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Vacuum stability in gauge theories
Volume 89B, number 2
PHYSICS LETTERS
14 January 1980
VACUUM STABILITY IN GAUGE THEORIES KS. VISWANATHAN Department of Phyws and Institute for Theoretwal Studies, Szmon Fraser Vnwerslty, Burnaby, Brttuh Columbia VSA IS6, Canada
Received 18 September 1979
The vacuum behaviour in a quantum SU(2) gauge theory is investigated by calculatmg the one-loop contrrbutrons to the effective action in a covarrant constant background field. It is found that the vacuum IS stable against decay, for a particular nonzero value of the electrrc field, indrcatmg a dynamical symmetry breakmg.
1. Introduction. Recently [l-3] the problem of quantizing a non-abelian gauge field m the temporal gauge, A, = 0, has been reinvestigated with a view to avoiding the Gribov ambiguities [4-61 arising in the Coulomb and covariant gauges. Untrl now actual calculations in the new gauge have been difficult to carry through due to the complicated nature of the effectrve lagrangian describmg the theory. In this article we reformulate this problem for an SU(2) gauge theory and express the generating functional in a form that is no more drffcult to handle than the usual perturbative QCD lagrangian. As a first application of this method we investigate the vacuum stability by calculatmg the one-loop contributrons to thq effective action following the method of Batalin et al. [7,8]. These authors use, m turn, a covariant constant field approximation (to be defined below) m conjunctron with Schwinger’s [9] proper time method. Recently Yildiz and Cox [IO] have reinvestigated this problem. The main conclusron of these authors [7,10] 1s that, at the one-loop level, the effective action has a positive definite imaginary part m the constant background field, which indicates the instability of the vacuum. In our new gauge-futing scheme we will see that stability of the vacuum requires a nonvanishing covariant constant color electric field whose value is determined, at the one-loop level, by setting the imaginary part of the lagrangian densrty to zero. The reason for the difference m qualitative behavior of the vacuum m the two schemes may be attrrbuted to the inadequacy of the Coulomb gauge (or
any gauge with ghosts) when nonvamshmg fields are present in the vacuum sector. This result may be taken as an indication that nonperturbative calculations may be done starting with the generating functional develop ed below. 2. Unconstrained temporal gauge. We briefly review the quantization of an SU(2) gauge field followingt the method in refs. [l-3]. The generating functional for a constrained system is given by [11,121 z
=JWl Ld-41 I-l %@(x)P(xot)> X!Y
(1)
X det II {G(x), x(v)) k”,
where S=-
I
(2)
d4x a FgvaF@‘,
(a(x), x0)} is the Poisson bracket of @ and x with respect to E and A as dynamical variables. (a(x) = 0 are the Gauss’ law constraints given by Q”(x) = $E,a +gcfbcAlbEic
= 0.
The constraint a(x) E (aa(x dependent gauge transformations.
I@(x), %(x)1 = 0,
(3)
generates time-inFurthermore (4)
where gY is the hamiltonian density. Hence the constraint a(x) = 0 at t = 0 will ensure its validity at all 215
Volume 89B, number 2
PHYSICS LETTERS
times. The choice of Xa(x) is arbitrary subject only to the requirement that det II(~(x), ×(v)} It4: 0.
(5)
We work in the AO a = 0 gauge. Then
We have dropped the irrelevant factor detx63(x - y). The integrals over Et a may be done as they involve moments of gaussian integrals. However, we find that the integral over E a yields the following:
fIdEl] det IlEta -
S = -fd4x[(aoEta)Aff + ½(Eft) 2 + ¼(Fqa)2],
X e x p [ - ~ f d 4 x ( E f f - F0aa) 2]
El a -- 11a' = 8 £?/8(boAt a)
(7)
= l-I det ItFol a - 6iaFok k U x
is the momentum conjugate to Aft. We choose x a ( x ) = e a i b E t b ( x ) - ~a(x),
(8)
where ~a(x) are arbitrary functions of space-time. Then {~a(x), xb(,v)} = g83(x - y ) ( M ( x ) - ~(x)),
+ ~x (FOkk)Sexp[-lifd4x(E11)2](E11)2[~11 (9)
where
Mab(X) = Ea b - 8 abEk k,
(10)
~b(x) = e . b S ( x ) .
