PRL 96, 081602 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending3 MARCH 2006

Topological Mass Generation in Four Dimensions

Gia Dvali,1 R. Jackiw,2 and So-Young Pi31Department of Physics, New York University, New York, New York 10003, USA

2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA3Department of Physics, Boston University, Boston, Massachusetts 02215, USA

(Received 17 November 2005; published 2 March 2006)

0031-9007=

We show that in a large class of physically interesting systems the mass-generation phenomenon can beunderstood in terms of topological structures, without requiring a detailed knowledge of the underlyingdynamics. This is first demonstrated by showing that Schwinger’s mechanism for mass generation relieson topological structures of a two-dimensional gauge theory. In the same manner, corresponding four-dimensional topological entities give rise to topological mass generation in four dimensions. Thisformulation offers a unified topological description of some seemingly unrelated phenomena, such astwo-dimensional superconductivity, and the generation of �0 and axion masses by QCD, and possibly bygravity.

DOI: 10.1103/PhysRevLett.96.081602 PACS numbers: 11.15.Tk, 04.50.+h

I. Introduction.—Generating a mass gap is a most im-portant phenomenon in systems with infinitely many de-grees of freedom, both in condensed matter and in quantumfield theory. In the present Letter we show that in a largeclass of such systems the mass-generation phenomenoncan be described by topological entities, requiring onlymild assumptions about the underlying dynamics.

When the bosonic sector of the Schwinger model [1](massless QED in 2D space-time) is presented by variablesdual to the usual ones, there arise well known topologicalentities: Chern-Pontryagin density P and Chern-Simonscurrent C�, @�C� � P . Consequently, the mass-generatingdynamics can be described in topological terms. Moreover,the same topological structures, when elevated to 4Dspace-time, provide a partial, 4D generalization of theSchwinger model, together with its mechanism for gener-ating a mass.

Our formulation suggests a common origin, in terms ofthe topological structures, of the mass-generation phe-nomena in seemingly unrelated systems, such as photonmass generation in 2D massless QED and generation of �0

mass in 4D QCD with massless quarks (or generation ofthe axion mass, if quarks are heavy). It also suggests thepossible generalization of this phenomenon to other inter-actions that possess similar topological entities, such asgravity. Thus gravity could contribute similarly to the �0

(or axion) masses, and this would have important implica-tion for the solution of the strong CP problem.

In Section II, we review the 2D model and emphasize thecentral role in the mass-generation scenario of the chiralanomaly, which is famously related to a topological term.This suggests employing topological entities when de-scribing the model’s dynamics. Such a topological formu-lation is given in Section III, which is then promoted tofour dimensions in Section IV. A commentary on ourresults comprises the last Section V.

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II. Schwinger model resume.—In the Schwinger model,a massless Abelian vector potential A� interacts with avector current J � constructed from massless Dirac fields . The Lagrange density reads

L � �1

4F��F�� � i � ���@� � ieA�� ; (1)

L I � �eJ�A�; J � � � �� : (2)

The solution to the quantized model exhibits mass genera-tion. Usually one says that the ‘‘photon’’ A� acquires amass, but in two dimensions A� can be decomposed asA� � @��� ���@

��0. The gauge part decouples; only thepseudoscalar �0 remains. So one could just as well say thata pseudoscalar excitation acquires the mass e=

�����p

.The axial vector current J 5

�, is dual to the vector cur-rent: J 5

� � ���J� This formula is a consequence of 2D

geometry: when J � is a vector, J 5� is an axial vector.

More explicitly, duality is seen in the 2D gamma matrixidentity ���5 � �����.

As is well known, J 5� possesses an anomalous diver-

gence.

@�J 5� � �

e2�

���F�� �e�F: (3)

In the second equality we have introduced the (pseudo)scalar F, dual in two dimensions to the antisymmetricF�� � ���F. The anomaly provides an immediate deriva-tion of the mass. We begin with the gauge field equation ofmotion that follows from (1).

���@�F � eJ � ) @�F � �eJ5�: (4)

A further divergence gives the d’Alembertian on the leftand the anomaly (3) on the right.

@2F�e2

�F � 0: (5)

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PRL 96, 081602 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending3 MARCH 2006

III. Topological entities in the Schwinger model.—The2D anomaly is proportional to �F � 1

2 ���F��, which is

recognized as the 2D Chern-Pontryagin density P 2.

