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Cylindrical gauge field configurations. A signal of confinement in QCD
Volume 71B, number 2
PHYSICS LETTERS
21 November 1977
CYLINDRICAL GAUGE FIELD CONFIGURATIONS. A SIGNAL OF CONFINEMENT IN QCD Peter MINKOWSKI Institute for Theoretical Physics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland Received 29 July 1977 An infrared approximation to the field equations of QCD is proposed which generates a mapping of configuration space into two dimensions (one time, one space dimension). The associated cylindrical gauge field configurations insure confinement whenever nontrivial interactions in two dimensions remain. This feature distinguishes abelian from nonabelian gauge theories. The association of scale breaking in strong interactions and the effective interactions for pseudoscalar mesons with the two central anometrics of QCD: energy momentum tensor trace anomaly [ 1 ]
O# - ~(g) ~(V.v 1 a r/-vva] " ]ren. + O(9?gq) g
(1)
axial vector current anomaly [e.g. 2] ju(5) =
~ -e c c r qbT'u75qb ' flavor (2) g2 1 a 3vJU(5) 321r2 (2nfl)~( Vta v VJfl e"V~fl)ren" + O(C~q),
Our is energy momentum tensor, vua = 3v vua OuVu a - g f a b c v u b v S , a = 1 .... , 8 is gluon field strengths, q~ is color triplet quark fields, c = 1, 2, 3 (color), b = 1 ..... nfl (flavor), cff~qb i s quark masses, g is strong coupling constant; fabc is structure constants of SU3 with fabc fabc = 24,/3(g) is Gell-MannLow-Callan-Symanzik function [3], establish the unbroken SU3 (color) gauge theory (QCD) as unique candidate for a field theory of strong interactions. The rescaling properties of QCD, summarized by the asymptotic equations
g2(p.)g2(,UO)(l+bg-2(p'O-i---)l(P' og21)-1 16rr 2
lgrr 2
167r2
\p21i
(4) cc
t
We remark first that the quark part of the Lagrangian in eq. (4) only gives rise to confining long range forces provided
£g =
1 a 74Via v
vu.a
+£gauge fixing
+
£ghost ,
(5)
primarily generates confining forces. Assuming the contrary we can tentatively consider £quark as a perturbation of the nonconfining gauge field configurations Of£g [7]
V l a ,a,(t-+ +~,~)-+6aA e-'PXe~(P) a --
(6)
(pU ev(p ) = 0 in a Lorentz-type gauge) . The associated action then becomes 1
A F#
•A = --7~Fur
vA
(> O)
" 2" " ,4/b C-~q(p) ~_ C~q(/.10) ( ' \ g tPO)Z
~i
392
+ •gauge fixing + £ghost "
'
P >> PO or p, PO large enough. b = 11 - ~ n f l
[e.g. 4], where p, P0 is quark mass function effective at short (euclidean) distances relative to flavor b, provide a viable explanation of approximate Bjorken scaling [e.g. 5] and the scaling of the e+e - annihilation cross section into hadrons at high energies [e.g. 6] We are interested here in the leading long range forces mediated by the Lagrangian 1 a g ~ va £ = aVuv +-~giTU(Duq)cb_-C qb C~b qbc
/'~ >~/d0'
(3)
--C
C
-- qbQ~bqb + £gauge fixing + £ghost ,
(7)
Volume 71B, number 2
PHYSICS LETTERS
(no summation over A). Choosing A = 3 is no loss of generality. The corresponding abelian gauge action according to eq. (7) exhibits an infrared stable (asymptotically free) fixed point equivalent to the situation in QED. For cf/~b = 0 the rescaling equations are K-
g2 16n 2
_
• =/a~-
K 4n
;
/(u
..~SnflK u 4 2 + O(K 3)
,
21 November 1977
The relativistic limit of the corresponding Schr6dinger equation
[1
- ~/'x
+ U(r)
]
~N.R. = E~N.R. ;
mQQ=ZmQ+E,
$IEI>>mQ,
E>>m Q,
[-21 x + ( m Q + 1U(r))2]ffrel. - m ~~ Q f r e l . -
(10)
mQ is mass of the heavy quark (e.g. mQ = m c ~ 1.5 GeV), gives rise to linearly rising Regge trajectories
,
~nflKuo
I~] l
(8)
rn~o~--l(n+u),
n=0,1,2
.....
