Vortex zero modes, large flux limit and

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Published for SISSA by Springer

Received: August 14, 2014 Revised: September 24, 2014 Accepted: September 24, 2014

Published: October 17, 2014

Vortex zero modes, large ux limit and Ambjrn-Nielsen-Olesen magnetic instabilities

Stefano Bolognesi,a,b Chandrasekhar Chatterjee,b,a Sven Bjarke Gudnasonc and Kenichi Konishia,b

aDepartment of Physics, E. Fermi, University of Pisa,

Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy

bINFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy

cNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: In the large ux limit vortices become ux tubes with almost constant magnetic eld in the interior region. This occurs in the case of non-Abelian vortices as well, and the study of such congurations allows us to reveal a close relationship between vortex zero modes and the gyromagnetic instabilities of vector bosons in a strong background magnetic eld discovered by Nielsen, Olesen and Ambjrn. The BPS vortices are exactly at the onset of this instability, and the dimension of their moduli space is precisely reproduced in this way. We present a unifying picture in which, through the study of the linear spectrum of scalars, fermions and W bosons in the magnetic eld background, the expected number of translational, orientational, fermionic as well as semilocal zero modes is correctly reproduced in all cases.

Keywords: Spontaneous Symmetry Breaking, Solitons Monopoles and Instantons, Nonperturbative E ects, Electromagnetic Processes and Properties

ArXiv ePrint: 1408.1572

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP10(2014)101

Web End =10.1007/JHEP10(2014)101

JHEP10(2014)101

Contents

1 Introduction 1

2 The Abelian vortex 2

3 The non-Abelian vortex 7

4 Discussion 16

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1 Introduction

We discuss some aspects of Abelian and non-Abelian vortex zero modes in the large magnetic ux limit, and their relationship with the magnetic instabilities rst studied in a series of papers by Nielsen, Olesen and Ambjrn [13].

Our quest begins with the following observation. The non-Abelian vortex is a generalization of the ordinary Abrikosov-Nielsen-Olesen (ANO) vortex that carries non-Abelian magnetic ux and supports internal orientational zero modes [47]. Basically it can be thought of as an ANO vortex embedded in a certain color-avor corner, even though their moduli spaces and the dynamics of their uctuations are found to be remarkably rich. On the other hand, it has been known for a long time that a non-Abelian magnetic eld can trigger an instability in the presence of charged W -bosons that can become e ectively tachyonic [13]. So the natural question is if these instabilities occur in the core of the non-Abelian vortex at all, and how they are related to the orientational zero modes of the latter.

It turns out that a natural setup to answer these questions is that of vortices in the large magnetic ux limit [811]. In this limit, the prole functions simplify drastically, and the vortex becomes essentially a tube with constant magnetic eld in the interior region separated from the vacuum by a thin domain wall. This solution resembles most the case of a constant magnetic eld background, which is the common situation considered in the early works of the magnetic instabilities. We show that, for BPS vortices, no magnetic instability occurs. The magnetic eld in the vortex interior is equal to the critical magnetic eld and thus the e ective mass of the lowest W -boson states is zero. This equivalence suggests that these states are related to the internal orientational zero modes. The counting of the number of zero modes, discussed below, conrms this conjecture.

It will be shown that a generic interpretation holds for vortex zero modes in the large ux limit. They can be interpreted as charged elds (scalars, fermions or vector bosons) trapped inside the vortex in the lowest Landau level. The mechanism behind the generation of the zero modes is the cancellation between di erent contributions to the energy squared: the term from the lowest Landau level, the gyromagnetic term (this one is present only for

vector bosons and fermions), and the bare mass squared. This analysis is applicable to all zero modes: translational, orientational, fermionic and semi-local.

The paper is organized as follows. In section 2, we review the large ux limit of Abelian vortices and compute the translational zero modes. We also show the existence of a domain wall separating the two phases. A related analysis of hole-vortex congurations nicely illustrates the relation between certain scalar zero modes in the linearized approximation and the exact translational zero modes of BPS vortices. In section 3, we discuss the non-Abelian vortex, its large ux limit, and analyze all types of vortex zero modes, gauge boson, scalar and fermion modes. It is shown that, on the one hand, they arise with exactly the same mechanism as in the onset of general Ambjrn-Nielsen-Olesen instabilities, and that, on the other, their total number coincides in all cases studied, with the known dimension of the BPS non-Abelian vortices or with the known index theorem. In section 4 we discuss the signicance of our results, and argue why the subtle relations found here between two seemingly unrelated phenomena of Ambjrn-Nielsen-Olesen instabilities and non-Abelian vortices, are nontrivial and interesting. As an example of implications of our analysis, we make a remark on some physics interpretation of Ambjrn and Olesen [2, 3].

2 The Abelian vortex

We rst review the large ux limit of Abelian vortices [8, 10]. We consider the Abelian-Higgs model

L =

with the following potential

2 ([notdef]q[notdef]2 )2 , (2.2)

whose minimum [notdef]q[notdef] = p [negationslash]= 0 is in the Higgs phase. The choice of = 1 corresponds to

having a BPS potential.

