- Email: [email protected]
Topological charge distribution in SU(N) gauge theories
Volume 129B, number 3,4
PHYSICS LETTERS
22 September 1983
TOPOLOGICAL CHARGE DISTRIBUTION IN SU(N) GAUGE THEORIES CaM. BISHOP 1 and P.V.D. SWIFT l
Physics Department, University of Edinburgh,Mayfield Road, Edinburgh EH9 3JZ, Scotland, UK Recewed 9 May 1983
The probabihty distribution function for topological charge contained m a sphere of ffmtte radius is calculated. We f'md no need to introduce an arbitrary cutoff on instanton scale sizes as this is controlled by the size of the sphere. Recent Monte Carlo calculatmns reflect qualitatively all the features of our results.
Calculations with instantons are frequently performed in the dilute gas approximation [1] and require an upper limit on instanton scale size determined by the point at which the approximation breaks down [ 1,2]. In this paper,we compute the probability P(q) dq that a measurement of the total topological charge q, contained inside a sphere o f radius R, lies in the range q to q + dq. We shall fired that instantons contribute to P(q) only if their scale size is less than some value determined by R. Thus by choosing R suitably we can ensure that such instantons are adequately described by the dilute gas approximation. Recent Monte Carlo studies o f SU(2) and SU(3) lattice gauge theories by Ishikawa et al. [3] produced results which were claimed to show the absence o f instantons on the lattice. In ref. [3] the topological charge * 1 Q = Zn F(n) F' (n) was measured over the whole lattice or a subvolume o f the lattice for a number o f equilibrium configurations. It was anticipated that the distribution of observed values of Q should be peaked around the integers (fig. 1 o f ref. [3]). The measured distributions showed a monotonic decrease away from the origin and this was interpreted as indicating the absence o f instantons. We shall fired, how1 Work supported by the Science and Engineering Research Council. ,1 We distingmsh between q and Q since q xs defined on a hypersphere in the continuum while Q ts defined on a hypercubical (sub)lattice. At most we expect this to produce small quantitative differences with no change in qualitative behaviour. 198
ever, that the distribution P(q) which we calculate is in good agreement with these Monte Carlo data. We begin by finding the topological charge q contained in a sphere [4] of radius R centred on the origin, due to a single instanton o f scale size p located at ou (which need not be inside the sphere). This is given by g2
q=
f
d4x F~v ~/~ua
sphere =6p4(
dax
if2 sphere [(X- tI)2+p2] 4'
(1)
where F~u is the field strength tensor o f the instanton. (Note that eq. (1)is gauge invariant.)Hence q=l
+ ~1 [4p2o-2 + (R 2 + p 2 _ 0-2)2] -3]2
X [(R 2 + 192 - 02) 3 - 2p 6 - 6p4R 2
+ 602R2o 2 - 602o 4] .
(2)
In fig. 1, q is plotted against o[R for various values of p. From eq. (2) it is easy to check that for ~- ~< q ~< 1 we have p ~
Volume 129B, number 3,4
EL 10~
PHYSICS LETTERS
22 September 1983 10-
p=O1R
~ SU(2) oo
O0
0'5
10
05-
~--
~/R
Fig. 1. The variation of topological charge with instanton location for instantons of various scale sizes. The broken line shows the 0 ~ 0 limit. 001 05
and thus for R less than the hadron scale size the gas of instantons which contribute to P(q) is dilute. It is straightforward to extend the calculation to q < 3 by allowing for instantons which have a larger range of scale sizes and which may be centred outside the sphere. Again the maximum scale size is controlled by q and R, and no arbitrary cutoff is needed. However, as q approaches zero we would have to restrict our attention to spheres of radius much smaller than the hadron scale size. The density of instantons having scale size in the range/9 to p + dp is [5,2,1]
, ~ 075
q,
10
Fig. 2. Distribution o f topological charge m SU (2) and SU (3) gauge theories.
