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Planar and non-planar Wilson loops in QCD
Nuclear Physacs B298 (1988) 613-635 North-Holland, Amsterdam
PLANAR AND NON-PLANAR WILSON L O O P S IN Q C D * Emil1 BAG,~N
Physws Department, Brookhaven Natwnal Laboratory, Upton, N Y 11973 USA Jos6 I L A T O R R E
Center for Theorettcal PhysmL Laboratory for Nuclear Scwnce and Department of Physics, Massachusetts Insmute of Technology, CambrMge, Massachusetts 02139, USA Recewed 2 June 1987
A general QCD sum rule calculation of vacuum expectation values of Wilson loops is presented and particularized to several specific planar and non-planar contours The s t u n g tension obtmned is ~/o = 0 50(5) GeV Within the errors of the approach, this result is shape-independent We comment on the posslbihty that corrections to the area and perimeter laws could be parametnzed by a geodesic curvature term
1. Introduction Wilson loops (WL's) behave in very different ways at short, intermediate and long distances. Perturbation theory correctly describes the ultraviolet regime once some divergences are renormahzed (see sect. 2). Nevertheless tins approach cannot be used to investigate intermediate energies since some non-perturbative technique is needed to deal with confinement. The situation worsens as we move to longer distances and screening takes place In principle, these three distinct behaviors are described by quantum chromodynamlcs (QCD) but such a hope would only become apparent upon solving the theory. The usual, pragmatic modus operan& consists in guessing the effect theory well-suited to describe a certain range of energies. The effective description of WL's at intermediate scales might correspond to a string theory. Within tins picture, a static effective quark-antiquark potential arises naturally for a wide class of strings [1]
V(r)
= or-
-
12 r -
-
+
.
.
-
(1)
* This work is supported in part by funds provided by the US Department of Energy (DOE) under contracts #DE-AC02-76ER03069 and DE-AC02-76CH00016 0550-3213/88/$03 50©Elsevier Scmnce Pubhshers B V (North-Holland Physms P u b h s h m g Division)
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E Bag6n, J Latorre / W t l s o n loops
where o is the string tension. The classical linear part in (1) leads to confinement whereas the correction l / r , universal Luscher term, comes from quantum fluctuations. Recently, Polyakov [2] suggested that the N a m b u - G o t o action for strings can be supplemented with a term proportional to the extrinsic curvature of the worldsheet. This new contribution as well as finite size effects modify eq. (1) (see ref. [3]). At this point, we still have to relate the string picture with QCD. Since, at present, there is very little hope of connecting them exactly, a natural way to proceed is to compute vacuum expectation values (vev) of WL's for different sizes and fit those parameters appearing in the phenomenological description. This program has been carried out for planar WL's within the contexts of lattice regularlzatlon [4] and Q C D sum rules. Although numerical simulations are more popular, the second method presents a main advantage, namely, it works in the continuum. In a series of previous papers [5], [6], we have analyzed rectangular and elliptic WL's using this technique. The results were very satisfactory. We found stable values for the string tension which turned out to be the same for the two shapes as was expected. It is important to notice, however, that the analysis of the elliptic case was ~mphc~tly based on the assumption that no divergence proportional to the perimeter arose - which is certainly true in dimensional regularlzation - and that the perimeter was a linear function of the axes a and b - which is only approximately true for a - b. In the last paper we also computed the mass gap. The result was m agreement with several other estimates. Recently, one of us [7] has applied s~rmlar techniques to the study of which string model is behind QCD. Taking into account boundary effects, sum rules seem to be sensitive enough so as to favor the modified stnng picture which includes the extrinsic curvature term as the effective theory providing the quark-antlquark static potentifil. In this paper we complete our study of planar WL's using a more general exploitation of the sum rule and we extend it to nonplanar contours (note that no roughening comphcations are encountered in our a p p r o a c h - we work in the continuum). An effort has been made to give as general a computation as possible. A priori, the classical analysis of three dimensional WL's becomes more involved since the corresponding effective theory may include a bunch of new geometric invariants. For instance the area law might be accompanied with some dependence on the intrinsic or extrinsic curvature. In the case of finite contours, we also have new lnvariants depending on its shape, namely perimeter or geodesic curvature laws. Actually, we d~spose of an Infinite set of potential dependences since we are searching an effecnve description. Yet, the accuracy of our approach is of the order of a 20%. Therefore we are forced to restrict the scope of our analysis to a certain order. It seems natural to only consider dependences on invariants of dimension L 2 (area), L (perimeter), and L ° (lntnnslc curvature, extrinsic curvature . . . . ), though quantum corrections might be relevant. We do not believe that Q C D sum rules are sensitive enough to analyze accurately L ° laws. Nevertheless we will comment on
E Bag6n, J Latorre / Wdson loops
615
some of these issues. We have organized our presentation in the following way. Sect. 2 is devoted to discuss the systematics of our approach. On one hand, QCD contributions are computed up to some final, shape-dependent integrals. On the other hand, effective lagranglan terms are discussed. Some problems show up from the very beginning. We will face a genuine type of divergences associated with nonlocal gauge invarlant color phases. These singularities will force us to get rid of perimeter laws through this work. In sect. 3 we particularize our calculations to planar WL's and exploit their respective sum rules. Stable, shape-independent values for the string tension are obtained. A similar analysis, though more complex, is carried out for nonplanar WL's in sect. 4. Again, the output shows a nice value for o. We summarize our conclusions in the last section and, finally, some technicalities are presented in the appendices.
2. General strategy WL's are the simplest nonlocal gauge lnvariant objects in QCD. Their vaccum expectation values are defined by euclidean space-time and for N colors as
Wc= ~
trPexp
--lgf)cdXt,A~(x)T ,
(2)
where C is a closed contour, P indicates path ordering and M stand for a N × N generalization of the Gell-Mann matrices of SU(3). The idea behind the sum rule approach consists in calculating the perturbative expansion of (2) adding nonperturbative corrections, e.g. condensate contributions, in order to have a fair description of WL's at intermediate energies. Then it will be possible to compare QCD with an effective theory valid at the same scale. Let us begin with the usual perturbative computation. Choosing the appropriate gauge the first non-trivial perturbatlve contribution to (2) corresponds to
Wc=I
a N2-1 4~r
~(~dx.C~dY~(x_y)2Jc Jc
(3)
The ultraviolet behavior of the above integral has been thoroughly investigated [8]. Depending on the contour, there are three kind of singularities: (i) A linear divergence proportional to the length of the contour C. Physically, it is understood as a mass renormahzatlon of the heavy quark that generates the WL. (ii) A logarithmic divergence for every cusp of C, corresponding to an infinite bremstrahlung of the test particle. (111) A logarithmic divergence whenever C self-intersects. Tins would mean that a quark and an antiquark pass through each other.
616
E Bagdn, J Latorre /
Wdson loops
Although the slngulariues of the first kind do not show up in dimensional regularization, they mess up a possible perimeter law. To be more precise, this hnear divergence exponentlates (see ref. [9] for a check up to two loops) and renders the perimeter law renormalization-scheme dependent. On the other hand, log divergences are somewhat Innocuous. They can be avoided by choosing smooth, nonself-intersecting contours. Each of the WL's analyzed m next sections will require some kind of normalization in order to get rid of any singularity before exploiting our results. Recall that (3) cannot describe non-perturbatlve effects. We have to compute sum rule corrections to this term. As a rule, we will consider contributions coming from condensates of dimension M4((aG2)) and M 6 ((g3fG3) and (~q)2). This kind of computation was carried out for the first time m ref. [10]. More technical details may be found in the literature [11-15]. Here we simply note that it is convenient to work in the Fock-Schwinger gauge [10]
x~,A~(x) = 0,
(4)
where all the non-perturbative integrals become trivial. After some straightforward algebra we obtain the gluon condensate contribution
W c = - 48-~(aG2)~)cdX~,X,,
dy~y¢(8,~8~- 8,,~8~,~).
(5)
The computation of order M 6 condensates becomes rather revolved and is presented in appendix A. Obviously, all the non-perturbative non-computable information is parametrized in the condensates, Numerically, we will take standard values for them a a ~v ) -- 0.04 GeV 4, (aG 2) =- (aG;~G
(g3fG3) =- /\ 6a 3 (J a b c l'7~a(Tbvf'g'colt\ V~v~p ~ /
=
(F/q) z -- ( - 0.25 GeV) 6 .
1.1 GeV2(aG 2)
(6)
A 25% uncertainty in (aG 2) will turn out to be the main source of error in our results which are strongly dominated by this condensate. This completes the QCD part of the sum rule. Note that our three equations (3), (5) and (A.2) only depend on contour integrals (path ordering is operatwe in the third one). Every specific shape will lead to different sum rule though, hopefully, to the same effectwe description. So far we have detailed how the QCD calculation is done. Now we need an effective theory to describe the vev of euclidean WL's around 1 GeV. Strong couphng suggests an exponential decay of the form exp ( - o. area},
(7)
E Bag~n, J Latorre / Wdson loops
617
o being the string tension. The area appearing in (7) is the area of the minimal surface S containing the contour C (we will keep this notaUon throughout this work). This behavior, as noted in the introduction, corresponds to a linear static quark-antiquark potential. A natural framework where (7) arises as a classical approximation is provided by N a m b u - G o t o strings, whose euchdean action reads
! = ors d2~y/~ .
