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Strong coupling expansion in random lattice gauge systems and interacting gaussian surfaces
Nuclear Physics B280 [FS 18] (1987) 1-12 North-Holland, Amsterdam
S T R O N G C O U P L I N G E X P A N S I O N IN R A N D O M LATYICE G A U G E S Y S T E M S A N D I N T E R A C T I N G G A U S S I A N SURFACES Jean-Michel DROUFFE Service de Physique Th~orique, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France
Received 10 July 1986
We present a strong coupling study for a pure gauge random system, where the interactions are cut off using a gaussian form factor (the nearest-neighbour concept is not used). Double series in the inverse Yang-MiUscoupling and in the density are obtained. We focus here on the SU(2) four-dimensional system. High and low density regions are investigated. We find that the high density limit is identical to recently proposed models of gaussian random surfaces, while finite density corrections introduce interactions between these surfaces, depending on the gauge group. Comparison with a Monte Carlo analysis is performed in the medium density region. We finally propose a phase diagram for this two-parameter model.
1. Introduction There has recently been proposed [1] a random lattice regularization of QCD, in which the corresponding action couples any two points on the random lattice, with a gaussian damping form factor of range b in the euclidean distance of the points. The introduction of the nearest-neighbour concept [2] is avoided. As a consequence, it is possible to compute practically some physical observables with various techniques. For instance, ref. [1] presents the estimation of the hadron spectrum in the strong coupling limit. The aim of this paper is to analyze the strong coupling expansions for a pure gauge system. N o matter fields are present, and series are calculated up to a rather high order. Two variables appear in this expansion. One is the usual inverse Yang-Mills coupling r , the other one is the number of lattice points within the interaction range o = ob d (p being the lattice site density). Varying this last p a r a m e t e r leads to several possible regimes. In an intermediate one, it is possible to tune the range b so that the number of interacting points is roughly the number of nearest neighbours. The model is thus expected to behave as the standard random system [2] with a nearest-neighbour interaction. We performed a Monte Carlo run for the SU(2) four-dimensional system which indeed shows a behaviour quite c o m p a r a b l e to that of the nearest-neighbour model [3]. In particular, curves are very 0619-6823/87/$03.50© Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
2
J.-M. Drouffe / Strong coupling expansion
regular, without any evidence of a structure as the one observed in regular lattice systems. It is therefore tempting to analyse strong coupling series, in order to realize a dream: to enter the physical region from the strong coupling limit. Such a calculation is presented here. Alas, straightforward extrapolations of the series can hardly fit the very beginning of these regular curves. The crossover phenomenon occurring in regular lattice systems at intermediate fl and preventing any extrapolation into the weak coupling region now seems to appear from the beginning. The reason is analyzed while resumming the series as a low density expansion. It is shown that the radius of convergence in fl is strongly limited by singularities on the imaginary axis, due to the gauge group fundamental functions. Such a behaviour is controlled after resummation as a functional series in o, and Monte Carlo results are then nicely fitted for any values of ft. Another interesting region is the high density limit. At the lowest order, the model coincides exactly with proposed models [4] of surfaces, represented by their gaussian random triangulations. The role of the gauge group disappears in this limit, which appears to be universal. Such models have been extensively studied [5-9] and are expected to present a non-trivial critical point. In the system studied here, corrections depending on the gauge group appear as o decreases from infinity. They can be interpreted as various interactions between the surfaces, leading to non-planar diagrams as well as interactions along a common boundary singular line. The strong coupling expansion allows us to track the transition observed at o = oo when it enters the ( f l - o) phase diagram, and to speculate about its exact nature and relevance. In sect. 2, the techniques used to derive the strong coupling expansions are described. Corresponding calculations and results for the SU(2) system are given in an appendix. Sect. 3 is devoted to the low density analysis, and presents our Monte Carlo simulation. The high density limit and the phase diagram are discussed in sect, 4.
2. Derivation of the strong coupling series The action of the model [1] is defined on a random lattice with mean density of sites 0 (x y z) S = ~ ~_, G Tr(U~yUy~U~) (1) {x,y,z} b ' b ' -b The summation runs over all (unordered) triplets of points x, y, z. Independent gauge fields Uxy = Uy-~1 are thus introduced for any pair of points, even largely separated. The cut-off form factor G restricts the interaction to a range b. In principle, any function, rotationally and translationally invariant, can be retained,
J.-M. Drouffe / Strongcouplingexpansion
3
provided it keeps the three points within a finite volume (so a cut-off built on the basis of the area of the xyz triangle cannot be retained [6]). For the sake of calculations, we restrict ourselves to a gaussian shape
G(X,Y,Z)=exp(-(X-
Y)2-(Y-Z)2-(Z-X)2).
(2)
This model is expected [1] to have a continuous limit b ~ 0 at fixed a = pb d which describes the Yang-Mills continuous theory. The following relation holds between the couplings 48 • 3d/2 b d-4
fi = ~d/~
O~g~ .
(3)
The observables should be averaged over all the possible random lattices with fixed density p (quenching procedure). To derive the strong coupling expansion, we follow the character expansion technique [10] applied for a fixed random lattice. The exponentiated action is developed in terms of irreducible characters Xr, using the Fourier coefficient functions fi,(t) defined by
exp(tTrU)= fi°(t)[l + ~d'fir(t)xr(U)]
(4)
where d r is the dimension of the irreducible representation r. This leads to terms diagrammatically represented by sets of triangles; to each of one a representation r has been assigned. The integration over gauge fields is done using the methods of group theory, and leaves only closed diagrams. The simplest topology encountered is homeomorphic to a triangulation of a closed surface, with the vertices on the random lattice sites; however, more complex diagrams with, e.g., n-fold singular lines should also be considered. The contribution of a given graph is to be summed over all different locations of its vertices on the lattice. The partition function is obtained by summing the contributions of all (connected and disconnected) diagrams. Using standard excluded volume expansion techniques [11], the logarithm of the partition function is then computed. It appears as a complicated function of the form factors fiG(X, Y, Z) of all the triangles entering in a connected diagram. It is this free energy, and not the partition function, which must be averaged over the random site locations (quenching procedure). This is done using the formula
E
¢(Xl
.....