The elds for an axially symmetric vortex of charge n can be parametrized by the following Ansatz

q =

pein q(r) , (2.3)

A = n

er A(r) .

The prole functions q(r) and A(r) are subject to the boundary conditions q(0) = 0, q(1) = p and A(0) = 0, A(1) = 1. The claim of [8, 9] is that, for every Higgs-like

potential V , in the large-n limit the prole for the scalar eld converges to a step function

lim

n!1

q(r) = H(r Rbag) , (2.4)

where H is the Heaviside step function and the vortex radius, Rbag, will be determined shortly. The gauge eld prole converges to the following limit

lim

n!1

[braceleftBigg]

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14F[notdef] F [notdef] + [notdef](@[notdef] ieA[notdef])q[notdef]2 V ([notdef]q[notdef]) , (2.1)

V = 2e2

A(r) =

R2bag r Rbag ,

1 r > Rbag .

(2.5)

The magnetic eld is zero outside the bag and constant inside

, B[notdef]r>Rbag = 0 . (2.6)

The total magnetic ux is xed by the boundary condition

B =

[contintegraldisplay]

A = 2n

e . (2.7)

This conjecture has been shown to hold numerically with great precision in [10]. The step function of the prole q(r) reveals the presence of a substructure: a domain wall interpolating between the Coulomb phase q = 0 and the Higgs phase [notdef]q[notdef] = p. This wall

has a physical thickness which is an O(1/pn) e ect with respect to the bag radius.

The radius of the bag is determined by minimization of the tension. The tension has two contributions, one from the magnetic eld and one from the potential energy at q = 0,i.e. inside the bag

T (R) = 2n2

e2R2 +

2e22R2

2 , (2.8)

and its minimization gives

R2bag = 2n

e2 . (2.9)

The tension of the vortex is then

Tbag = 2n . (2.10)

The value of the magnetic eld B inside the bag is

B = e . (2.11)

Note that B is independent of n.

We now want to study the spectrum of uctuations around this solution. The phase outside the vortex is gapped, with the photon mass ep2 and the scalar mass ep2. The phase inside is more interesting. Here we have a massless gauge eld with a background constant magnetic eld (2.11) which is coupled to a charged scalar eld q. To compute the mass of the eld we have to expand around the tip of the potential:

V = 2e22

2 2e2[notdef]q[notdef]2 +

2e2

2 [notdef]q[notdef]4 . (2.12)

This is a tachyon with negative mass squared

m2 = 2e2 . (2.13)

The quartic term can be neglected in the limit of small uctuations

p . (2.14)

B[notdef]r Rbag =

2n eR2bag

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Tachyons are in general a signal of instabilities, but here we also have to take into account the e ect of the background magnetic eld before jumping to conclusions.

Inside the bag we choose the symmetric gauge for the gauge eld, viz. Ak =

(By/2, +Bx/2). The scalar eld equation, in the limit of small uctuations (2.14)),

is the linear equation

@2t (@x ieAx)2 (@y ieAy)2 + m2

Substituting

q = eiEt(x, y) , (2.16)

the energy-squared operator is then given by

E2 = (@x ieAx)2 (@y ieAy)2 + m2

, (2.18)

with the operators a = z/2 + 2@z and a = z/2 2@z satisfying the commutation relation

[a, a] = 2. The eigenstates are then

n1,n2 = an2(zn1e[notdef]z[notdef]2/4) . (2.19)

The energy spectrum is then

E2n1,n2 = (2n2 + 1)eB + m2 . (2.20)

The ground state, which is the lowest Landau level, has the energy E0 = peB + m2. If the scalar eld is allowed to have a tachyonic mass, then the ground state becomes massless at the critical value m2 = eB. Below this point, two zeros of (2.20) disappear

in the complex plane and the eld becomes really tachyonic. This situation is precisely realized for vortices with large ux. Using (2.11) and (2.20), the energy of the ground state is

E0 = e

p (1 ) . (2.21) Thus the spectrum is gapped for < 1, massless for = 1 and tachyonic for > 1. This result has a nice physical interpretation. Type I vortices, < 1, are known to attract each other and this is manifested, in the large n limit, by the stability of the spectrum. For the type II vortices, > 1, there is repulsion between the vortices and this is manifested in the tachyonic instability of the multi-vortex. We may then want to interpret the massless state for = 1 as the zero modes of BPS vortices.

q = 0 . (2.15)

. (2.17)

The operator on the right-hand-side is the same as that of the non-relativistic Landau level problem, and the same technique can be used for its diagonalization. Changing to complex coordinates: z = peB (x + iy) and z = peB (x iy), the spectrum operator can

be rewritten as

E2 = eB(aa + 1) + m2

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We check that the number of zero modes is correctly reproduced. For BPS vortex we have 2n zero modes corresponding to translations in the transverse plane. The ground state Landau level, n2 = 0, is not isolated, but come with a degeneracy proportional to the area. The states (2.19) are concentric rings localized at radius Rn1 [similarequal]

p2n1/eB so the density of zero modes per unit of area is eB/2. The number of zero modes in the area spanned by the bag is then

#zero modes = R2bag eB

2 = n . (2.22)

It is thus natural to associate them with the n translational zero modes of the BPS equations. Note that for a BPS vortex of winding number n, the dimension of its moduli space can be found conveniently by going to the limit of far-distant n minimal vortices, whose translational moduli are simply given by Cn. This approach has basically neglected the back reaction of the zero modes on the gauge elds and on themselves via the quartic interaction. The approximation is thus valid in the linear approximation of small uctuations (2.14).