(8)
Here, d2~ corresponds to the surface differential and g - d e t g~b, where g u t ,=Oux, Obx~ is the induced metric on the surface or first fundamental form. However m a n y other effective lagrangians are valid. We could add a whole new collection of geometric invariants which become relevant for the quantum theory a n d / o r at the classical level when the WL lives in more than two dimensions. In sect 4 we will need to make up our mind about the kind of dimensionless law Of any) we should introduce in the non-planar Wilson loop sum rules. A detailed analysis of renormahzable, parametrlzation m v a n a n t lagrangians for WL's has been carned out by Alvarez [6]. Classically we may add to (8) the following term
a fcdsk,
(9)
where a is a new fundamental parameter which we will try to determine in sect. 4, ds is the arclength differential of C and k is its geodesic curvature. More precisely, consider the loop C and then find the unit tangent (t a) and normal (n ~) vectors to the contour using the metric g,,h. Then k is defined as
taW~t b = kn b .
(10)
One may think that we are missing a term proportional to the intrinsic curvature, R. This is not the case. For a finite surface, the Gauss-Bonnet theorem states that
½fs d2~v~-R + fas d s k = 2 ~ r x ( S ) '
(11)
where x ( S ) = 2 - 2 h - b ( h = # of holes, b = # of boundaries) is the Euler characteristic of S. We still have two other well-known geometric invariants, namely the extrinsic curvature
~/g- (Kata) 2
and
vg"bG-~ l,-,u~,b..u,
(12)
E Bagdn, J Latorre /
618
Wdson loops
built out of the second fundamental form K~h which is defined by
O~Obx ~ = F2bOcx ~ + K~bn ~ ,
(13)
!
where n, are D - 2 vectors normal to the surface and laCb are the Christoffel symbols of this surface. Nevertheless, a fundamental relation in analytic geometry links the two extrinsic curvature structures to the intrinsic curvature on the surface,
R = (K;o)2_
*x b
*x a
.
(14)
Furthermore, we note that for surfaces of mimmal area
= 0.
(15)
This amounts to saying that the Gauss-Bonnet theorem relates the non-trivial combination ,,e tc-~a~ ,,~~b to the geodesic curvature for minimal surfaces. We conclude that a classical description of WL's may require a dimensionless law related to the geodesic curvature of the contour or, equivalently, to the extrinsic curvature of the surface. We have not commented on the perimeter law yet. Due to the genuine divergence associated to the length of any color phase, a perimeter law would become renormalization-scheme dependent. The easiest way of coping with this problem consists in only considering appropnate ratios of WL's so as to cancel this ominous perimeter law. As a bonus, constants also drop out and systematic uncertainties are reduced. From now on we wall omit any reference to such terms.
3. Planar Wilson loops In order to get numbers, all the machinery of the previous section has to be applied to specific contours. It is natural to start considenng WL's which lie on a plane. Such a class of contours introduce a good deal of simplification. For instance, independently of the shape of the two-dimensional WL, the (aG 2) contribution turns out to be q7
Wc = - 1 2 ~ (°LG2)S2'
(16)
where S c stands for the area circled by C This may be verified by using Stokes theorem in (5). Likewise, no direct (g3fG3) condensation takes place (which is the only term where path ordering is kept). On the phenomenological side, leaving aside the perimeter law and constants we expect exp{ - o. area}
(17)
619
E Bagdn, J Latorre / Wdson loops
as no other non-trivial geometric invariants exist in two dimensions. Note that, classically, the extrinsic curvature of a flat surface vanishes and the integral of the geodesic curvature on a two-dimensional contour amounts to 2~r. Although we do not explicitly consider quantum corrections to (17) we will be able to have some control on them (see below). The simplest WL instance is a rectangular contour, L times T. Computing all the integrals and writing T = )~L we find c~ N 2 - 1
Wre~t(L,X)=l+
~r
- N
12N
[ 2 + 21nl*L + lnX + 2 + f()t) + f( ~)]
(aG2))~eL4
+ 12~-N
96
+ -9 (a~r(qq))2Nv ~ N
)t2(1 + ~2)L6'
(18)
where f ( x ) = x arctanx + ½1n(1 + x 2) and /* stands for some subtraction scale. Since we have dimensionally regularized (D = 4 - e) the perturbative part, we do not have a hnear divergence. However we find a pole 1/e coming from the cusps of the rectangle. Notice that condensates are providing power corrections to the first two terms. They are enlarging the range of distances where (18) gives a correct description of the WL, hence our hope of capturing non-perturbative effects. Now we can go on in &fferent ways. In ref. [5] we considered a useful combination of WL's and fitted the value of the string tensmn. Here we proceed differently. In order to get rid of a perimeter law, we take the expansmn
in msq .... ( L ' )
~ ~
+ -12N -
[ln
42,
f(X) -f
(aG2)L4~2 1 1
12N
+ 2f(1)
96
z
N2-1]
+ ~(a~r(~q)) U v -N-T
XL6 [~.2(1 + ~k2) - 2( L' ] 6]
g-jj'
(19)
where L' = ~(L + T) = 1L(1 + X) and thus both WL's have the same perimeter This normalization also eliminates the 1/t and /, factors in (18). We further
E Bagdn, J Latorre / Wdson loops
620
r e n o r m a h z a t i o n group improve the perturbatlve contribution by introducing a running coupling constant 2~r 1311n(L,2A2) ,
~(L')
where [171 A = ½AMNexp(yE), " Y E = 0 . 5 7 7 . . ,
A~g=0.12
(20) G e V and 131 = -
~-N
1 + ~ N F.
At intermediate energms and according to our expectations, eq. (19) as a function of L 2 for a fixed X should behave as a straight line passing by the origin, namely o-(SrectSsquare ) = -- ¼oL2(1 - X) 2. This is indeed so. Fig. l a shows eq. (19) v e r s u s L 2 for )~ = 1.1 (sohd hne) as well as the best fit to Q C D for 2.7 GeV -~ ,%
intercept = 0.
(21)
T h e solid hnes in fig. l b and fig. l c are respectively the slope and intercept of the straxght line that best fits Q C D m the same L region as before but now for a given value of )t. The dashed hnes correspond to the global fit (i.e. slope = - ¼o(1 - )t) 2, taking o f r o m (21); and intercept = 0). Note the good matching for ] 1 - )t t < 0.5 w h m h should be interpreted as a measure of the stability of our output. The result (21) for the string tension is m complete agreement with our previous analysis of ref. [5], W h a t is new and non-trivial is finding a stable null intercept. A shift of the straight line would mean that some corrections are present. We did not expect classmal m o d i f i c a t m n s but q u a n t u m corrections nnght have entered into the game. In fact, our result states that fluctuations of the finite world-sheet 0.e. taking into account b o u n d a r y effects) appearing in the " N a m b u - G o t o + &menslonless laws" action are small for )t = 1 or, at least, that the present analysis ~s insensitive to them. It is of capital importance to check whether these results depend on the specific shape we have considered. Consequently, we take a second and non-pathological c o n t o u r - an ellipse with axes a = ?tR and b = R. After some work we find N2-1 Wen(R, X) = 1 + a ~ r - - 2N
219
+ 4T
l+)t 2
2~,
(g3fG3) + ~
¢r
12N (°tG2)(rr~'R2)2 --NF(~r(oq))
~k2(1 -'~ ~k2)R6" (22)
O f course, no singularities are encountered since the contour 1s smooth and we work in dimensional regularlzation. Again, to avoid a perimeter and constant law, we
E Bagdn, J Latorre /
(a)
O O
O
ig
621
Wdson loops
-
'
'
0.0
5.0
|
O O
10.0
(b)
O
d
I
I
15.0
20.0
L 2 ( GeV -2)
(c)
{X/
I
~d
25.0
°
{
{
I
oI
1
I
I
(D
).
0 O0
I......-I LO 0
I
d I
0.5
I
I
1.0
1.5
0.5
X
1.0
1.5
X
Fig 1 In (a) we plot ln(WrL~t/W~quar~ ) (sohd lane) and its best phenomenological fit (dashed straight hne) at X = 1 10 The sohd curves m (b) and (c) are obtmned by fitting, for a fixed .~, slope × L 2 + intercept to - ln( Wr,~,/~,q,,~rc ) Again, dashed lines are the best phenomenologlcal fit
consider the r a n o In
We.(R, X)
(23)
mclrcle ( R , ) '
where the W L in the denominator corresponds to a circle of radius 2 R ' = - - R E ( I / i - - 7t2 )
(24)
7r
and E(x) is the complete elliptic function of second kind. Fig. 2 depicts the a p p r o x i m a t e hnear behavior of (23). The output still shows stability for o and zero intercept. O u r best fit yields fo- --- 0.52 G e V ,
intercept -- 0.
(25)
E Bagdn, J Latorre /
622
Wdson loops
(a)
0 LO 0
d I
I
I
I .....
1
.
Z~o ° o
1 80 0
-'"'" 0.0
5.0
10.0
(a v
15.0
( )
0 (D
I
I
I
0 O0
I---I
I
0
0.0
I
q
I
0.5
*,-'4
I
1.0
0.0
0.5
1.0
),
X
Fig 2 Comparison between -ln(W~n/W~,~d~)and its phenomenologlcal counterpart We have kept the same notatmn as m fig 1
Both r e c t a n g u l a r and elliptic sum rules are affected b y errors b u t we p o s t p o n e this d i s c u s s i o n to the conclusions• A n y sum rule contains a certain a m o u n t of m f o r m a U o n which b e c o m e s manifest u p o n p e r f o r m i n g some m a m p u l a t i o n s , as we showed above. Different e l a b o r a t m n s l e a d to n u m e r i c a l l y similar but not equal results• A c o m m o n i m p r o v e d sum rule is o b t a i n e d b y a p p l y i n g a Borel t r a n s f o r m a t i o n [18]. O u r a p p r o a c h does not need such a trick b e c a u s e we work in configurataon space. Nevertheless we could look for nice or simple final expressions. Let us briefly c o m m e n t one possible e x p l o l t a u o n of the e l h p t i c case. T a k e (22) a n d consider
W
OR 2
OR } "
E Bagcin, J Latorre / W t i s o n loops
623
L )xL
Fig 3 The nibbled contour is obtained by cutting out a httle (;kL) 2 square at one of the corners of the blg [(1 + )t)L] x square
The resulting expression presents a minimum at ,a-(a~G2> A=I,
o = 40N
R 2=
1011/32 + (IN
1)/3U2)N;I =l2] --
90eV
[1/32(g3fG3) + ( ( N 2 - 1)/3N2)NF(a~r)2] -~ 0.26 GeV 2.
