### Table 1 Data for the Wilson loops

1983

### Table 2: Coe cients for the perturbative expansions, in powers of P(qm;n), of small Wilson loops. Scale qm;n is the average momentum carried by the gluon in the rst-order correction.

1997

"... In PAGE 11: ... The scale qm;n is the average gluon momentum in the rst-order contribution to Wm;n, computed directly from the Feynman diagrams as described in [17, 18]. In Table2 we list the perturbative coe cients through third order for nf =0, and through second order for nf =2 [19]. Unfortunately, the nf dependence of the third-order coe cients has not yet been computed.... ..."

### Table 4: The t parameters from ts to the nite T approximants to the lattice Yukawa potential, obtained from lowest order Wilson loops with a massive gauge boson propaga- tor, for =3:0.

1996

### Table 9: Wilson Coe cients of the operators Q1 to Q6 at two-loops in the NDR scheme. (GeV)

"... In PAGE 29: ... For C3, C4, C5, and C6 we set the small imaginary part due to the top loop to zero. The results to one-loop accuracy are in Table 8 and for two-loops are in Table9 . In the two-loop case we are using the NDR scheme.... In PAGE 30: ...1) term which can be found in the NDR scheme in [7, 19] for instance, are in Table 10. Here we give the one-loop results in Table 8, two-loop results with5 r1 = 0 at two-loops in Table9 and the one with the scheme dependence properly removed, including r1, in Table 10. It can be seen that the change from one to two- loops in the NDR scheme is not so large but inclusion of the r1 makes a large change.... ..."

### Table 4: The nal results for the three O(p2) couplings using the one-loop Wilson coe cients. The numbers in brackets refer to using Q1, Q2, and Q6 only. (GeV)

### Table 1: The comparison, as a function of r=a, between the two-quark potentials aVa;b(2; r) of Eqs. 3 and 4 (rows 2 and 5) and those given by Wilson loops in Refs. [8] and [7] (rows 1 and 4). Row 3 shows the modi ed lattice values when an additive constant is removed from row 1 by matching to aVa(2; r) at r=a = 5. Row 6 is the same as row 5 but with the additive constant in Eq. 4 removed. Row 7 is the same as row 4 but with the additive constant removed. In row 4, the r=a = 21=2 entry is taken from row 1 { since this was not calculated in [7].

"... In PAGE 3: ... Upto an additive constant (V0), the outcome was a potential (in fm?1) of the form Va(2; r) = ? 12r + bsr; (3) where r is in fermis and the string energy bs is (425 MeV)2 for the appropriate lattice spacing of a = 0:18 fm. In Table1 , values are given for the basic lattice potentials of Ref. [8] (row 1) and for the t in Eq.... In PAGE 4: ...levels (2,13). The basic data are in row 4 of Table1 . In row 5, a t with a 2/dof=1.... ..."

### Table 8: Wilson Coe cients of the operators Q1 to Q6 at one-loop. of one-loop running labelled One-loop, two-loop running with r1 = 0, labelled Two- loops, r1 at its value, labelled Scheme-Independent (SI), and a version where we use the exact solution of the two-loop running with r1 included so as to cancel the full scheme-dependence there too, i.e. ^ BK = 1 + 2 1 s( )

"... In PAGE 29: ... For C3, C4, C5, and C6 we set the small imaginary part due to the top loop to zero. The results to one-loop accuracy are in Table8 and for two-loops are in Table 9. In the two-loop case we are using the NDR scheme.... In PAGE 30: ...1) term which can be found in the NDR scheme in [7, 19] for instance, are in Table 10. Here we give the one-loop results in Table8 , two-loop results with5 r1 = 0 at two-loops in Table 9 and the one with the scheme dependence properly removed, including r1, in Table 10. It can be seen that the change from one to two- loops in the NDR scheme is not so large but inclusion of the r1 makes a large change.... ..."