P 2 �1

2���F��: (6)

Furthermore, the gauge potential A� is dual to the Chern-Simons current C�2 ,

C �2 � ���A�; (7)

whose divergence forms the Chern-Pontryagin density.

@�C�2 � ���@�A� �

1

2���F�� � P 2 � �F: (8)

The bosonic portion of the Lagrange density for theSchwinger model may be written in terms of these topo-logical entities.

L2 � �1

4F��F�� � eJ �A�

�1

2F2 � eA��

��J 5�

�1

2P 2

2 � eC�2 J

5�: (9)

Moreover, since C�2 and A� are linearly related, it makes nodifference which one is the fundamental variable. Thusvarying C�2 in (9) gives (4) directly as the equation ofmotion. A further divergence and the anomaly Eq. (3)reproduce (5), since P 2 � �F.

It is this last, topological reformulation of the Schwingermodel that we shall take to four dimensions. However, wemust still address an important point that will arise in the4D theory. Observe that the equation of motion (4) entailsan integrability condition: Since the (axial) vector J 5

� is setequal to a gradient of (the pseudoscalar) P 2, it must be thatthe curl of the axial vector vanishes. Equivalently, the dualof the axial vector must be divergence-free; viz. the vectorcurrent must be conserved. Of course the same integra-bility condition is seen in the original vector formulation ofthe model, with equation of motion (4), which entailsconservation of the vector current (dual to the axial vectorcurrent).

But let us suppose that we have dynamical informationonly about the topological variables, and do not knowwhether the current dual to the axial vector current isconserved. (This is the situation that we shall meet infour dimensions.) Then we must reformulate our theoryin such a way that the integrability condition is avoided.

This reformulation proceeds by introducing twoStuckelberg fields p and q into L2.

L 02 �

1

2P 2

2 � e�C�2 � �

�@p��J5� � ���@

�q�: (10)

Upon varying C�2 , (4) becomes replaced by

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�@�P 2 � e�J5� � ���@

�q� � 0: (11)

Additionally, variation of p and q give, respectively,

@���J 5

� @2q � 0; (12)

@���C2 � @

2p � 0: (13)

The integrability condition on (11) demands that the curlof J 5

� � ���@�q vanish, but this is secured by (12). Thisequation determines a nontrivial value for q if the curl ofJ 5� is nonvanishing, while (13) fixes an innocuous value

for p. The divergence of (11) annihilates the q-dependentterm, leaving in the end the previous Eq. (5). Alternatively,eliminating the Stuckelberg fields in (11) with the help of(12) and the anomaly equation (3) gives

@�

�P 2 �

e2=�

@2 P 2

�� 0: (14)

This is equivalent to (4), but carries no integrability con-dition. Thus we see that the Stuckelberg modification over-comes difficulties, which arise when the current dual to theaxial vector is not conserved.

We may understand the role of the Stuckelberg fields byreverting to the original vector variables. Then the inter-action part of L02 in (10) reads

L 02I � �e�J

� � @�q��A� � @�p�: (15)

Eliminating p and q from (15) with the help of (12) and(13) leaves

L 02I � �eJ

���� �

@�@�

@2

�A�: (16)

This shows that the Stuckelberg fields ensure that theinteraction occurs only between transverse componentsof J � and A�.

IV. 4D model with topological mass generation.—For a4D generalization of the previous, the above 2D topologi-cal entities are promoted to four dimensions: P 4 and C�4 ,with the latter coupling to an axial vector current J 5

�whose divergence is anomalous. These are constructedfrom gauge potentials, which we take to be Abelian ornon-Abelian; in either case P 4 and C�4 remain gaugesinglets.

P 4 �1

2����Fa�F

a�� �

�F��aFa��; (17)

C �4 � 2����!�Aa�@�Aa! �

1

3fabcAa�Ab�Ac!�; (18)

@�C�4 � P 4: (19)

Unlike in the 2D case, the Chern-Simons current is notlinear in the gauge vector potential; nevertheless we remainwith the potential as the fundamental dynamical variable,

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PRL 96, 081602 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending3 MARCH 2006

and the variation of the Chern-Simons current reads

C�4 � 4�F��aAa� � 2���!�@��Aa!A

a��: (20)

A further difference from the Schwinger model is that thereis no reason to suppose that the dual to the 4D axial vectorcurrent is conserved. On the level of 4D gamma matrices,the duality relation is

���!����5 � g���! � g�!�� � g�!�� � �����!:

(21)

It is improbable that fermion dynamics (here unspecified)would leave conserved the current dual to the axial vectorcurrent. But this is not an obstacle to our construction,because we can employ the Stuckelberg formalism, toovercome the difficulty.