=,
(27)
This result is just a different exploitation of (22), yieldmg a snmlar result (compare (25) and (27)), as expected. In this case we were lucky since the mlmn~zation procedure could be carried out analytically. Nevertheless, these equalmes have to be understood only numerically, they are not fundamental relations. The last planar loop we have studied is depmted in fig. 3. A straightforward calculation yields ~N2-1 mn,bbled ( L , ~k ) = 1 +
Tr
N
[
3+31n(/~L)+lnX+ln(l+X)+3
+f(l +A) +f()t) +f( l ~ ) -f( l ~ ) ] gr
12N (aG25L4(1 + 2x)2 ~G
1
N 2- 1
2]
1 (g ') + - 9 N F ~ (a~r(glq))] +6-U ×L6(1 + 6~t + llA 2 + 10)k3 + 5 • 4 ) .
(28)
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E, Bagfn, Y Latorre / W t l s o n loops
(a)
0
I
I
I j
°'.....
~o ~,-4
0
""/ 0.0
I.
I
5.0
10.0
15.0
20.0
(c v
(b)
0
..1
I
(c)
O
,.4
I
tO
25.0
I
I
~C3
C3
C)
c5 0.0
0.5
1.0
0.0
0.5
1.0
), Fig 4 Compar:son between -In( Wmbbled/ Ws3q/uu2re)and :ts phenomenologlcal counterpart We have kept the same notation as in fig 1 except that ~/~ is plotted instead of the slope :n (b)
In order to remove the divergences coming from the six cusps of the loop we consider the following ratio
-In
Wmbbled( L,
h)
(29)
[Wsquare(L')] 3 / 2 '
where L' = ~(1 + X)L. Again, L ' is pxcked out in such a way that the perimeter law also cancels. The results are summarized in fig. 4. Figure 4a shows once more a race
E Bag~Jn,J Latorre / Wilson loops
625
hnear behavaor. The values of ~ that lead to the best fit for 3 GeV G E V - 1 and fixed ~ are represented by the sohd line in fig. 4b (v~ is used the slope for better illustration). The fact that eq. (29) does not present X--* 0 hmit results in the smaller stability regions for both v~- and Nevertheless our results
~/o- -- 0.48 GeV,
i< L ~ 4 instead of a smooth intercept.
intercept = 0
(30)
are still m agreement w~th (21) and (25) (as a matter of fact, the intercept is not strictly null, as fig. 4c shows, but compatible with zero when uncertainties are considered). We have seen that planar loops are correctly described by an area law plus a possible irrelevant constant. Moreover, quantum fluctuations do not disturb too much this result since the intercept of (19), (23) and (29) are practically null. In this way we achieve a control on undetermined corrections to the leading behavior.
4. Non-planar Wilson loops Our next step consists in studying three-dlmensxonal WL's. For the sake of simplicity we have chosen a contour which is obtained by folding a rectangle (see fig. 5a), This loop has six singular points where the contour bends 7r/2 and is a natural candidate for lattice computations [19]. A simple but lengthy calculation
(,)
~q
l
X .(
L
1
,. jf<~;-
Fig 5 Mammal surfaces defined by the folded Wilson loop (a) and by the wavy one (b)
626
E Bagdn, J Latorre / Wtlson loops
yields the Q C D result
aN2-1 Wfolded(L , ~k) = 1 ÷ ~r - N
(1)
+31n(luL)+lnX+f
~2
~
+/(1)+/(2t)
]
(g3fG3)
12N(o~G2)(1 ÷)k2)L4+ 5 7 6 ~ ( N2-1 2 1÷ + " ~ -NF(°L'/r(~q)) (
1 --?k2+ 1)k4)L6
57k2+4
~k4) L6,
(31)
where f(x) was defined in sect. 2. Notice that we find the expected cusps singularities in addition to the familiar ill-defined perimeter law. Therefore it turns out to be advantageous to consider the quantity
-ln
Wfolded(g, ~) [Wsq.... (L')] 3/2 '
(32)
where L' = ~L(2 + X). This normalization cancels all cusps poles as well as perimeter dependences. Let us now turn to the phenomenologlcal side of the sum rule. Dropping the perimeter law as usual, the expected behavior for Wc corresponds to exp { - o. area - dimensionless laws },
(33)
It seems to us natural to try the geodesic curvature as a classical dimensionless law although we bear in mind that the accuracy of our technique might not be high enough to reach a clear conclusion. For contours made up of a finite number of smooth segments one must add to (9) the angles of the cusps - at which expressions like (10) make no sense - measured according to the metric on S. In particular, the geodesic curvature term for Wfolaea amounts to 37r (the straight lines give no contribution while each cusp adds a 7r/2 factor). As a result the dimensionless law drops out in (32) and straight lines with null intercepts are again expected. In this case we do not have an exact analytic expression for the minimal surface defined by the folded WL. Instead, we have written a program of discretlzed surfaces and looked numerically for the solution. The precision attained depends on the number of points which define the surface. We have run the program for increasing number of points until reaching an accuracy of 1%. Now, we are ready to perform the numerical analysis. We find a fair range of stable values for the string tension around ~ - = 0.48 GeV but nothing similar for the intercept (see fig 6).
E Bag6n, J Latorre / Wdson loops
627
(a)
o
eJ
t
I
--
I
[
[
),= 0.50
I
--
ed) 0
I qQ 0.0
5.0
10.0
15.0
20.0
~5.0
L 2 (GeV -e)
(b)
c]
(o)
o
I
I
I
I
I
I
~
[
I
(
I
I
o
o
c5 0.0
0.5
~I 1.0
I 0.0
k
0.5
1.0
k
Fig 6 Comparison between - I n ( Wfold~a/ l,Vs~/~) and its phenomenologlcal counterpart We have kept the same notatmn as m fig 1 except that f g as plotted instead of the slope in (b)
Actually, as for (29), eq. (32) does not have a smooth X ~ 0 llmat. Tbas turns out to be a drawback since stability is presumably spoded on this region Actually, our normalization (32) did not take into account that the six cusps of the folded contour reduce to four in the k ~ 0 hm]t. A neater and clearer analysis of non-planar WL's may be achieved studying smooth contours. Therefore we consider the second non-planar loop depxcted m fig. 5b. (wavy loop) which is defined by x = R c o s o~,
y =RSlna,
z --- ) t R c o s 2 o ~ ,
aE
[0,27r].
(34)
The c o m p u t a t m n of QCD contributions takes some time. In appendix B, where
628
E Bagdn, J Latorre /
Wdson Ioop~
some notation is defined, we present two calculations of the perturbative part with different regularlzanons. On the other hand, non-perturbanve corrections are easily handled. Collecting all the pieces and writing Q2 = 1 3- 4~ 2, we find
Wwavy(R,X)=l
2N a ~K --~ -3QE -~ 573
71"2
lZN@~G2)R4+4608~---~g3fG3)(43 +
- 19Q2)R 6
(35)
,z.2(N2- 1) 288N 3 NF(a~r(~/q))2( 17 -- Q2) R6.
First of all, we need the minimal surface defined by (34) It is good enough to take the following approximation solution for small X, Z(X, y) =
~(X 2 __y2).
(36)
Now we can compute the geometric invarlants we are interested in, area
__2q'/'R2Q2 3- Q 3- 1 3
Q+I
(37)
It turns out to be convenient to consider the ratio In Wwavy(R' X)
Wcircle( R, )
(39)
where we have normalized with a planar circular WL whose radius
IS chosen in order to cancel perimeter laws. Because of eq. (38) now we expect hnear trajectories but with the non-null intercept 2rra(1 - Q )
(41)
E Bag6n, J Latorre / Wdsonloops
629
(a)
O
tel
I
X=
I
0.30
.... ""
)g ¢5 0.0
I
I
5.0
10.0
(cev-9
R
(b)
tO
0
I
15.0
(c) I I I
O
N
©d © Or)
I
O I
0.0
I
X
0.5
I
I
1.0
0.0
X
0.5
1.0
Fig 7 Comparison between - l n ( W,~vv/ We,role) and its phenomenologlcal counterpart We have kept the same notataon as an fig 1 except that in (a) R has been changed to R' for better illustration Note the agreement for any value of ), lower than 0 5
c o m i n g f r o m the subtraction of the geodesic curvatures of the wavy contour and the circle. Actually, we find a nice and stable (see fig. 7) result v~- = 0.53 G e V ,
a = - 0.18.
(42)
It seems that a better description of non-planar W L ' s is obtained if a is different f r o m zero. The graphac fig. 7c shows an astonishing, nice matching between Q C D and the effective terms discussed above. Apparently, a classic geodesic curvature d e p e n d e n c e IS favored by the sum rule output. Nevertheless, we think that this result has to be taken with a grain of salt.
630
E Bagdn, J Latorre / Wilson loops
5. Conclusions A systematic Q C D sum rule computation of vev's of Wilson loops yields the same output for the string tension
--- 0.50(5) GeV.