Ot
We conclude that in 4 dimensions even massless fermions do not generate an infrared instability by themselves, when coupled to gauge bosons. We therefore neglect the fermion part in eq. (4) in the following * ~. Moreover one can realize an approximately external color source by the color current of a heavy quark flavor Q (either a fictitious flavor or a real one as e.g. c,b .... ) 4-2
for
mQo>>mQ,
exactly if [e.g. 91.
U(r) rises linearly with
U(r) r ~ c2r, L- 1
1
geff(r)
distance for r -+ o~
r ~ > Zr ' .
--1.8X10 -14cm
for
c~'"~ 1 (GeV) - 2 .
The corresponding nonrelativistic potential acting e.g. in the c-d-channel [e.g. 9] is o f the form
In the limit m Q 0" >> mQ, (U>mQQ N mQ eq. (10) becomes
41 UQ~(r) = - - f i r K(r)
[--Ax+C4X2]~rel'~
1 2 > 37rr geff(r) '
K(r) - g2(120) 1 +~( _ _sin(qr) d [ q2,122]
an
n d Clqq
(9)
Duv (q,. 120) =8
c'
V-J+q~qv 1 1 d(q2 ,t22)
ququ q---~ 1 dL(q 2, 122),
- r/ q ~
evaluated in the gauge where Z 3 = Z 1 [e.g. 10].
~--8c2n •
(11)
c 2 = 1/8c( .
This implies L = (8a/3n) 1/2 ~-- 1.8 X 10 -14 cm. The situation in eq. (1 1) can only be generated by cylindrical gauge field configurations. We introduce the corresponding cylindrical infrared cutoff in momentum space in order to show that this is indeed the case
d4k-+
f -~
+1 Of course if the number of quark flavors with identically vanishing mass (which is presumably zero) exceeds 16 the fermions do dominate the long range forces and thus no confinement occurs. 4-2 Recently [8] certain infrared problems in extracting this potential in the limit mQ ~ ~ have been noted. They originate from keeping external quarks on shell and have no bearing on the arguments presented here.
~rel.,
yielding the asymptotic spectrum m~
In eq. (9) d(q 2, 122) denotes the transverse part of the gluon propagator
4
f
dk0 dk 3
d2k±;
Ik±i
k±=(kx, ky). (12)
In this manner a mapping o f the gauge fields in four dimensions on to corresponding two dimensional configurations results [1 1] , 3 : g ° , 3 -+ l}2,3(Xo,X3)- A ~ A 2Qrl/2) :1:3 See footnote on next page.
fd2x2 gaO'3(Xo,X3;X,), (13)
393
Volume v1,2 a
71 B, number
;: %1’2(x0,
= ~
A
2(#)
s
2
PHYSICS
x3)
d2x1
(14)
The action in four dimensions is accordingly on the two dimensional action -f t;,“u J%”
)
mapped /.l,v= 0,3,
+d: gauge fixing ’ Lghost (15)
The fact that nontrivial interactions between the transverse modes in faur dimensions (P~,~)~ and the electric field E, = pi3 remain after the cylindrical approximation has been performed signals the confining nature of the long range forces generated by nonabelian (unbroken) Yang-Mills interactions. We note that 5,) in eq. (15) describes free fields in the abelian case. This clear distinction between
*3 In ref. [ 111 an extended transverse momentum cutoff corresponding to a lattice in transverse configuration space has been discussed. This approach does not demonstrate the origin of the cylindrical gauge field configurations but rather applies their dynamics to the computation of hadron masses. *’ The renormalization in 4 dimensions carries over to the two dimensional action precisely because there is only one (independent) renormalization constant (23 = Zt), gFen + IJ 3 I.L*. g(4)ren(p*) ’ P*; P* = P exp(-8n2/bg2(d),
394
LETTERS
21 November
1977
abelian- and nonabelian lang range forces exhibited by the cylindrical gauge field configurations in the infrared approximation is rather different from the lattice approximation discussed e.g. by Wilson and also Kogut and Susskind [ 121. It is a pleasure to thank J. Gasser, M. Ltischer and H. Joos for rnteresting discussions.