.... x . ) d x l . . . d x ~.
(5)
Xl,...~X n all different
The contribution of each graph is developed as a series in /3 and, thanks to the
J.-M. Drouffe / Strong coupling expansion
gaussian shape (2), the spatial integration (5) can be performed for each coefficient of this expansion. Details and results for the gauge group SU(2) are gathered in the appendix. Note that the result appears as a double series in f l - counting the number of fundamental interactions- and o = pb d- counting the number of vertices. There are two possible ways to perform a resummation: as an expansion in o at fixed fl (resumming all graphs with a given number of vertices), or in 1 / a at fixed fl 2a. These two procedures are examined in the following sections.
3. Comparison with simulations and low density expansions In fact, nothing is known about the present system, and in particular about its phase structure. This paper is intended to answer some questions about its low and high density limit, but we also decided to perform a Monte Carlo simulation to look into possible unexpected effects and to compare the expansions to experiment. This numerical analysis has not been done on an extensive scale; a mini-computer has been sufficient (200 hours on Prime 550). A priori, the model does not seem appropriate for a Monte Carlo simulation. As all pairs of sites bear a gauge field, there are - N 2 quantities to update, each one involving all other lattice points. Hence the sweep time behaves as N 3 instead of N. Moreover, there are storage problems for N 2 fields, plus numerous large arrays needed to access information about the geometry in a reasonable time. Hence it has been necessary to introduce a cut-off, modifying the form factors in such a way that only links of length less than L contribute
(xyz) (xyz)
G b'b'b
-* G
b'b'-b
O(L-
Ix-yl)O(L-
lY-ZI)O(L-
Iz - x [ ) ,
(6)
0 being the step function. The effect of this truncation can be estimated by the deficit on the mean ground state energy
~1( L ) = 1 - ff lx'-xjl
(7)
In a four-dimensional simulation, this deficit is limited to 2% as soon as L/b is greater than 2.4, which is sufficient for our purpose. On the other hand, we have to limit the mean number of gauge fields linked on a particular lattice site, 2¢rd/2a(L/b)d/dF(½d), in order to achieve a reasonable computing time; hence we are limited to rather low values of a. We take a lattice of N = 1000 sites and work with a = 0.25 and L/b = 2.4; the mean number of considered neighbours is then of order 40 (to be compared with the 8 on an hypercubical regular lattice). If the density parameter o cannot be chosen too high, it should be noticed that it cannot also be made too low. Indeed, the density fluctuations induce fluctuations on
J.-M. Drouffe / Strongcouplingexpansion
the action density, which should be correctly averaged on the sample size. A measure of this effect is shown by the fluctuations of the maximal action relative to one point
~,=
~/(1/N)Ex(Ey,~G(x, y, z)) 2- ((1/N)F.,x,y,~G(x, y,
= (~)a/4 /
z)) 2
(1/N)F.x,y,~G(x , y, z) 4
2 +
.
(8)
The quantity ~ / f N will characterize the fluctuations between several samples, and should be sufficiently low. With our choice, it lies around 3%. We have used 4 different lattices (measured quantities are auto-averaged, and few lattices are thus necessary) and a statistical study of their results agrees with the expectation (8). We have measured the normalized energy
(s) Sm=
E = 1 - --
(9)
while heating and cooling a sample from fl = 1 to fl = 200. The gauge group SU(2) has been replaced by its 120 dement discrete subgroup. The resulting curve is displayed in fig. 1. It appears to be very regular, and is quite comparable to the one obtained with a nearest-neighbour interaction [3]. There is no trace of some structure at intermediate fl, such as the one observed with regular lattices, and no trace of a transition. In order to avoid misinterpretation of the unusual high values of fl, we also indicate the corresponding values (quite normal) of the Yang-Mills coupling constant g2. The fit by the strong coupling series and their extrapolations (e.g. Pad6 approximants) is not indicated on the figure. The reason is that they fit the curve badly, except in the very beginning (up to fl - 3). This fact may seem strange when one acknowledges the regularity of the curve; in the regular lattice case, series fit very well up to the crossover region, dearly visible as an accident on the experimental data. The reason for this poor fit can be analysed in the low density limit. We already indicated that the expansion may be reorganized as a series in o, according to the number of vertices in the diagrammatic representation. The first two terms have been explicitly displayed in the appendix (formulae (A.4) and (A.6)). Singularities are easily found in these integral representations and correspond to the singularities of the Fourier coefficients (4). They are analytic on the real axis, but are singular on the imaginary axis. The first one lies at the first zero of I1(2fl), viz fl - + 1.9159i. It limits the radius of convergence far below the required values.
J.-M. Drouffe / Strong coupling expansion 1.0,
50
10
5
3
2
I
I
I
I
I
1.5 I
-(g~)
0.5
i
I
I
50
100
150
-(~) 200
Fig. 1. The mean energy, from Monte Carlo simulation, fitted by the low density expansion. Dashed curve is the zero densitylimit, plain curveinclude the first correction(a = 0.25).
There is nevertheless a way to obtain a reasonable fit, by using this a expansion, which realizes a summation in ft. The first two terms have been used to fit nicely the data of fig. 1 within statistical errors, and on the whole range of ft.