(43)
This result does not depend on the shape of the WL for we have analyzed rectangular, elliptic, nibbled, folded and wavy contours. Our procedure seems qmte effective since we have considered ratios of WL's that tend to cancel or reduce errors. This point is supported by the large stability regions appearing in the final sum rules. We would like to stress that, to our knowledge, tins is the first time that non-planar loops are studied at intermediate energies within a continuum approach. Since all our expressions have been worked out for an arbitrary number of colors, we can consider the large N limit. It turns out that the numerical value of remains practically unchanged, we fred a small 4% increase. As expected, confinement survives when N--* ~ . All our sum rules are affected by errors. We would like to distingmsh between uncertainties inherent to our method and those coming from the values of condensates, A M and higher order contributions. While the former are certainly small since we have repetitively found the same stable output, the later are systematic, uncontrollable and dominated by the value of {aG2). The error quoted above is a large, conservative estimate of the first kind of uncertainties. Although we have very little control on the second source of inaccuracies, experience from other kind of sum rules tells us that whenever stabdlty shows up and ( a G 2) is the dominant contribution (as happens in our case) the output may be safely trusted within 20%. Besides, one maght think that the string tension values (21) and (25) are a bit too high compared to Regge slopes data (yielding an approximate value of 0.42 GeV). Our result Is pointing at an overestimation of (ctG2). This could well be possable though the present understanding of condensates values seems to be unclear [20]. Subleadlng laws become an interesting issue in our approach winch needs some more work. The nice, linear behavior with null intercept obtained for planar loops strongly indicates that any correction to the area law is numerically small. For nonplanar contours this is not so. We find straight lines passing away from the origin winch is a hint at classical or quantum corrections. A classical analysis (including boundary conditions) may attribute that correction to a geodesic (or, eqmvalently, extrinsic) curvature law. It is curious to note that, for the Polyakov model, even starting w~thout a geodesic curvature law at the classical level, quantum fluctuations generate such a term with a finite coefficient [16], namely - D / 8 T r , very close to our result. However, sum rule accuracy, we feel, may be too poor to reach any trustable conclusion as far as subleading laws are concerned.
E Bag(in, J Latorre / Wilson loops
631
All in all, we conclude that extended objects like WL's are well-suited to be described by means of Q C D sum rules. We would like to thank D. Espriu for ins help and advice at different stages of this work. We also acknowledge R. Koul's comments on differential geometry as well as discussions with P. Orland.
Appendix A DIMENSION M 6
CONDENSATES
condensates receive two kind of contributions. The first one comes from expanding W(C) up to order g< whereas the condensation (A~,(x)A,(O)) picks up a second term. The former case, or direct condensation, is given by Order
M 6
w(c) -
< g768N ? a ~ > P~cdX["X'~l~cdY['sY'~l~cdZ['sz~'l'
(A.I)
where d x m x ol means ( d x , x ~ - d x ~ x , ) . Note that path ordering is functional in this contribution. The second term, or non-local condensation, has a more complicated expression,
14/('-
2N
A -
C)(I..xI[o.lX)
+ B(2Iu~°Iox x - I~,o,,I~,xx- l~,olxxo) ]
1 (A + B - c)(/~oli~.lxx)}
8
(A.2)
where
I~,. = f ) c d x u x . ,
I.oxo = ~cdx~,xoxxxo
(A.3)
632 and
E Bagdn,J Latorre / Wdsonloops 1 [3 N2-1 2] A = 144 [ 2 (g3fG3) + 8 ~ T-NF(aqr(~lq)) ]' B=~
1 [ 3 N2-1 [-- -~(g3fG3) - 8 ~ N F ( a r r ( q q )
2] ) ],
C= -2A.
(A.4)
In obtaining this result we have made use of QCD sum rules standard techniques (see for instance refs. [10], [13], [14]). All the above expressions are general in the sense that all the shape dependence is parametrized in the integrals (A.3). However, some drastic simplifications occur for especially contours. For instance, no direct condensation takes place for planar loops due to the antisymmetric properties of the integral (A.1). In the case of loops spanned m three dimensions, eq. (A.2) may also be slmplfied using Stokes' theorem which transforms contour into surface integrals. Introducing the totally antisymmetric tensor in three dimensions e,j k we can rewrite (A.3) as follows Ibt o ~
--~p.oala
,
(A.5)
I..xx = - e~,°alxx, ~ -- 2e~,xalox, ~, where
/~1 = fsdx2dx3,
[2 = fsdx3 d x l ,
I3 = fsdXl dx2,
I~x,1 = fsXx dx2dx3 .... ,
= fsX
(A.6)
Xx d x 2 d x 3 . . . . .
Then, in the particular case of three-dimensional contours, the final expression (A.2) can be cast into the form
1{1
- __
2N -2 ( A + B - C)[ L L b ' h - 2LI~bb'~ + 2L'bL'h]
1
3 (A - C)[~ • ~Ib , b -
Ola,bib, a)" }
(A.7)
E Bag&l,J Latorre /Wtlson loops
633
The advantage of this expression over (A.2) is simply that one has to calculate a smaller number of integrals.
Appendix B PERTURBATIVE INTEGRAL
The perturbative integral appearing in the study of the contour C,
x=Rcosa,
y=Rsina,
z=XRcos2a,
(B.1)
IS divergent. We will regularize ~t in two different ways. First we introduce a cut-off a m configuration space, 8,~
'=-~cdXt'~cdY~(x_y)Z+a
(B .2)
2
Parametrizing (B.2) with x . = R(cos a, sin a, h cos 2a, 0), y~ = R(cos/3, sm ]~, h cos 2]3, 0) and changing to the variables r = a - ]3, s = fi we can isolate the singular behavior f02~
R2(1 +4x2sm22s)
[2~
dr,o
47rRE(2tX)+O(R)
dSa2+2Ra(l+4)t2sin22s)(1-cosr)
=
(B.3)
a
where E(x) is the elliptic integral of the second kind. Then we compute the fimte part (no a remains) by integrating If2~drf2~dst
2 o
o
cosr+4~2sIn2s(sin2rcos2s+cos2rsln2s) ~ 1-cosr+2Xasm2r(smrcos2s+cosrsin2s) 2
= 2w{ K(2/~k)
--
1
}
1-cosr
3E(2iX)},
(B.4)
where K(x) is the elliptic integral of the first kind. Transforming imaginary arguments into real ones, vxa
K(21X) =
--2-K
(we have introduced the notation
, Q2 = l +
E(2th) =
QE
(B.5)
4he), we obtain the final result
[ 2QR { 2h ~ 1K 2~ ) 3QE(2h
(B.6)
E Bagdn,J Latorre /Wtlson loops
634
Let us now consider dimensional regularizaUon. The integral (B.1) is defined as
~v I=~cdX~,~cdY,,[(x_y)2]D/2
1.
(B.7)
After some rearrangements of the integration variables, this expression can be recast into the form
[(°2) (°2)
1 = 4 ~6-°'/2 I 2 - D , ~ - - - , 0
+4(Q 2-1)1
(°2) ( °_2)]
+2(1-2Q2)I
6-D,~,0
4-D,---~,0
+(Q2-1)I
2-D,-~---,2
, (B.8)
where slnt~o~sin73
(B.9)
I(la, v , ' r ) - fo~/2d°~fo~/2d~ (l+4~2cos2otsln2fl)~ "
The a lntegraUon is easily done using (ref. [21]; eq. (3.681)) yielding hypergeometrlc functions. The last step (ref. [21]; eq. (7.512.12)) gives F(/z+l I(/~'v'T):
4 F( " + 2 ] F]( ~ ~' + 2 21 2
]
'
T+I
#+2
• +2
2
2
2
• _ 43k2 '
)
(B.10) When particularizing to D = 4 we obtain
I=2~r[QK(2-~)-3QE(2-~)],
(B.11)
which reproduces the finite part of (B.6). As expected, dimensional regularizaUon sweeps out the hnear divergence which appears as a pole in three dimensions.