In the large-n limit the radius of the vortex (2.9) goes to innity while the magnetic eld in the interior region (2.11) remains xed. This suggests that the domain wall separating the two phases should exists also in isolation, and as a proper wall it should be translational invariant in one direction. We will now show that indeed this object exists in isolation for the BPS theory.

The Bogomolnyi completion of the static energy density is

H =

. (2.23)

We take the domain wall to be extended in the y direction. Furthermore, we choose to work in the analogue of the vortex singular gauge (the singularity for the wall is pushed to y ! [notdef]1), thus the scalar eld is a function of x only with no winding and we can set

Ax = 0. Writing down the BPS equations for (x) and Ay(x) we have

A[prime]y + e(2 ) = 0 ,

[prime] + eAy = 0 . (2.24)

Solving for Ay and plugging the second BPS equation into the rst, we get

(log )[prime][prime] = e2(2 ) . (2.25)

In the rst row of gure 1 are shown two numerical solutions to this equation. There are two domain walls separating the Higgs phase from a Coulomb phase with constant magnetic eld. Note that the two walls are both solutions to the same BPS equation, they are related by parity and charge conjugation. Solutions with arbitrary separation between the two walls are also possible and are displayed in the second row of gure 1.

Let us consider a nal conguration, which claries the relation between the domain wall solutions of gure 1 and the linear zero modes previously discussed. We consider a

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Fxy + e([notdef][notdef]2 ) 2 + [notdef](Dx + iDy)[notdef]2 + e Fxy i"ij@i [bracketleftBig]

(Dj)

[bracketrightBig]

1.0

0.8

0.6

0.4

0.2

-10 -5 5 10

1.0

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0.8

0.6

0.4

0.2

-10 -5 5 10

[Minus]10 [Minus]5 5 10

Figure 1. Top row: the two domain wall solutions of eq. (2.25) with e = = 1. Bottom row: a one-parameter family of solutions with two walls at various distances: (left) the solutions and (right) the corresponding magnetic elds Fxy.

= 1 B = 0

Rbag

= 0

B = 1

= 0

B = 1

= 1

Rhole

B = 0

Figure 2. Vortex bag (left) compared with the hole-vortex (right).

1.0

0.8

0.6

0.4

0.2

2 4 6 8 10 12

Figure 3. Examples of a one-parameter family of solutions for the hole-vortex of eq. (2.28): (left) the solutions and (right) the corresponding magnetic elds Fxy.

hole-vortex, which is a region of zero magnetic eld in a background of constant magnetic eld (see gure 2). The axial-symmetric Ansatz is

A = er2

1r f(r) , Ar = 0 , = (r) , (2.26)

with boundary conditions (1) = 0 and [prime](0) = 0 for the Higgs eld. The value of

(0) is left to be arbitrary. Note one di erence between the vortex and the hole-vortex. The missing ux inside the hole vortex, which is 2

[integraltext]

0 drf[prime] and is related to (0), is a continuous parameter: it is not quantized. Inserting the Ansatz into the BPS equations we obtain

f[prime]

r + e2 = 0 ,

[prime] + e

er2

f r

= 0 . (2.27)

From this we obtain a second-order equation for :

1r (r(log )[prime])[prime] = e2(2 ) . (2.28)

Both from analytic inspection of the equation, and from the shape of the numerical solutions, we can detect two di erent regimes. When (0) is very small, the 2 term on the right-hand side of eq. (2.28) is negligible and the solution is thus

[similarequal] ee2r2/4 ; Fxy = e O(2) : (2.29)

this is exactly the rst Landau level (2.19) with n1 = n2 = 0. The magnetic eld does not receive any correction to linear order in . When (0) [similarequal] p the hole-vortex is well approx

imated by a ring of domain wall as equation (2.28) becomes almost equivalent to (2.25). Examples are shown in gure 3.1

3 The non-Abelian vortex

For a generic particle with spin S and gyromagnetic ratio gS the spectrum in a constant magnetic eld is:

E2n1, = (2n1 + 1)eB + gSe

[vector]B [notdef]

+ m2 , (3.1)

where gSe [vector]B [notdef]

is the Zeeman term. For Dirac fermions we have S = 1/2 and gS = 2 and

the spectrum is

E2n1,"# = (2n1 + 1)eB [notdef] eB + m2F . (3.2)

1In contrast to the vortex (the left of gure 2), the hole-vortex (the right gure) does not represent a minimum-tension conguration, as it stands. For its stability, it is necessary to consider the external region with B [negationslash]= 0 as a part of a vortex with a xed quantized total ux. This makes perfect sense, as the

tiny hole-vortex (2.29) can then be thought of as a germ of the instability of the vortex itself, occurring anywhere inside the vortex

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The Zeeman term, for the right choice of spin orientation, cancels exactly the rst Landau level term. This is the reason for the existence of fermionic zero modes, whenever mF = 0.