4. High density expansion We turn now to the high density limit. The expansion can be reorganized, according to the topologies of the diagrams, as a 1 / o expansion at fixed fl2a. Indeed, any connected diagrams can be constructed (in a non-unique way), starting from a simple closed surface, by adding simple opened surfaces sharing their boundary vertices and links. We show that the more simple components, the more subdominant the diagram is. For any of these surfaces, topological relations can be written. With the following notations no nl n2 P k g
number of non-boundary vertices, number of non-boundary links, number of triangles, number of boundary vertices (or links), number of boundary components, genus of the surface ( >/0, equal to twice the number of handles for an orientable surface),
J.-M. Drouffe / Strongcouplingexpansion I
I
7
I
1/o t 1.0
[I11]
0.5
,/\/ 5
,
10
15
20
Fig. 2. Nearest real positivesingularityfor Pad4 [1/1], [1/2] and [2/3] at fixed o in the (fl20,1/o)-plane
the relations are 3n 2 = 2n 1 + p , n 2+ k - n 1 + n 0 = 2 - g .
(10)
It follows that the multiplicative factor fl n2o n0 involved in the diagram contribution can be recast as (flv~)n~(1/o) k+p/z+g-2. Any of the open-component surfaces satisfies k >i 1 and p >/3 and is therefore subdominant in a 1 / o expansion at fixed f12o. Also, the higher the genus is, the more subdominant the diagram is. Hence dominant diagrams appear topologically as a triangulation of the sphere (g = 0). Note that, in this limit o ~ oo, the model becomes independent of the gauge group. N o t e also that this limiting case can be obtained in a different way, N ~ oo. The diagrammatic rules are identical to those of gaussian random surface models [6, 7] which have been proposed as a possible regularization of non-interacting string systems. A careful strong coupling analysis has been done [7] and shows evidence for a singularity. Our series, which also contain 1 / o corrections, are shorter in this limiting case, but lead to the same conclusion. We use a Pads analysis at fixed o for the specific heat to locate and follow the singularity in the phase diagram (fl2o, 1 / o ) . Fig. 2 displays the result. Pads [2/2] gives unstable results and has been discarded. A line of singularities separates the plane into two regions. It is not clear, however, that the line extends up to infinity. We have tried to answer this question using a 2-variable generalized Pads approximant [12]. The position of the singularity line is confirmed, but the position of the end-point, when it exists, wildly fluctuates from a minimum fl2o--15, according to the chosen approximant. No serious conclusion can be drawn from this analysis.
8
J.-M. Drouffe / Strong coupling expansion
Several hypotheses concerning the exact phase structure remain possible. (1) The line may be a second-order transition line, near which a continuous limit exists. This is hope for the random surface models, for which a sensible continuous limit is looked for, as a realization of the string model. In our interpretation, the renormalization flow goes to a fixed point at fl = ~ , which describes the Yang-Mills continuous theory. (2) This line may also be the extremity of a metastable region, with a first-order transition occurring at smaller ft. On regular lattices, this phenomenon is always observed as soon as there is a singularity. If this is the case here, random gaussian surface models (o = oo) will not have a sensible continuous limit. Continuous limits are expected at fl = o0 (hopefully Yang-Mills theory), or at the second-order end-point (if any) of this first-order line. It is not yet possible to decide between these two possibilities. Maybe the second one is favored, because of an analogy with regular lattice theory. Numerical simulations like the one described in this paper do not allow us to check the order of the transition (o is too high), and a numerical study [13] of the gaussian random surface model has not yet been applied to this problem.
Appendix The scheme of the calculations has been indicated in sect. 2. We describe only some details and list the results. Let us consider first the influence of the term/3o(t) in eq. (4). It contributes to l n Z a factor ln/~o(flG ) for each triangle, so the quenched average reads, according to eq. (5)
f f f d"xd yd ln °( fla( b ' b '
"
(A.1)
The integrand is expanded in series of r , using ln/~0(t ) =
Y'~bnt".
(A.2)
Each term appears as a gaussian integral; one factorizes the volume V for the zero mode corresponding to translation invariance. Hence lnZ ]
-
"~--2"2] t~iangle
bnfln 6. 3d/~-------~ ~-, n d O2~rd
(A.3)
The series on the r.h.s, can be recast in an integral representation, as for instance
In Z /
o2u d
1
Vb-d]triangle- 6~'~2 f0 (-lnt)d-llnfl°(flt)dt'(d-1)!
(A.4)
J.-M. Drouffe / Strong coupling expansion
9
Let us now turn to the diagrammatic corrections. We call diagram ~ any subset of the triangles of the random lattice. If one restricts the action to only these triangles, one can compute, using the character expansion technique [10], the restricted partition function Z ( ~ ) . One writes, according to the excluded volume expansion method [11], In Z ( ~ ) as the sum of contributions C(D) of all connected subdiagrams D In Z ( ~ ) =
E
C(D).
(A.5)
D_c~ I) connected
These formulae are solved recursively in C ( ~ ) for larger and larger diagrams, then used for the whole lattice to obtain the free energy. For instance, at the lowest order, the tetrahedron has a partition function 1 + Zrd2/34 and has no subdiagrams (only diagrams such that each link belongs to at least two plaquettes have to be considered, other ones leading to a vanishing contribution). Hence
dZflr(flG(S,Y,Z))~r(flG(Y,S,O))
(ln'Z-)tetrahedron = 1 o 4 f l n [ 1 + • r=~0
×fir(fiG(Z,
Y,O))i~r(t~G(X,
Z,0))]
daXdaYdaZ.
