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Anti-screening of large magnetic fields by vector bosons
Volume 214, n u m b e r 4
PHYSICS LETTERS B
1 D e c e m b e r 1988
A N T I - S C R E E N I N G OF LARGE M A G N E T I C FIELDS BY VECTOR BOSONS J. AMBJORN and P. OLESEN The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Received 2 July 1988
In the SO ( 3 ) m o d e l with m a s s i v e vector bosons we show that for m a g n e t i c fields exceeding m 2 / e there is c o n d e n s a t i o n of W's. T h i s c o n d e n s a t i o n is c h a r a c t e r i z e d by anti-screening. N e a r the critical field we show t h a t the c o n d e n s a t e is a lattice o f vortex lines.
The usual vacuum of the electroweak theory becomes unstable when a magnetic field exceeding the critical value m 2 / e ~ 1024 G is applied [ 1,2]. This instability is related to asymptotic freedom, which is relevant for large magnetic fields in the electroweak theory [ 3 ]. In present day physics it is hard to obtain such large magnetic fields, and the only known example where this is feasible is Witten's superconducting string [ 4 ] discussed in ref. [ 2 ]. However, if such fields could be constructed one would have the possibility of studying the electroweak theory in a form which is very different from the one found in present day accelerators. In this paper we shall study the stabilization of a homogeneous magnetic field in a model which is somewhat simpler than the electroweak theory, namely the SO (3) model with a massive W. This is the same as the Georgi-Glashow model, except that we do not consider the backreaction of the W field on the Higgs field. We hope to come back to the full electroweak theory later. The stabilization found by us is a W condensate producing domains in the magnetic field, which means that the vacuum reacts back on this field and destroys homogeneity. The resulting configuration is a lattice of magnetic vortex lines with non-abelian flux quantization. In many ways this state looks like the lattice of vortex lines in a type II superconductor near the critical field [ 5 ]. However, there is one crucial difference: in a type II superconductor the condensate produces currents which tend to decrease the size of the magnetic field, so this field is screened (this is
basically the Meissner effect). In our case we have anti-screening: the W's form a condensate which set up currents which run around in such a way that the field is increased. Since the original instability is closely related to asymptotic freedom one can say that the enhancement of the field is due to the fact that the vacuum is paramagnetic (or perhaps even ferromagnetic) in asymptotically free theories. However, we feel that it would in a sense be more appropriate to say that the vacuum behaves like an anti-type II superconductor. In any case, for fields larger than ~ 1024 G the vacuum behaves like an interesting material, which we call "asymptotic freeon". In the SO (3) model we have the isovector field A~. Introducing
w.= ~l
( a .1+ i A' . )2,
3_
A I, =Ap ,
(1)
and considering At, as the electromagnetic field with field strength f ~ = OuA. - OVA.,
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(2)
the lagrangian can be written . W = - ~Fu~ , ~ -__ -
~1f u . 2
-
m~wW*W~ ½
ID. W . - D .
_iefu. W,uW.+½e2[W,u2
Du=Ou-ieAu.
W u 12 - - H~ ~2 W .rv 1,
/~vr/t ,~,.
2 w.-(w~w~):]
(3)
The coupling - i e f u . W*u W. is due to the magnetic moment of the W's and causes the instability [ 1,2 ] if 565
Volume 214, number 4
PHYSICS LETTERS B
the non-linear terms are ignored. For a homogeneous field the instability occurs for momenta k which are infrared, k2
W2 .~--i W,
W3 -- Wo -~-0 .
(4)
Of course, it cannot be excluded that as Wgrows new instabilities can develop. However, their nature would be different from the instability discussed above. From previous analysis [ 1,3 ] we also know that the unstable mode develops under the condition D~, W~,= 0, so in our ansatz we include this condition, D,,W~,=0
or
(D~ + i D 2 ) W = 0 .
