Revisiting critical vortices in three-dimensional supersymmetric QED

S. Olmez1 and M. Shifman2,3

1Department of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA2William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA

3Laboratoire de Physique Theorique* Universite de Paris-Sud XI Batiment 210, F-91405 Orsay Cedex, France(Received 25 August 2008; published 18 December 2008)

We consider renormalization of the central charge and the mass of theN ¼ 2 supersymmetric Abelian

vortices in 2þ 1 dimensions. We obtain N ¼ 2 supersymmetric theory in 2þ 1 dimensions by dimen-

sionally reducing the N ¼ 1 supersymmetric QED in 3þ 1 dimensions with two chiral fields carrying

opposite charges. Then we introduce a mass for one of the matter multiplets without breaking N ¼ 2

supersymmetry. This massive multiplet is viewed as a regulator in the large mass limit. We show that the

mass and the central charge of the vortex get the same nonvanishing quantum corrections, which preserves

Bogomol’nyi-Prasad-Sommerfield saturation at the quantum level. Comparison with the operator form of

the central extension exhibits fractionalization of a global U(1) charge; it becomes �1=2 for the minimal

vortex. The very fact of the mass and charge renormalization is due to a ‘‘reflection’’ of an unbalanced

number of the fermion and boson zero modes on the vortex in the regulator sector.

DOI: 10.1103/PhysRevD.78.125021 PACS numbers: 11.27.+d, 12.60.Jv

I. INTRODUCTION

N ¼ 2 supersymmetric QED with the Fayet-Iliopoulosterm in 2þ 1 dimensions supports Abrikosov-Nielsen-Olesen (ANO) vortices [1,2]. These classical solutionsare 1=2-BPS-saturated (two out of four supercharges areconserved). Quantum corrections to the vortex mass andcentral charge were discussed in the literature more thanonce. It is firmly established [3] that there are two fermionzero modes on the vortex implying that the supermultipletto which the vortex belongs is two-dimensional. This is ashort supermultiplet. Hence, the classical Bogomol’nyi-Prasad-Sommerfield (BPS) saturation cannot be lost inloops.

Particular implementation of the vortex BPS saturationturned out to be a contentious issue, almost to the sameextent as it had happened with two-dimensional kinks inN ¼ 1 models (for reviews see [4], Sec. 3.1 in [5], and[6]). The authors of [1,7] obtained a vanishing quantumcorrection to the vortex mass using the following eigen-value densities:

nBðwÞ � nFðwÞ / �ðwÞ; (1)

where nBðFÞ is the bosonic (fermionic) density of states.

The vanishing mass correction ensues since

�Mv /Z

dwðnBðwÞ � nFðwÞÞw ¼ 0: (2)

Since the vortex mass Mv is proportional to the Fayet-Iliopoulos (FI) parameter �, and � is renormalized in oneloop, the above result caused a problem.

Later new calculations of the vortex mass were under-taken and a nonvanishing one-loop correction to the vortexmass was reported in [8,9]. It was shown [3] that the centralcharge also gets a correction, so that the BPS saturation ofthe vortex persists at the one-loop level. However, the(dimensional) regularization that was used in the mostdetailed paper [3], expressly written to discuss three-dimensional supersymmetric vortices, does not allow oneto treat in a straightforward manner the Chern-Simons (CS)term, whose role in the problem at hand is important. Inthis paper we use another regularization method in whichthe CS term naturally appears in the limit of large regulatormass. This mass is also crucial in the operator form of thecentrally extended algebra which we derive at one loop.Our operator expression for the central extension includesthe Noether charge (20).Here we would like to close these gaps. In this paper we

revisit the issue using a physically motivated regularizationwhich is absolutely transparent. We recalculate the renor-malization of the vortex mass at one loop

Mv;R ¼ 2�

��R � m

4�

�(3)

and the one-loop effect in the central charge. (Here �R isthe renormalized value of the FI parameter, m is the matterfield mass,

m ¼ effiffiffiffiffiffiffiffi2�R

p;

and the subscript R stands for renormalized.) The aboveresult is in agreement with the previous calculations [3,8].Needless to say, our direct calculation confirms BPS satu-ration,Mv;R ¼ jZRj. Moreover, it demonstrates that, in the

limit of the large regulator mass, regulator’s role is taken*Unite Mixte de Recherche du CNRS (UMR 8627).

PHYSICAL REVIEW D 78, 125021 (2008)

1550-7998=2008=78(12)=125021(10) 125021-1 � 2008 The American Physical Society

over by the Chern-Simons term. A new finding obtained bycomparing the central charge calculation with the operatorform of the central extension is a U(1) global chargefractionalization. The operator expression for the centralextension which we derived in our regularization is pre-sented in Eqs. (19) and (20). Then we discuss the centralcharge/vortex mass renormalization to all orders in pertur-bation theory [see Eq. (55)].

N ¼ 2 supersymmetric QED (SQED) Lagrangian in2þ 1 dimensions (four supercharges) can be obtained bydimensional reduction of N ¼ 1 supersymmetricLagrangian in 3þ 1 dimensions. In order to have a well-defined anomaly-free SQED in four dimensions, one has to

have two matter superfields, say � and ~�, with the oppo-site charges. Since there is no chirality in three dimensions,in three-dimensional SQED, in principle, it is sufficient to

keep a single superfield (say, �), while ~� can be elimi-nated. This is a minimal setup which is routinely consid-ered. The four-dimensional anomaly is reflected in threedimensions in the form of a ‘‘parity anomaly’’ [10,11] andthe emergence of the Chern-Simons term, as will be ex-plained momentarily.