(a.6)
To obtain the (/3, o) expansion, the group coefficients are replaced by their explicit values /~0(t) = I1(2t) -
-
t,
flj(t)
I2J +l(2t) i1(2t)
(A.7)
1 ,
j=
2,
1, . . . .
From the practical point of view the risk of calculation errors is avoided by using computer algebra [14]. However, the determination of relevant diagrams and the form of their contributions have been controlled by hand in this work; an automatic procedure would certainly allow longer series. In the SU(2) case, the number of relevant diagrams and terms is reduced by a parity condition ( Z 2 is a subgroup of SU(2)). Namely, the sum of powers of fl on triangles adjacent to a given link should be even. Table 1 lists the relevant diagrams up to 6 vertices. The total number of diagrams (including those with 7 and 8 vertices, not displayed on the table) is 58 up to and including order 12. Only 23 of them are pure closed surfaces and contribute to the high density leading term. No closed surfaces with higher Euler characteristic than the sphere (as e.g. the torus) are present at this order. Note that there appears at order 10 a new topology for the corrections: a closed surface with two coinciding points. Due to the quenching procedure, it contributes to a different power of o than if the two points did not coincide. It can be interpreted as a contact interaction
J.-M. Drouffe/ Strongcouplingexpansion
10
TABLE1 List of unequivalent diagrams (set of triangles) occurring in the 12th order expansion, up to 6 vertices 1 1 1 2 1 1 1 2 2 3 1 1 1 1 2 2 2 3 3 4 Tfian~es 2 2 3 3 2 3 4 3 4 4 2 3 4 5 3 4 5 4 5 5 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 0 • • 0 0 0 0 0
Remarks
• •
• • • 0 • 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0
*tetr~edron *hex~edron t
t O0 O0 • O0 0 0 0 0 O0 O0 • O0 O0 000 • O0 O0
•
* octahedron
000
•
t
•
•
t
•
•
•
•
•
•
•
diagrams O0 •
O0
•
•
*
•
•
t
•
•
•
•
t
•
•
t
• •
• •
**
*Simple closed surfaces. **Closed surfaces with two coinciding points. *B-configurations.
between two strings, without exchanging parts of strings. 13 such diagrams are present (1 at order 10 displayed in table 1 with a mark **, 12 at order 12). Table 2 displays the result in any dimension up to 10th order in /3. To these 39 terms should be added 116 more terms at order 12. The table displays coefficients, according to the notation
In Z
02~ d/2 ~ [ q'/'d/20/32 ~ ni
)
2d/2 I p~ ( ~--~o ] FiXi d/2"
(A.8)
Finally, table 3 contains the additional numerical coefficients in dimension 4 at order 12 in r , which cannot be obtained from table 2.
I a m very grateful to H. Kluberg-Stern and F. David for their interest in this work. I had stimulating discussions with members of the Niels Bohr Institute, where part of this w o r k has been completed.
J.-M. Drouffe / Strongcouplingexpansion
11
TABLE 2 Coefficients for SU(2) gauge system up to 10th order in fl, according to eq. (A.8) notations n
p
F
X
n
p
1 2 2 3 3
0 0 1 0 1
4 2 - ~ 4 - 6x
3 16 12 75 49
5
1
3 4
2 0
8 12 1 - 8 - 16 1~3
27 336 384 192 208 100
163 289
1 2
9
F
X
t36 48 16 16 32
729 801 851 873 908
1~ 24 ~ 16 32 32
1014 1152 363 407 451 469
120
2~
486
128
16
495
3~ 3
544
3
_ 45 3A
48
0
8 12
1445 1463
24
1488
8
1682
1A5
1805
-
2
3
5
4
- 32 2456
567 169
32 3 160 9
221
~7 41~ 675
243 75
242
TABLE 3 Numerical coefficients for the four-dimensional SU(2) system at 12th order in fl p=0
1
2
3
4
n ~ 65.9578398110 -6 -2.7261343210 -59.4364586610 -56.5726467010 -5 -5.34784402105
5 -4.7902105510 -5
Lower order coefficients can be obtained from table 2.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9]
J-M. Drouffe and H. Kluberg-Stern, Nucl. Phys. B260 (1985) 253 N.H. Christ, R. Friedberg and T.D. Lee, Nucl. Phys. B202 (1982) 89; B210 [FS6] (1982) 310, 337 H.C. Ren, Nucl. Phys. B235 [FSll] (1984) 321 J-M. Drouffe, Nucl. Phys. B218 (1983) 89; V.A. Kasakov, Phys. Lett. 150B (1985) 282 F. David, Nucl. Phys. B257 (1985) 45 J. A m b j o m , B. Durhuus and J. FrShlich, Nucl. Phys. B257 (1985) 433 F. David, Nucl. Phys. B257 (1985) 543 V.A. Kasakov, I.K. Kostov and A.A. Migdal, Phys. Lett. 157B (1985) 295 J. Ambjom, B. Durhuus, J. FrShlich and P. Orland, Nucl. Phys. B270 [FS16] (1986) 457
12
J.-M. Drouffe / Strong coupling expansion
[10] J-M. Drouffe and J-B. Zuber, Phys. Reports 102 (1983) 1 [11} S. McKenzie, in Phase transitions, Proc. Carg6se 1980 Summer School, vol. B72, eds. D. Levy et al (Plenum, New York 1982), p 247 [12] M.E. Fisher and J.H. Chen, in Phase transitions, Proc. Carg6se 1980 Summer School, vol. B72, ed~, D. Levy et al. (Plenum, New York 1982), p 169 [13] A. Billoire and F. David, Phys. Lett. 168B (1986) 279 [14] J-M. Drouffe, AMP user's manual, Note CEA N2297 (1982)
S T R O N G C O U P L I N G E X P A N S I O N IN R A N D O M LATYICE G A U G E S Y S T E M S A N D I N T E R A C T I N G G A U S S I A N SURFACES Jean-Michel DROUFFE Service de Physique Th~orique, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France
Received 10 July 1986
We present a strong coupling study for a pure gauge random system, where the interactions are cut off using a gaussian form factor (the nearest-neighbour concept is not used). Double series in the inverse Yang-MiUscoupling and in the density are obtained. We focus here on the SU(2) four-dimensional system. High and low density regions are investigated. We find that the high density limit is identical to recently proposed models of gaussian random surfaces, while finite density corrections introduce interactions between these surfaces, depending on the gauge group. Comparison with a Monte Carlo analysis is performed in the medium density region. We finally propose a phase diagram for this two-parameter model.