(5)
To discuss the full problem radiative corrections must be included (to compute these one needs evidently to generate mw through the Higgs mechanism), which makes the problem extremely hard. However, as already mentioned for large magnetic fields one has asymptotic freedom and hence radiative correction can be ignored. This is expected to be valid when e//>> (all masses) 2, i.e. for eI~>>m 2, where/7 is some typical mean field. Thus, close to the critical point e l l = m~v one can actually have important radiative corrections which may even shift the value of the critical field. However, for larger values of the field one can simply solve the problem by solving the classical equations of motion. This is what we shall do in the following. Also, since these equations are anyhow rather interesting we shall allow ourselves to consider the solution for all values of/q, although this can only be rigorously justified for very large/4. The equations of motion resulting from the lagrangian (3) clearly have the solution fj2 = n ,
W=0,
(6)
which becomes unstable for H > m2w/e. Here and in the following we take the magnetic field to point in the 3-direction. We now look for a solution which is different from (6), and we start by considering the 566
1 December 1988
backreaction of W# 0 on the electromagnetic field. We have the equation of motion O,f,,u = __jinduced, j~nduced =
(7)
- i e [ W~(I), - 5 ~ ) W~]
+ie[ W~(D~ Wu) - W~(D~ W~,)*] - ieO~( Wf, W~ - Wu W~ ) .
(8)
Using the ansatz (4) and (5) one finds the simple result •induced__ • W~Wv_WuW¢u) J~, - -xe0~(
(9)
which implies that the equation of motion (7) can be trivially solved, f i 2 = - f i t = C + 2 e l WI 2
(10)
with all other fur's equal to zero. In eq. (10) C is an arbitrary constant (to be fixed later). The next step is to consider the equation of motion for W, [ - (O-ieA)2 +m2w-2ef~2] W =-2e21WI2W.
(11)
Before doing that it is convenient to eliminate the potentials A u in terms of I WI. To do this we transform the induced current (9) by use of D~ W u = 0 to the form jinduced = ,u
2ieW*~, W + 4e2A~, [ WI 2
(12)
This follows from eq. (9) by performing the differentiation and using D u W u = O. At this stage it is clear why asymptotic freeon is like an anti-superconductor (type II), since eq. (12) is precisely the same as the equation for the current in the Landau-Ginzburg theory except for the sign (identify I WI with the order parameter). Thus, the current (12) tends to increase the magnetic field (anti-screening) whereas the Landau-Ginzburg current tends to decrease the magnetic field, since superconductors do not like magnetism. Eq. (12) allows us to eliminate A u for simply connected geometries. Making a gauge transformation Au ~Au =Au + 10uZ, e
we have from eqs. (7) and (12)
(13)
Volume 214, number 4
PHYSICS LETTERSB
-O,, f,~, = 4e2~,, I WI 2
(14)
Using the solution (10) f o r f ~ we finally obtain A,=eO-le %lnF WI •
(15)
In this gauge the equation of motion ( 11 ) becomes [ - 0 5 - 0 2 + (01 lnl WI )2+ (02 lnl WI
-2ef~2] l WI = -2e21WI 3.
(16)
eft2 = e(O,A: - O2A, )
-
-1
I WI
+ ~
1
"l- 02 ~ )
(051Wl + 0 2 1 W l )
(17)
where we used eqs. ( 15 ) and (10). From eq. (17) we see that the kinetic operator in eq. (16) is cancelled by - efl 2, and we are left with (m2w- efl2)[ ~,I/'1=-2e21 W[ 3 .
(18)
Inserting the solution (10) forfl2 we end up with the simple result
( m2w - e f ) l Wl =O .