When we speak of eliminating ~� we should be careful.Eliminating does not mean discarding. As was brieflydiscussed in [5] (Sec. III B), a perfectly safe method of

getting rid of ~� is to make the tilded fields heavy. Then thecorresponding supermultiplet decouples and does not ap-pear in the low-energy theory. It leaves a trace, however, inthe form of the Chern-Simons term [10,11], as shown inSec. IV.

There is a well-known method of making the tildedfields heavy without altering the masses of the untildedfields. It works in three dimensions. One can introduce a‘‘real’’ mass ~m [12] (a three-dimensional analog of thetwisted mass in two dimensions [13]) without breakingN ¼ 2 supersymmetry of three-dimensional SQED. Thereal mass corresponds to a constant background vectorfield along the reduced direction.

When the masses of the tilded and untilded fields areequal, the renormalization of the FI term vanishes [14], andso do quantum corrections to the vortex mass. When wemake the tilded fields heavy, ~m � e

ffiffiffi�

p, effectively they

become physical regulators. As long as we keep their mass~m large but finite it acts as an ultraviolet cutoff in loopintegrals. All one-loop corrections, including the linearlydivergent part, become well-defined and perfectly trans-parent. We have a smooth transition as we eventually send~m to infinity.Our analysis is organized as follows. In Sec. II we

describe our basic model obtained from four-dimensionalSQED by reducing one of the spatial dimensions. Weintroduce the real mass ~m, to be treated as a free parameter,for the ‘‘second’’ chiral superfield. Section III, carrying themain weight of this work, is devoted to quantum correc-tions to the central charge and vortex mass. The operator

form of the central extension is discussed in detail in thissection. In Sec. IV we consider a global charge fractional-ization and a related question of Chern-Simons.

II. DESCRIPTION OF THE MODEL ANDCLASSICAL RESULTS

Our starting point isN ¼ 1 SQED in 3þ 1 dimensions

with two chiral matter superfields � and ~� and the Fayet-Iliopoulos term. It has four conserved supercharges. Thecorresponding Lagrangian is

L ¼�1

4e2

Zd2�W�W

� þ H:c:

�þZ

d4���eV�

þZ

d4� ~��e�V ~�� �Z

d2�d2�yVðx; �; �yÞ; (4)

where W� is the gauge field multiplet

W� ¼ 1

8�D2D�V ¼ �� � ��D� i��F�� þ i�2@� _��

y _�:

(5)

In order to get N ¼ 2 supersymmetry in 2þ 1 dimen-sions we compactify one of the dimensions, say the thirdaxis, keeping the zero Kaluza-Klein modes and discardingnonzero ones. To introduce the tilded field mass we intro-duce a constant background gauge field along the compac-tified axis Vbg where the subscript bg means background.

In terms of the components we have

Vbg ¼ �y�0��Vbg ; (6)

� matrices are defined in Eq. (31) below. The backgroundvector field is chosen to be a constant field along thecompactified axis, i.e. V

bg ¼ 2 ~m�

3 . It is important to

note that this is a new auxiliary field, rather than theexpectation value of the original photon field. This back-

ground is coupled to ~� only, with the charge�1. Then theLagrangian takes the form

L ¼�1

4e2

Zd2�W�W

� þ H:c:

�þZ

d4���eV�

þZ

d4� ~��e�V�Vbg ~�� �Z

d2�d2�yVðx; �; �yÞ:(7)

Upon introduction of the constant background field, ~�multiplet becomes massive whereas � multiplet is notaffected, since it is chosen to be neutral with respect tothe background field. It is clear that the kinetic term for thegauge multiplet is not affected, and similarly, the Fayet-Iliopoulos term remains the same since the superspaceintegral

Rd4�V does not vanish only for the last compo-

nent of the superfield V.After compactification of the third axis, we get the

following bosonic and fermionic Lagrangians in terms ofthe component fields (in the Wess-Zumino gauge):

S. OLMEZ AND M. SHIFMAN PHYSICAL REVIEW D 78, 125021 (2008)

125021-2

LB ¼ � 1

4e2FF

þD ~��D~�þD��D�

þ 1

2e2ð@NÞ2 þ 1

2e2D2 � �DþDð���� ~�� ~�Þ

� N2���� ð ~mþ NÞ2 ~�� ~�;

LF ¼ 1

e2��i@6 �þ �c iD6 c þ �~c iD6 ~c þ N �c c

� ð ~mþ NÞ �~c ~c þiffiffiffi2

p ½ð ��c�� � �c��Þ�� i

ffiffiffi2

p ½ð �� ~c Þ ~�� � ð �~c�Þ ~��;

(8)

where N ¼ �A3 is a real pseudoscalar field, and

iD� ¼ ði@ þ AÞ�; iD~� ¼ ði@ � AÞ ~�:

Moreover, D is an auxiliary field, which can be eliminatedvia its equation of motion. The Lagrangian (8) is invariantunder the following supersymmetry transformations:

�� ¼ ffiffiffi2

p��c ;

�c ¼ ffiffiffi2

p ðiD6 �� eN�Þ�;� ~� ¼ ffiffiffi

2p

�� ~c ;

� ~c ¼ ffiffiffi2

p ðiD6 ~�þ eðN þ ~mÞ ~�Þ�;�A ¼ ið ����� ����Þ;�� ¼ ���ð@N � fÞ þ i�

D

e;

(9)

where

f ¼ � i

2���F

��; D ¼ e2ðj�j2 � j ~�j2 � �Þ;

and � ¼ ð�1; �2Þ is a complex spinor. The correspondingsupersymmetry current is

j ¼ ffiffiffi2

p ðD6 �� þ ieN��Þ�c

þ ffiffiffi2

p ðD6 ~�� � ieðN þ ~mÞ ~��Þ� ~c

þ ði@6 N � if6 þDÞ��: (10)

The centrally extended algebra of the supercharges is dis-cussed below, in Sec. III B, see Eq. (19). After eliminationof the auxiliary D field via equation of motion, we get thefollowing scalar potential:

V ¼ e2

2½�� ð���� ~�� ~�Þ�2 þ N2���þ ð ~mþ NÞ2 ~�� ~�:

(11)

If � is positive (and we will assume � > 0) the theory is inthe Higgs regime and supports the BPS-saturated vortices.We will assume ~m to be positive too. If ~m � 0, the vacuumconfiguration is as follows:

~� ¼ 0; N ¼ 0; ��� ¼ �: (12)

Vortices with nonvanishing winding number correspond towindings of the� field [15]. The fermionic fields are set tozero in the classical approximation.We are interested in static solutions; the relevant part of

the Lagrangian, upon the Bogomol’nyi completion [16],takes the form

LBPS ¼ � 1

2e2B2 � jDi�j2 � e2

2½������2

¼ �jDþ�j2 � 1

2e2½B� e2ðj�j2 � �Þ�2

� �B� i@kð�kl��Dl�Þ; (13)

where B ¼ @1A2 � @2A1 is the magnetic field and Dþ �D1 þ iD2.Since the solution is static we have H ¼ �LBPS. We

will label the fields minimizing H by the subscript (orsuperscript) v. They satisfy the following first-order BPSequations:

Bv � e2ðj�vj2 � �Þ ¼ 0; Dvþ�v ¼ 0: (14)

The boundary conditions are self-evident. Solutions tothese BPS equations in different homotopy classes arelabeled by the winding number n. Needless to say, theyare well-known. A vortex with the winding number n hasthe mass

Mv ¼ 2�n�; (15)

where, at the classical level, the parameter � on the right-hand side is that entering the Lagrangian (8). At this levelthe central charge

jZvj ¼ �Z

d2xB ¼ 2�n�: (16)

The vortex solution breaks 1=2 of supersymmetry. Moreprecisely, the vortex solution is invariant under the super-symmetry transformations (9) restricted to � ¼ ð0; �2Þ. InSec. III we will show that this residual symmetry betweenbosons and fermions is strong enough to preserve the BPSsaturation at the quantum level.

III. QUANTUM CORRECTIONS

In this section we will calculate quantum corrections tothe Fayet-Iliopoulos parameter, the vortex mass, and thecentral charge, using the regularization outlined in Sec. I.We will keep ~m large but finite, taking the limit ~m ! 1 atthe very end. In order to calculate one-loop corrections tothe classical results we will expand the fields around thebackground solutions

� ¼ �v þ ; A ¼ Av þ a; (17)

REVISITING CRITICAL VORTICES IN THREE-. . . PHYSICAL REVIEW D 78, 125021 (2008)

125021-3

keeping the terms quadratic in , a. The fields and ahave the mass m ¼ e

ffiffiffiffiffiffi2�

p, while ~� and ~c have the mass

~m.1

A. Fayet-Iliopoulos parameter at one loop

As was mentioned, the Fayet-Iliopoulos parameter re-ceives no corrections if ~m ¼ m. If ~m � m, there is a one-loop quantum correction. The simplest way to compute therenormalization of � is to consider the Lagrangian beforeeliminating the auxiliary field D, i.e. the bosonic part inEq. (8). In this exercise we treat D as a constant back-ground field. Figure 1 shows the tadpole diagrams arising

from the couplings Dð���� ~�� ~�Þ, which renormalize �,

�R � �þ �� ¼ �þZ d3k

ð2�Þ3�

i

k2 � ~m2� i

k2 �m2

�

¼ �þm� ~m

4�: (18)

We see that ~m plays the role of the ultraviolet cutoff, as wasexpected. Needless to say, the finite part of the correctionm=4� depends on the definition of the renormalized FIparameter. In fact, it has an infrared origin (otherwise, oddpowers of m could not have entered). The renormalized FIparameter is defined as the coefficient in front of the Dterm in �one-loop. Here we note a couple of differences

between the result in Eq. (18) and the results in [3,8].The first difference is that ~m, which represents the lineardivergence of �, is absent in the previous results since theauthors used dimensional and zeta-function regularization,respectively. Another difference is the sign of the m

4� term.

The calculation of the vortex mass renormalization in [3,8]was phrased as a counterterm calculation; therefore, theresult [3,8] �� ¼ � m

4� which superficially has the sign

opposite to that in Eq. (18) is in full accord with our resultand with the central charge renormalization.

B. Central charge

The nonvanishing (and linearly divergent) correction to� implies that the classical central charge in Eq. (16) mustbe corrected too, in accordance with Eq. (18), so that � isconverted to �R in the central charge. Now we will explainwhere this correction comes from.

The centrally extended superalgebra is

fQ; ðQyÞ�0g ¼ 2ðP0�0 þ P1�

1 þ P2�2Þ

� 2

�P3 þ �

Zd2xB

�; (19)

where our conventions for the gamma matrices are sum-marized in Eq. (31) and P3 is the ‘‘momentum’’ along thereduced direction,

P3 ¼ � ~mZ

d2xði ~��@$t~�þ �~c�0

~c Þ � ~mq: (20)

Here q is the Noether charge of the vortex,

q ¼Z

d2x~J0; ~J ¼ �ði ~��@$~�þ �~c�

~c Þ: (21)

The current ~J defines a global U(1) symmetry acting in

the regulator sector. Below we will show that the corre-sponding charge fractionalizes. (In the low-energy sector itis related to the occurrence of the Chern-Simons term afterthe tilded fermion is integrated out.)It is rather obvious that the P3 term is in one-to-one

correspondence with the fact that integrating out massivefermions in 2þ 1 dimensions generates the Chern-Simonsterm in the Lagrangian [10,11] which, in turn, makes thevortex electrically charged [17]. Since our theory is fullyregularized, the superalgebra (19) presents the exact op-erator equality in an explicit representation (which issometimes elusive in other regularizations). The secondline in Eq. (19) is �2Zv. Although the coefficient of theChern-Simons term in the Lagrangian is dimensionless,integrating out heavy fermions in the central charge pro-duces a term which has mass dimension 1. In fact, inSec. IV (see also Appendix) we will calculate the valueof the Noether charge q (at one loop) and will show thatq ¼ � n