1. Introduction There has recently been proposed [1] a random lattice regularization of QCD, in which the corresponding action couples any two points on the random lattice, with a gaussian damping form factor of range b in the euclidean distance of the points. The introduction of the nearest-neighbour concept [2] is avoided. As a consequence, it is possible to compute practically some physical observables with various techniques. For instance, ref. [1] presents the estimation of the hadron spectrum in the strong coupling limit. The aim of this paper is to analyze the strong coupling expansions for a pure gauge system. N o matter fields are present, and series are calculated up to a rather high order. Two variables appear in this expansion. One is the usual inverse Yang-Mills coupling r , the other one is the number of lattice points within the interaction range o = ob d (p being the lattice site density). Varying this last p a r a m e t e r leads to several possible regimes. In an intermediate one, it is possible to tune the range b so that the number of interacting points is roughly the number of nearest neighbours. The model is thus expected to behave as the standard random system [2] with a nearest-neighbour interaction. We performed a Monte Carlo run for the SU(2) four-dimensional system which indeed shows a behaviour quite c o m p a r a b l e to that of the nearest-neighbour model [3]. In particular, curves are very 0619-6823/87/$03.50© Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
2
J.-M. Drouffe / Strong coupling expansion
regular, without any evidence of a structure as the one observed in regular lattice systems. It is therefore tempting to analyse strong coupling series, in order to realize a dream: to enter the physical region from the strong coupling limit. Such a calculation is presented here. Alas, straightforward extrapolations of the series can hardly fit the very beginning of these regular curves. The crossover phenomenon occurring in regular lattice systems at intermediate fl and preventing any extrapolation into the weak coupling region now seems to appear from the beginning. The reason is analyzed while resumming the series as a low density expansion. It is shown that the radius of convergence in fl is strongly limited by singularities on the imaginary axis, due to the gauge group fundamental functions. Such a behaviour is controlled after resummation as a functional series in o, and Monte Carlo results are then nicely fitted for any values of ft. Another interesting region is the high density limit. At the lowest order, the model coincides exactly with proposed models [4] of surfaces, represented by their gaussian random triangulations. The role of the gauge group disappears in this limit, which appears to be universal. Such models have been extensively studied [5-9] and are expected to present a non-trivial critical point. In the system studied here, corrections depending on the gauge group appear as o decreases from infinity. They can be interpreted as various interactions between the surfaces, leading to non-planar diagrams as well as interactions along a common boundary singular line. The strong coupling expansion allows us to track the transition observed at o = oo when it enters the ( f l - o) phase diagram, and to speculate about its exact nature and relevance. In sect. 2, the techniques used to derive the strong coupling expansions are described. Corresponding calculations and results for the SU(2) system are given in an appendix. Sect. 3 is devoted to the low density analysis, and presents our Monte Carlo simulation. The high density limit and the phase diagram are discussed in sect, 4.
2. Derivation of the strong coupling series The action of the model [1] is defined on a random lattice with mean density of sites 0 (x y z) S = ~ ~_, G Tr(U~yUy~U~) (1) {x,y,z} b ' b ' -b The summation runs over all (unordered) triplets of points x, y, z. Independent gauge fields Uxy = Uy-~1 are thus introduced for any pair of points, even largely separated. The cut-off form factor G restricts the interaction to a range b. In principle, any function, rotationally and translationally invariant, can be retained,
J.-M. Drouffe / Strongcouplingexpansion
3
provided it keeps the three points within a finite volume (so a cut-off built on the basis of the area of the xyz triangle cannot be retained [6]). For the sake of calculations, we restrict ourselves to a gaussian shape
G(X,Y,Z)=exp(-(X-
Y)2-(Y-Z)2-(Z-X)2).
(2)
This model is expected [1] to have a continuous limit b ~ 0 at fixed a = pb d which describes the Yang-Mills continuous theory. The following relation holds between the couplings 48 • 3d/2 b d-4
fi = ~d/~
O~g~ .
(3)
The observables should be averaged over all the possible random lattices with fixed density p (quenching procedure). To derive the strong coupling expansion, we follow the character expansion technique [10] applied for a fixed random lattice. The exponentiated action is developed in terms of irreducible characters Xr, using the Fourier coefficient functions fi,(t) defined by
exp(tTrU)= fi°(t)[l + ~d'fir(t)xr(U)]
(4)
where d r is the dimension of the irreducible representation r. This leads to terms diagrammatically represented by sets of triangles; to each of one a representation r has been assigned. The integration over gauge fields is done using the methods of group theory, and leaves only closed diagrams. The simplest topology encountered is homeomorphic to a triangulation of a closed surface, with the vertices on the random lattice sites; however, more complex diagrams with, e.g., n-fold singular lines should also be considered. The contribution of a given graph is to be summed over all different locations of its vertices on the lattice. The partition function is obtained by summing the contributions of all (connected and disconnected) diagrams. Using standard excluded volume expansion techniques [11], the logarithm of the partition function is then computed. It appears as a complicated function of the form factors fiG(X, Y, Z) of all the triangles entering in a connected diagram. It is this free energy, and not the partition function, which must be averaged over the random site locations (quenching procedure). This is done using the formula
E
¢(Xl
.....