References [1] M Luscher. K Symanzlkand P Welsz,Nucl Phys B173(1980) 365, M Ltischer,Nucl Phys B180(1981) 317 [2] A M Polyakov,Nucl Phys B268(1986) 406 [3] P Olesen and S-K Yang, B283 (1987) 73, E Braaten, R D Plsarskl and S-M Tse, Phys Rev Lett 58 (1987) 93
E Bag6n, J Latorre / Wilson loops [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21]
635
M Flensburg and C Peterson, Nucl Phys B283 (1987) 141 E Bag~n, J I Latorre and R Tarrach, Phys Lett 152B (1985) 113, Phys Lett 158B (1985) 145 E BagSn and J I Latorre, Phys Lett 176B (1986) 449 E BagSn, Phys Lett 192B (1987)420 R A Brandt et al, Phys Rev D24 (1981) 879 A Bassetto and R Soldatl, Nucl Phys B276 (1986) 517, G P Korchemsky and A V Radyusktnn, Nucl Phys B283 (1987) 342 M A Shlfman, Nucl Phys B173 (1980)13 S Nanson, CERN-TH 4624/86 L J Remders, H Rubmsteln and S Yazakl, Phys Reports 127 (1985) 1 Contributions in the Workshop on Non-perturbatxve methods (Montpelher) (World Scientific 1986) P Pascual and R Tarrach, QCD Renormahzatlon for the practatloner (Springer, 1984) E Baghn, J I Latorre and P Pascual, Z Plays C32 (1986) 43 O Alvarez, Nucl Phys B216 (1983)125 R Karschner et al, Nucl Phys B210 (1982) 567 M A Shlfman, A Vamshtem and V Zakharov, Nucl Phys B147 (1979) 385, 448 D Arnaudon and C Roxesnel, Phys Lett 187B (1987) 153 C A Dominguez, DESY 87-002; H G Dosch, Phys Lett 190B (1987) 177, R A Bertlmann, UWThPh-1986-25 I S Gradshteyn and I M Ryzhlk, eds., Tables of integrals and formulas (Academic Press, 1980)
PLANAR AND NON-PLANAR WILSON L O O P S IN Q C D * Emil1 BAG,~N
Physws Department, Brookhaven Natwnal Laboratory, Upton, N Y 11973 USA Jos6 I L A T O R R E
Center for Theorettcal PhysmL Laboratory for Nuclear Scwnce and Department of Physics, Massachusetts Insmute of Technology, CambrMge, Massachusetts 02139, USA Recewed 2 June 1987
A general QCD sum rule calculation of vacuum expectation values of Wilson loops is presented and particularized to several specific planar and non-planar contours The s t u n g tension obtmned is ~/o = 0 50(5) GeV Within the errors of the approach, this result is shape-independent We comment on the posslbihty that corrections to the area and perimeter laws could be parametnzed by a geodesic curvature term
1. Introduction Wilson loops (WL's) behave in very different ways at short, intermediate and long distances. Perturbation theory correctly describes the ultraviolet regime once some divergences are renormahzed (see sect. 2). Nevertheless tins approach cannot be used to investigate intermediate energies since some non-perturbative technique is needed to deal with confinement. The situation worsens as we move to longer distances and screening takes place In principle, these three distinct behaviors are described by quantum chromodynamlcs (QCD) but such a hope would only become apparent upon solving the theory. The usual, pragmatic modus operan& consists in guessing the effect theory well-suited to describe a certain range of energies. The effective description of WL's at intermediate scales might correspond to a string theory. Within tins picture, a static effective quark-antiquark potential arises naturally for a wide class of strings [1]
V(r)
= or-
-
12 r -
-
+
.
.
-
(1)
* This work is supported in part by funds provided by the US Department of Energy (DOE) under contracts #DE-AC02-76ER03069 and DE-AC02-76CH00016 0550-3213/88/$03 50©Elsevier Scmnce Pubhshers B V (North-Holland Physms P u b h s h m g Division)
614
E Bag6n, J Latorre / W t l s o n loops
where o is the string tension. The classical linear part in (1) leads to confinement whereas the correction l / r , universal Luscher term, comes from quantum fluctuations. Recently, Polyakov [2] suggested that the N a m b u - G o t o action for strings can be supplemented with a term proportional to the extrinsic curvature of the worldsheet. This new contribution as well as finite size effects modify eq. (1) (see ref. [3]). At this point, we still have to relate the string picture with QCD. Since, at present, there is very little hope of connecting them exactly, a natural way to proceed is to compute vacuum expectation values (vev) of WL's for different sizes and fit those parameters appearing in the phenomenological description. This program has been carried out for planar WL's within the contexts of lattice regularlzatlon [4] and Q C D sum rules. Although numerical simulations are more popular, the second method presents a main advantage, namely, it works in the continuum. In a series of previous papers [5], [6], we have analyzed rectangular and elliptic WL's using this technique. The results were very satisfactory. We found stable values for the string tension which turned out to be the same for the two shapes as was expected. It is important to notice, however, that the analysis of the elliptic case was ~mphc~tly based on the assumption that no divergence proportional to the perimeter arose - which is certainly true in dimensional regularlzation - and that the perimeter was a linear function of the axes a and b - which is only approximately true for a - b. In the last paper we also computed the mass gap. The result was m agreement with several other estimates. Recently, one of us [7] has applied s~rmlar techniques to the study of which string model is behind QCD. Taking into account boundary effects, sum rules seem to be sensitive enough so as to favor the modified stnng picture which includes the extrinsic curvature term as the effective theory providing the quark-antlquark static potentifil. In this paper we complete our study of planar WL's using a more general exploitation of the sum rule and we extend it to nonplanar contours (note that no roughening comphcations are encountered in our a p p r o a c h - we work in the continuum). An effort has been made to give as general a computation as possible. A priori, the classical analysis of three dimensional WL's becomes more involved since the corresponding effective theory may include a bunch of new geometric invariants. For instance the area law might be accompanied with some dependence on the intrinsic or extrinsic curvature. In the case of finite contours, we also have new lnvariants depending on its shape, namely perimeter or geodesic curvature laws. Actually, we d~spose of an Infinite set of potential dependences since we are searching an effecnve description. Yet, the accuracy of our approach is of the order of a 20%. Therefore we are forced to restrict the scope of our analysis to a certain order. It seems natural to only consider dependences on invariants of dimension L 2 (area), L (perimeter), and L ° (lntnnslc curvature, extrinsic curvature . . . . ), though quantum corrections might be relevant. We do not believe that Q C D sum rules are sensitive enough to analyze accurately L ° laws. Nevertheless we will comment on
E Bag6n, J Latorre / Wdson loops
615
some of these issues. We have organized our presentation in the following way. Sect. 2 is devoted to discuss the systematics of our approach. On one hand, QCD contributions are computed up to some final, shape-dependent integrals. On the other hand, effective lagranglan terms are discussed. Some problems show up from the very beginning. We will face a genuine type of divergences associated with nonlocal gauge invarlant color phases. These singularities will force us to get rid of perimeter laws through this work. In sect. 3 we particularize our calculations to planar WL's and exploit their respective sum rules. Stable, shape-independent values for the string tension are obtained. A similar analysis, though more complex, is carried out for nonplanar WL's in sect. 4. Again, the output shows a nice value for o. We summarize our conclusions in the last section and, finally, some technicalities are presented in the appendices.
2. General strategy WL's are the simplest nonlocal gauge lnvariant objects in QCD. Their vaccum expectation values are defined by euclidean space-time and for N colors as
Wc= ~
trPexp
--lgf)cdXt,A~(x)T ,
(2)
where C is a closed contour, P indicates path ordering and M stand for a N × N generalization of the Gell-Mann matrices of SU(3). The idea behind the sum rule approach consists in calculating the perturbative expansion of (2) adding nonperturbative corrections, e.g. condensate contributions, in order to have a fair description of WL's at intermediate energies. Then it will be possible to compare QCD with an effective theory valid at the same scale. Let us begin with the usual perturbative computation. Choosing the appropriate gauge the first non-trivial perturbatlve contribution to (2) corresponds to
Wc=I
a N2-1 4~r
~(~dx.C~dY~(x_y)2Jc Jc
(3)
The ultraviolet behavior of the above integral has been thoroughly investigated [8]. Depending on the contour, there are three kind of singularities: (i) A linear divergence proportional to the length of the contour C. Physically, it is understood as a mass renormahzatlon of the heavy quark that generates the WL. (ii) A logarithmic divergence for every cusp of C, corresponding to an infinite bremstrahlung of the test particle. (111) A logarithmic divergence whenever C self-intersects. Tins would mean that a quark and an antiquark pass through each other.
616
E Bagdn, J Latorre /
Wdson loops
Although the slngulariues of the first kind do not show up in dimensional regularization, they mess up a possible perimeter law. To be more precise, this hnear divergence exponentlates (see ref. [9] for a check up to two loops) and renders the perimeter law renormalization-scheme dependent. On the other hand, log divergences are somewhat Innocuous. They can be avoided by choosing smooth, nonself-intersecting contours. Each of the WL's analyzed m next sections will require some kind of normalization in order to get rid of any singularity before exploiting our results. Recall that (3) cannot describe non-perturbatlve effects. We have to compute sum rule corrections to this term. As a rule, we will consider contributions coming from condensates of dimension M4((aG2)) and M 6 ((g3fG3) and (~q)2). This kind of computation was carried out for the first time m ref. [10]. More technical details may be found in the literature [11-15]. Here we simply note that it is convenient to work in the Fock-Schwinger gauge [10]
x~,A~(x) = 0,
(4)
where all the non-perturbative integrals become trivial. After some straightforward algebra we obtain the gluon condensate contribution
W c = - 48-~(aG2)~)cdX~,X,,
dy~y¢(8,~8~- 8,,~8~,~).
(5)
The computation of order M 6 condensates becomes rather revolved and is presented in appendix A. Obviously, all the non-perturbative non-computable information is parametrized in the condensates, Numerically, we will take standard values for them a a ~v ) -- 0.04 GeV 4, (aG 2) =- (aG;~G
(g3fG3) =- /\ 6a 3 (J a b c l'7~a(Tbvf'g'colt\ V~v~p ~ /
=
(F/q) z -- ( - 0.25 GeV) 6 .
1.1 GeV2(aG 2)
(6)
A 25% uncertainty in (aG 2) will turn out to be the main source of error in our results which are strongly dominated by this condensate. This completes the QCD part of the sum rule. Note that our three equations (3), (5) and (A.2) only depend on contour integrals (path ordering is operatwe in the third one). Every specific shape will lead to different sum rule though, hopefully, to the same effectwe description. So far we have detailed how the QCD calculation is done. Now we need an effective theory to describe the vev of euclidean WL's around 1 GeV. Strong couphng suggests an exponential decay of the form exp ( - o. area},
(7)
E Bag~n, J Latorre / Wdson loops
617
o being the string tension. The area appearing in (7) is the area of the minimal surface S containing the contour C (we will keep this notaUon throughout this work). This behavior, as noted in the introduction, corresponds to a linear static quark-antiquark potential. A natural framework where (7) arises as a classical approximation is provided by N a m b u - G o t o strings, whose euchdean action reads
! = ors d2~y/~ .