2 . Note that for odd n the charge is fractional, a

well-known phenomenon of charge fractionalization [18].Assembling two terms in the central charge and using

the fact that q ¼ � n2 we get

jZn;vj ¼ 2�n�þ ~mq ¼ 2�n�� ~mn

2¼ n

�2��R �m

2

�;

(22)

where we used Eq. (18) to convert � into �R. The contri-bution due to P3 comes precisely in the combination

D D

η ϕ

FIG. 1. Tadpole diagrams determining one-loop correctionto �.

1The superpartners c and � do not have definite masses; themass matrix for these fields can be diagonalized providing us

with two diagonal combinations, c 0 ¼ cþi�ffiffi2

p and �0 ¼ c�i�ffiffi2

p . The

latter have masses effiffiffiffiffiffi2�

p. Note that both parameters e and

ffiffiffi�

phave dimensions ½m�1=2.

S. OLMEZ AND M. SHIFMAN PHYSICAL REVIEW D 78, 125021 (2008)

125021-4

ensuring that the bare parameter � is converted into therenormalized �R. Equation (22) demonstrates the emer-gence of the quantum correction �mn=2.

C. Renormalization of the vortex mass

To calculate the one-loop contribution to the vortexmass, we expand the Lagrangian (8) around the back-ground field, in the quadratic order, using the definitions(17). It is convenient to introduce the following gauge-fixing term2:

L gf ¼ � 1

2

�1

e@a

þ ieð�v � ���

v Þ�2: (23)

Note that under this gauge choice, a0 becomes a dynamicalfield and one has to take its loop contribution into account.The corresponding ghost Lagrangian is

L gh ¼ �c

�� 1

e2@@

� ð2j�vj2 þ�v � þ��

v Þ�c;

(24)

where �c and c are spin-zero complex fields with fermionstatistics. We will drop the last two terms in Eq. (24) sincethey show up only in higher-order corrections. Assemblingall the bosonic contributions, we get the following bosonicLagrangian (at the quadratic order):

Lð2ÞB ¼ Lð2Þ

gf þLð2ÞB þLð2Þ

gh

¼ jDv j2 � e2ð3j�vj2 � �Þj j2 þ 1

2e2ð@amÞ2 � j�vj2a2m � 2iamð �Dv

m�v � Dvm�

�vÞ þ jDv

~�j2 þ ½e2ðj�vj2

� �2Þ � ~m2�j ~�j2 � 1

2e2ð@a0Þ2 þ j�vj2a20 þ

1

2e2ð@NÞ2 � N2j�vj2 þ �c

�� 1

e2@@

� 2j�vj2�c;

(25)

where ¼ 0; 1; 2 and m ¼ 1; 2 [the fields and a aredefined in Eq. (17)]. The last two lines in Eq. (25) includeone complex scalar field with the fermion statistics and tworeal scalar fields with the boson statistics, satisfying thesame equations of motion. If we impose the same boundaryconditions on the fields a0, N, �c, and c (and we do), theyproduce the same determinants and their contributions tothe vortex mass cancel each other [8]. With this observa-tion in mind, we will drop this line in what follows.

The transverse components of the gauge field, a1 and a2,can be combined into complex fields by defining

a� ¼ a1 � ia2ffiffiffi2

pe

: (26)

By the same token, we define Dv� ¼ Dv1 � iDv

2 . Withthese definitions Eq. (25) can be rewritten as follows:

Lð2ÞB ¼ jDv

j2 � e2ð3j�vj2 � �Þj j2 þ @aþ@a�

� 2e2j�vj2aþa� � ffiffiffi2

pieð �aþDv��v

� a�Dvþ��vÞ þ jDv

~�j2 þ ðe2ðj�vj2 � �2Þ

� ~m2Þj ~�j2: (27)

Note that, at the quadratic order, the tilded bosonic sector is

decoupled from the fluctuations of the untilded one, i.e. ~�is coupled to the background fields only. (We will soon seethat the same decoupling occurs for the fermionic sector.)

This allows us to consider the contributions of tilded anduntilded fields separately.

1. One-loop contribution from the untilded sector

In the first part of this subsection we will compute theclassical Hamiltonian (density) of the fluctuations. In thesecond part we will quantize the Hamiltonian by imposingcanonical (anti)commutation relations. Finally we willcompute the sum of the energies, which turns out to bevanishing. We first start with the bosonic Hamiltoniancorresponding to the untilded part of the Lagrangian (27),which can be written in the matrix form,

H ð2ÞB ¼ ð _ ; i _aþÞ� _

i _aþ

� �þ ð ; iaþÞ�D2

B

iaþ

� �; (28)

where we defined the quadratic bosonic operator

D2B ¼ �ðDv

k Þ2 þ e2ð3j�vj2 � �Þ ffiffiffi2

peDv��vffiffiffi

2p

eðDv��vÞ� �@2k þ 2e2j�vj2 !

:

(29)

Equation (28) gives the classical Hamiltonian for the bo-sonic fields. The fermionic Lagrangian (8) is already qua-dratic in the fermionic fields. Setting the bosonic fields totheir background values gives the following quadraticLagrangian for the untilded fermionic fields:

L ð2ÞF ¼ 1

e2��i@6 �þ �c iD6 c þ i

ffiffiffi2

p ½ð ��c�� � �c��Þ�:(30)

We choose the following set of � matrices:

2This gauge-fixing term is chosen to cancel the terms ð ��vÞ2and ð ��

vÞ2 originating from the scalar potential (11) as well asthe term @a

ð ��v � ��vÞ arising from the term

D��D� in Eq. (8).