.... x . ) d x l . . . d x ~.
(5)
Xl,...~X n all different
The contribution of each graph is developed as a series in /3 and, thanks to the
J.-M. Drouffe / Strong coupling expansion
gaussian shape (2), the spatial integration (5) can be performed for each coefficient of this expansion. Details and results for the gauge group SU(2) are gathered in the appendix. Note that the result appears as a double series in f l - counting the number of fundamental interactions- and o = pb d- counting the number of vertices. There are two possible ways to perform a resummation: as an expansion in o at fixed fl (resumming all graphs with a given number of vertices), or in 1 / a at fixed fl 2a. These two procedures are examined in the following sections.
3. Comparison with simulations and low density expansions In fact, nothing is known about the present system, and in particular about its phase structure. This paper is intended to answer some questions about its low and high density limit, but we also decided to perform a Monte Carlo simulation to look into possible unexpected effects and to compare the expansions to experiment. This numerical analysis has not been done on an extensive scale; a mini-computer has been sufficient (200 hours on Prime 550). A priori, the model does not seem appropriate for a Monte Carlo simulation. As all pairs of sites bear a gauge field, there are - N 2 quantities to update, each one involving all other lattice points. Hence the sweep time behaves as N 3 instead of N. Moreover, there are storage problems for N 2 fields, plus numerous large arrays needed to access information about the geometry in a reasonable time. Hence it has been necessary to introduce a cut-off, modifying the form factors in such a way that only links of length less than L contribute
(xyz) (xyz)
G b'b'b
-* G
b'b'-b
O(L-
Ix-yl)O(L-
lY-ZI)O(L-
Iz - x [ ) ,
(6)
0 being the step function. The effect of this truncation can be estimated by the deficit on the mean ground state energy
~1( L ) = 1 - ff lx'-xjl
(7)
In a four-dimensional simulation, this deficit is limited to 2% as soon as L/b is greater than 2.4, which is sufficient for our purpose. On the other hand, we have to limit the mean number of gauge fields linked on a particular lattice site, 2¢rd/2a(L/b)d/dF(½d), in order to achieve a reasonable computing time; hence we are limited to rather low values of a. We take a lattice of N = 1000 sites and work with a = 0.25 and L/b = 2.4; the mean number of considered neighbours is then of order 40 (to be compared with the 8 on an hypercubical regular lattice). If the density parameter o cannot be chosen too high, it should be noticed that it cannot also be made too low. Indeed, the density fluctuations induce fluctuations on
J.-M. Drouffe / Strongcouplingexpansion
the action density, which should be correctly averaged on the sample size. A measure of this effect is shown by the fluctuations of the maximal action relative to one point
~,=
~/(1/N)Ex(Ey,~G(x, y, z)) 2- ((1/N)F.,x,y,~G(x, y,
= (~)a/4 /
z)) 2
(1/N)F.x,y,~G(x , y, z) 4
2 +
.
(8)
The quantity ~ / f N will characterize the fluctuations between several samples, and should be sufficiently low. With our choice, it lies around 3%. We have used 4 different lattices (measured quantities are auto-averaged, and few lattices are thus necessary) and a statistical study of their results agrees with the expectation (8). We have measured the normalized energy
(s) Sm=
E = 1 - --
(9)
while heating and cooling a sample from fl = 1 to fl = 200. The gauge group SU(2) has been replaced by its 120 dement discrete subgroup. The resulting curve is displayed in fig. 1. It appears to be very regular, and is quite comparable to the one obtained with a nearest-neighbour interaction [3]. There is no trace of some structure at intermediate fl, such as the one observed with regular lattices, and no trace of a transition. In order to avoid misinterpretation of the unusual high values of fl, we also indicate the corresponding values (quite normal) of the Yang-Mills coupling constant g2. The fit by the strong coupling series and their extrapolations (e.g. Pad6 approximants) is not indicated on the figure. The reason is that they fit the curve badly, except in the very beginning (up to fl - 3). This fact may seem strange when one acknowledges the regularity of the curve; in the regular lattice case, series fit very well up to the crossover region, dearly visible as an accident on the experimental data. The reason for this poor fit can be analysed in the low density limit. We already indicated that the expansion may be reorganized as a series in o, according to the number of vertices in the diagrammatic representation. The first two terms have been explicitly displayed in the appendix (formulae (A.4) and (A.6)). Singularities are easily found in these integral representations and correspond to the singularities of the Fourier coefficients (4). They are analytic on the real axis, but are singular on the imaginary axis. The first one lies at the first zero of I1(2fl), viz fl - + 1.9159i. It limits the radius of convergence far below the required values.
J.-M. Drouffe / Strong coupling expansion 1.0,
50
10
5
3
2
I
I
I
I
I
1.5 I
-(g~)
0.5
i
I
I
50
100
150
-(~) 200
Fig. 1. The mean energy, from Monte Carlo simulation, fitted by the low density expansion. Dashed curve is the zero densitylimit, plain curveinclude the first correction(a = 0.25).
There is nevertheless a way to obtain a reasonable fit, by using this a expansion, which realizes a summation in ft. The first two terms have been used to fit nicely the data of fig. 1 within statistical errors, and on the whole range of ft.