(19)
Obviously this equation has two solutions, C = an arbitrary constant, I WI = 0 and C = m---~-2w= critical field,
e
4e21 WI 2f12 = _ (0~2 + 02)f~2 2
(22)
If I WI could be taken constant (which it cannot) this produces one of the London equations with the "wrong" sign, which again indicates that vacuum is like an anti-superconductor. Eq. (22) is of course not useful for us, since we knowfl2 in terms of I WI, and the simpler eq. (21 ) should be used. The energy of the system can easily be determined from the lagrangian (3), since the system is static. After some partial integrations one obtains the simple expression
(20)
(23)
for the energy density. This corresponds to the production of a magnetic field f~2 with a negative condensate energy. Inserting the solution forfi2 eq. (23) can be written 1 m 4 +2rn2wl WI2 e = ~ e----5-
(24)
which is larger than the critical energy ~~,~, ~4, w //~2 t; , as is to be expected. The average energy can be expressed in terms of the flux through some area A. Because of the instability, which is an infrared problem, we shall use periodic boundary conditions. Thus [ WI 2 and the current j/z /induced are periodic. From eq. (12) this implies
_
/induced
At,= 1 0,Z+ ~ e
[ WI ~ 0. The first solution clearly means that C is equal to the constant homogeneous field, which is unstable above the critical strength (20). Thus, the second solution is the condensate solution which corresponds to asymptotic freeon. With I WI ~ 0 we still need to satisfy eq. (17), which can be written - (02 +02 ) lnl W[ =m2w+2e2l W[2
2
+ ~ - ~ 011 WI 01Am+ ~-~-~ 021Wl 02f12 •
e=!z]"22--2e21W[ 4
[(0, I Wl )2+(02 I W[)2]
=eC+2e21WI 2 ,
quantity we see that eq. (21 ) is of the Liouville-type. The negative sign for the kinetic operator is due to anti-screening. This can e.g. be seen by rewriting the equation in the somewhat complicated form
)2+m:w
This equation can be considerably simplified by using
= -- (01 ~
1 December 1988
(25)
4e21 WI 2 •
Thus, because of periodicity of I WI 2 a n d j u-induced,
flux= f fl2d2x=~ A, d x ~ - 2nx cell
cell
e
(26)
with x=O, _+ 1, + 2, ..., and where the integration is over one fundamental cell. The average I WI 2 is then
(21)
This is thus the equation to be satisfied by the W condensate. If one considers ln l W[ as the unknown
2e f I WI 2 d2x= 2nx d e
cell
m~v A. e
(27)
567
Volume 214, number 4
PHYSICS LETTERS B
From this and eq. (24) we then obtain g - ~1 ~ e d Z x = mZw (2.~Ax .--
m~w) .
(28)
e cell
Furthermore, integrating eq. ( 2 1 ) f o r IWI we also obtain by use ofeq. (26)
1 December 1988
the solution. As discussed in ref. [ 6 ] the solution (33) with periodicity imposed corresponds to a lattice of vortex lines, which are quantized as discussed by 't Hooft [ 7]. Selecting a suitable gauge and defining A u =A ~z a, one has for a rectangular lattice with sides al and a2
A~,(x~, a2) =A~,(x,, O), --
f 02 lnl WI d 2 x = 2 n x .
(29)
cell
Eq. (28) can be used to express the area of a fundamental periodicity cell by means of the energy,
2nxm~v A= e2(g+m~v/2e2 ) ,
Au(al, x2) =t2(xz)Au(O, x2)f2 -~ (x2) i f2(x2)0uK2_ l (x2) e
(34)
where (30)
ff~(X2) = e x p ( - i m 2 a l x 2 z 3 ) .
(35)
which shows that the larger the energy becomes, the smaller the area becomes. Near the critical field g..~ ½m~v/e 2, so
This gauge function satisfies
2nx A ~ m 2w (near critical field) .
where Z is a center element (+_ 1 ). Eqs. ( 3 4 ) - ( 3 6 ) correspond to 't Hooft's flux definition. If we write W= I WI exp(iz) then it can easily be seen [6] that
(31 )
These features are somewhat similar to the behavior of a superconductor with x = 1/,e/2 except for the anti-screening. An approximate solution to eq. (20) can be obtained if I WI is very small. We then have - ( 0 2 + 0 2 ) lnl WI ~ m 2 ,
(32)
t2(0) = t 2 ( a 2 ) Z ,
(36)
Z(x~ +a~, x2) =Z(Xl, x2) +rn~valx2,
X(x~, xz +a2) =Z(x~, x2) •
(37)
The lattice area must satisfy [ 6 ] [ see also eq. (31 ) ] 2nx ala2 =lattice a r e a = m2w , x = l , 2, 3, ...,
(38)
which has the general solution
W(xl, x2) ~ e x p ( - ½rn~vx2)F(x~ + i x 2 ) ,
(33)
where x is related to the center by 12(a2)g2-t (0) = exp(2~Kr3) .