REVISITING CRITICAL VORTICES IN THREE-. . . PHYSICAL REVIEW D 78, 125021 (2008)

125021-5

�0 ¼ �3; �1 ¼ i�2; �2 ¼ i�1: (31)

With the chosen representation of � matrices the Hamiltonian corresponding to the Lagrangian (30) reads

H ð2ÞF ¼ �i

c 1

c 2�1=e�2=e

0BB@

1CCA

y 0 Dvþ � ffiffiffi2

pe�v 0

Dv� 0 0ffiffiffi2

pe�vffiffiffi

2p

e��v 0 0 @þ

0 � ffiffiffi2

pe��

v @� 0

0BBB@

1CCCA

c 1

c 2�1=e�2=e

0BB@

1CCA ¼ U

V

� �y 0 �iDF

iDyF 0

� �UV

� �; (32)

where we regrouped the components of � and c ,

fU;Vg ¼ fðc 1; �2=eÞ; ðc 2; �1=eÞg; (33)

and defined the fermionic operator,

DF � Dvþ � ffiffiffi2

pe�v

� ffiffiffi2

pe��

v @�

!;

DyF ¼ �Dv� � ffiffiffi

2p

e�v

� ffiffiffi2

pe��

v �@þ

!:

(34)

Supersymmetry of the Lagrangian reveals itself when wecalculate the following quadratic fermionic operator:

DyFDF ¼ �ðDv

k Þ2 þ e2ð3j�vj2 ��Þ ffiffiffi2

peDv��vffiffiffi

2p

eðDv��vÞ� �@2k þ 2e2j�vj2 !

;

(35)

which coincides with D2B defined in Eq. (29),

D2B ¼ Dy

FDF: (36)

By virtue of this identification we rewrite the fullHamiltonian for untilded fields in terms of the operatorsDF and Dy

F,

H ð2Þ ¼ ð _ ; i _aþÞ�_

i _aþ

!þ ð ; iaþÞ�Dy

FDF

iaþ

!

� iUyDFV þ iVyDyFU: (37)

To quantize the Hamiltonian (37) we will follow methodsworked out long ago (e.g. [19]). First, we impose boundaryconditions which are compatible with the residual super-symmetry. We place the system into a spherical two-dimensional ‘‘box’’ of radius R, with the assumption thatR is much larger than any length scale in the model at hand.To ensure that the energy associated with the boundaryvanishes, we require all the fields to vanish at r ¼ R. Thiscondition does not break the residual supersymmetry sinceit is compatible with the transformations defined in Eq. (9).Then we expand the fields in Eq. (37) in eigenmodes of theoperators D2

B and the associated operator D20B which is

defined as follows:

D20B ¼ DFD

yF: (38)

The eigenvalue equations for these operators are

D2B�n;� � w2

n�n;�; D20B�

0n;� � w2

n�0n;�: (39)

The eigenvalues for both operators are the same: the ei-genfunctions can be related to each other by

�0n;� ¼ 1

wn

DF�n;�; �n;� ¼ 1

wn

DyF�

0n;�: (40)

For eachw2n there are two independent solutions, which are

labeled by subscript �. The above statement excludes thezero modes wn ¼ 0 which occur only in one of theseoperators, namely D2

B, reflecting the translational invari-ance in the problem at hand. Usually, they are referred to astranslational. Their fermion counterparts, the zero modesof DF, are supertranslational modes. Dy

F has no zeromodes.The eigenfunctions �n;� form an orthonormal and com-

plete basis, in which we expand the fields in Eq. (37)

ðt;xÞiaþðt;xÞ

!¼ X

n�0�¼1;2

an;�ðtÞ�n;�ðxÞ;

Vðt;xÞ ¼ Xn�0�¼1;2

vn;�ðtÞ�n;�ðxÞ;

Uðt;xÞ ¼ Xn�0�¼1;2

un;�ðtÞ�0n;�ðxÞ:

(41)

Note that the zero modes do not enter in the expansion (41),nor do they appear in the Hamiltonian (37). For nonzeromodes the ratio of the bosonic to fermionic modes is 1:2,i.e. we have two complex expansion coefficients an;�ðtÞ forbosons and four complex expansion coefficients vn;�ðtÞand un;�ðtÞ for fermions, for each value of w2

n. As we

will see below, this is precisely what is needed for cancel-lation. Let us note in passing that for zero modes the ratio is1:1. We have one complex bosonic modulus and onefermionic.Using the above mode decompositions in Eq. (37), we

arrive at an infinite set of oscillators,3

3To be accurate we should note that here we use integration byparts in the last term, which means that Eq. (42) is valid up to afull spatial derivative.

S. OLMEZ AND M. SHIFMAN PHYSICAL REVIEW D 78, 125021 (2008)

125021-6

H ð2Þ ¼ Xn;n0�0�;�0

�_a�n;� _an0;�0�y

n;��n0;�0

þ a�n;�an0;�0�yn;�D

yFDF�n0;�0

� i1

wn0v�n;�un0;�0�y

n;�DyFDF�n0;�0

þ iwnu�n;�vn0;�0�y

n;��n0;�0

�: (42)