4. High density expansion We turn now to the high density limit. The expansion can be reorganized, according to the topologies of the diagrams, as a 1 / o expansion at fixed fl2a. Indeed, any connected diagrams can be constructed (in a non-unique way), starting from a simple closed surface, by adding simple opened surfaces sharing their boundary vertices and links. We show that the more simple components, the more subdominant the diagram is. For any of these surfaces, topological relations can be written. With the following notations no nl n2 P k g
number of non-boundary vertices, number of non-boundary links, number of triangles, number of boundary vertices (or links), number of boundary components, genus of the surface ( >/0, equal to twice the number of handles for an orientable surface),
J.-M. Drouffe / Strongcouplingexpansion I
I
7
I
1/o t 1.0
[I11]
0.5
,/\/ 5
,
10
15
20
Fig. 2. Nearest real positivesingularityfor Pad4 [1/1], [1/2] and [2/3] at fixed o in the (fl20,1/o)-plane
the relations are 3n 2 = 2n 1 + p , n 2+ k - n 1 + n 0 = 2 - g .
(10)
It follows that the multiplicative factor fl n2o n0 involved in the diagram contribution can be recast as (flv~)n~(1/o) k+p/z+g-2. Any of the open-component surfaces satisfies k >i 1 and p >/3 and is therefore subdominant in a 1 / o expansion at fixed f12o. Also, the higher the genus is, the more subdominant the diagram is. Hence dominant diagrams appear topologically as a triangulation of the sphere (g = 0). Note that, in this limit o ~ oo, the model becomes independent of the gauge group. N o t e also that this limiting case can be obtained in a different way, N ~ oo. The diagrammatic rules are identical to those of gaussian random surface models [6, 7] which have been proposed as a possible regularization of non-interacting string systems. A careful strong coupling analysis has been done [7] and shows evidence for a singularity. Our series, which also contain 1 / o corrections, are shorter in this limiting case, but lead to the same conclusion. We use a Pads analysis at fixed o for the specific heat to locate and follow the singularity in the phase diagram (fl2o, 1 / o ) . Fig. 2 displays the result. Pads [2/2] gives unstable results and has been discarded. A line of singularities separates the plane into two regions. It is not clear, however, that the line extends up to infinity. We have tried to answer this question using a 2-variable generalized Pads approximant [12]. The position of the singularity line is confirmed, but the position of the end-point, when it exists, wildly fluctuates from a minimum fl2o--15, according to the chosen approximant. No serious conclusion can be drawn from this analysis.
8
J.-M. Drouffe / Strong coupling expansion
Several hypotheses concerning the exact phase structure remain possible. (1) The line may be a second-order transition line, near which a continuous limit exists. This is hope for the random surface models, for which a sensible continuous limit is looked for, as a realization of the string model. In our interpretation, the renormalization flow goes to a fixed point at fl = ~ , which describes the Yang-Mills continuous theory. (2) This line may also be the extremity of a metastable region, with a first-order transition occurring at smaller ft. On regular lattices, this phenomenon is always observed as soon as there is a singularity. If this is the case here, random gaussian surface models (o = oo) will not have a sensible continuous limit. Continuous limits are expected at fl = o0 (hopefully Yang-Mills theory), or at the second-order end-point (if any) of this first-order line. It is not yet possible to decide between these two possibilities. Maybe the second one is favored, because of an analogy with regular lattice theory. Numerical simulations like the one described in this paper do not allow us to check the order of the transition (o is too high), and a numerical study [13] of the gaussian random surface model has not yet been applied to this problem.
Appendix The scheme of the calculations has been indicated in sect. 2. We describe only some details and list the results. Let us consider first the influence of the term/3o(t) in eq. (4). It contributes to l n Z a factor ln/~o(flG ) for each triangle, so the quenched average reads, according to eq. (5)
f f f d"xd yd ln °( fla( b ' b '
"
(A.1)
The integrand is expanded in series of r , using ln/~0(t ) =
Y'~bnt".
(A.2)
Each term appears as a gaussian integral; one factorizes the volume V for the zero mode corresponding to translation invariance. Hence lnZ ]
-
"~--2"2] t~iangle
bnfln 6. 3d/~-------~ ~-, n d O2~rd
(A.3)
The series on the r.h.s, can be recast in an integral representation, as for instance
In Z /
o2u d
1
Vb-d]triangle- 6~'~2 f0 (-lnt)d-llnfl°(flt)dt'(d-1)!
(A.4)
J.-M. Drouffe / Strong coupling expansion
9
Let us now turn to the diagrammatic corrections. We call diagram ~ any subset of the triangles of the random lattice. If one restricts the action to only these triangles, one can compute, using the character expansion technique [10], the restricted partition function Z ( ~ ) . One writes, according to the excluded volume expansion method [11], In Z ( ~ ) as the sum of contributions C(D) of all connected subdiagrams D In Z ( ~ ) =
E
C(D).
(A.5)
D_c~ I) connected
These formulae are solved recursively in C ( ~ ) for larger and larger diagrams, then used for the whole lattice to obtain the free energy. For instance, at the lowest order, the tetrahedron has a partition function 1 + Zrd2/34 and has no subdiagrams (only diagrams such that each link belongs to at least two plaquettes have to be considered, other ones leading to a vanishing contribution). Hence
dZflr(flG(S,Y,Z))~r(flG(Y,S,O))
(ln'Z-)tetrahedron = 1 o 4 f l n [ 1 + • r=~0
×fir(fiG(Z,
Y,O))i~r(t~G(X,
Z,0))]
daXdaYdaZ.