(I l)
Here E ~ represents the symmetric part of Eab, i.e we write Ei a = ELq + eiabEb . In (9) we have evaluated the Poisson bracket at the constraint X a = 0. Using the delta function 8(Xa), we may integrate over the antisymmetric part of Eft. We set E b = ~b everywhere. Next we write
l-Ix8(~'a(x))=f[~°a(x)] exp[ifd4xAOa(x)~a(x)] " (12) Since the physical quantities should be independent of ~a(x) defined in (8) we may integrate the generating functional over these fields. Let Et a = Eia + eiab~b . We can write the generating functional as follows:
8,aEkk(x)II
(13)
X exPl-ifd4x[½(Eia) 2 - ' i a F O i a + ~(Ftia)2] } , where we have defined
Foi a = OoAi a + a.r40 a + geabCAobAi c.
(14)
detQ refers to the determinant of the 3 X 3 matrix. 216
8taEk k I1
(6)
where
z =ft .q t Pl H d~t IIEa(x)-
14 January 1980
]. (15)
By usmg the discrete version of the functional integrals we notice that the term involving Fokk(x) is multiplied by 1/e 12 while the determinant term is multiplied by lie 4 where e 4 is the volume of the space-time cells. We will have to take the e ~ 0 limit. Since physical quantities are generated from an external source we will actually be interested in Z [J]/Z [0] where J is the external source. When we take e -+ 0 of this ratio we notice that only the trace term survives. Thus our generating functional is
Z:f[dAu](~x
Fogk(x))e iS,
(16)
where
S = -¼fFuuaFauvd4x.
(17)
The field configurations Au a in (16) are chosen such that Fok k :/=O. This implies that all states of the theory correspond to nonvanishing color electric fields. We note that the t e r m Fok k as well as the determinant term in (15) are invarlant under simultaneous rotation SO(3)j × SO(3)T m three-dtmenslonal space and in lsospace.
3. Covariant constant field approximation and the stability o f the vacuum. We will use the generating functional in (16) to reinvestigate the vacuum stability studied in refs. [7,10]. Since all the difficult calculations have been done in refs. [7] and [10] we merely summarize the relevant steps and point out the cor-
Volume 89B, number 2
PHYSICS LETTERS
rection terms due to the Fokk term. The effectwe action at the one-loop level [7] may be written as Serf = S[A] - ifln
+ ½i Trln(32S/SASA).
Fokk(X)d4x
(18)
The second term on the right-hand side m (18)has been computed in refs. [7] and [10] in the covariant constant field approximation using Schwinger's proper time method. The covariant constant field satisfies
DpabF~a = 0.
(19)
One can readily see that
(20)
satisfies (19). nana = 1, na is an arbitrary unit vector m isospace. Furthermore A a = - ~1, ~- x
vn a .
(21)
In (20) and ( 2 1 ) F u r is an abelian tensor field. One finds that the effective lagrangian density dx given by the second and third terms in (18) is AZ?= i In E
1 f ds
eflS)f~f2s)
+ 4"2 so s--3 sinh(gfls) sin(gf2s)
+ ~3(gfls)(gf2s)47r2 1 [ sin(gfls)
xLs
)-
sin(gf2s)l sinh(gfl s ) J
(22) '
where f l = [F + (F 2 + G2)1/21//2, f2 = I - F + (F 2 +
G2)1/2] 1[2,
F -- 4l - wL '/av#"wtw=½(H2 G= ± p
field pointing along the third direction in isospace. This is analogous to coupling an lsotropic ferromagnet to an infiniteslmal uniform field in a gwen direction to produce a nonzero value of the magnetization. In the purely electric field background we have f l = 0 a n d f 2 = E and we find that the effective lagranglan has an imaginary part given by
Im Z?= g2E2 /48rr + lnE.
E2),
.~uv = ½E" H.
(23) (24) (25)
(26)
Here, *F~v is the dual of F~w In (22)s 0 is the ultraviolet cut-off. In arriving at (22) the quantity Fokk = E . h is replaced by E by choosing fi to point in the third direction in isospace. This is achieved by coupling the system to an external
(27)
By requiring that the vacuum be stable in a nonvanishing background field we arrive at an E 0 determined from
g2E2/487r + l n E 0 = 0.