where F is an arbitrary analytic function. To ensure stability of the system we need a suitable boundary condition. [Any analytic function F ( z ) does not imply stability; e.g. F=const. is analytic, but the instability remains for this choice. ] As already mentioned the instability is an infrared problem, and hence we need domains. Taking a regular lattice of domains this amounts to requiring that I W(Xl, x2) I be periodic in the Xl-X2 plane ~1. This situation has previously been analyzed in detail by the authors [ 6 ]. We just mention that this implies that F(Xl+ix2) can be expressed in terms of generalized O functions. Since the resulting expression is rather lengthy, we shall not write it down explicitly. Instead let us discuss the topological properties of ~ There may be other reasonable choices o f F ( z ) .
568
(39)
Therefore, the phase difference o f z one picks up by going around along the lattice boundary is 2nx, as one can see from eqs. (37) and (38). Topologically this means that W has x simple zeros in a fundamental lattice cell, just like in the abelian case. In the abelian case, however, x is proportional to the magnetic flux, whereas in our case it is x modulus 2 which is the magnetic flux in the sense defined by 't Hooft [ 7 ]. Thus, the solution for small W has interesting nonabelian topological properties. However, it should also be mentioned that an "observer" who measures the abelian flux ~ = f A u dx u actually finds the usual abelian result (obtained by integrating around the boundary of a fundamental lattice cell) known from Abrikosov's paper [ 5 ]. This can be seen by going back to the "old" gauge A u, where the current is given by
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
eq. (12). Using that the current is periodic and I WI 2 is periodic one simply obtains by integrating around the boundary of a fundamental lattice cell [ see eqs. (25) and (26)]
vide an alternative to particle accelerators as far as W and Z production is concerned, if large magnetic fields could be established in laboratories.
• =~A,,dx.=l~o,,zdx~2e r t x -
Note added. After submitting the paper we have been informed that the possibility of having a vortex structure above the critical field Hc has also been discussed by Skalozub [ 8 ].
e
(40)
Thus an "observer" who can only measure electromagnetism would not discover that the vortex condensate is actually non-abelian. From the point of view of energy considerations presumably x = 1 is preferred [ 6 ], although this has not been proved. We can sum up by saying that the vacuum has very interesting properties in the model given by the lagrangian (3). To make more contact with physics these considerations should be extended to the proper electroweak theory, which implies that the backreaction of the W condensate (there will also be a Z condensate) on the Higgs field should be taken into account. If it turns out that the qualitative features of our results remain valid in this case ~2, it would pro~2 In any case it should be remembered that the usual electroweak vacuum becomes unstable for large magnetic fields, so something must happen.
References [ 1 ] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376; J. Ambjorn, R.J. Hughes and N.K. Nielsen, Ann. Phys. (NY) 150 (1983) 92. [2] J. Ambjorn, N.K. Nielsen and P. Olesen, preprint NBI-HE88-18, Nucl. Phys. B, to be published. [3] J. Ambjorn, N.K. Nielsen and P. Olesen, Nucl. Phys. B 152 (1979) 75. [4] E. Witten, Nucl. Phys. B 249 (1985) 557. [5] A.A. Abrikosov, Sov. Phys. JETP 5 (1957) 1174. [ 6 ] J. Ambjorn and P. Olesen, Nucl. Phys. B 170 (1980) 60, 265. [7] G. 't Hooft, Nucl. Phys. B 153 (1979) 141. [ 8 ] V.V. Skalozub, Soy. J. Nucl. Phys. 43 (1986) 665; 45 ( 1987 ) 1058.