(a.6)
To obtain the (/3, o) expansion, the group coefficients are replaced by their explicit values /~0(t) = I1(2t) -
-
t,
flj(t)
I2J +l(2t) i1(2t)
(A.7)
1 ,
j=
2,
1, . . . .
From the practical point of view the risk of calculation errors is avoided by using computer algebra [14]. However, the determination of relevant diagrams and the form of their contributions have been controlled by hand in this work; an automatic procedure would certainly allow longer series. In the SU(2) case, the number of relevant diagrams and terms is reduced by a parity condition ( Z 2 is a subgroup of SU(2)). Namely, the sum of powers of fl on triangles adjacent to a given link should be even. Table 1 lists the relevant diagrams up to 6 vertices. The total number of diagrams (including those with 7 and 8 vertices, not displayed on the table) is 58 up to and including order 12. Only 23 of them are pure closed surfaces and contribute to the high density leading term. No closed surfaces with higher Euler characteristic than the sphere (as e.g. the torus) are present at this order. Note that there appears at order 10 a new topology for the corrections: a closed surface with two coinciding points. Due to the quenching procedure, it contributes to a different power of o than if the two points did not coincide. It can be interpreted as a contact interaction
J.-M. Drouffe/ Strongcouplingexpansion
10
TABLE1 List of unequivalent diagrams (set of triangles) occurring in the 12th order expansion, up to 6 vertices 1 1 1 2 1 1 1 2 2 3 1 1 1 1 2 2 2 3 3 4 Tfian~es 2 2 3 3 2 3 4 3 4 4 2 3 4 5 3 4 5 4 5 5 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 0 • • 0 0 0 0 0
Remarks
• •
• • • 0 • 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0
*tetr~edron *hex~edron t
t O0 O0 • O0 0 0 0 0 O0 O0 • O0 O0 000 • O0 O0
•
* octahedron
000
•
t
•
•
t
•
•
•
•
•
•
•
diagrams O0 •
O0
•
•
*
•
•
t
•
•
•
•
t
•
•
t
• •
• •
**
*Simple closed surfaces. **Closed surfaces with two coinciding points. *B-configurations.
between two strings, without exchanging parts of strings. 13 such diagrams are present (1 at order 10 displayed in table 1 with a mark **, 12 at order 12). Table 2 displays the result in any dimension up to 10th order in /3. To these 39 terms should be added 116 more terms at order 12. The table displays coefficients, according to the notation
In Z
02~ d/2 ~ [ q'/'d/20/32 ~ ni
)
2d/2 I p~ ( ~--~o ] FiXi d/2"
(A.8)
Finally, table 3 contains the additional numerical coefficients in dimension 4 at order 12 in r , which cannot be obtained from table 2.
I a m very grateful to H. Kluberg-Stern and F. David for their interest in this work. I had stimulating discussions with members of the Niels Bohr Institute, where part of this w o r k has been completed.
J.-M. Drouffe / Strongcouplingexpansion
11
TABLE 2 Coefficients for SU(2) gauge system up to 10th order in fl, according to eq. (A.8) notations n
p
F
X
n
p
1 2 2 3 3
0 0 1 0 1
4 2 - ~ 4 - 6x
3 16 12 75 49
5
1
3 4
2 0
8 12 1 - 8 - 16 1~3
27 336 384 192 208 100
163 289
1 2
9
F
X
t36 48 16 16 32
729 801 851 873 908
1~ 24 ~ 16 32 32
1014 1152 363 407 451 469
120
2~
486
128
16
495
3~ 3
544
3
_ 45 3A
48
0
8 12
1445 1463
24
1488
8
1682
1A5
1805
-
2
3
5
4
- 32 2456
567 169
32 3 160 9
221
~7 41~ 675
243 75
242
TABLE 3 Numerical coefficients for the four-dimensional SU(2) system at 12th order in fl p=0
1
2
3
4
n ~ 65.9578398110 -6 -2.7261343210 -59.4364586610 -56.5726467010 -5 -5.34784402105
5 -4.7902105510 -5
Lower order coefficients can be obtained from table 2.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9]
J-M. Drouffe and H. Kluberg-Stern, Nucl. Phys. B260 (1985) 253 N.H. Christ, R. Friedberg and T.D. Lee, Nucl. Phys. B202 (1982) 89; B210 [FS6] (1982) 310, 337 H.C. Ren, Nucl. Phys. B235 [FSll] (1984) 321 J-M. Drouffe, Nucl. Phys. B218 (1983) 89; V.A. Kasakov, Phys. Lett. 150B (1985) 282 F. David, Nucl. Phys. B257 (1985) 45 J. A m b j o m , B. Durhuus and J. FrShlich, Nucl. Phys. B257 (1985) 433 F. David, Nucl. Phys. B257 (1985) 543 V.A. Kasakov, I.K. Kostov and A.A. Migdal, Phys. Lett. 157B (1985) 295 J. Ambjom, B. Durhuus, J. FrShlich and P. Orland, Nucl. Phys. B270 [FS16] (1986) 457
12
J.-M. Drouffe / Strong coupling expansion
[10] J-M. Drouffe and J-B. Zuber, Phys. Reports 102 (1983) 1 [11} S. McKenzie, in Phase transitions, Proc. Carg6se 1980 Summer School, vol. B72, eds. D. Levy et al (Plenum, New York 1982), p 247 [12] M.E. Fisher and J.H. Chen, in Phase transitions, Proc. Carg6se 1980 Summer School, vol. B72, ed~, D. Levy et al. (Plenum, New York 1982), p 169 [13] A. Billoire and F. David, Phys. Lett. 168B (1986) 279 [14] J-M. Drouffe, AMP user's manual, Note CEA N2297 (1982)