FJ = F # , , n a
14 January 1980
(28)
This determines the covariant constant electric field in a consistent way. That this result is reasonable follows from the starting point (16) where the functional integral describes the correct quantization only if Fokk :I=O, 1.e., for nonvanishing electric fields. In passing, we may further note that in the covariant constant field approximation the determinant (15) is identically zero. In conclusion, we demonstrated the vacuum stability of an SU(2) theory in the one-loop approximation in a constant nonvanishing background color electric field. The electric field is determined from (28). Extension of this result to quantum chromodynamics based on SU(3) is not necessarily straightforward [3]. The author wishes to thank the Natural Sciences and Engineering Research Council, Canada, for partial financial support. Thanks are also due to Dr. C. Ragiadakos for fruitful discussions. [1] J. Goldstone and R. Jackiw, Phys. Lett. 74B (1978) 81. [2] A.G.Izergin, V.E. Korepin, M.A. Sememov-Tian-Shansky and L.D. Faddeev, Theor. Math. Phys. 38 (1979) 3. [3] Ch. Ragiadakos and K.S. Viswanathan, Phys. Lett. 86B (1979) 288. [4] V.N. Gribov, SLAC Trans. 176 (1977), unpublished. [5] V.N. Gribov, Nucl. Phys. B139 (1978) 1. [6] Ch. Ragladakos and K.S. Vlswanathan, Phys. Rev. D, to be published. [7] I.A. Batalin, S.G. Matinyan and G.K. Sawidi, Soy. J. Nucl. Phys. 26 (1977) 2. [8] S.G.Matmyan and G.K. Sawidi, Soy. J. Nucl. Phys. 25 (1977) 118. [9[ J. Schwinger, Phys. Rev. 82 (1951) 664. [10] A.Yfldiz and P.H. Cox, preprint. [11] L.D. Faddeev, Theor. Math. Phys. 1 (1969) 3. [12] L.D. Faddeev, in: Methods in field theory (Les Houches, 1975), eds. R. Balian and J. Zinn Justin (North-Holland). 217
PHYSICS LETTERS
14 January 1980
VACUUM STABILITY IN GAUGE THEORIES KS. VISWANATHAN Department of Phyws and Institute for Theoretwal Studies, Szmon Fraser Vnwerslty, Burnaby, Brttuh Columbia VSA IS6, Canada
Received 18 September 1979
The vacuum behaviour in a quantum SU(2) gauge theory is investigated by calculatmg the one-loop contrrbutrons to the effective action in a covarrant constant background field. It is found that the vacuum IS stable against decay, for a particular nonzero value of the electrrc field, indrcatmg a dynamical symmetry breakmg.
1. Introduction. Recently [l-3] the problem of quantizing a non-abelian gauge field m the temporal gauge, A, = 0, has been reinvestigated with a view to avoiding the Gribov ambiguities [4-61 arising in the Coulomb and covariant gauges. Untrl now actual calculations in the new gauge have been difficult to carry through due to the complicated nature of the effectrve lagrangian describmg the theory. In this article we reformulate this problem for an SU(2) gauge theory and express the generating functional in a form that is no more drffcult to handle than the usual perturbative QCD lagrangian. As a first application of this method we investigate the vacuum stability by calculatmg the one-loop contributrons to thq effective action following the method of Batalin et al. [7,8]. These authors use, m turn, a covariant constant field approximation (to be defined below) m conjunctron with Schwinger’s [9] proper time method. Recently Yildiz and Cox [IO] have reinvestigated this problem. The main conclusron of these authors [7,10] 1s that, at the one-loop level, the effective action has a positive definite imaginary part m the constant background field, which indicates the instability of the vacuum. In our new gauge-futing scheme we will see that stability of the vacuum requires a nonvanishing covariant constant color electric field whose value is determined, at the one-loop level, by setting the imaginary part of the lagrangian densrty to zero. The reason for the difference m qualitative behavior of the vacuum m the two schemes may be attrrbuted to the inadequacy of the Coulomb gauge (or
any gauge with ghosts) when nonvamshmg fields are present in the vacuum sector. This result may be taken as an indication that nonperturbative calculations may be done starting with the generating functional develop ed below. 2. Unconstrained temporal gauge. We briefly review the quantization of an SU(2) gauge field followingt the method in refs. [l-3]. The generating functional for a constrained system is given by [11,121 z
=JWl Ld-41 I-l %@(x)P(xot)> X!Y
(1)
X det II {G(x), x(v)) k”,
where S=-
I
(2)
d4x a FgvaF@‘,
(a(x), x0)} is the Poisson bracket of @ and x with respect to E and A as dynamical variables. (a(x) = 0 are the Gauss’ law constraints given by Q”(x) = $E,a +gcfbcAlbEic
= 0.