Now, for each oscillator, the coefficients a, _a, v, u, andtheir complex conjugated must be represented as linearcombinations of the corresponding creation and annihila-tion operator subject to the standard (anti)commutationrelations. This procedure parallels that discussed in detailin Ref. [4]. The only difference is that in [4] for each modeone has an oscillator for one real degree of freedom, whilein the case at hand we deal with a complex degree offreedom which is equivalent to two real degrees of free-dom. We will not dwell on details, referring the reader toRef. [4]. Imposing the appropriate (anti)commutation rela-tions on the creation and annihilation operators, we get forexpectation values of bilinears in the vortex ground state

ha�n;�an0;�0 ivor ¼ 1

2wn

�nn0���0 ;

h _a�n;� _an0;�0 ivor ¼ wn

2�nn0���0 ;

hu�n;�vn0;�0 ivor ¼ i

2�nn0���0 ;

hv�n;�un0;�0 ivor ¼ � i

2�nn0���0 ;

(43)

where the angular brackets mark the vortex expectationvalue. Expectation values of all other bilinears vanish. Ifwe substitute these results in Eq. (42) we immediately seethat the one-loop correction in the untilded sector vanisheslocally, i.e. in the Hamiltonian density. Needless to say, it

vanishes in the integralRd2xH ð2Þ too.

Thus, we demonstrated the cancellation of the bosonicand fermionic contributions mode by mode, for each givenn. This vanishing result shows that the vortex mass receivesno correction from the untilded sector. If we did not have

the ~� multiplet, this would be the final answer. However,the theory per se is ill-defined without the tilded sector.

From Eq. (18) we see that in the absence of ~�, the FIparameter � would be linearly divergent at one loop. With~� included, the theory is regularized; cancellation of loopsin Fig. 1 takes place. The linear divergence is replaced bythe linear dependence of � on ~m. The latter parameter iskept large but finite till the very end. It is only natural thatthe linear dependence ofMv;R on ~mwill be provided by the

tilded sector contribution (Sec. III C 2).

2. The tilded sector (regulator) contribution in Mv

The Lagrangian for the tilded sector is

~Lð2Þ ¼ jDv~�j2 þ ðe2ðj�vj2 � �2Þ � ~m2Þj ~�j2

þ �~c iD6 ~c � ~m �~c ~c : (44)

The corresponding Hamiltonian density then takes theform

~H ð2Þ ¼ j _~�j2 þ ~��ð�DvþDv� þ ~m2Þ ~�

þ ~c 1~c 2

� � ~m �iDvþ�iDv� � ~m

� � ~c 1~c 2

!; (45)

where Dv� ¼ Dv1 � iDv

2 and we used Eq. (14). For whatfollows it is important to know that the operator DvþDv�has no zero modes.If we denote the eigenvalues of the bosonic operator

�DvþDv� þ ~m2 (46)

by � (� is strictly larger than ~m2), for each given � wehave two eigenmodes of the associated fermion equation

~m �iDvþ�iDv� � ~m

� � ~c 1~c 2

!¼ �

ffiffiffiffi�

p ~c 1~c 2

!: (47)

The eigenfunctions have the following structure. If ~c 1 isthe normalized eigenfunction of the operator �DvþDv�,then ~c 2 is the corresponding eigenfunction of the conju-gated operator �Dv�Dvþ timesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� ffiffiffiffi�

p � ~m

� ffiffiffiffi�

p þ ~m

vuutdepending on the sign in the eigenvalue equation (47).Thus, for each complex boson mode with the eigenvalue� we have two complex fermion modes with the eigenval-

ues � ffiffiffiffi�

p. This balance of modes guarantees that the

corresponding quantum corrections to Mv vanish.This is not the end of the story, however. There is one

additional (complex) fermion mode with � exactly equalto ~m2. (The above statement refers to the elementary vortexwith the unit winding number. Generalization to higherwinding numbers is straightforward.) Let us focus on thisunbalanced mode which will be solely responsible for thecontribution of the tilded sector in Mv.From Eqs. (14) and (47) it is clear that this fermion mode

has the form

0~c ð0Þ2

� �; (48)

where the eigenvalue on the right-hand side of Eq. (47) is� ~m. This gives rise to the following contribution in theenergy density:

E ð0Þ ¼ � ~mð ~c ð0Þ2 Þ� ~c ð0Þ

2 : (49)

REVISITING CRITICAL VORTICES IN THREE-. . . PHYSICAL REVIEW D 78, 125021 (2008)

125021-7

We proceed to quantization in the standard manner. To thisend we represent

~c ð0Þ2 ¼ �yðtÞ’ðxÞ; (50)

where ’ðxÞ is the normalized c-numerical part of the zeromode while �y is the operator part with the appropriateanticommutation relation implying

h��yi ¼ 1

2: (51)

Now, the contribution of the tilded sector to Mv obviouslyreduces to

�M �Z

d2xhEð0Þi ¼ � ~mh��yi ¼ � ~m

2: (52)

Equation (52) gives the only nonvanishing quantum cor-rection,

MR � Mþ �M ¼ 2��� ~m

2¼ 2��R �m

2; (53)

where we again used Eq. (18) to convert � to �R.Comparing this result with the renormalization of thecentral charge in Eq. (22), we conclude that the BPSsaturation does indeed hold at the quantum level.

D. Higher orders

Let us discuss now what changes as we pass to higherorders of perturbation theory. Returning to Sec. III B and,in particular, to Eq. (20), it is not difficult to understand thatthe relation Z ¼ 2��� 1

2~m (for the elementary vortex)

remains exact to all orders. Indeed, q is half-integer andthe relation q ¼ � 1

2 for the elementary vortex cannot

receive corrections4 in e=ffiffiffi�

p. If we define ~� as

~� ¼ �� ~m

4�; (54)

where � and ~m are bare parameters, then the statement that

Mv ¼ Z ¼ 2�~� (55)

is valid to all orders. The term ~m comes from the ultraviolet

and, therefore, it is natural to refer to ~� as to an ‘‘effectiveultraviolet parameter.’’ Equation (55) is akin to theNovikov-Shifman-Vainshtein-Zakharov theorem for thegauge coupling renormalization in four dimensions [20]:being expressed in terms of the ultraviolet (bare) parame-ters the gauge coupling renormalization is limited to oneloop (see also the second paper in [14]).