The generalization for charged spin-1 W bosons will be of interest in the rest of this section.

We now consider the theory of non-Abelian vortices [4]. Stripped to its basic constituents, the model consists of a U(N) gauge theory coupled to N avors of fundamental quarks

L =

2TrN(F[notdef] F [notdef] ) + TrN(D[notdef]q)(D[notdef]q)

qq 1N[notdef]N[parenrightBig]2. (3.3)

with D[notdef] = @[notdef] igA[notdef]. For the moment we consider the case of equal couplings for the

U(1)- and SU(N)-part of the gauge group. The vacuum is the color-avor locked phase

q =

...

2g2

4 TrN

. (3.4)

The color-avor diagonal U(N) symmetry is unbroken by the vacuum. The mass of the gauge bosons in the vacuum is M2 = g2, and this is true for all the generators of the U(N) gauge group.

The SU(N)[notdef]U(1) gauge symmetry is completely broken and, as 1(SU(N)[notdef]U(1)) = Z,

the system supports vortices. To build a vortex conguration we embed the ordinary Abelian U(1) vortex in this theory. A minimum individual vortex conguration breaks the residual symmetry to SU(N 1) [notdef] U(1) SU(N) and the vortex acquires orientational

zeromodes of the coset CP N1. A possible Ansatz for a multi-vortex of charge n is

q =

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ein pq(r)

...

[epsilon1]klnrlA(r)/gr

...

Ak =

. (3.5)

This corresponds to having n non-Abelian vortices in the same spatial position and in the same internal orientation. It is only a special point in the big moduli space of n non-Abelian vortices, but for the moment is the one we shall focus on. Since this is just an embedding of the ANO axial-symmetric vortex (2.3), the same considerations about the large-n limit discussed in the previous section hold (this is valid only in the case of equal couplings for U(1) and SU(N)). In particular the large-n limit of the prole functions is (2.4) and (2.5). So we may use all the formulae of the previous section by replacing A[notdef] ! p2A[notdef] and e with g/p2 to account for the di erent normalization of the generators.

We are interested in the spectrum around this multi-vortex which is sketched in the left of gure 4. Outside the bag radius the scalar elds take the form of eq. (3.4) and all

...

q =

0 B

1 C

Rbag1

Rbag

Rbag2

...

q =

0 B

1 C

q =

0 B

1 C

q =

0 B

...

1 C

...

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Figure 4. Two possible congurations of large-n multi-vortices.

the states, gluons and scalars, are massive. Inside the bag, however, the scalar quarks are

q =

. (3.6)

...

The radius of the bag is given by

R2bag = 4n

g2 , (3.7)

and the value of the B eld is constant inside the bag and given by

Fxy =

g/2

...

. (3.8)

The eld q11 has a negative mass squared, and its spectrum is the same as in the Abelian case. In particular, for = 1 this eld gives n complex zero modes to be associated with the translational zero modes of the vortex. All the elds in the reduced sector (N 1)2

are massive, as they are in the vacuum state.

The interesting thing happens for the N 1, W bosons in the following matrix com

ponents of the gauge elds

. . .

...

. (3.9)

These are charged particles and thus they couple to the magnetic eld inside the vortex. We are interested in computing the spectrum for those. We can consider the problem of N = 2 where we have to deal with one W boson only. So we denote

A[notdef] = A[notdef] W[notdef] W [notdef] B[notdef]

[parenrightBigg]

, q = q11 q12 q21 q22

[parenrightBigg]

. (3.10)

The terms in the Lagrangian which contributes to the W mass are

g2[notdef]W[notdef][notdef]2([notdef]q11[notdef]2 + [notdef]q22[notdef]2) = 8

g2[notdef]W[notdef][notdef]2 r Rbag ,

2g2[notdef]W[notdef][notdef]2 r > Rbag .

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(3.11)

The W boson is massive everywhere, but the mass squared inside the bag is reduced by half since only q22 contributes to the mass term: this fact will be very important below.

The Lagrangian, reduced to the W -boson sector, is

L =

2F[notdef] F [notdef] [notdef]D[notdef]W D W[notdef][notdef]2

2 i g F[notdef] W [notdef] W + 2 m2W [notdef]W[notdef][notdef]2 + O(g2W 4) , (3.12)

where

2 . (3.13)

The quartic term can be neglected for small uctuations

p . (3.14)

The linear equation for the W boson, in the gauge D[notdef]W [notdef] = 0, is

(DD + m2W ) [notdef] 2igF[notdef] [parenrightBig]

W = 0 . (3.15)

It is important to note that the W boson is not minimally coupled to the gauge eld A[notdef].

For minimally coupled elds the gyromagnetic factor is gS = 1/S while for the W boson, gS = 2 and not 1. This is due to the last term in (3.15). We consider the magnetic eld directed in the third direction F12 = B.