The constraint a(x) E (aa(x dependent gauge transformations.
I@(x), %(x)1 = 0,
(3)
generates time-inFurthermore (4)
where gY is the hamiltonian density. Hence the constraint a(x) = 0 at t = 0 will ensure its validity at all 215
Volume 89B, number 2
PHYSICS LETTERS
times. The choice of Xa(x) is arbitrary subject only to the requirement that det II(~(x), ×(v)} It4: 0.
(5)
We work in the AO a = 0 gauge. Then
We have dropped the irrelevant factor detx63(x - y). The integrals over Et a may be done as they involve moments of gaussian integrals. However, we find that the integral over E a yields the following:
fIdEl] det IlEta -
S = -fd4x[(aoEta)Aff + ½(Eft) 2 + ¼(Fqa)2],
X e x p [ - ~ f d 4 x ( E f f - F0aa) 2]
El a -- 11a' = 8 £?/8(boAt a)
(7)
= l-I det ItFol a - 6iaFok k U x
is the momentum conjugate to Aft. We choose x a ( x ) = e a i b E t b ( x ) - ~a(x),
(8)
where ~a(x) are arbitrary functions of space-time. Then {~a(x), xb(,v)} = g83(x - y ) ( M ( x ) - ~(x)),
+ ~x (FOkk)Sexp[-lifd4x(E11)2](E11)2[~11 (9)
where
Mab(X) = Ea b - 8 abEk k,
(10)
~b(x) = e . b S ( x ) .
(I l)
Here E ~ represents the symmetric part of Eab, i.e we write Ei a = ELq + eiabEb . In (9) we have evaluated the Poisson bracket at the constraint X a = 0. Using the delta function 8(Xa), we may integrate over the antisymmetric part of Eft. We set E b = ~b everywhere. Next we write
l-Ix8(~'a(x))=f[~°a(x)] exp[ifd4xAOa(x)~a(x)] " (12) Since the physical quantities should be independent of ~a(x) defined in (8) we may integrate the generating functional over these fields. Let Et a = Eia + eiab~b . We can write the generating functional as follows:
8,aEkk(x)II
(13)
X exPl-ifd4x[½(Eia) 2 - ' i a F O i a + ~(Ftia)2] } , where we have defined
Foi a = OoAi a + a.r40 a + geabCAobAi c.
(14)
detQ refers to the determinant of the 3 X 3 matrix. 216
8taEk k I1
(6)
where
z =ft .q t Pl H d~t IIEa(x)-
14 January 1980
]. (15)
By usmg the discrete version of the functional integrals we notice that the term involving Fokk(x) is multiplied by 1/e 12 while the determinant term is multiplied by lie 4 where e 4 is the volume of the space-time cells. We will have to take the e ~ 0 limit. Since physical quantities are generated from an external source we will actually be interested in Z [J]/Z [0] where J is the external source. When we take e -+ 0 of this ratio we notice that only the trace term survives. Thus our generating functional is
Z:f[dAu](~x
Fogk(x))e iS,
(16)
where
S = -¼fFuuaFauvd4x.
(17)
The field configurations Au a in (16) are chosen such that Fok k :/=O. This implies that all states of the theory correspond to nonvanishing color electric fields. We note that the t e r m Fok k as well as the determinant term in (15) are invarlant under simultaneous rotation SO(3)j × SO(3)T m three-dtmenslonal space and in lsospace.