References S.L. Adler, J.C. Collins and A. Duncan, Phys. Rev. D15 (1977) 1712 and Institute for Advanced Study, Princeton, preprint (1976); N.K. Nielsen, Nordita preprint 1976; P. Minkowski, University of Bern preprint 1976. PI H. Fritzsch, H. Leutwyler and M. Gell-Mann, Phys. Lett. 47B (1973) 365; A.A. Belavin, A.M. Polyakov, A.S. Schwarz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85; G. ‘t Hooft, Phys. Rev. D14 (1976) 3432. 131 M. Gell-Mann and F.E. Low, Phys. Rev. 9.5 (1954) 1300; C. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227. [41 H. Georgi and H.D. Politzer, Phys. Rev. D14 (1976) 1829 and related remarks in ref. [lo]. [51 H.D. Politzer, Physics Reports 14C (1974) 130; D. Gross and F. Wilczek, Phys. Rev. D8 (1973) 3622 and D9 (1974) 980; H. Fritzsch and P. Minkowski, Nucl. Phys. 76B (1974) 365. [61 T. Applequist and H.D. Politzer, Phys. Rev. Lett. 34 (1975) 43. 171 S. Coleman, There are no classical glueballs, Harvard University preprint HUTP-77/A027, for a relevant remark on the classical stability of such configurations. [81 W. Fischler, CERN preprint TH-2321 (1977); T. Applequist et al., Yale University preprint (1977). PI E. Eichten, K. Gottfried, T. Kinoshita, J. Kogut, K.D. Lane and T.M. Yan, Phys. Rev. Lett. 34 (1975) 369; B. Harrington, S.Y. Park and A. Yildiz, Phys. Rev. Lett. 34 (1975) 169. 1101 P. Minkowski, in: Lectures delivered at the Finnish Spring Workshop on QCD, Tva?minne, Finland 1977. 1111 W.A. Bardeen and R.B. Pearson, Phys. Rev. D14 (1976) 547. 1121 K.G. Wilson, Phys. Rev. DlO (1974) 2445 and CLNS327 preprint (Erice Summer School Lectures 1975); J. Kogut and L. Susskind, Phys. Rev. Dll (1975) 395.
PHYSICS LETTERS
21 November 1977
CYLINDRICAL GAUGE FIELD CONFIGURATIONS. A SIGNAL OF CONFINEMENT IN QCD Peter MINKOWSKI Institute for Theoretical Physics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland Received 29 July 1977 An infrared approximation to the field equations of QCD is proposed which generates a mapping of configuration space into two dimensions (one time, one space dimension). The associated cylindrical gauge field configurations insure confinement whenever nontrivial interactions in two dimensions remain. This feature distinguishes abelian from nonabelian gauge theories. The association of scale breaking in strong interactions and the effective interactions for pseudoscalar mesons with the two central anometrics of QCD: energy momentum tensor trace anomaly [ 1 ]
O# - ~(g) ~(V.v 1 a r/-vva] " ]ren. + O(9?gq) g
(1)
axial vector current anomaly [e.g. 2] ju(5) =
~ -e c c r qbT'u75qb ' flavor (2) g2 1 a 3vJU(5) 321r2 (2nfl)~( Vta v VJfl e"V~fl)ren" + O(C~q),
Our is energy momentum tensor, vua = 3v vua OuVu a - g f a b c v u b v S , a = 1 .... , 8 is gluon field strengths, q~ is color triplet quark fields, c = 1, 2, 3 (color), b = 1 ..... nfl (flavor), cff~qb i s quark masses, g is strong coupling constant; fabc is structure constants of SU3 with fabc fabc = 24,/3(g) is Gell-MannLow-Callan-Symanzik function [3], establish the unbroken SU3 (color) gauge theory (QCD) as unique candidate for a field theory of strong interactions. The rescaling properties of QCD, summarized by the asymptotic equations
g2(p.)g2(,UO)(l+bg-2(p'O-i---)l(P' og21)-1 16rr 2
lgrr 2
167r2
\p21i
(4) cc
t
We remark first that the quark part of the Lagrangian in eq. (4) only gives rise to confining long range forces provided