(8)
Here, d2~ corresponds to the surface differential and g - d e t g~b, where g u t ,=Oux, Obx~ is the induced metric on the surface or first fundamental form. However m a n y other effective lagrangians are valid. We could add a whole new collection of geometric invariants which become relevant for the quantum theory a n d / o r at the classical level when the WL lives in more than two dimensions. In sect 4 we will need to make up our mind about the kind of dimensionless law Of any) we should introduce in the non-planar Wilson loop sum rules. A detailed analysis of renormahzable, parametrlzation m v a n a n t lagrangians for WL's has been carned out by Alvarez [6]. Classically we may add to (8) the following term
a fcdsk,
(9)
where a is a new fundamental parameter which we will try to determine in sect. 4, ds is the arclength differential of C and k is its geodesic curvature. More precisely, consider the loop C and then find the unit tangent (t a) and normal (n ~) vectors to the contour using the metric g,,h. Then k is defined as
taW~t b = kn b .
(10)
One may think that we are missing a term proportional to the intrinsic curvature, R. This is not the case. For a finite surface, the Gauss-Bonnet theorem states that
½fs d2~v~-R + fas d s k = 2 ~ r x ( S ) '
(11)
where x ( S ) = 2 - 2 h - b ( h = # of holes, b = # of boundaries) is the Euler characteristic of S. We still have two other well-known geometric invariants, namely the extrinsic curvature
~/g- (Kata) 2
and
vg"bG-~ l,-,u~,b..u,
(12)
E Bagdn, J Latorre /
618
Wdson loops
built out of the second fundamental form K~h which is defined by
O~Obx ~ = F2bOcx ~ + K~bn ~ ,
(13)
!
where n, are D - 2 vectors normal to the surface and laCb are the Christoffel symbols of this surface. Nevertheless, a fundamental relation in analytic geometry links the two extrinsic curvature structures to the intrinsic curvature on the surface,
R = (K;o)2_
*x b
*x a
.
(14)
Furthermore, we note that for surfaces of mimmal area
= 0.
(15)
This amounts to saying that the Gauss-Bonnet theorem relates the non-trivial combination ,,e tc-~a~ ,,~~b to the geodesic curvature for minimal surfaces. We conclude that a classical description of WL's may require a dimensionless law related to the geodesic curvature of the contour or, equivalently, to the extrinsic curvature of the surface. We have not commented on the perimeter law yet. Due to the genuine divergence associated to the length of any color phase, a perimeter law would become renormalization-scheme dependent. The easiest way of coping with this problem consists in only considering appropnate ratios of WL's so as to cancel this ominous perimeter law. As a bonus, constants also drop out and systematic uncertainties are reduced. From now on we wall omit any reference to such terms.
3. Planar Wilson loops In order to get numbers, all the machinery of the previous section has to be applied to specific contours. It is natural to start considenng WL's which lie on a plane. Such a class of contours introduce a good deal of simplification. For instance, independently of the shape of the two-dimensional WL, the (aG 2) contribution turns out to be q7
Wc = - 1 2 ~ (°LG2)S2'
(16)
where S c stands for the area circled by C This may be verified by using Stokes theorem in (5). Likewise, no direct (g3fG3) condensation takes place (which is the only term where path ordering is kept). On the phenomenological side, leaving aside the perimeter law and constants we expect exp{ - o. area}
(17)
619
E Bagdn, J Latorre / Wdson loops
as no other non-trivial geometric invariants exist in two dimensions. Note that, classically, the extrinsic curvature of a flat surface vanishes and the integral of the geodesic curvature on a two-dimensional contour amounts to 2~r. Although we do not explicitly consider quantum corrections to (17) we will be able to have some control on them (see below). The simplest WL instance is a rectangular contour, L times T. Computing all the integrals and writing T = )~L we find c~ N 2 - 1
Wre~t(L,X)=l+
~r
- N
12N
[ 2 + 21nl*L + lnX + 2 + f()t) + f( ~)]
(aG2))~eL4
+ 12~-N
96
+ -9 (a~r(qq))2Nv ~ N
)t2(1 + ~2)L6'
(18)
where f ( x ) = x arctanx + ½1n(1 + x 2) and /* stands for some subtraction scale. Since we have dimensionally regularized (D = 4 - e) the perturbative part, we do not have a hnear divergence. However we find a pole 1/e coming from the cusps of the rectangle. Notice that condensates are providing power corrections to the first two terms. They are enlarging the range of distances where (18) gives a correct description of the WL, hence our hope of capturing non-perturbative effects. Now we can go on in &fferent ways. In ref. [5] we considered a useful combination of WL's and fitted the value of the string tensmn. Here we proceed differently. In order to get rid of a perimeter law, we take the expansmn
in msq .... ( L ' )
~ ~
+ -12N -
[ln
42,
f(X) -f
(aG2)L4~2 1 1
12N
+ 2f(1)
96
z
N2-1]
+ ~(a~r(~q)) U v -N-T
XL6 [~.2(1 + ~k2) - 2( L' ] 6]
g-jj'
(19)
where L' = ~(L + T) = 1L(1 + X) and thus both WL's have the same perimeter This normalization also eliminates the 1/t and /, factors in (18). We further
E Bagdn, J Latorre / Wdson loops
620
r e n o r m a h z a t i o n group improve the perturbatlve contribution by introducing a running coupling constant 2~r 1311n(L,2A2) ,
~(L')
where [171 A = ½AMNexp(yE), " Y E = 0 . 5 7 7 . . ,
A~g=0.12
(20) G e V and 131 = -
~-N
1 + ~ N F.
At intermediate energms and according to our expectations, eq. (19) as a function of L 2 for a fixed X should behave as a straight line passing by the origin, namely o-(SrectSsquare ) = -- ¼oL2(1 - X) 2. This is indeed so. Fig. l a shows eq. (19) v e r s u s L 2 for )~ = 1.1 (sohd hne) as well as the best fit to Q C D for 2.7 GeV -~ ,%
intercept = 0.
(21)
T h e solid hnes in fig. l b and fig. l c are respectively the slope and intercept of the straxght line that best fits Q C D m the same L region as before but now for a given value of )t. The dashed hnes correspond to the global fit (i.e. slope = - ¼o(1 - )t) 2, taking o f r o m (21); and intercept = 0). Note the good matching for ] 1 - )t t < 0.5 w h m h should be interpreted as a measure of the stability of our output. The result (21) for the string tension is m complete agreement with our previous analysis of ref. [5], W h a t is new and non-trivial is finding a stable null intercept. A shift of the straight line would mean that some corrections are present. We did not expect classmal m o d i f i c a t m n s but q u a n t u m corrections nnght have entered into the game. In fact, our result states that fluctuations of the finite world-sheet 0.e. taking into account b o u n d a r y effects) appearing in the " N a m b u - G o t o + &menslonless laws" action are small for )t = 1 or, at least, that the present analysis ~s insensitive to them. It is of capital importance to check whether these results depend on the specific shape we have considered. Consequently, we take a second and non-pathological c o n t o u r - an ellipse with axes a = ?tR and b = R. After some work we find N2-1 Wen(R, X) = 1 + a ~ r - - 2N
219
+ 4T
l+)t 2
2~,
(g3fG3) + ~
¢r
12N (°tG2)(rr~'R2)2 --NF(~r(oq))
~k2(1 -'~ ~k2)R6" (22)
O f course, no singularities are encountered since the contour 1s smooth and we work in dimensional regularlzation. Again, to avoid a perimeter and constant law, we
E Bagdn, J Latorre /
(a)
O O
O
ig
621
Wdson loops
-
'
'
0.0
5.0
|
O O
10.0
(b)
O
d
I
I
15.0
20.0
L 2 ( GeV -2)
(c)
{X/
I
~d
25.0
°
{
{
I
oI
1
I
I
(D
).
0 O0
I......-I LO 0
I
d I
0.5
I
I
1.0
1.5
0.5
X
1.0
1.5
X
Fig 1 In (a) we plot ln(WrL~t/W~quar~ ) (sohd lane) and its best phenomenological fit (dashed straight hne) at X = 1 10 The sohd curves m (b) and (c) are obtmned by fitting, for a fixed .~, slope × L 2 + intercept to - ln( Wr,~,/~,q,,~rc ) Again, dashed lines are the best phenomenologlcal fit
consider the r a n o In
We.(R, X)
(23)
mclrcle ( R , ) '
where the W L in the denominator corresponds to a circle of radius 2 R ' = - - R E ( I / i - - 7t2 )
(24)
7r
and E(x) is the complete elliptic function of second kind. Fig. 2 depicts the a p p r o x i m a t e hnear behavior of (23). The output still shows stability for o and zero intercept. O u r best fit yields fo- --- 0.52 G e V ,
intercept -- 0.
(25)
E Bagdn, J Latorre /
622
Wdson loops
(a)
0 LO 0
d I
I
I
I .....
1
.