The solution we are mainly interested in is given by the following

W[notdef] = eiEt

m2W = g2

0 w(x, y) iw(x, y) 0

, (3.16)

which is the negative eigenstate of the spin in the magnetic eld direction

(S3)[notdef] W = W[notdef] .

For these states the spectrum is

E2w = (@x igAx)2 (@y igAy)2 2gB + m2W

and the gauge xing condition becomes

D[notdef]W [notdef] = eiEt

@x + i@y + gB2 (x + iy)

w , (3.17)

w = 0 . (3.18)

The solution is then given by the lowest Landau level states

w = f(z)e[notdef]z[notdef]2/4 , (3.19)

with f(z) any holomorphic function, and the spectrum for those states is

E2 = gB + m2W . (3.20)

For a generic state, with eigenvalue of S3 which can be [epsilon1] = [notdef]1, 0 and Landau level n1, the

spectrum is

E2n1,[epsilon1] = (2n1 + 1)gB + 2[epsilon1]gB + m2W . (3.21) Note that the non-minimal coupling of the W boson is responsible for the anomalous gyromagnetic factor gs = 2. Now the Zeeman term is twice the rst Landau level term, and so the ground state energy is En,1 =

qgB + m2W . This is somehow similar to the scalar eld story of section 2, except for the fact that the critical value for the existence of zero modes is now a positive mass squared, m2W = B, and not a negative one. For m2W < B we have an instability; the ground state becoming tachyonic is the signal of a phase transition which can be driven by the W condensate. For pure Yang-Mills (i.e.

mW = 0) the ground state is always tachyonic for any B [negationslash]= 0. This is the instability

discussed by Nielsen, Olesen and Ambjrn [13].

The ground state energy for the W bosons, taking into account the value of the magnetic eld (3.8) and the mass inside the bag (3.11), is given by

E0 = g

r2p1 . (3.22)

This is sub-critical for < 1 (type I vortices) and above-critical for > 1 (type II vortices). This nicely ts with the expectation that for type I the ground state is given by vortices all in the same orientation state while for type II this is an unstable point. For the BPS case = 1 we have exactly B = Bcr. The number of zero modes, including the scalars (2.22), n, and the W bosons ((N 1)n), in total is

#zero modes = NR2bag gB

2 = Nn , (3.23)

which is in agreement with the dimension of the moduli space of the winding number n, BPS non-Abelian vortices (i.e. the number of the zero modes). Indeed, even though the structure of the moduli space of higher-winding non-Abelian vortices is quite rich and has

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been studied only for some simplest cases [1317], its dimension is known from the index theorem [5, 16]. Alternatively it can be deduced from the limiting case where the n minimal vortices are well separated. The moduli space approaches in that limit the form [18]

C [notdef] CPN1

n /Sn , (3.24)

where Sn is a permutation of n vortices. Its dimension is given by

n(N 1 + 1) = Nn . (3.25)

The determination of the number of zero modes (3.23) was made by studying the properties of the uctuation of the particular vortex solution (3.5). Around that point the structure of the vortex moduli space is certainly more complicated than (3.24), but since the dimension of a manifold is the same at any point, the agreement between (3.23) and (3.25) shows that the zero modes related to the orientational and translational zero modes of the BPS non-Abelian vortices have indeed the same origin as the zero (or negative) modes which trigger the Ambjrn-Nielsen-Olesen instabilities.

In a supersymmetric extension of our model, the fermions get mass through the Yukawa term,

LYukawa = p2g qA A + h.c. , (3.26)

where are gauge fermions in the adjoint representation of the color gauge group, SU(N)[notdef]

U(1) and A = 1, 2, . . . , Nf = N is the avor index. The scalar VEV (3.6) inside the vortex implies that the nonvanishing Dirac mass terms are

p2g

XA=2

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Xi=1( )iA Ai + h.c. ; (3.27)

note that the fermions ( )11 and 1i, i = 1, 2, . . . , N do not appear; they can be thought of as N massless Dirac fermions. We see from eq. (3.2) that the number of the fermionic zero modes is then Nn, as expected.

As a further nontrivial check, we consider another multi-vortex conguration, i.e. the one sketched on the right of gure 4. It consists of two multi-vortices, one with radius Rbag 1

and the other with radius Rbag 2, in mutually orthogonal internal orientations. Hence, in the theory (3.3) which has equal couplings for the U(1) and the SU(N) parts, they can overlap with no modication of their prole functions. We take the two vortices to have respectively n1 and n2 units of ux, so we expect to recover a total of N(n1 + n2) complex zero modes. We take the second vortex to be completely immersed in the other one, as in gure 4, with n1 > n2 and

R2bag 1 = 2n1

g2 , R2bag 2 =

2n2g2 . (3.28)

We now consider only the case = 1. In the ring between Rbag 2 and Rbag 1 the scalar eld

and magnetic eld are the same of the previous example, (3.6) and (3.8), and the counting of zero modes is unchanged. We have one mode from the scalar eld

#zero modes ring = N R2bag 1 R2bag 2

[parenrightbig]

2 = N(n1 n2) . (3.29)

In the internal disk we have instead the following elds

q =

, Fxy =

g/2

...

