integrates only over a finite interval, the coefficients dCe_¢Z¢4n < 2A4n for any A, so in that case the perturbation expansion converges. Thus one may call the cause of the divergence of the perturbative expansion of (39) a "large field problem" [61], and its solution is obtained by splitting the integration domain into different regions. The analysis is of course much more complicated in field theory than in this simple example, but again, the small field region is chosen such that perturbation theory is a good approximation there and that its complement, the large field region, only contributes exponentially small terms. The starting point of MRS is obtained by the following preparations. First the axial gauge A0 -- 0 is chosen, and a term e x p ( - a f A 4) is introduced into the integral. For a given configuration the large field region shall consist of those x for which At,(x ) >_ A -1/2-~, c > 0, and its complement is the small field region. The functional integral is then split into a sum of integrals with fixed large field region. One of the results of the analysis of MRS is that the occurrence of large large field regions is very improbable. The actual definition of the large field region is, technically seen, very different from the description given above: the size of A is determined by the axial propagator. This is a tech-

fA

X

into the integral. In this formula, 7 is Lie-algebra valued, g = e i~'y, and ~ is the gauge parameter. (40) is a formal identity. In the rigorous version of it which MRS use, the formal flat measure is again replaced by a Gaussian one which restricts the size of 7 so that the transformed field is again small. Moreover, there is a polynomial interaction term for the 7, and the gauge transformation is replaced by a truncation derived from taking the first two terms in the expansion g(x) = e iv(~:)~ in powers of A. Thus what is really done is a change of variables which is only an approximation to a gauge transformation. The advantage is that the so defined change of variables can then explicitly shown to be invertible. However, the gauge symmetry is further violated by this truncation. These violations must be shown to disappear in the limit where the cutoff is removed; this is another reason why the Slavnov-Taylor identities have to be checked in the end. That they are satisfied is the ultimate condition that the Gribov problem can really be avoided by chosing the small field region and the approximate gauge transformation. The gauge parameter ¢ is chosen to have a value close to 3/13~ which defines the "homothetic" gauge. The advantage of this gauge is that no wave function renormalization is needed. Now perturbation theory is done, and MRS show that the one-loop gauge restoring A4-counterterm is positive. Moreover, with a particular choice of the cutoff function it can be made as large as one needs it. The A4-term inserted at the beginning is now chosen to have the coefficient given by perturbation theory, and so the solution of problem 2 in the small field region also solves 1 everywhere. MRS state that this counterterm can also be used to stabilize the large field region although its flow

M. Salmhofer/Analyticresults- latticeroutcontinuum

94

(i.e. its dependence on j) is determined by the small field region and perturbation theory.

6

5.4. C o n c l u d i n g Q u e s t i o n s Many points are not explicit in [57], and I have not been able to go through the proofs. Also, some simple questions remain open to me, namely 1. how A0 = 0 is done on a torus, 2. how the anisotropy of the axial gauge is overcome at the end, 3. how large the volume is. Really imposing A0 = 0 on a torus would fix physical degrees of freedom. However, MRS do not strictly work in ax!al gauge because of the various modifications which they make to get their starting point. There is no detailed discussion of this problem in [57]. The question how large the volume is in physical units is of course very important. Since it has been shown that the Gribov copies become important when the volume increases, one may suspect that the way it is avoided by MRS in the small field region indirectly restricts the size of the volume [63]. Technically, such a restriction may arise from the smallness condition on which enters together with properties of the axial propagator into the determination of the large field region. I would like to repeat that these questions may be due to my incomplete understanding of [57]. I hope that I have at least given a motivation for reading it, so that anyone who does that can decide about the physical implications of this important work.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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tion of YM4 with an infrared cutoff, Eeole Polytechnique preprint February 1992 T.Balaban, Comm. Math. Phys. 122 (1989) 175, 355, and reference~ therein T.Hurd, Comm. Math. Phys. 125 (1989) 515 V.Rivasseau, From perturbative to constructive renormalization, Princeton University Press, 1991 J.Feldman, in Proceedings of the 9th International Congress in Mathematical Physics, Adam Hilger, 1983 C.Bender, S.Orszag, Advanced mathematical methods for scientists and engineers, McGrawHill, 1978 I thank Pierre van Baal for pointing this out to me