where CN depends on the definition of the coupling constant g R ~ ) . We keep the N of SU(N) explicit as we shall find a marked difference in behaviour between SU(2) and SU(3). For given q, the contribution to the distribution function P ( q ) d q due to instantons having o in the range o to o + do is
and o' = o/R, p' = p/R. Since we cannot invert eq. (2) analytically we resort to numerical methods for the evaluation of (6). The distributionP(q) is shown for SU(2) and SU(3) in fig. 2. This clearly has the same qualitative behavlour as the Monte Carlo results (fig. 3 ofref. [3]). The integrated probability distribution cannot be normalized to one since the whole distribution has not been calculated. Therefore, in fig. 2 we display n (q).Note, however, that from eq. (5) we expect the contribution to P(q) due to single instanton effects to scale like R 1IND. This is reflected in the SU (2) Monte Carlo data [3] since in going from an 84 to a 44 lattice the distribution will be scaled down by a factor ~ 160. For q near 1 we can fred the approximate form of P(q) analytically. Expanding eq. (2) for p small we get
D(p) dpS4 03 d o ,
q = 1 - ( p 4 / R 4 ) f ( o / R ) + O(p6) ,
D (p) d p = CN exp [ - Srr2/g2 ~ ) ] [STr2/g2 (~)] 2N X (p/a) IIN/3
p-5 d p ,
(3)
(4)
where S 4 = 2rr 2 and p = P(q, o) is obtained by inverting eq. (2). Hence R
P(q) = S 4
f d,,o3D (t9) lap/Oql ~
(R#) 11N/3n (q) , (5)
0 ~vhere 1
n (q) =
fo'3p '11N/3-S lap'/aq I d o ' , 0
(6)
(7)
hence from eq. (6) P(q) ~ (1 - q)llN]12-2 ,
(8)
and thus we find a significant difference between SU(2) and S U 0 ) . This behaviour is observed around 1 - q ~ 10 -4 (i.e. P ~ R/10) in our numerical evaluation of eq. (6). On the lattice we do not expect to observe instantons whose scale size is much smaller than a lattice spacing. We can take this into account in our calculations by imposing a minimum instanton scale size Pmln = a/2, where a is the lattice spacing. Clearly 199
Volume 129B, number 3,4
PHYSICS LETTERS
10
22 September 1983
gralwas evaluated numerically. The results are changed significantly only in the region o f q close to 1, since this is where the effect of small instantons dominates. We expect P(q) to fall to zero at qmax given, from eq. (2), by
SU(2) x 05 \ \
00
qmax = q(Pmin, o = 0 , R ) = 0.990,
\ SU(3)
20-
\\--.\
0 o
10-
00
\'\
90
,
0 95
1 00
Fig. 3. Distribution of topological charge near q = 1 (soLid lines). The broken lines show the effect of impo~ng a minimum Jnstanton scale size PmJn = R/4.
the behaviour shown in eq. (8) will not be seen on the lattice since it comes from the effects of small instantons. Since, for fixed q, p decreases as o is increased, a minimum cutoff on p implies a maximum value of o, given by Omax = O(0min ,q) < R and found by inverting eq. (2). Thus the only change required is to set the upper limit o f integration in eq. (5) equal to Omax. For a 44 sublattice, such as is used in ref. [3] we take R = 2a and hence Pmin =R[4. Again the hate-
200
(9)
which is in agreement with the numerical results (fig. 3). Thus in this region the dominant contribution will come from two-instanton effects. In conclusion: calculations of the topological charge distribution in gauge theories can be performed in the dilute gas approximation without requiring an arbitrary cutoff on instanton scale sizes. Monte Carlo studies give results which agree with this calculated distribution. The authors are grateful to D J . Wallace for useful discussions and to C.N. Smith for assistance with the numerical work.
References [1] C.G. Callan et al., Phys. Rev. D17 (1978) 2717. [2] S. Coleman, The uses ofinstantons, Erice lectures (1977). [3] K. Ishikawa, G. Schierholz, K. Schneider and M. Teper, Phys. Lett. 128B (1983) 309. [4] E.B. Bogomolny and V.A. Fateyev, Phys. Lett. 71B
(1977) 93. [5] G. 't Hooft, Phys. Rev. D14 (1976) 3432.