. (3.30)

The zero modes are 4 scalars and 2(N 2) W bosons in the following components

q =

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, W =

. . . . . .

... ...

, (3.31)

so a total of 2N. The number of zero modes in the internal disk is then

#zero modes disk = 2NRbag 2 gB

2 = 2Nn2 . (3.32)

The sum of the disk and the ring gives indeed the correct answer, Nn.

Yet another check is provided by studying the U(N) theory with the number of fundamental scalars Nf larger than N. The scalar potential is of the form,

V = g2

4 TrN

qq 1N[notdef]N[parenrightBig]2, (3.33)

as a natural extension of (3.3), where q now is an N [notdef] Nf matrix. Inside the vortex bag,

the scalar elds take the form,

q =

0 0 . . .

p ...

... ...p 0 . . .

. (3.34)

Expansion of the potential V around such values of q determines the masses of the scalar elds inside the vortex. It is obvious that the negative mass squared terms can only arise from the part

g2 4

XA=1qA1qA1 1

qA1qA1 + . . . (3.35)

2, as all other terms contain positive coe cients.

However, the terms A = 2, . . . , N in (3.35) are exactly canceled by terms arising from the

in the (11) element of qq 1N[notdef]N

product of nondiagonal elements

g2 4

XA,B

Xj=2

hqA1qAjqBj qB1 + (1 $ j)[bracketrightBig] !

g24N

XA=2

hqA1(

p + qAA)(

p + qAA)qA1 + . . .

= g2 2

XA=2qA1q1A + . . . . (3.36)

so that the tachyonic scalars, with mass squared g

22 are q11 and qA1, A = N + 1, . . . , Nf.

According to the discussion at the beginning of this section, taking into account the magnetic eld inside the bag, g

2 , (we consider the BPS case, = 1) the number of the scalar zero modes is then 1 + Nf N. Adding the vector zero modes which are unchanged:

(N 1), one nds a total of Nf, or by taking into account the Landau level degeneracy:

Nf n zero modes.

BPS non-Abelian vortices for Nf > N are semilocal vortices: the moduli contain the vortex transverse size moduli, and their structure is very rich and interesting [5, 12, 19] (see for instance [19] for a new, Seiberg-like duality in pairs of systems of di erent (Nf, N)s

having closely related moduli spaces). In any event, the dimension of the moduli space can be deduced very generally e.g., from an index theorem or from the symplectic quotient construction of the moduli space [5, 19]:

{Z, , ~ [notdef]D = 0[notdef]/U(n) ; D = [Z, Z] +

JHEP10(2014)101

~ , (3.37)

are n[notdef]n, N [notdef]n, and n[notdef](Nf N) matrices, respectively. Its (complex)

dimension is therefore given by

n2 + nN + n(Nf N) n2 = nNf, (3.38) in agreement with the zero-mode counting.

Our last example of nontrivial checks refers to the cases with di erent coupling constants for the Abelian and non-Abelian gauge group factors. From now on we focus on the case with N = 2. The U(2) gauge eld can be decomposed as

A[notdef] = a[notdef]

2 1 +

Aa[notdef]

2 a , (3.39)

and the covariant derivative is

D[notdef] = @[notdef] i

ea[notdef]

2 1 i

gAa[notdef]

where Z, and ~

2 a . (3.40)

The choice of di erent couplings is very natural, especially if one considers the fact that g has a quantum mechanical running distinct from that of the Abelian one, e, and can be tuned to be equal to the latter only at a specic energy scale. This case was considered in the very rst paper [4].

The BPS Lagrangian for arbitrary e and g is

L =

1 4f[notdef] f[notdef]

14F a[notdef] F [notdef] a + Tr (D[notdef]q)(D[notdef]q)

8 [notdef]q[notdef]2 2

g2 8

qaq

2, (3.41)

1.0

fxy

F3xy

0.8

0.6

0.4

0.2

[Minus]5 5

Figure 5. Domain wall solution of the equations (3.46) for di erent couplings. The gure refers to the values e = 1 and = 2.

where [notdef]q[notdef]2 = Tr (qq). The BPS equations are

fxy + e2 [notdef]q[notdef]2 2

[parenrightbig]

= 0 ;

F a[notdef] + g2Tr qaq = 0 ;(Dx + iDy)q = 0 . (3.42)

We derive things in a di erent order than we did before. First we search the stable vacuum which would then correspond to the interior phase of the multi-vortex. A solution of the BPS equations is the following magnetic phase

q = 0 e [radicalBig]

2 e2+g2

JHEP10(2014)101

[parenrightBigg]

, fxy = eg2

e2 + g2 , F 3xy =

e2g

e2 + g2 . (3.43)

This is the internal phase of the non-Abelian vortex for generic couplings. For e = g this reduces to (3.6) and (3.8).