Thus the Lagrange density that we adopt is (a dimen-sionful normalization is omitted)

L 04 �

1

2P 2

4 ��2�C�4 � @p���J 5

� � @�q���: (22)

The Stuckelberg fields p� and q�� are antisymmetric intheir indices; �2 carries a mass-squared dimension; theaxial vector current possesses an anomalous divergence.

@�J 5� � �N

�F��aFa�� � �NP 4: (23)

N is a numerical coupling constant, taken positive.Variation of the L04 action with respect to Aa� gives, with

the help of (20), the equation of motion.

2�� @�P 4 ��2�J 5� � @

�q�����F��a

� ����!Aa�@!�2�J 5� � @

�q��� � 0: (24)

Variation of the two Stuckelberg fields yields the equations

@��J5 � @

�q�� � � ! � 0; (25)

@��C4 � @�p�� � � ! � 0: (26)

The first of these allows setting to zero the second memberof (24), while in the first member of that equation we maystrip away �F��a with the help of the identity �F��F�� �� 1

4��P 4. Consequently, (provided P 4 � 0) we are left

with

�@�P 4 ��2�J 5� � @�q��� � 0: (27)

The integrability condition on this equation is satisfiedby virtue of (25). Taking another divergence of (27) anni-hilates the Stuckelberg field because of its antisymmetry,while (23) provides the divergence for J 5

�. Thus we are leftwith

@2P 4 � N�2P 4 � 0: (28)

This shows that the pseudoscalar P 4 has acquired the mass,m2 � N�2.

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By taking the divergence of (25), we find from (23)

J 5 � @

�q� � �N

@2 @P 4: (29)

Inserting this in (27) yields

@�

�P 4 �

N�2

@2 P 4

�� 0; (30)

which is equivalent to (28), but does not entail integrabilityconditions. Here, and below, 1=@2 is the inversed’Alembertian, with proper accounting of its zero modes(massless excitations).

V. Conclusion.—While the 4D transposition of the 2DSchwinger model succeeds in generating a mass for apseudoscalar, just as in the 2D case, the model must becompleted in various aspects, to which we now callattention.

Dynamics that should produce the anomaly for the axialvector current is not specified. In the Schwinger model, thesame dynamics and the same degrees of freedom that gen-erate the mass are also responsible for the anomaly (3). Inthe 4D theory we must posit the anomaly (23) separatelyfrom the mass-generating dynamics. Moreover, our finalresult is that P 4 propagates as a free massive field. Addi-tional dynamics must be specified to describe interactions.

A related question concerns the role in physical theoryfor our Lagrangian (22). Since it involves dimension eight(P 2

4) and dimension six (C�4 J5�) operators, it should be

viewed as an effective Lagrangian. In this connection,observe that the Born-Infeld action and the radiativelyinduced Euler-Heisenberg action both contain theAbelian ��F��F���2 quantity in a weak-field expansion[also accompanied by an �F��F���2 term]. In the latter,the topological quantity enters with coefficient 7=360m4,for every virtually exchanged spin-1=2 fermion of massm.

The U�1� character of our anomalous current and thepresence in our theory of the Chern-Pontryagin quantitysuggest that here we are dealing with the problems of theunwanted axial U�1� symmetry and the mass of the �0

meson. Conventionally these issues are resolved by instan-tons [2]. Here we offer a phenomenological description.We relate the axial vector current to the �0 field,

J 5� � Z@��

0=�; (31)

(Z is a normalization) and add an �0 kinetic term to (22).

L �0 �1

2P 2

4 � Z�C�4 @��0 �

1

2@��0@��0: (32)

[We dispense with the Stuckelberg fields because the dualof the current in (31) is conserved.] Observe that the �0

field enjoys a constant shift symmetry, as befits the qua-dratic portion of a Goldstone field Lagrangian. The equa-tions that follow from varying Aa� and �0, respectively, are

@��P 4 � Z��0��F��a � 0; (33)

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FIG. 1. Anomaly diagram that correlates C�4 and J 5.