3. Covariant constant field approximation and the stability o f the vacuum. We will use the generating functional in (16) to reinvestigate the vacuum stability studied in refs. [7,10]. Since all the difficult calculations have been done in refs. [7] and [10] we merely summarize the relevant steps and point out the cor-
Volume 89B, number 2
PHYSICS LETTERS
rection terms due to the Fokk term. The effectwe action at the one-loop level [7] may be written as Serf = S[A] - ifln
+ ½i Trln(32S/SASA).
Fokk(X)d4x
(18)
The second term on the right-hand side m (18)has been computed in refs. [7] and [10] in the covariant constant field approximation using Schwinger's proper time method. The covariant constant field satisfies
DpabF~a = 0.
(19)
One can readily see that
(20)
satisfies (19). nana = 1, na is an arbitrary unit vector m isospace. Furthermore A a = - ~1, ~- x
vn a .
(21)
In (20) and ( 2 1 ) F u r is an abelian tensor field. One finds that the effective lagrangian density dx given by the second and third terms in (18) is AZ?= i In E
1 f ds
eflS)f~f2s)
+ 4"2 so s--3 sinh(gfls) sin(gf2s)
+ ~3(gfls)(gf2s)47r2 1 [ sin(gfls)
xLs
)-
sin(gf2s)l sinh(gfl s ) J
(22) '
where f l = [F + (F 2 + G2)1/21//2, f2 = I - F + (F 2 +
G2)1/2] 1[2,
F -- 4l - wL '/av#"wtw=½(H2 G= ± p
field pointing along the third direction in isospace. This is analogous to coupling an lsotropic ferromagnet to an infiniteslmal uniform field in a gwen direction to produce a nonzero value of the magnetization. In the purely electric field background we have f l = 0 a n d f 2 = E and we find that the effective lagranglan has an imaginary part given by
Im Z?= g2E2 /48rr + lnE.
E2),
.~uv = ½E" H.
(23) (24) (25)
(26)
Here, *F~v is the dual of F~w In (22)s 0 is the ultraviolet cut-off. In arriving at (22) the quantity Fokk = E . h is replaced by E by choosing fi to point in the third direction in isospace. This is achieved by coupling the system to an external
(27)
By requiring that the vacuum be stable in a nonvanishing background field we arrive at an E 0 determined from
g2E2/487r + l n E 0 = 0.
FJ = F # , , n a
14 January 1980
(28)
This determines the covariant constant electric field in a consistent way. That this result is reasonable follows from the starting point (16) where the functional integral describes the correct quantization only if Fokk :I=O, 1.e., for nonvanishing electric fields. In passing, we may further note that in the covariant constant field approximation the determinant (15) is identically zero. In conclusion, we demonstrated the vacuum stability of an SU(2) theory in the one-loop approximation in a constant nonvanishing background color electric field. The electric field is determined from (28). Extension of this result to quantum chromodynamics based on SU(3) is not necessarily straightforward [3]. The author wishes to thank the Natural Sciences and Engineering Research Council, Canada, for partial financial support. Thanks are also due to Dr. C. Ragiadakos for fruitful discussions. [1] J. Goldstone and R. Jackiw, Phys. Lett. 74B (1978) 81. [2] A.G.Izergin, V.E. Korepin, M.A. Sememov-Tian-Shansky and L.D. Faddeev, Theor. Math. Phys. 38 (1979) 3. [3] Ch. Ragiadakos and K.S. Viswanathan, Phys. Lett. 86B (1979) 288. [4] V.N. Gribov, SLAC Trans. 176 (1977), unpublished. [5] V.N. Gribov, Nucl. Phys. B139 (1978) 1. [6] Ch. Ragladakos and K.S. Vlswanathan, Phys. Rev. D, to be published. [7] I.A. Batalin, S.G. Matinyan and G.K. Sawidi, Soy. J. Nucl. Phys. 26 (1977) 2. [8] S.G.Matmyan and G.K. Sawidi, Soy. J. Nucl. Phys. 25 (1977) 118. [9[ J. Schwinger, Phys. Rev. 82 (1951) 664. [10] A.Yfldiz and P.H. Cox, preprint. [11] L.D. Faddeev, Theor. Math. Phys. 1 (1969) 3. [12] L.D. Faddeev, in: Methods in field theory (Les Houches, 1975), eds. R. Balian and J. Zinn Justin (North-Holland). 217