Z~o ° o
1 80 0
-'"'" 0.0
5.0
10.0
(a v
15.0
( )
0 (D
I
I
I
0 O0
I---I
I
0
0.0
I
q
I
0.5
*,-'4
I
1.0
0.0
0.5
1.0
),
X
Fig 2 Comparison between -ln(W~n/W~,~d~)and its phenomenologlcal counterpart We have kept the same notatmn as m fig 1
Both r e c t a n g u l a r and elliptic sum rules are affected b y errors b u t we p o s t p o n e this d i s c u s s i o n to the conclusions• A n y sum rule contains a certain a m o u n t of m f o r m a U o n which b e c o m e s manifest u p o n p e r f o r m i n g some m a m p u l a t i o n s , as we showed above. Different e l a b o r a t m n s l e a d to n u m e r i c a l l y similar but not equal results• A c o m m o n i m p r o v e d sum rule is o b t a i n e d b y a p p l y i n g a Borel t r a n s f o r m a t i o n [18]. O u r a p p r o a c h does not need such a trick b e c a u s e we work in configurataon space. Nevertheless we could look for nice or simple final expressions. Let us briefly c o m m e n t one possible e x p l o l t a u o n of the e l h p t i c case. T a k e (22) a n d consider
W
OR 2
OR } "
E Bagcin, J Latorre / W t i s o n loops
623
L )xL
Fig 3 The nibbled contour is obtained by cutting out a httle (;kL) 2 square at one of the corners of the blg [(1 + )t)L] x square
The resulting expression presents a minimum at ,a-(a~G2> A=I,
o = 40N
R 2=
1011/32
1)/3U2)N;I =
90eV
[1/32(g3fG3) + ( ( N 2 - 1)/3N2)NF(a~r
=,
(27)
This result is just a different exploitation of (22), yieldmg a snmlar result (compare (25) and (27)), as expected. In this case we were lucky since the mlmn~zation procedure could be carried out analytically. Nevertheless, these equalmes have to be understood only numerically, they are not fundamental relations. The last planar loop we have studied is depmted in fig. 3. A straightforward calculation yields ~N2-1 mn,bbled ( L , ~k ) = 1 +
Tr
N
[
3+31n(/~L)+lnX+ln(l+X)+3
+f(l +A) +f()t) +f( l ~ ) -f( l ~ ) ] gr
12N (aG25L4(1 + 2x)2 ~G
1
N 2- 1
2]
1 (g ') + - 9 N F ~ (a~r(glq))] +6-U ×L6(1 + 6~t + llA 2 + 10)k3 + 5 • 4 ) .
(28)
624
E, Bagfn, Y Latorre / W t l s o n loops
(a)
0
I
I
I j
°'.....
~o ~,-4
0
""/ 0.0
I.
I
5.0
10.0
15.0
20.0
(c v
(b)
0
..1
I
(c)
O
,.4
I
tO
25.0
I
I
~C3
C3
C)
c5 0.0
0.5
1.0
0.0
0.5
1.0
), Fig 4 Compar:son between -In( Wmbbled/ Ws3q/uu2re)and :ts phenomenologlcal counterpart We have kept the same notation as in fig 1 except that ~/~ is plotted instead of the slope :n (b)
In order to remove the divergences coming from the six cusps of the loop we consider the following ratio
-In
Wmbbled( L,
h)
(29)
[Wsquare(L')] 3 / 2 '
where L' = ~(1 + X)L. Again, L ' is pxcked out in such a way that the perimeter law also cancels. The results are summarized in fig. 4. Figure 4a shows once more a race
E Bag~Jn,J Latorre / Wilson loops
625
hnear behavaor. The values of ~ that lead to the best fit for 3 GeV G E V - 1 and fixed ~ are represented by the sohd line in fig. 4b (v~ is used the slope for better illustration). The fact that eq. (29) does not present X--* 0 hmit results in the smaller stability regions for both v~- and Nevertheless our results
~/o- -- 0.48 GeV,
i< L ~ 4 instead of a smooth intercept.
intercept = 0
(30)
are still m agreement w~th (21) and (25) (as a matter of fact, the intercept is not strictly null, as fig. 4c shows, but compatible with zero when uncertainties are considered). We have seen that planar loops are correctly described by an area law plus a possible irrelevant constant. Moreover, quantum fluctuations do not disturb too much this result since the intercept of (19), (23) and (29) are practically null. In this way we achieve a control on undetermined corrections to the leading behavior.
4. Non-planar Wilson loops Our next step consists in studying three-dlmensxonal WL's. For the sake of simplicity we have chosen a contour which is obtained by folding a rectangle (see fig. 5a), This loop has six singular points where the contour bends 7r/2 and is a natural candidate for lattice computations [19]. A simple but lengthy calculation
(,)
~q
l
X .(
L
1
,. jf<~;-
Fig 5 Mammal surfaces defined by the folded Wilson loop (a) and by the wavy one (b)
626
E Bagdn, J Latorre / Wtlson loops
yields the Q C D result
aN2-1 Wfolded(L , ~k) = 1 ÷ ~r - N
(1)
+31n(luL)+lnX+f
~2
~
+/(1)+/(2t)
]
(g3fG3)
12N(o~G2)(1 ÷)k2)L4+ 5 7 6 ~ ( N2-1 2 1÷ + " ~ -NF(°L'/r(~q)) (
1 --?k2+ 1)k4)L6
57k2+4
~k4) L6,
(31)
where f(x) was defined in sect. 2. Notice that we find the expected cusps singularities in addition to the familiar ill-defined perimeter law. Therefore it turns out to be advantageous to consider the quantity
-ln
Wfolded(g, ~) [Wsq.... (L')] 3/2 '
(32)
where L' = ~L(2 + X). This normalization cancels all cusps poles as well as perimeter dependences. Let us now turn to the phenomenologlcal side of the sum rule. Dropping the perimeter law as usual, the expected behavior for Wc corresponds to exp { - o. area - dimensionless laws },
(33)
It seems to us natural to try the geodesic curvature as a classical dimensionless law although we bear in mind that the accuracy of our technique might not be high enough to reach a clear conclusion. For contours made up of a finite number of smooth segments one must add to (9) the angles of the cusps - at which expressions like (10) make no sense - measured according to the metric on S. In particular, the geodesic curvature term for Wfolaea amounts to 37r (the straight lines give no contribution while each cusp adds a 7r/2 factor). As a result the dimensionless law drops out in (32) and straight lines with null intercepts are again expected. In this case we do not have an exact analytic expression for the minimal surface defined by the folded WL. Instead, we have written a program of discretlzed surfaces and looked numerically for the solution. The precision attained depends on the number of points which define the surface. We have run the program for increasing number of points until reaching an accuracy of 1%. Now, we are ready to perform the numerical analysis. We find a fair range of stable values for the string tension around ~ - = 0.48 GeV but nothing similar for the intercept (see fig 6).
E Bag6n, J Latorre / Wdson loops
627
(a)
o
eJ
t
I
--
I
[
[
),= 0.50
I
--
ed) 0
I qQ 0.0
5.0
10.0
15.0
20.0
~5.0
L 2 (GeV -e)
(b)
c]
(o)
o
I
I
I
I
I
I
~
[
I
(
I
I
o
o
c5 0.0
0.5
~I 1.0
I 0.0
k
0.5
1.0
k
Fig 6 Comparison between - I n ( Wfold~a/ l,Vs~/~) and its phenomenologlcal counterpart We have kept the same notatmn as m fig 1 except that f g as plotted instead of the slope in (b)
Actually, as for (29), eq. (32) does not have a smooth X ~ 0 llmat. Tbas turns out to be a drawback since stability is presumably spoded on this region Actually, our normalization (32) did not take into account that the six cusps of the folded contour reduce to four in the k ~ 0 hm]t. A neater and clearer analysis of non-planar WL's may be achieved studying smooth contours. Therefore we consider the second non-planar loop depxcted m fig. 5b. (wavy loop) which is defined by x = R c o s o~,
y =RSlna,
z --- ) t R c o s 2 o ~ ,
aE
[0,27r].
(34)
The c o m p u t a t m n of QCD contributions takes some time. In appendix B, where
628
E Bagdn, J Latorre /
Wdson Ioop~
some notation is defined, we present two calculations of the perturbative part with different regularlzanons. On the other hand, non-perturbanve corrections are easily handled. Collecting all the pieces and writing Q2 = 1 3- 4~ 2, we find
Wwavy(R,X)=l
2N a ~K --~ -3QE -~ 573
71"2
lZN@~G2)R4+4608~---~g3fG3)(43 +
- 19Q2)R 6
(35)
,z.2(N2- 1) 288N 3 NF(a~r(~/q))2( 17 -- Q2) R6.
First of all, we need the minimal surface defined by (34) It is good enough to take the following approximation solution for small X, Z(X, y) =
~(X 2 __y2).
(36)
Now we can compute the geometric invarlants we are interested in, area
__2q'/'R2Q2 3- Q 3- 1 3
Q+I
(37)
It turns out to be convenient to consider the ratio In Wwavy(R' X)
Wcircle( R, )
(39)
where we have normalized with a planar circular WL whose radius
IS chosen in order to cancel perimeter laws. Because of eq. (38) now we expect hnear trajectories but with the non-null intercept 2rra(1 - Q )
(41)
E Bag6n, J Latorre / Wdsonloops
629
(a)
O
tel
I
X=
I
0.30
.... ""
)g ¢5 0.0
I
I
5.0
10.0
(cev-9
R
(b)
tO
0
I
15.0
(c) I I I
O
N
©d © Or)
I
O I
0.0
I
X
0.5
I
I
1.0
0.0
X
0.5
1.0
Fig 7 Comparison between - l n ( W,~vv/ We,role) and its phenomenologlcal counterpart We have kept the same notataon as an fig 1 except that in (a) R has been changed to R' for better illustration Note the agreement for any value of ), lower than 0 5
c o m i n g f r o m the subtraction of the geodesic curvatures of the wavy contour and the circle. Actually, we find a nice and stable (see fig. 7) result v~- = 0.53 G e V ,
a = - 0.18.
(42)
It seems that a better description of non-planar W L ' s is obtained if a is different f r o m zero. The graphac fig. 7c shows an astonishing, nice matching between Q C D and the effective terms discussed above. Apparently, a classic geodesic curvature d e p e n d e n c e IS favored by the sum rule output. Nevertheless, we think that this result has to be taken with a grain of salt.
630
E Bagdn, J Latorre / Wilson loops
5. Conclusions A systematic Q C D sum rule computation of vev's of Wilson loops yields the same output for the string tension
--- 0.50(5) GeV.