Corrections in powers of e=ffiffiffi�

parise if we decide to

express the result in terms of �R, a parameter defined in theinfrared; the expression of �R in terms of � does contain aninfrared contribution (otherwise, odd powers of e could not

have entered; see Sec. III A). Generalizing the argumentsof [14] we can write, instead of (18)

~� ¼ �R

�1� 1

2ffiffiffi2

p�

effiffiffi�

pR

�: (56)

Equations (55) and (56) assembled together present aperturbatively exact result for Mv ¼ Z.

IV. CALCULATION OF THE NOETHER CHARGE q

In Sec. III B we used the fact that the Noether U(1)charge of the elementary vortex is �1=2. The Noethercharge is saturated by the fermion term in Eq. (20)

q ¼ �Z

d2x �~c�0 ~c : (57)

Here we will explore this issue in more detail. The vortexNoether charge can be calculated in a number of ways. Themost straightforward calculation is that of the Feynmandiagram depicted in Fig. 2, using the background fieldexpansion. This expansion is justified because the back-ground photon field is small compared to the value of ~m (inthe very end we want to tend ~m to infinity). For ourpurposes it is sufficient to limit ourselves to the leadingterm (proportional to F��). Using �

in the upper vertex in

Fig. 2 (denoted by the closed circle) we get the Noethercurrent in the background field in the form

�~c� ~c ! ~mF�����

Z d3p

ð2�Þ31

ðp2 þ ~m2Þ2

¼ 1

8�F���

��: (58)

The current in (58) couples to the gauge field A, giving a

term of the form AF�����, which is nothing but the

Chern-Simons term. Now, if we set ¼ 0 in (58) andinvoke the standard value of the magnetic fluxZ

d2xB ¼ 2�;

we immediately get

ψψ ∼∼

γ

FIG. 2. Calculation of the vortex Noether charge. We trace theterm which is linear in the momentum k of the produced photon(assuming k to be small). Terms of the zeroth order vanishbecause of the gauge invariance. Quadratic and higher orderterms in k are irrelevant since they are suppressed by powers of1= ~m.

4A simple dimensional analysis shows that perturbative cor-rections run in powers of e=

ffiffiffi�

p.

S. OLMEZ AND M. SHIFMAN PHYSICAL REVIEW D 78, 125021 (2008)

125021-8

hqi ¼ �1=2: (59)

V. CONCLUSION

In this paper we showed that the mass and the centralcharge of the N ¼ 2 vortices in 2þ 1 dimensions, beingexpressed in terms of �R, get a quantum correction�mn=2where m is the mass of the charged bosons (fermions) andn is the winding number of the vortex. The equality of thecorrections to the vortex mass/central charge shows that theBPS saturation persists at the quantum level. Our result isin agreement with the previous ones [6,8].

New elements of our work (compared to [6,8]) are asfollows. We use a more straightforward and physicallytransparent regularization scheme which captures linearlydivergent terms invisible in the regularization methodsused in the previous papers. In our scheme we have amassive regulator multiplet acting in loops as an ultravioletcutoff. In the limit of infinitely large regulator mass, theregulator’s role is taken over by the Chern-Simons term.We establish a contact between one-loop calculations andthe general operator expression for the central charge(obtained within the same regularization scheme).Analyzing both, in a single package, we are able to reveala simple physical interpretation behind the occurrence ofthe �mn=2 shift, and obtain all-order results (Sec. III D).

ACKNOWLEDGMENTS

Wewould like to thank A. Vainshtein, M. Voloshin, A. S.Goldhaber, A. Rebhan, P. van Nieuwenhuizen, and D.V.Vassilevich for useful discussions. This work is supportedin part by DOE Grant No. DE-FG02-94ER-40823. M. S. issupported in part by Chaire Internationalle de RechercheBlaise Pascal de l’Etat et de la Regoin d’Ille-de-France,geree par la Fondation de l’Ecole Normale Superieure.

APPENDIX

It is instructive to illustrate calculations of the charge qby inspecting the fermion mode decomposition discussedin Sec. III C 2. It is important to note that the mode decom-position in Sec. III C 2 is not the canonical expansion. Asimilar charge calculation by virtue of the canonical ex-pansion was first performed in [21]. We will discuss bothmethods.

First, we expand the fields ~c 1 and ~c 2 in terms of theeigenfunctions of the operators �DvþDv� and �Dv�Dvþ,namely, in n;� and 0

n;�, respectively:

~c 1 ¼Xn�0�¼1;2

vn;�ðtÞ n;�ðxÞ;

~c 2 ¼ ~c ð0Þ2 þ X

n�0�¼1;2

un;�ðtÞ 0n;�ðxÞ;

(A1)

where� labels two independent solutions corresponding to

the same eigenvalue, and ~c ð0Þ2 is the zero mode defined in

Eq. (50). The nonvanishing bilinears constructed fromun;�ðtÞ and vn;�ðtÞ are given in Eq. (43). With the expan-

sion in Eq. (A1) in hand, it is easy to see that the onlynonvanishing contribution to q comes from the zero modeof the operatorDvþ. This statement is a consequence of thefollowing expansion of q:

hqi ¼ �Z

d2xh ~c y ~c i

¼ �h��yi � Xn�0�¼1;2

hu�n;�un;� þ v�n;�vn;�i: (A2)