PRL 96, 081602 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending3 MARCH 2006

@2�0 � Z�P 4 � 0: (34)

Together the two imply

@2P 4 � Z2�2P 4 � 0: (35)

As before, a mass is generated.In the case of 4D QCD with massless quark flavor(s),

Eq. (35) can be obtained without any assumptions aboutthe dependence of the effective Lagrangian on the �0

meson. We only need to assume that the effectiveLagrangian contains the first P 2

4 term in (32). The C4-J 5

coupling is automatically generated from the anomalydiagram (Fig. 1) that correlates C�4 and J 5

�. The diagramgenerates the operator

�2C�4@�@

@2 J 5; (36)

where �2 arises as a momentum cutoff. This expression isalso what one obtains from (22) after eliminating theStuckelberg fields p� and q�� through their equationsof motion (25) and (26). Thus massless quark dynamicsdue to the anomaly substitute the effect of the Stuckelbergfields. Variation with respect to Aa� yields the analog ofEq. (27).

�@�P 4 ��2 @�@2 @

J 5 � 0: (37)

Using the anomalous divergence relation (23), we arrive atEq. (30), which is equivalent to (35). Because P 4 acquiresa mass, its expectation value in the QCD vacuum mustvanish. This explains why QCD solves both U�1� and thestrong CP problems in the zero quark mass limit [3].

Because of the universality of the topological formula-tion, the topological mass-generation phenomenon in prin-ciple can be generalized to other interactions, that possessimilar topological entities. The obvious interesting candi-date for such a generalization is gravity. In order to see howgravity could contribute to the �0 mass, we need to assumethat effective Lagrangian contains a term bilinear in gravi-tational Cherm-Pontryagin density P gr. The entire reason-ing leading to (37) then could be repeated for thegravitational case, provided we substitute the topologicalquantities with their gravitational counterparts, P gr and

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C�gr. The gravitational analog of (36) is then generatedfrom the gravitational chiral anomaly. Notice, however,that since in the presence of both (QCD and gravity) chiralanomalies, the anomalous divergence of the axial current(23) is modified as

@�J 5� � �NP 4 � NgrP gr; (38)

only one linear combination of NP 4 and NgrP gr willpropagate a massive pseudoscalar, whereas the orthogonalcombination will remain massless. Of course, the latterterminology implies that in effective theory P gr shouldbehave as a composite field. This is something that inreality will be determined by the underlying gravitationaldynamics.

In case of massive QCD the very similar topologicalmass generation takes place for the axion field, if one isintroduced into the theory. The topological formulation ofthe axion mass, in particular, makes it clear how gravitycould spoil the axion solution of the strong CP problem[3]. Similar effects should be present in all even dimen-sions, but the singularity structure and the required dimen-sional parameter (analog of the 2D and 4D e and �) willchange.

We conclude by commenting about the connection withthe Witten-Veneziano [4] relation (m2

�0 / �, where � is thetopological susceptibility). There is the obvious fact thatboth are bilinear in P . Perhaps, some connection can bedrawn from the fact that the topological susceptibility isrelated with the second derivative of the free-energy withrespect to �, taken at � � 0. This derivative in our casetranslates as the second derivative of the �0 potential at itsminimum, and thus, by default, represents the �0 mass.This conclusion is insensitive to the precise P dependenceof the effective Lagramgian (32). We also remark that ourapproach to the �0 mass has some connection with [5].

We thank S. Deser, G. Gabadaze, and R. L. Jaffe foruseful discussions. The work of G. D. is supported in partby the David and Lucile Packard Foundation, and by NSFGrant No. PHY-0245068; that of R. J. and S.-Y. Pi by DOEGrants No. DE-FCO2-94ER-40818 and No. DE-FG02-91ER-40676, respectively.

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[1] J. S. Schwinger, Phys. Rev. 125, 397 (1962); Phys. Rev.128, 2425 (1962).

[2] G. ’t Hooft, Phys. Rep. 142, 357 (1986).[3] G. Dvali, hep-th/0507215.[4] E. Witten, Nucl. Phys. B156, 269 (1979); G. Veneziano,

Nucl. Phys. B159, 213 (1979); Phys. Lett. B 95, 90 (1980).[5] A. Aurilia, Y. Takahashi, and P. K. Townsend, Phys. Lett.

B 95, 265 (1980); C. Rosenzweig, J. Schechter, andC. Trahern, Phys. Rev. D 21, 3388 (1980).