569
PHYSICS LETTERS B
1 D e c e m b e r 1988
A N T I - S C R E E N I N G OF LARGE M A G N E T I C FIELDS BY VECTOR BOSONS J. AMBJORN and P. OLESEN The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Received 2 July 1988
In the SO ( 3 ) m o d e l with m a s s i v e vector bosons we show that for m a g n e t i c fields exceeding m 2 / e there is c o n d e n s a t i o n of W's. T h i s c o n d e n s a t i o n is c h a r a c t e r i z e d by anti-screening. N e a r the critical field we show t h a t the c o n d e n s a t e is a lattice o f vortex lines.
The usual vacuum of the electroweak theory becomes unstable when a magnetic field exceeding the critical value m 2 / e ~ 1024 G is applied [ 1,2]. This instability is related to asymptotic freedom, which is relevant for large magnetic fields in the electroweak theory [ 3 ]. In present day physics it is hard to obtain such large magnetic fields, and the only known example where this is feasible is Witten's superconducting string [ 4 ] discussed in ref. [ 2 ]. However, if such fields could be constructed one would have the possibility of studying the electroweak theory in a form which is very different from the one found in present day accelerators. In this paper we shall study the stabilization of a homogeneous magnetic field in a model which is somewhat simpler than the electroweak theory, namely the SO (3) model with a massive W. This is the same as the Georgi-Glashow model, except that we do not consider the backreaction of the W field on the Higgs field. We hope to come back to the full electroweak theory later. The stabilization found by us is a W condensate producing domains in the magnetic field, which means that the vacuum reacts back on this field and destroys homogeneity. The resulting configuration is a lattice of magnetic vortex lines with non-abelian flux quantization. In many ways this state looks like the lattice of vortex lines in a type II superconductor near the critical field [ 5 ]. However, there is one crucial difference: in a type II superconductor the condensate produces currents which tend to decrease the size of the magnetic field, so this field is screened (this is
basically the Meissner effect). In our case we have anti-screening: the W's form a condensate which set up currents which run around in such a way that the field is increased. Since the original instability is closely related to asymptotic freedom one can say that the enhancement of the field is due to the fact that the vacuum is paramagnetic (or perhaps even ferromagnetic) in asymptotically free theories. However, we feel that it would in a sense be more appropriate to say that the vacuum behaves like an anti-type II superconductor. In any case, for fields larger than ~ 1024 G the vacuum behaves like an interesting material, which we call "asymptotic freeon". In the SO (3) model we have the isovector field A~. Introducing
w.= ~l
( a .1+ i A' . )2,
3_
A I, =Ap ,
(1)
and considering At, as the electromagnetic field with field strength f ~ = OuA. - OVA.,
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(2)
the lagrangian can be written . W = - ~Fu~ , ~ -__ -
~1f u . 2
-
m~wW*W~ ½
ID. W . - D .
_iefu. W,uW.+½e2[W,u2
Du=Ou-ieAu.
W u 12 - - H~ ~2 W .rv 1,
/~vr/t ,~,.
2 w.-(w~w~):]
(3)
The coupling - i e f u . W*u W. is due to the magnetic moment of the W's and causes the instability [ 1,2 ] if 565
Volume 214, number 4
PHYSICS LETTERS B
the non-linear terms are ignored. For a homogeneous field the instability occurs for momenta k which are infrared, k2
W2 .~--i W,
W3 -- Wo -~-0 .
(4)
Of course, it cannot be excluded that as Wgrows new instabilities can develop. However, their nature would be different from the instability discussed above. From previous analysis [ 1,3 ] we also know that the unstable mode develops under the condition D~, W~,= 0, so in our ansatz we include this condition, D,,W~,=0
or
(D~ + i D 2 ) W = 0 .
