Physics Letters B 529 (2002) 261–264

www.elsevier.com/locate/npe

A note on theta dependence

Frank Ferraria,b,1

a Institut de Physique, Université de Neuchâtel, rue A.-L. Bréguet 1, CH-2000 Neuchâtel, Switzerlandb Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA

Received 14 November 2001; accepted 31 January 2002

Editor: L. Alvarez-Gaumé

Abstract

The dependence on the topologicalθ angle term in quantum field theory is usually discussed in the context of instantoncalculus. There the observables are 2π periodic, analytic functions ofθ . However, in strongly coupled theories, the semi-classical instanton approximation can break down due to infrared divergences. Instances are indeed known where analyticity inθ can be lost, while the 2π periodicity is preserved. In this short Letter we exhibit a simple two-dimensional example where the2π periodicity is lost. The observables remain periodic under the transformationθ �→ θ + 2kπ for somek � 2. We also brieflydiscuss the case of four-dimensionalN = 2 supersymmetric gauge theories. 2002 Elsevier Science B.V. All rights reserved.

A topological θ angle termLθ can be added tothe Lagrangian in various quantum field theories. Themost important and well-known example is the case offour-dimensional gauge theories, for which

(1)L4Dθ = θ

32π2 εµνρκ trFµνFρκ .

Other examples include two-dimensional gauge theo-ries, for which

(2)L2Dθ = θ

4πεµνFµν,

or two-dimensional non-linearσ models with targetspaceM, for which

(3)Lσθ = θ

4πεµνBij (φ)∂µφ

i∂νφj ,

E-mail address: frank.ferrari@unine.ch (F. Ferrari).1 On leave of absence from Centre National de la Recherche Sci-

entifique, Laboratoire de Physique Théorique de l’École NormaleSupérieure, Paris, France.

whereB ∈H 2(M,Z) has integer periods (for Kählerσ models like theCPN model in which we willbe interested in,B is the suitably normalized Kählerform). All the θ angle terms can be written locallyas total derivatives, and are normalized in such away that with classical boundary conditions at infinity(determined by the requirement of finite classicalaction),

(4)∫Lθ/θ ∈ Z.

These two properties have two important consequencesin weakly coupled field theories. The fact thatLθ isa total derivative implies that there is noθ depen-dence in perturbation theory. The fact that

∫Lθ/θ is

quantized implies that topologically non-trivial clas-sical field configurations, called instantons, that cancontribute to the path integral in a semiclassical ap-proximation, may induce a 2π periodic and smoothθdependence. More precisely, ak instanton ork anti-

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)01261-3

262 F. Ferrari / Physics Letters B 529 (2002) 261–264

instanton contribution,k ∈ Z, is proportional to

(5)e−8π2k/g2e±ikθ ,

where g is the conventionally normalized couplingconstant. In theories where dimensional transmutationtakes place, the running couplingg is replaced by ascale|Λ|. It is convenient to introduce a complexifiedscale

(6)Λ= |Λ|eiθ/β,whereβ is the coefficient of the one-loopβ function.Contributions fromk instantons andk anti-instantonsin the one-loop approximation are then, respectively,proportional toΛkβ andΛ̄kβ .

Understanding theθ dependence instrongly cou-pled field theories is a much more difficult problem.Boundary conditions at infinity, or equivalently thestructure of the vacuum, should be determined by us-ing the (generally unknown) quantum effective action,not the classical action, and quantization laws such as(4) can be invalidated. Moreover, subtle infrared ef-fects can induce aθ dependence in Feynman diagrams.The conclusion is that analyticity or 2π periodicity arenot ensured a priori. A typical example where the de-pendence inθ is not consistent with instanton calcu-lus is the two-dimensional quantum electrodynamics,or equivalently the purely bosonicCPN non-linearσ model whose effective action is QED2 [1]. Thereit is well-known that the term (2) induces a constantbackground electric fieldE = e2θ/2π , wheree is theelectric charge [2]. The phenomenon of pair creation,which is energetically favoured for|E| > e2/2, en-sures 2π periodicity in θ , but analyticity atθ = ±πis lost. Discussions of similar effects in various con-texts can be found for example in [3].

An interesting generalization of the standard QED2discussed above is theN = 2 supersymmetric QED2with a twisted superpotentialW , or, equivalently, theN = 2 supersymmetricCPN model with twistedmassesmi [4–6]. There, in addition to the gaugefield E = F 01, one has a Dirac fermionλ and acomplex scalarσ . All these fields belong to the samesupersymmetry multiplet and are packed up in a single(twisted) chiral superfield

(7)Σ = σ − 2i(θ−λ̄+ + θ̄+λ−

) + 2θ̄+θ−(D − iE).The real auxiliary fieldD appears on the same footingas the gauge fieldE, consistently with the fact that

gauge fields do not propagate in two dimensions. Theequation for the background electric field is in thiscaseE = −2e2 ImW ′(σ ), which shows that in thesupersymmetric vacua for whichW ′(σ ) = 0, there isactually no background electric field, whateverθ maybe. The phenomenon of pair creation never occurs,since the supersymmetric vacua have zero vacuumenergy and are thus stable. What is left then of the2π periodicity inθ? For theCPN model, the twistedsuperpotential is [5–7]

W(σ)= σ

4πlnN+1∏i=1

σ +mieΛ

(8)+ 1

4π

N+1∑i=1

mi lnσ +mieΛ

,

whereΛ is given by (6) withβ = N + 1. The fieldσis chosen so that

∑N+1i=1 mi = 0. The vacuum equation

W ′(〈σ 〉)= 0 reduces to

(9)N+1∏i=1

(〈σ 〉 +mi) =ΛN+1.