Using Eqs. (43) and (51) we get

hqi ¼ �1=2; (A3)

in perfect agreement with the previous result (59).(The fact that q ¼ �1=2 on the vortex is in one-to-one

correspondence with the fact that integrating out the mas-

sive fermion ~c we generate the Chern-Simons term with� ¼ e

4� [11]. It is well-known that self-dual n vortices with

the Chern-Simons term have charge q ¼ � 2�n�e ¼ � n

2

where n is the winding number [17,22].)We can carry out a slightly different calculation of the q

charge by expanding the tilded fermion field in the canoni-cal basis. However, we should remember that, generallyspeaking, the U(1) charge of the vacuum is infinite in theabsence of proper regularization.5 The same ‘‘vacuum’’infinity then shows up in q. In fact, we are interested inthe difference between the values of q on the vortex and inthe vacuum.This problem is automatically solved if, instead of the

charge �Rd2x ~c y ~c , one uses the following definition:

q ¼ � 1

2

Zd2xð ~c y ~c � ~c y

c~c cÞ; (A4)

where ~c c ¼ �ið ~c y�2ÞT is the charge-conjugated fermion

field. We now expand the fermionic field ~c in the canoni-cal basis,

~c ¼ ay00’0

� �þ X

n�0�¼1;2

�e�iwnt

an;�ffiffiffi2

p ’n;� þ eiwntbyn;�ffiffiffi2

p ’�n;�

�;

(A5)

where ’n;� are the energy eigenfunctions of the fermionic

Hamiltonian with the eigenvalues wn. The operators a0,an;�, and bn;� obey the canonical anticommutation rela-

tions6

5This infinity does not show up in Eq. (A3) because a regu-larized definition (A4) of the q charge is built in to the expansioncoefficients.

6Needless to say, all other anticommutators, not indicated in(A6), vanish.

REVISITING CRITICAL VORTICES IN THREE-. . . PHYSICAL REVIEW D 78, 125021 (2008)

125021-9

fa0; ay0 g ¼ 1; fan;�; ayn0;�0 g ¼ �n;n0��;�0 ;

fbn;�; byn0;�0 g ¼ �n;n0��;�0 :(A6)

The operators an;� and byn;� are the annihilation and crea-

tion operators associated with the positive and negativeenergy solutions.7 The first term in the expansion (A5) isthe zero mode. Inserting the expansion (A5) into Eq. (A4),we get

hqi ¼ � 1

2ha0ay0 � ay0a0i

� Xn�0�¼1;2

hayn;�an;� � byn;�bn;� � an;�ayn;� þ bn;�b

yn;�i:

(A7)

The condition we impose on a is ajvori ¼ 0. With thiscondition we get

hqi ¼ �1=2; (A8)

which again agrees with the previous results.

[1] J. R. Schmidt, Phys. Rev. D 46, 1839 (1992).[2] J. D. Edelstein, C. Nunez, and F. Schaposnik, Phys. Lett. B

329, 39 (1994).[3] A. Rebhan, P. van Nieuwenhuizen, and R. Wimmer, Nucl.

Phys. B679, 382 (2004).[4] M.A. Shifman, A. I. Vainshtein, and M.B. Voloshin, Phys.

Rev. D 59, 045016 (1999).[5] M. Shifman and A. Yung, Rev. Mod. Phys. 79, 1139

(2007).[6] A. S. Goldhaber, A. Rebhan, P. van Nieuwenhuizen, and

R. Wimmer, Phys. Rep. 398, 179 (2004).[7] B. H. Lee and H. Min, Phys. Rev. D 51, 4458 (1995).[8] D. V. Vassilevich, Phys. Rev. D 68, 045005 (2003).[9] K. Shizuya, Phys. Rev. D 71, 065006 (2005).[10] A. J. Niemi and G.W. Semenoff, Phys. Rev. Lett. 51, 2077

(1983).[11] A. N. Redlich, Phys. Rev. Lett. 52, 18 (1984).[12] O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg, and

M. J. Strassler, Nucl. Phys. B499, 67 (1997).[13] L. Alvarez-Gaume and D. Z. Freedman, Commun. Math.

Phys. 91, 87 (1983); S. J. Gates, Nucl. Phys. B238, 349(1984); S. J. Gates, C.M. Hull, and M. Rocek, Nucl. Phys.B248, 157 (1984).

[14] W. Fischler, H. P. Nilles, J. Polchinski, S. Raby, and L.Susskind, Phys. Rev. Lett. 47, 757 (1981); M.A. Shifmanand A. I. Vainshtein, Nucl. Phys. B277, 456 (1986).

[15] A. A. Penin, V.A. Rubakov, P. G.Tinyakov, and S. V.Troitsky, Phys. Lett. B 389, 13 (1996).

[16] E. B. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976).[17] H. J. de Vega and F.A. Schaposnik, Phys. Rev. Lett. 56,

2564 (1986); Phys. Rev. D 34, 3206 (1986).[18] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986

(1981); Solitons and Particles, edited by C. Rebbi and G.Soliani (World Scientific, Singapore, 1984), p. 601.

[19] H. Yamagishi, Phys. Lett. 147B, 425 (1984).[20] V. A. Novikov, M.A. Shifman, A. I.Vainshtein, and V. I.

Zakharov, Nucl. Phys. B229, 381 (1983); Phys. Lett.166B, 329 (1986); M.A. Shifman, A. I.Vainshtein, andV. I. Zakharov, Sov. J. Nucl. Phys. 43, 1028 (1986);Phys. Lett. 166B, 334 (1986); M.A. Shifman and A. I.Vainshtein, Sov. J. Nucl. Phys. 44, 321 (1986).

[21] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398(1976).

[22] S. K. Paul and A. Khare, Phys. Lett. B 174, 420 (1986);177, 453(E) (1986).

7The operators a0 and ay0 are not necessarily required to beparticle annihilation and creation operators; see Ref. [21] fordetails.

S. OLMEZ AND M. SHIFMAN PHYSICAL REVIEW D 78, 125021 (2008)

125021-10