PHYSICS LETTERS
22 September 1983
TOPOLOGICAL CHARGE DISTRIBUTION IN SU(N) GAUGE THEORIES CaM. BISHOP 1 and P.V.D. SWIFT l
Physics Department, University of Edinburgh,Mayfield Road, Edinburgh EH9 3JZ, Scotland, UK Recewed 9 May 1983
The probabihty distribution function for topological charge contained m a sphere of ffmtte radius is calculated. We f'md no need to introduce an arbitrary cutoff on instanton scale sizes as this is controlled by the size of the sphere. Recent Monte Carlo calculatmns reflect qualitatively all the features of our results.
Calculations with instantons are frequently performed in the dilute gas approximation [1] and require an upper limit on instanton scale size determined by the point at which the approximation breaks down [ 1,2]. In this paper,we compute the probability P(q) dq that a measurement of the total topological charge q, contained inside a sphere o f radius R, lies in the range q to q + dq. We shall fired that instantons contribute to P(q) only if their scale size is less than some value determined by R. Thus by choosing R suitably we can ensure that such instantons are adequately described by the dilute gas approximation. Recent Monte Carlo studies o f SU(2) and SU(3) lattice gauge theories by Ishikawa et al. [3] produced results which were claimed to show the absence o f instantons on the lattice. In ref. [3] the topological charge * 1 Q = Zn F(n) F' (n) was measured over the whole lattice or a subvolume o f the lattice for a number o f equilibrium configurations. It was anticipated that the distribution of observed values of Q should be peaked around the integers (fig. 1 o f ref. [3]). The measured distributions showed a monotonic decrease away from the origin and this was interpreted as indicating the absence o f instantons. We shall fired, how1 Work supported by the Science and Engineering Research Council. ,1 We distingmsh between q and Q since q xs defined on a hypersphere in the continuum while Q ts defined on a hypercubical (sub)lattice. At most we expect this to produce small quantitative differences with no change in qualitative behaviour. 198
ever, that the distribution P(q) which we calculate is in good agreement with these Monte Carlo data. We begin by finding the topological charge q contained in a sphere [4] of radius R centred on the origin, due to a single instanton o f scale size p located at ou (which need not be inside the sphere). This is given by g2
q=
f
d4x F~v ~/~ua
sphere =6p4(
dax
if2 sphere [(X- tI)2+p2] 4'
(1)
where F~u is the field strength tensor o f the instanton. (Note that eq. (1)is gauge invariant.)Hence q=l
+ ~1 [4p2o-2 + (R 2 + p 2 _ 0-2)2] -3]2
X [(R 2 + 192 - 02) 3 - 2p 6 - 6p4R 2
+ 602R2o 2 - 602o 4] .
(2)
In fig. 1, q is plotted against o[R for various values of p. From eq. (2) it is easy to check that for ~- ~< q ~< 1 we have p ~
Volume 129B, number 3,4
EL 10~
PHYSICS LETTERS
22 September 1983 10-
p=O1R
~ SU(2) oo
O0
0'5
10
05-
~--
~/R
Fig. 1. The variation of topological charge with instanton location for instantons of various scale sizes. The broken line shows the 0 ~ 0 limit. 001 05
and thus for R less than the hadron scale size the gas of instantons which contribute to P(q) is dilute. It is straightforward to extend the calculation to q < 3 by allowing for instantons which have a larger range of scale sizes and which may be centred outside the sphere. Again the maximum scale size is controlled by q and R, and no arbitrary cutoff is needed. However, as q approaches zero we would have to restrict our attention to spheres of radius much smaller than the hadron scale size. The density of instantons having scale size in the range/9 to p + dp is [5,2,1]
, ~ 075
q,
10
Fig. 2. Distribution o f topological charge m SU (2) and SU (3) gauge theories.