(5)
To discuss the full problem radiative corrections must be included (to compute these one needs evidently to generate mw through the Higgs mechanism), which makes the problem extremely hard. However, as already mentioned for large magnetic fields one has asymptotic freedom and hence radiative correction can be ignored. This is expected to be valid when e//>> (all masses) 2, i.e. for eI~>>m 2, where/7 is some typical mean field. Thus, close to the critical point e l l = m~v one can actually have important radiative corrections which may even shift the value of the critical field. However, for larger values of the field one can simply solve the problem by solving the classical equations of motion. This is what we shall do in the following. Also, since these equations are anyhow rather interesting we shall allow ourselves to consider the solution for all values of/q, although this can only be rigorously justified for very large/4. The equations of motion resulting from the lagrangian (3) clearly have the solution fj2 = n ,
W=0,
(6)
which becomes unstable for H > m2w/e. Here and in the following we take the magnetic field to point in the 3-direction. We now look for a solution which is different from (6), and we start by considering the 566
1 December 1988
backreaction of W# 0 on the electromagnetic field. We have the equation of motion O,f,,u = __jinduced, j~nduced =
(7)
- i e [ W~(I), - 5 ~ ) W~]
+ie[ W~(D~ Wu) - W~(D~ W~,)*] - ieO~( Wf, W~ - Wu W~ ) .
(8)
Using the ansatz (4) and (5) one finds the simple result •induced__ • W~Wv_WuW¢u) J~, - -xe0~(
(9)
which implies that the equation of motion (7) can be trivially solved, f i 2 = - f i t = C + 2 e l WI 2
(10)
with all other fur's equal to zero. In eq. (10) C is an arbitrary constant (to be fixed later). The next step is to consider the equation of motion for W, [ - (O-ieA)2 +m2w-2ef~2] W =-2e21WI2W.
(11)
Before doing that it is convenient to eliminate the potentials A u in terms of I WI. To do this we transform the induced current (9) by use of D~ W u = 0 to the form jinduced = ,u
2ieW*~, W + 4e2A~, [ WI 2
(12)
This follows from eq. (9) by performing the differentiation and using D u W u = O. At this stage it is clear why asymptotic freeon is like an anti-superconductor (type II), since eq. (12) is precisely the same as the equation for the current in the Landau-Ginzburg theory except for the sign (identify I WI with the order parameter). Thus, the current (12) tends to increase the magnetic field (anti-screening) whereas the Landau-Ginzburg current tends to decrease the magnetic field, since superconductors do not like magnetism. Eq. (12) allows us to eliminate A u for simply connected geometries. Making a gauge transformation Au ~Au =Au + 10uZ, e
we have from eqs. (7) and (12)
(13)
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PHYSICS LETTERSB
-O,, f,~, = 4e2~,, I WI 2
(14)
Using the solution (10) f o r f ~ we finally obtain A,=eO-le %lnF WI •
(15)
In this gauge the equation of motion ( 11 ) becomes [ - 0 5 - 0 2 + (01 lnl WI )2+ (02 lnl WI
-2ef~2] l WI = -2e21WI 3.
(16)
eft2 = e(O,A: - O2A, )
-
-1
I WI
+ ~
1
"l- 02 ~ )
(051Wl + 0 2 1 W l )
(17)
where we used eqs. ( 15 ) and (10). From eq. (17) we see that the kinetic operator in eq. (16) is cancelled by - efl 2, and we are left with (m2w- efl2)[ ~,I/'1=-2e21 W[ 3 .
(18)
Inserting the solution (10) forfl2 we end up with the simple result
( m2w - e f ) l Wl =O .