This equation hasN + 1 solutions that correspondgenerically toN +1 physically inequivalent vacua|i〉,1 � i � N + 1. The question we want to address isthen the following: can we trust instanton calculus, andin particular the quantization condition (4), in thosevacua?

An interesting property of the theory describedby (8), shared more generally by asymptotically freenon-linearσ models with mass terms [8], is that any ofthe vacua can be made arbitrarily weakly coupled bysuitably choosing the mass parameters. If|mi−mk| |Λ| for all i �= k, then the vacuum|k〉 characterized by〈k|σ |k〉 � −mk is weakly coupled because the coordi-nate fields onCPN have large masses|mi −mk|.2 Theexact formula (9) then predicts that〈k|σ |k〉 is given byan instanton series of the form

(10)〈k|σ |k〉 = −mk +∞∑j=1

c(k)j Λ

j(N+1),

2 The massesmi play the same role as Higgs vevs in four-dimensional gauge theories, and the coordinate fields are like Wbosons [8].

F. Ferrari / Physics Letters B 529 (2002) 261–264 263

where thej -instanton contributionc(k)j is a calcu-

lable function of the masses, for example,c(k)1 =1/

∏j �=k(mj − mk). Obviously, in the regime|mi −

mk| |Λ|, instanton calculus is valid: the expectationvalue〈k|σ |k〉 is a smooth, 2π periodic function ofθ ,

(11)〈k|σ |k〉(θ + 2π)= 〈k|σ |k〉(θ),as are the other correlators in thekth vacuum.

When the masses|mi −mk| decrease, the couplinggrows, and we enter a regime where the semiclas-sical instanton calculus is no longer reliable. In ourmodel, there is actually a sharp transition between aninstanton dominated semiclassical regime and a purelystrongly coupled regime. Mathematically, the transi-tion occurs because series inΛN+1 like (10) have afinite radius of convergenceR(mi). When|Λ|N+1 >

R(mi), all the deductions based on a naive discussionof instanton series can be invalidated.3 To be more spe-cific, let us consider the caseN = 1,m1 = −m2 =m.For |m| > |Λ|, the expectation values are given byconvergent instanton series deduced from (9),

〈1|σ |1〉 = −〈2|σ |2〉

(12)= −m∞∑j=0

(−1)j+1Γ (j − 1/2)

2√π j !

(Λ

m

)2j

.

However, for |m| < |Λ|, one must use the analyticcontinuation to (12), given by

(13)〈1|σ |1〉 = −〈2|σ |2〉 = −√m2 +Λ2.

A very important property of the analytic continua-tions is that they have branch cuts. Eq. (11) is no longervalid and is replaced by

(14)〈σ 〉1(θ + 2π)= 〈σ 〉2(θ).

The statement for arbitraryN is that the vacua arepermuted forθ → θ + 2π at strong coupling. Sincefor a generic choice of the massesmi theN + 1 vacuaare physically inequivalent, we see that in generalthephysics is not 2π periodic in θ .4 Let us note that ifwe sum up over all the vacua, as would be the case

3 For an analysis of the largeN limit of the analytic continua-tions, see [9,10].

4 In the caseN = 1, the vacua|1〉 and |2〉 are physicallyequivalent due to a specialZ2 symmetry of the theory, but forN � 2this does not occur.

in a calculation of the path integral in finite volume,then the 2π periodicity is restored. However, in infinitevolume, cluster decomposition implies that we shouldrestrict ourselves to a given vacuum. In that case,though the path integral is not 2π periodic, we stillhave

(15)〈σ 〉k(θ + 2π(N + 1)

) = 〈σ 〉k(θ).This is consistent with a modified quantization law

(16)(N + 1)∫Lθ/θ ∈ Z

that is to be compared with the classical quantizationlaw (4). Eq. (16) would correspond to so-called “frac-tional instantons”, field configurations with fractionaltopological charge 1/(N+1). The fractional instantonpicture remains elusive, however, because we are notable to exhibit the corresponding field configurations.5

A precise way to state the result obtained above isto say that, at strong coupling, the transformationθ →θ +2π corresponds to a non-trivial monodromy of thevacua, and that, interestingly,this monodromy is not asymmetry transformation. Another context where sim-ilar phenomena take place is four-dimensionalN = 2gauge theories. The analogy with theCPN model dis-cussed above was emphasized in [6], and fractional in-stanton contributions were computed in the largeNlimit in [9]. The formulas of [9] clearly show thatsome observables transform in a complicated way un-der θ → θ + 2π . The transformationθ → θ + 2π isactually implemented inN = 2 gauge theories by anon-trivial duality transformation, which is a symme-try of the low energy effective action. Unlike the mon-odromy transformation of our two-dimensional model,it could be a symmetry of the whole theory. To checkthis explicitly is however a highly non-trivial issue.

Acknowledgements

I would like to thank Adel Bilal and Edward Wittenfor useful comments. This work was supported in partby the Swiss National Science Foundation and by aRobert H. Dicke fellowship.

5 They cannot be smooth configurations of the microscopicfields, but might be naturally described in terms of composite fieldsthat enter in the quantum effective action.

264 F. Ferrari / Physics Letters B 529 (2002) 261–264

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