We construct the domain wall between the Higgs phase and the magnetic phase using the following Ansatz

ay(x) , A3y(x) , q = q1(x) q2(x) [parenrightBigg]

, (3.44)

and the BPS equations become

ay[prime] + e

2 q21 + q22 2

[parenrightbig]

= 0 ,

A3y[prime] + g2 q21 q22

[parenrightbig]

= 0 ,

q1[prime] +

e2ay +g2A3y

[parenrightBig]

q1 = 0 ,

q2[prime] +

e2ay

g2A3y

[parenrightBig]

q2 = 0 . (3.45)

These then reduce to the following two coupled second-order equations:

(log q1)[prime][prime] = e24 (1 + )q21 + (1 )q22 2

[parenrightbig]

;

(log q2)[prime][prime] = e2

4 (1 )q21 + (1 + )q22 2

[parenrightbig]

, (3.46)

where = g2/e2. The magnetic elds are related to the scalar elds by

fxy = e2 2 q21 q22

[parenrightbig]

, F 3xy = g2 q22 q21

[parenrightbig]

. (3.47)

A domain wall solution interpolating between the Higgs and magnetic phases is given by the numerical solution in gure 5 for the case = 2. The case of equal couplings = 1 is simpler because q2 = p and fxy = F 3xy. Another simplication occurs in the non-Abelian strong coupling limit ! 1 for which it can be seen that a solution is given by q1 = q2 and F 3xy = 0.

The multi-vortex is an area of the magnetic phase (3.43) separated from the Higgs phase by the previously found domain wall. The W bosons, when expanded around the vacuum of the magnetic phase, have the mass squared

m2W = g2e2

e2 + g2 , (3.48)

which is the generalization of (3.13) to unequal U(1) and SU(N) gauge couplings. Given the non-Abelian magnetic eld F 3xy in (3.43), this is exactly the value for the lowest level to be marginal.

As for the scalars, expansion of the scalar potentials in eq. (3.41) around the value of q in eq. (3.43) gives the quadratic terms

e2g2

e2 + g2 q11(q11) +

JHEP10(2014)101

4 (q22 + (q22) )2 . (3.49)

The only tachyonic scalar is q11. Now by making the replacement

eB !

efxy + gF 3xy

e2 + g2 , (3.50)

in eq. (2.15) and eq. (2.20), where we used the values of the magnetic elds (3.43), one gets for the spectrum of q11

En1,n2 = e2g2

e2 + g2 (2 n2 + 1) + m2 : (3.51)

we see that the negative mass squared m2 = e

2g2

e2+g2 in (3.49) is precisely the value which

gives the zero energy modes.

4 Discussion

A close relationship is thus found to exist between the general vortex zero modes and magnetic instabilities of the type discussed by Ambjrn, Nielsen and Olesen. The large ux limit, in which the vortex interior has an almost constant magnetic eld, is an ideal setup for disclosing such a connection. We used the W -boson gyromagnetic instability and similar ones for the scalar and fermion elds. The counting of zero modes obtained this

2 =

e2g2

way and the dimension of the known moduli spaces of BPS non-Abelian vortices match precisely in all cases and provides a unifying picture, valid for translational, orientational, fermionic or semilocal zero modes. This seems to be particularly remarkable in view of the fact that the way the Landau-level zero-point energy, the Zeeman term and the mass term add up to zero is di erent for various types of elds. We conclude that there is a universal mechanism for the generation of the vortex zero modes, which encompasses both the onset of Ambjrn, Nielsen, Olesen magnetic instabilities in the electroweak theory (or in QCD), and all sorts of vortex zero modes inherent in Abelian and non-Abelian vortices.

Let us clarify that the fact that our counting of the vortex zero modes coincides, in the case of BPS vortices, with the known dimension of the vortex moduli space as well as with the known index theorem, just shows that our analysis is correct and consistent. Even though it is quite nontrivial to show how things work out, leading to such a consistent picture, this is not the main purpose of our analysis.

Most importantly, it was shown for the rst time that the BPS vortex conguration reduces, in the large winding limit, precisely to the critical situation envisaged by Olesen and Ambjrn, which corresponds to the onset of magnetic instabilities of the broken phase of e.g., standard Weinberg-Salam theory.

This was quite unexpected and surprising, as the magnetic instability analyses in [13] were made in the partially broken phase of e.g., SU(2)L [notdef] UY (1) theory with unbroken

UEM(1) gauge group. The authors of [13] then considered some external magnetic source which produces a strong external magnetic eld of the unbroken UEM(1). This is quite in contrast to the standard setting of non-Abelian vortices, where one considers the vacuum in a fully Higgsed phase, i.e., with no massless gauge bosons in the bulk. The orientational zero modes arise in the latter due to the presence of the global color-avor diagonal symmetry (absent in systems considered in [13]), broken by individual vortex solutions. Therefore the two classes of systems look quite distinct and it would seem hardly possible to nd any contact between the two.

What was shown here is that actually the two seemingly unrelated physics phenomena, the Nielsen-Olesen-Ambjrn magnetic instabilities and non-Abelian vortices, are deeply related by the universal mechanism of charged zero modes in the presence of magnetic elds. To prove such a connection, the consideration of the large winding limit of the latter turned out to be particularly useful.