(19)
Obviously this equation has two solutions, C = an arbitrary constant, I WI = 0 and C = m---~-2w= critical field,
e
4e21 WI 2f12 = _ (0~2 + 02)f~2 2
(22)
If I WI could be taken constant (which it cannot) this produces one of the London equations with the "wrong" sign, which again indicates that vacuum is like an anti-superconductor. Eq. (22) is of course not useful for us, since we knowfl2 in terms of I WI, and the simpler eq. (21 ) should be used. The energy of the system can easily be determined from the lagrangian (3), since the system is static. After some partial integrations one obtains the simple expression
(20)
(23)
for the energy density. This corresponds to the production of a magnetic field f~2 with a negative condensate energy. Inserting the solution forfi2 eq. (23) can be written 1 m 4 +2rn2wl WI2 e = ~ e----5-
(24)
which is larger than the critical energy ~~,~, ~4, w //~2 t; , as is to be expected. The average energy can be expressed in terms of the flux through some area A. Because of the instability, which is an infrared problem, we shall use periodic boundary conditions. Thus [ WI 2 and the current j/z /induced are periodic. From eq. (12) this implies
_
/induced
At,= 1 0,Z+ ~ e
[ WI ~ 0. The first solution clearly means that C is equal to the constant homogeneous field, which is unstable above the critical strength (20). Thus, the second solution is the condensate solution which corresponds to asymptotic freeon. With I WI ~ 0 we still need to satisfy eq. (17), which can be written - (02 +02 ) lnl W[ =m2w+2e2l W[2
2
+ ~ - ~ 011 WI 01Am+ ~-~-~ 021Wl 02f12 •
e=!z]"22--2e21W[ 4
[(0, I Wl )2+(02 I W[)2]
=eC+2e21WI 2 ,
quantity we see that eq. (21 ) is of the Liouville-type. The negative sign for the kinetic operator is due to anti-screening. This can e.g. be seen by rewriting the equation in the somewhat complicated form
)2+m:w
This equation can be considerably simplified by using
= -- (01 ~
1 December 1988
(25)
4e21 WI 2 •
Thus, because of periodicity of I WI 2 a n d j u-induced,
flux= f fl2d2x=~ A, d x ~ - 2nx cell
cell
e
(26)
with x=O, _+ 1, + 2, ..., and where the integration is over one fundamental cell. The average I WI 2 is then
(21)
This is thus the equation to be satisfied by the W condensate. If one considers ln l W[ as the unknown
2e f I WI 2 d2x= 2nx d e
cell
m~v A. e
(27)
567
Volume 214, number 4
PHYSICS LETTERS B
From this and eq. (24) we then obtain g - ~1 ~ e d Z x = mZw (2.~Ax .--
m~w) .
(28)
e cell
Furthermore, integrating eq. ( 2 1 ) f o r IWI we also obtain by use ofeq. (26)
1 December 1988
the solution. As discussed in ref. [ 6 ] the solution (33) with periodicity imposed corresponds to a lattice of vortex lines, which are quantized as discussed by 't Hooft [ 7]. Selecting a suitable gauge and defining A u =A ~z a, one has for a rectangular lattice with sides al and a2
A~,(x~, a2) =A~,(x,, O), --
f 02 lnl WI d 2 x = 2 n x .
(29)
cell
Eq. (28) can be used to express the area of a fundamental periodicity cell by means of the energy,
2nxm~v A= e2(g+m~v/2e2 ) ,
Au(al, x2) =t2(xz)Au(O, x2)f2 -~ (x2) i f2(x2)0uK2_ l (x2) e
(34)
where (30)
ff~(X2) = e x p ( - i m 2 a l x 2 z 3 ) .
(35)
which shows that the larger the energy becomes, the smaller the area becomes. Near the critical field g..~ ½m~v/e 2, so
This gauge function satisfies
2nx A ~ m 2w (near critical field) .
where Z is a center element (+_ 1 ). Eqs. ( 3 4 ) - ( 3 6 ) correspond to 't Hooft's flux definition. If we write W= I WI exp(iz) then it can easily be seen [6] that
(31 )
These features are somewhat similar to the behavior of a superconductor with x = 1/,e/2 except for the anti-screening. An approximate solution to eq. (20) can be obtained if I WI is very small. We then have - ( 0 2 + 0 2 ) lnl WI ~ m 2 ,
(32)
t2(0) = t 2 ( a 2 ) Z ,
(36)
Z(x~ +a~, x2) =Z(Xl, x2) +rn~valx2,
X(x~, xz +a2) =Z(x~, x2) •
(37)
The lattice area must satisfy [ 6 ] [ see also eq. (31 ) ] 2nx ala2 =lattice a r e a = m2w , x = l , 2, 3, ...,
(38)
which has the general solution
W(xl, x2) ~ e x p ( - ½rn~vx2)F(x~ + i x 2 ) ,
(33)
where x is related to the center by 12(a2)g2-t (0) = exp(2~Kr3) .