where CN depends on the definition of the coupling constant g R ~ ) . We keep the N of SU(N) explicit as we shall find a marked difference in behaviour between SU(2) and SU(3). For given q, the contribution to the distribution function P ( q ) d q due to instantons having o in the range o to o + do is
and o' = o/R, p' = p/R. Since we cannot invert eq. (2) analytically we resort to numerical methods for the evaluation of (6). The distributionP(q) is shown for SU(2) and SU(3) in fig. 2. This clearly has the same qualitative behavlour as the Monte Carlo results (fig. 3 ofref. [3]). The integrated probability distribution cannot be normalized to one since the whole distribution has not been calculated. Therefore, in fig. 2 we display n (q).Note, however, that from eq. (5) we expect the contribution to P(q) due to single instanton effects to scale like R 1IND. This is reflected in the SU (2) Monte Carlo data [3] since in going from an 84 to a 44 lattice the distribution will be scaled down by a factor ~ 160. For q near 1 we can fred the approximate form of P(q) analytically. Expanding eq. (2) for p small we get
D(p) dpS4 03 d o ,
q = 1 - ( p 4 / R 4 ) f ( o / R ) + O(p6) ,
D (p) d p = CN exp [ - Srr2/g2 ~ ) ] [STr2/g2 (~)] 2N X (p/a) IIN/3
p-5 d p ,
(3)
(4)
where S 4 = 2rr 2 and p = P(q, o) is obtained by inverting eq. (2). Hence R
P(q) = S 4
f d,,o3D (t9) lap/Oql ~
(R#) 11N/3n (q) , (5)
0 ~vhere 1
n (q) =
fo'3p '11N/3-S lap'/aq I d o ' , 0
(6)
(7)
hence from eq. (6) P(q) ~ (1 - q)llN]12-2 ,
(8)
and thus we find a significant difference between SU(2) and S U 0 ) . This behaviour is observed around 1 - q ~ 10 -4 (i.e. P ~ R/10) in our numerical evaluation of eq. (6). On the lattice we do not expect to observe instantons whose scale size is much smaller than a lattice spacing. We can take this into account in our calculations by imposing a minimum instanton scale size Pmln = a/2, where a is the lattice spacing. Clearly 199
Volume 129B, number 3,4
PHYSICS LETTERS
10
22 September 1983
gralwas evaluated numerically. The results are changed significantly only in the region o f q close to 1, since this is where the effect of small instantons dominates. We expect P(q) to fall to zero at qmax given, from eq. (2), by
SU(2) x 05 \ \
00
qmax = q(Pmin, o = 0 , R ) = 0.990,
\ SU(3)
20-
\\--.\
0 o
10-
00
\'\
90
,
0 95
1 00
Fig. 3. Distribution of topological charge near q = 1 (soLid lines). The broken lines show the effect of impo~ng a minimum Jnstanton scale size PmJn = R/4.
the behaviour shown in eq. (8) will not be seen on the lattice since it comes from the effects of small instantons. Since, for fixed q, p decreases as o is increased, a minimum cutoff on p implies a maximum value of o, given by Omax = O(0min ,q) < R and found by inverting eq. (2). Thus the only change required is to set the upper limit o f integration in eq. (5) equal to Omax. For a 44 sublattice, such as is used in ref. [3] we take R = 2a and hence Pmin =R[4. Again the hate-
200
(9)
which is in agreement with the numerical results (fig. 3). Thus in this region the dominant contribution will come from two-instanton effects. In conclusion: calculations of the topological charge distribution in gauge theories can be performed in the dilute gas approximation without requiring an arbitrary cutoff on instanton scale sizes. Monte Carlo studies give results which agree with this calculated distribution. The authors are grateful to D J . Wallace for useful discussions and to C.N. Smith for assistance with the numerical work.
References [1] C.G. Callan et al., Phys. Rev. D17 (1978) 2717. [2] S. Coleman, The uses ofinstantons, Erice lectures (1977). [3] K. Ishikawa, G. Schierholz, K. Schneider and M. Teper, Phys. Lett. 128B (1983) 309. [4] E.B. Bogomolny and V.A. Fateyev, Phys. Lett. 71B
(1977) 93. [5] G. 't Hooft, Phys. Rev. D14 (1976) 3432.