£g =
1 a 74Via v
vu.a
+£gauge fixing
+
£ghost ,
(5)
primarily generates confining forces. Assuming the contrary we can tentatively consider £quark as a perturbation of the nonconfining gauge field configurations Of£g [7]
V l a ,a,(t-+ +~,~)-+6aA e-'PXe~(P) a --
(6)
(pU ev(p ) = 0 in a Lorentz-type gauge) . The associated action then becomes 1
A F#
•A = --7~Fur
vA
(> O)
" 2" " ,4/b C-~q(p) ~_ C~q(/.10) ( ' \ g tPO)Z
~i
392
+ •gauge fixing + £ghost "
'
P >> PO or p, PO large enough. b = 11 - ~ n f l
[e.g. 4], where p, P0 is quark mass function effective at short (euclidean) distances relative to flavor b, provide a viable explanation of approximate Bjorken scaling [e.g. 5] and the scaling of the e+e - annihilation cross section into hadrons at high energies [e.g. 6] We are interested here in the leading long range forces mediated by the Lagrangian 1 a g ~ va £ = aVuv +-~giTU(Duq)cb_-C qb C~b qbc
/'~ >~/d0'
(3)
--C
C
-- qbQ~bqb + £gauge fixing + £ghost ,
(7)
Volume 71B, number 2
PHYSICS LETTERS
(no summation over A). Choosing A = 3 is no loss of generality. The corresponding abelian gauge action according to eq. (7) exhibits an infrared stable (asymptotically free) fixed point equivalent to the situation in QED. For cf/~b = 0 the rescaling equations are K-
g2 16n 2
_
• =/a~-
K 4n
;
/(u
..~SnflK u 4 2 + O(K 3)
,
21 November 1977
The relativistic limit of the corresponding Schr6dinger equation
[1
- ~/'x
+ U(r)
]
~N.R. = E~N.R. ;
mQQ=ZmQ+E,
$IEI>>mQ,
E>>m Q,
[-21 x + ( m Q + 1U(r))2]ffrel. - m ~~ Q f r e l . -
(10)
mQ is mass of the heavy quark (e.g. mQ = m c ~ 1.5 GeV), gives rise to linearly rising Regge trajectories
,
~nflKuo
I~] l
(8)
rn~o~--l(n+u),
n=0,1,2
.....
Ot
We conclude that in 4 dimensions even massless fermions do not generate an infrared instability by themselves, when coupled to gauge bosons. We therefore neglect the fermion part in eq. (4) in the following * ~. Moreover one can realize an approximately external color source by the color current of a heavy quark flavor Q (either a fictitious flavor or a real one as e.g. c,b .... ) 4-2
for
mQo>>mQ,
exactly if [e.g. 91.
U(r) rises linearly with
U(r) r ~ c2r, L- 1
1
geff(r)
distance for r -+ o~
r ~ > Zr ' .
--1.8X10 -14cm
for
c~'"~ 1 (GeV) - 2 .
The corresponding nonrelativistic potential acting e.g. in the c-d-channel [e.g. 9] is o f the form
In the limit m Q 0" >> mQ, (U>mQQ N mQ eq. (10) becomes
41 UQ~(r) = - - f i r K(r)
[--Ax+C4X2]~rel'~
1 2 > 37rr geff(r) '
K(r) - g2(120) 1 +~( _ _sin(qr) d [ q2,122]
an
n d Clqq
(9)
Duv (q,. 120) =8
c'
V-J+q~qv 1 1 d(q2 ,t22)
ququ q---~ 1 dL(q 2, 122),
- r/ q ~
evaluated in the gauge where Z 3 = Z 1 [e.g. 10].
~--8c2n •
(11)
c 2 = 1/8c( .
This implies L = (8a/3n) 1/2 ~-- 1.8 X 10 -14 cm. The situation in eq. (1 1) can only be generated by cylindrical gauge field configurations. We introduce the corresponding cylindrical infrared cutoff in momentum space in order to show that this is indeed the case
d4k-+
f -~
+1 Of course if the number of quark flavors with identically vanishing mass (which is presumably zero) exceeds 16 the fermions do dominate the long range forces and thus no confinement occurs. 4-2 Recently [8] certain infrared problems in extracting this potential in the limit mQ ~ ~ have been noted. They originate from keeping external quarks on shell and have no bearing on the arguments presented here.