(43)
This result does not depend on the shape of the WL for we have analyzed rectangular, elliptic, nibbled, folded and wavy contours. Our procedure seems qmte effective since we have considered ratios of WL's that tend to cancel or reduce errors. This point is supported by the large stability regions appearing in the final sum rules. We would like to stress that, to our knowledge, tins is the first time that non-planar loops are studied at intermediate energies within a continuum approach. Since all our expressions have been worked out for an arbitrary number of colors, we can consider the large N limit. It turns out that the numerical value of remains practically unchanged, we fred a small 4% increase. As expected, confinement survives when N--* ~ . All our sum rules are affected by errors. We would like to distingmsh between uncertainties inherent to our method and those coming from the values of condensates, A M and higher order contributions. While the former are certainly small since we have repetitively found the same stable output, the later are systematic, uncontrollable and dominated by the value of {aG2). The error quoted above is a large, conservative estimate of the first kind of uncertainties. Although we have very little control on the second source of inaccuracies, experience from other kind of sum rules tells us that whenever stabdlty shows up and ( a G 2) is the dominant contribution (as happens in our case) the output may be safely trusted within 20%. Besides, one maght think that the string tension values (21) and (25) are a bit too high compared to Regge slopes data (yielding an approximate value of 0.42 GeV). Our result Is pointing at an overestimation of (ctG2). This could well be possable though the present understanding of condensates values seems to be unclear [20]. Subleadlng laws become an interesting issue in our approach winch needs some more work. The nice, linear behavior with null intercept obtained for planar loops strongly indicates that any correction to the area law is numerically small. For nonplanar contours this is not so. We find straight lines passing away from the origin winch is a hint at classical or quantum corrections. A classical analysis (including boundary conditions) may attribute that correction to a geodesic (or, eqmvalently, extrinsic) curvature law. It is curious to note that, for the Polyakov model, even starting w~thout a geodesic curvature law at the classical level, quantum fluctuations generate such a term with a finite coefficient [16], namely - D / 8 T r , very close to our result. However, sum rule accuracy, we feel, may be too poor to reach any trustable conclusion as far as subleading laws are concerned.
E Bag(in, J Latorre / Wilson loops
631
All in all, we conclude that extended objects like WL's are well-suited to be described by means of Q C D sum rules. We would like to thank D. Espriu for ins help and advice at different stages of this work. We also acknowledge R. Koul's comments on differential geometry as well as discussions with P. Orland.
Appendix A DIMENSION M 6
CONDENSATES
condensates receive two kind of contributions. The first one comes from expanding W(C) up to order g< whereas the condensation (A~,(x)A,(O)) picks up a second term. The former case, or direct condensation, is given by Order
M 6
w(c) -
< g768N ? a ~ > P~cdX["X'~l~cdY['sY'~l~cdZ['sz~'l'
(A.I)
where d x m x ol means ( d x , x ~ - d x ~ x , ) . Note that path ordering is functional in this contribution. The second term, or non-local condensation, has a more complicated expression,
14/('-
2N
A -
C)(I..xI[o.lX)
+ B(2Iu~°Iox x - I~,o,,I~,xx- l~,olxxo) ]
1 (A + B - c)(/~oli~.lxx)}
8
(A.2)
where
I~,. = f ) c d x u x . ,
I.oxo = ~cdx~,xoxxxo
(A.3)
632 and
E Bagdn,J Latorre / Wdsonloops 1 [3 N2-1 2] A = 144 [ 2 (g3fG3) + 8 ~ T-NF(aqr(~lq)) ]' B=~
1 [ 3 N2-1 [-- -~(g3fG3) - 8 ~ N F ( a r r ( q q )
2] ) ],
C= -2A.
(A.4)
In obtaining this result we have made use of QCD sum rules standard techniques (see for instance refs. [10], [13], [14]). All the above expressions are general in the sense that all the shape dependence is parametrized in the integrals (A.3). However, some drastic simplifications occur for especially contours. For instance, no direct condensation takes place for planar loops due to the antisymmetric properties of the integral (A.1). In the case of loops spanned m three dimensions, eq. (A.2) may also be slmplfied using Stokes' theorem which transforms contour into surface integrals. Introducing the totally antisymmetric tensor in three dimensions e,j k we can rewrite (A.3) as follows Ibt o ~
--~p.oala
,
(A.5)
I..xx = - e~,°alxx, ~ -- 2e~,xalox, ~, where
/~1 = fsdx2dx3,
[2 = fsdx3 d x l ,
I3 = fsdXl dx2,
I~x,1 = fsXx dx2dx3 .... ,
= fsX
(A.6)
Xx d x 2 d x 3 . . . . .
Then, in the particular case of three-dimensional contours, the final expression (A.2) can be cast into the form
1{1
- __
2N -2 ( A + B - C)[ L L b ' h - 2LI~bb'~ + 2L'bL'h]
1
3 (A - C)[~ • ~Ib , b -
Ola,bib, a)" }
(A.7)
E Bag&l,J Latorre /Wtlson loops
633
The advantage of this expression over (A.2) is simply that one has to calculate a smaller number of integrals.
Appendix B PERTURBATIVE INTEGRAL
The perturbative integral appearing in the study of the contour C,
x=Rcosa,
y=Rsina,
z=XRcos2a,
(B.1)
IS divergent. We will regularize ~t in two different ways. First we introduce a cut-off a m configuration space, 8,~
'=-~cdXt'~cdY~(x_y)Z+a
(B .2)
2
Parametrizing (B.2) with x . = R(cos a, sin a, h cos 2a, 0), y~ = R(cos/3, sm ]~, h cos 2]3, 0) and changing to the variables r = a - ]3, s = fi we can isolate the singular behavior f02~
R2(1 +4x2sm22s)
[2~
dr,o
47rRE(2tX)+O(R)
dSa2+2Ra(l+4)t2sin22s)(1-cosr)
=
(B.3)
a
where E(x) is the elliptic integral of the second kind. Then we compute the fimte part (no a remains) by integrating If2~drf2~dst
2 o
o
cosr+4~2sIn2s(sin2rcos2s+cos2rsln2s) ~ 1-cosr+2Xasm2r(smrcos2s+cosrsin2s) 2
= 2w{ K(2/~k)
--
1
}
1-cosr
3E(2iX)},
(B.4)
where K(x) is the elliptic integral of the first kind. Transforming imaginary arguments into real ones, vxa
K(21X) =
--2-K
(we have introduced the notation
, Q2 = l +
E(2th) =
QE
(B.5)
4he), we obtain the final result
[ 2QR { 2h ~ 1K 2~ ) 3QE(2h
(B.6)
E Bagdn,J Latorre /Wtlson loops
634
Let us now consider dimensional regularizaUon. The integral (B.1) is defined as
~v I=~cdX~,~cdY,,[(x_y)2]D/2
1.
(B.7)
After some rearrangements of the integration variables, this expression can be recast into the form
[(°2) (°2)
1 = 4 ~6-°'/2 I 2 - D , ~ - - - , 0
+4(Q 2-1)1
(°2) ( °_2)]
+2(1-2Q2)I
6-D,~,0
4-D,---~,0
+(Q2-1)I
2-D,-~---,2
, (B.8)
where slnt~o~sin73
(B.9)
I(la, v , ' r ) - fo~/2d°~fo~/2d~ (l+4~2cos2otsln2fl)~ "
The a lntegraUon is easily done using (ref. [21]; eq. (3.681)) yielding hypergeometrlc functions. The last step (ref. [21]; eq. (7.512.12)) gives F(/z+l I(/~'v'T):
4 F( " + 2 ] F]( ~ ~' + 2 21 2
]
'
T+I
#+2
• +2
2
2
2
• _ 43k2 '
)
(B.10) When particularizing to D = 4 we obtain
I=2~r[QK(2-~)-3QE(2-~)],
(B.11)
which reproduces the finite part of (B.6). As expected, dimensional regularizaUon sweeps out the hnear divergence which appears as a pole in three dimensions.
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[21]
635
M Flensburg and C Peterson, Nucl Phys B283 (1987) 141 E Bag~n, J I Latorre and R Tarrach, Phys Lett 152B (1985) 113, Phys Lett 158B (1985) 145 E BagSn and J I Latorre, Phys Lett 176B (1986) 449 E BagSn, Phys Lett 192B (1987)420 R A Brandt et al, Phys Rev D24 (1981) 879 A Bassetto and R Soldatl, Nucl Phys B276 (1986) 517, G P Korchemsky and A V Radyusktnn, Nucl Phys B283 (1987) 342 M A Shlfman, Nucl Phys B173 (1980)13 S Nanson, CERN-TH 4624/86 L J Remders, H Rubmsteln and S Yazakl, Phys Reports 127 (1985) 1 Contributions in the Workshop on Non-perturbatxve methods (Montpelher) (World Scientific 1986) P Pascual and R Tarrach, QCD Renormahzatlon for the practatloner (Springer, 1984) E Baghn, J I Latorre and P Pascual, Z Plays C32 (1986) 43 O Alvarez, Nucl Phys B216 (1983)125 R Karschner et al, Nucl Phys B210 (1982) 567 M A Shlfman, A Vamshtem and V Zakharov, Nucl Phys B147 (1979) 385, 448 D Arnaudon and C Roxesnel, Phys Lett 187B (1987) 153 C A Dominguez, DESY 87-002; H G Dosch, Phys Lett 190B (1987) 177, R A Bertlmann, UWThPh-1986-25 I S Gradshteyn and I M Ryzhlk, eds., Tables of integrals and formulas (Academic Press, 1980)