where F is an arbitrary analytic function. To ensure stability of the system we need a suitable boundary condition. [Any analytic function F ( z ) does not imply stability; e.g. F=const. is analytic, but the instability remains for this choice. ] As already mentioned the instability is an infrared problem, and hence we need domains. Taking a regular lattice of domains this amounts to requiring that I W(Xl, x2) I be periodic in the Xl-X2 plane ~1. This situation has previously been analyzed in detail by the authors [ 6 ]. We just mention that this implies that F(Xl+ix2) can be expressed in terms of generalized O functions. Since the resulting expression is rather lengthy, we shall not write it down explicitly. Instead let us discuss the topological properties of ~ There may be other reasonable choices o f F ( z ) .
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(39)
Therefore, the phase difference o f z one picks up by going around along the lattice boundary is 2nx, as one can see from eqs. (37) and (38). Topologically this means that W has x simple zeros in a fundamental lattice cell, just like in the abelian case. In the abelian case, however, x is proportional to the magnetic flux, whereas in our case it is x modulus 2 which is the magnetic flux in the sense defined by 't Hooft [ 7 ]. Thus, the solution for small W has interesting nonabelian topological properties. However, it should also be mentioned that an "observer" who measures the abelian flux ~ = f A u dx u actually finds the usual abelian result (obtained by integrating around the boundary of a fundamental lattice cell) known from Abrikosov's paper [ 5 ]. This can be seen by going back to the "old" gauge A u, where the current is given by
Volume 214, number 4
PHYSICS LETTERS B
1 December 1988
eq. (12). Using that the current is periodic and I WI 2 is periodic one simply obtains by integrating around the boundary of a fundamental lattice cell [ see eqs. (25) and (26)]
vide an alternative to particle accelerators as far as W and Z production is concerned, if large magnetic fields could be established in laboratories.
• =~A,,dx.=l~o,,zdx~2e r t x -
Note added. After submitting the paper we have been informed that the possibility of having a vortex structure above the critical field Hc has also been discussed by Skalozub [ 8 ].
e
(40)
Thus an "observer" who can only measure electromagnetism would not discover that the vortex condensate is actually non-abelian. From the point of view of energy considerations presumably x = 1 is preferred [ 6 ], although this has not been proved. We can sum up by saying that the vacuum has very interesting properties in the model given by the lagrangian (3). To make more contact with physics these considerations should be extended to the proper electroweak theory, which implies that the backreaction of the W condensate (there will also be a Z condensate) on the Higgs field should be taken into account. If it turns out that the qualitative features of our results remain valid in this case ~2, it would pro~2 In any case it should be remembered that the usual electroweak vacuum becomes unstable for large magnetic fields, so something must happen.
References [ 1 ] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376; J. Ambjorn, R.J. Hughes and N.K. Nielsen, Ann. Phys. (NY) 150 (1983) 92. [2] J. Ambjorn, N.K. Nielsen and P. Olesen, preprint NBI-HE88-18, Nucl. Phys. B, to be published. [3] J. Ambjorn, N.K. Nielsen and P. Olesen, Nucl. Phys. B 152 (1979) 75. [4] E. Witten, Nucl. Phys. B 249 (1985) 557. [5] A.A. Abrikosov, Sov. Phys. JETP 5 (1957) 1174. [ 6 ] J. Ambjorn and P. Olesen, Nucl. Phys. B 170 (1980) 60, 265. [7] G. 't Hooft, Nucl. Phys. B 153 (1979) 141. [ 8 ] V.V. Skalozub, Soy. J. Nucl. Phys. 43 (1986) 665; 45 ( 1987 ) 1058.
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