~rel.,
yielding the asymptotic spectrum m~
In eq. (9) d(q 2, 122) denotes the transverse part of the gluon propagator
4
f
dk0 dk 3
d2k±;
Ik±i
k±=(kx, ky). (12)
In this manner a mapping o f the gauge fields in four dimensions on to corresponding two dimensional configurations results [1 1] , 3 : g ° , 3 -+ l}2,3(Xo,X3)- A ~ A 2Qrl/2) :1:3 See footnote on next page.
fd2x2 gaO'3(Xo,X3;X,), (13)
393
Volume v1,2 a
71 B, number
;: %1’2(x0,
= ~
A
2(#)
s
2
PHYSICS
x3)
d2x1
(14)
The action in four dimensions is accordingly on the two dimensional action -f t;,“u J%”
)
mapped /.l,v= 0,3,
+d: gauge fixing ’ Lghost (15)
The fact that nontrivial interactions between the transverse modes in faur dimensions (P~,~)~ and the electric field E, = pi3 remain after the cylindrical approximation has been performed signals the confining nature of the long range forces generated by nonabelian (unbroken) Yang-Mills interactions. We note that 5,) in eq. (15) describes free fields in the abelian case. This clear distinction between
*3 In ref. [ 111 an extended transverse momentum cutoff corresponding to a lattice in transverse configuration space has been discussed. This approach does not demonstrate the origin of the cylindrical gauge field configurations but rather applies their dynamics to the computation of hadron masses. *’ The renormalization in 4 dimensions carries over to the two dimensional action precisely because there is only one (independent) renormalization constant (23 = Zt), gFen + IJ 3 I.L*. g(4)ren(p*) ’ P*; P* = P exp(-8n2/bg2(d),
394
LETTERS
21 November
1977
abelian- and nonabelian lang range forces exhibited by the cylindrical gauge field configurations in the infrared approximation is rather different from the lattice approximation discussed e.g. by Wilson and also Kogut and Susskind [ 121. It is a pleasure to thank J. Gasser, M. Ltischer and H. Joos for rnteresting discussions.
References S.L. Adler, J.C. Collins and A. Duncan, Phys. Rev. D15 (1977) 1712 and Institute for Advanced Study, Princeton, preprint (1976); N.K. Nielsen, Nordita preprint 1976; P. Minkowski, University of Bern preprint 1976. PI H. Fritzsch, H. Leutwyler and M. Gell-Mann, Phys. Lett. 47B (1973) 365; A.A. Belavin, A.M. Polyakov, A.S. Schwarz and Yu.S. Tyupkin, Phys. Lett. 59B (1975) 85; G. ‘t Hooft, Phys. Rev. D14 (1976) 3432. 131 M. Gell-Mann and F.E. Low, Phys. Rev. 9.5 (1954) 1300; C. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227. [41 H. Georgi and H.D. Politzer, Phys. Rev. D14 (1976) 1829 and related remarks in ref. [lo]. [51 H.D. Politzer, Physics Reports 14C (1974) 130; D. Gross and F. Wilczek, Phys. Rev. D8 (1973) 3622 and D9 (1974) 980; H. Fritzsch and P. Minkowski, Nucl. Phys. 76B (1974) 365. [61 T. Applequist and H.D. Politzer, Phys. Rev. Lett. 34 (1975) 43. 171 S. Coleman, There are no classical glueballs, Harvard University preprint HUTP-77/A027, for a relevant remark on the classical stability of such configurations. [81 W. Fischler, CERN preprint TH-2321 (1977); T. Applequist et al., Yale University preprint (1977). PI E. Eichten, K. Gottfried, T. Kinoshita, J. Kogut, K.D. Lane and T.M. Yan, Phys. Rev. Lett. 34 (1975) 369; B. Harrington, S.Y. Park and A. Yildiz, Phys. Rev. Lett. 34 (1975) 169. 1101 P. Minkowski, in: Lectures delivered at the Finnish Spring Workshop on QCD, Tva?minne, Finland 1977. 1111 W.A. Bardeen and R.B. Pearson, Phys. Rev. D14 (1976) 547. 1121 K.G. Wilson, Phys. Rev. DlO (1974) 2445 and CLNS327 preprint (Erice Summer School Lectures 1975); J. Kogut and L. Susskind, Phys. Rev. Dll (1975) 395.