Such a close connection found here then brings us to comment on some physics interpretation emphasized by Ambjrn and Olesen. In a somewhat unrealistic BPS saturated version of electroweak theory, with = g2

8 sin2 , where is the Weinberg angle and is the quartic Higgs coupling, these authors nd the rst order (BPS) equation [2] ,

f12 = g2 sin 20 + 2 sin [notdef]w[notdef]2, (4.1)

and an analogous equation for Z12, where f12 is the UEM(1) magnetic eld, 0 is the Higgs VEV. The second term on the right-hand side is then interpreted as an antiscreening e ect, where the condensate of the W bosons tends to increase the applied magnetic eld f12, in contrast to what happens in the ANO vortex (where the scalar condensate tends

JHEP10(2014)101

to diminish the magnetic eld - screening e ect, or Lenzs law). It is then natural to ask2 whether the non-Abelian vortices show screening or antiscreening e ect.

As a non-Abelian vortex is in a sense simply an ANO vortex embedded in a particular color-avor corner, the standard screening e ect is certainly there. As for the antiscreening e ect, eq. (4.1), a color-avor rotation (orientational zero modes) is accompanied by the excitation of the W [notdef] boson components of the vortex conguration, see eqs. (3.9)(3.21), in exactly the same mechanism that brings us to eq. (4.1). Therefore one might conclude that the non-Abelian vortices possess both screening (scalar condensates) and anti-screening e ect (W boson condensates).

These considerations, at the same time, lead us to an alternative interpretation of the second term of eq. (4.1). Namely, the fact that the W bosons become massless at the critical magnetic eld due to the Zeeman e ect means that the SUL(2) symmetry is (at least locally around the vortex) restored. Now the electromagnetic gauge eld is

A[notdef] = sin W 3[notdef] + cos B[notdef], (4.2)

where W (a)[notdef] and B[notdef] stand for SUL(2) and UY (1) gauge bosons, respectively. In the broken SUL(2) phase, the UEM(1) magnetic eld is then

f12 = sin (@1W 32 @2W 31) + cos (@1B2 @2B1) , (4.3) whereas in the unbroken phase the SUL(2) eld tensor is given by

W 312 = @1W 32 @2W 31 + [epsilon1]3abW a1W b2 = @1W 32 @2W 31 2 [notdef]w[notdef]2, (4.4)

where the form of the condensate eq. (3.16) for

W = 1p2(W 1 iW 2) (4.5)

has been used. At this point it is quite clear that eq. (4.1) simply signals the fact that the equation of motion is being satised by f12, in which the Abelian tensor @1W 32 @2W 31

is replaced by a non-Abelian SUL(2) tensor, @1W 32 @2W 31 + [epsilon1]3abW a1W b2. The analogous

term on the Z12 equation can also be understood as the restoration of non-Abelian nature of SUL(2) elds.

Such a reinterpretation is very much in line with the result of Ambjrn and Olesen [3] that the magnetic instability and vortex formation at the critical UEM(1) magnetic eld is actually nothing but the onset of phase transition to the unbroken SUL(2) [notdef] UY (1)

symmetric phase of the electroweak theory.

Acknowledgments

The authors thank Poul Olesen for raising interesting questions about the non-Abelian vortices and the Nielsen-Olesen-Ambjrn magnetic instabilities, which triggered the present

2We thank Poul Olesen for raising this question to us (private communication).

JHEP10(2014)101

investigation. We thank Jarah Evslin for useful discussions. The work of S.B. is funded by the Grant Rientro dei Cervelli Rita Levi Montalcini of the Italian government. S.B. wishes to thank E. Rabinovici for discussions on magnetic instabilities. S.B.G. thanks Institute of Modern Physics, Lanzhou, China, for hospitality. The present research is supported by the INFN special project GAST (Gauge and String Theories).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Abstract

In the large flux limit vortices become flux tubes with almost constant magnetic field in the interior region. This occurs in the case of non-Abelian vortices as well, and the study of such configurations allows us to reveal a close relationship between vortex zero modes and the gyromagnetic instabilities of vector bosons in a strong background magnetic field discovered by Nielsen, Olesen and Ambjørn. The BPS vortices are exactly at the onset of this instability, and the dimension of their moduli space is precisely reproduced in this way. We present a unifying picture in which, through the study of the linear spectrum of scalars, fermions and W bosons in the magnetic field background, the expected number of translational, orientational, fermionic as well as semilocal zero modes is correctly reproduced in all cases.

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Title

Vortex zero modes, large flux limit and Ambjørn-Nielsen-Olesen magnetic instabilities

Author

Bolognesi, Stefano; Chatterjee, Chandrasekhar; Gudnason, Sven Bjarke; Konishi, Kenichi

Pages

1-21

Publication year

2014

Publication date

Oct 2014

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP10(2014)101

ProQuest document ID

1708020318

SISSA, Trieste, Italy 2014

Vortex zero modes, large flux limit and Ambjørn-Nielsen-Olesen magnetic instabilities

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