### Table 5: The nal results for the three O(p2) couplings using the two-loop Wilson coe cients with the inclusion of the r1 factors. The numbers in brackets refer to using Q1, Q2, and Q6 only. we get for the one-loop short-distance is not bad, and there is some minimum around 0.7 GeV for the two-loop running. We get in general too large values for this ratio compared to the experimental 16.4 value (2.5) due to the somewhat small value of G27 we get.In order to show the improvement with previous results and the quality of the matching we have shown in Figure 6 for G27( ) the lowest order result Eq. (5.2), the ENJL result for the same quantity and the nal result for G27 with the two-loop short (GeV) One-Loop Two-Loops

### Table 1: A CC-Loop

2000

"... In PAGE 2: .... L. Wilson, Jr. [19]. For p = 3, this loop is displayed in Table1... In PAGE 3: ... The Moufang loops, whose structure is already widely discussed in the literature [1][4], are always diassociative (that is, every two elements generate a group) by Moufang apos;s Theorem. The CC-loops need not even be power associative (that is, every single element generates a group); for example, in Table1 , the single element 4 generates the whole loop. It is shown in [11] that the CC-loops which are diassociative (equivalently, Moufang) are the extra loops studied by Fenyves [7][8].... In PAGE 3: ...6), this implies that if G is any nite CC-loop, then for some prime p dividing jGj, G has a subloop H isomorphic to the cyclic group of order p. In Table1... In PAGE 4: ... SEM is used to construct nite examples. For example, the CC-loops given in Table1 and Example 2.20 were constructed using SEM.... In PAGE 4: ...20 were constructed using SEM. Once one has such an example, it is usually possible to describe it in a more conceptual way; for example, the loop in Table1 can be recognized as the one already constructed by Wilson (see [19] or Theorem 4.15), and we have described the one in Example 2.... In PAGE 10: ... Proof: Since the only automorphism of N(G; ) is the identity, it follows that for each a 2 G, and each x 2 N(G; ), xLaR?1 a = x, so ax = xa. Table1 is an example of a CC-loop in which the nucleus has size 3 and the center has size 1. Nevertheless, we shall see later (Lemma 4.... In PAGE 26: ...7), these cannot have prime order, and it is easy to see by inspection that all loops of order four are commutative, and hence groups if they are CC, so that leaves order six. In that case, we have the CC-loop from Table1 , and that is the only one, as is true in general for orders 2q, where q is an odd prime, by the following theorem: Theorem 4.15 If q is an odd prime, then there are exactly three CC-loops of order 2q, exactly two of which are groups.... ..."

Cited by 7

### Table 1: A CC-Loop

2000

"... In PAGE 2: ...ne constructed by R. L. Wilson, Jr. [19]. For p = 3, this loop is displayed in Table1 . Also (Theorem 4.... In PAGE 3: ... The Moufang loops, whose structure is already widely discussed in the literature [1][4], are always diassociative (that is, every two elements generate a group) by Moufang apos;s Theorem. The CC-loops need not even be power associative (that is, every single element generates a group); for example, in Table1 , the single element 4 generates the whole loop. It is shown in [11] that the CC-loops which are diassociative (equivalently, Moufang) are the extra loops studied by Fenyves [7][8].... In PAGE 3: ...5), this implies that if G is any nite CC-loop, then for some prime p dividing jGj, G has a subloop H isomorphic to the cyclic group of order p. In Table1 , jGj = 6, p = 3, and H = f1; 2; 3g; there are no subloops of order 2, as one might have hoped from group theory. Our structure theory succeeds through the study of loop automorphisms.... In PAGE 4: ...omorphisms. SEM is used to construct nite examples. For example, the CC-loops given in Tables 1 and 2 (see Section 3) were constructed using SEM. Once one has such an example, it is usually possible to describe it in a more conceptual way; for example, the loop in Table1 can be recognized as the one already constructed by Wilson (see [19] or Theorem 4.13), and the one in Table 2 can be described in terms of cosets of a 4-element nucleus.... In PAGE 9: ... Proof: Since the only automorphism of N(G; ) is the identity, it follows that for each a 2 G, and each x 2 N(G; ), xLaR?1 a = x, so ax = xa. Table1 is an example of a CC loop in which the nucleus has size 3 and the center has size 1. Nevertheless, we shall see later (Lemma 4.... In PAGE 23: ...6), these cannot have prime order, and it is easy to see by inspection that all loops of order four are commutative, and hence groups if they are CC, so that leaves order six. In that case, we have the CC-loop from Table1 , and that is the only one, as is true in general for orders 2q,... ..."

Cited by 7