PHYSICAL REVIEW D 103, 025016 (2021)
Sigma model on a squashed sphere with a Wess-Zumino term
Daniel Schubring1 and Mikhail Shifman 2 1Physics Department, University of Minnesota, Minneapolis, Minnesota 55455, USA 2William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
(Received 18 December 2020; accepted 4 January 2021; published 15 January 2021)
A class of two-dimensional sigma models interpolating between CP1 and the SUð2Þ principal chiral model is discussed. We add the Wess-Zumino-Novikov-Witten term and examine the renormalization group flow of the two coupling constants which characterize the model under consideration. The model flows to the SUð2Þ Wess-Zumino-Novikov-Witten conformal field theory in the IR limit. There is an ordinary phase in which the model flows from the asymptotically free UV limit of the CP1 model coupled to an extra massless degree of freedom. At higher-loop order we find evidence that there is also a phase in which the model can flow from nontrivial fixed points in the UV. A nonperturbative confirmation of these extra fixed points would be desirable.
DOI: 10.1103/PhysRevD.103.025016
I. INTRODUCTION natural candidate for an effective field theory of a system with such global symmetry. For this reason, it has been well studied Nonlinear sigma models are often used as toy models in in the condensed matter community beginning with the N ¼ high energy physics to illuminate aspects of more realistic 2 case in a 1989 paper by Dombre and Read on quantum theories like QCD (see, e.g., [1]). But even seemingly antiferromagnets on a two-dimensional spatial triangular abstract models can turn out to be useful effective field lattice [3]. A 1995 paper by Azaria, Lecheminant, and theory descriptions of phenomena in both high energy Mouhanna study this model in both the weak coupling and physics and condensed matter. Some examples of the latter large N limits and contains many more references to work on which directly pertain to the target space under consideration this model from this era [4]. A more recent 2018 paper here are found in the study of frustrated spin systems [2]. considers this model as an effective field theory for anti- The sigma model we are considering here is a deforma- −1 ferromagnets on a two-dimensional triangular lattice with tion of both the Oð2NÞ model and the CPN model. The noncoplanar ordering, and also on three-dimensional pyro- Lagrangian is given by chlore lattices [5]. From the high-energy point of view, this is also an 1 † † 2 Lκ ∂n ∂n − κ n ∂n ; ¼ 2λ2 ð j j Þ ð1Þ interesting toy model, especially in two spacetime dimen- sions, which is connected to the high temperature limit of where n is an N-dimensional complex unit vector, jnj2 ¼ 1. the condensed matter models in three spacetime dimen- When κ ¼ 0, this is just the Oð2NÞ model, i.e., the sigma sions. So far the introduction of an interpolating parameter κ 2N−1 κ 1 1 in the Lagrangian seems rather ad hoc, but it arises model on S . When ¼ , the model becomes Uð Þ N−1 N−1 naturally by coupling the CP model to massless fields. gauge invariant and is the sigma model on CP .For N−1 intermediate values of κ, the target space is a “squashed To see this, note that the ordinary CP Lagrangian can 2 −1 be written with an auxiliary gauge field Aμ in order to make sphere” that is topologically equivalent to S N , but which has a deformed metric along the Uð1Þ fibers of the fiber the gauge symmetry obvious, bundle defined by the natural map from S2N−1 → CPN−1. 1 The κ deformation breaks the global Oð2NÞ symmetry of ∂ † ∂μ − μ 2 ð μ þ iAμÞn ð iA Þn: S2N−1 down to SUðNÞ × Uð1Þ, and so this sigma model is a 2λ
The auxiliary gauge field can then be coupled to a massless Published by the American Physical Society under the terms of Dirac fermion ψ, the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1 1 ’ † μ μ μ the author(s) and the published article s title, journal citation, ð∂μ þ iAμÞn ð∂ − iA Þn þ ψγ¯ ði∂μ þ AμÞψ; ð2Þ and DOI. Funded by SCOAP3. 2λ2 2α
2470-0010=2021=103(2)=025016(11) 025016-1 Published by the American Physical Society DANIEL SCHUBRING and MIKHAIL SHIFMAN PHYS. REV. D 103, 025016 (2021) where α is an arbitrary parameter we will eventually the impact of the WZNW term on the squashed sphere connect to κ. This Lagrangian in turn is connected to a sigma model, which is in some sense interpolating to the Stueckelberg field ϕ, through the bosonization map in two CP1 model in which the WZNW term does not make sense dimensions (2D), in two spacetime dimensions. This is not the first time the SUð2Þ PCM and the CP1 1 1 † μ μ 2 model have been connected through topological terms. In ð∂μ þ iAμÞn ð∂ − iA Þn þ ð∂μϕ − AμÞ : ð3Þ 2λ2 4πα the so-called Haldane conjecture [10], an antiferromagnetic Heisenberg spin chain with a half-integer spin is shown to For an example of bosonizing a Dirac fermion in a be equivalent to the CP1 model with a theta term set to background gauge field see, e.g., [6]. The gauge can θ π ϕ 0 ¼ . The antiferromagnetic Heisenberg spin chain in turn now be fixed so that ¼ , and the additional term flows to a massless Dirac fermion in the IR, which can be becomes a mass term for A, which breaks the confinement 2 N−1 equivalently represented as SUð Þ PCM with a WZNW of the CP model. When auxiliary A field is integrated 1 term at level k ¼ 1. So the CP model at θ ¼ π flows to the out we recover the squashed sphere Lagrangian (1) if we SUð2Þ WZNW model at level k ¼ 1 [11,12]. make the identification, However the squashed sphere model considered in the 1 κλ2 following is not the same as this flow. In the UV the model α looks like CP1 coupled to a massless fermion, as in (2).As ¼ 2π 1 − κ : ð4Þ usual this massless fermion will wash out any dependence So this formulation shows that the squashed sphere on a theta term we might try to define. Even so, we can still define a WZNW term for the squashed sphere model or Lagrangian is something rather natural to consider, just 1 being equivalent to a Higgsed CPN−1. But going back to the equivalently the Higgsed CP model, and it will have some original formulation (1) in terms of the parameter κ interesting consequences. emphasizes that it is somehow interpolating between the Various generalizations of this squashed sphere model, −1 both with and without a WZNW term, have been consid- CPN and Oð2NÞ models, which is interesting because the properties of these models are very different. In particular ered by a number of authors, particularly in regards to its −1 integrability (see below) and applications to AdS=CFT CPN has a nontrivial second homotopy group and can be (see, e.g., [13]). The classical integrability of the squashed modified by a θ term, whereas this cannot be defined in the sphere model was first shown by Cherednik in 1981 [14].A case of the Oð2NÞ models. On the other hand, for N ¼ 2, later rediscovery of the integrability [15] involves modi- the two components of the complex unit vector n can be fying the SUð2Þ current by a topological current so as to packaged into a matrix U ∈ SUð2Þ where preserve the flatness condition [16]. This result was also extended to the squashed sphere with a WZNW term, as we n1 n0 U ¼ : consider here, and the RG flow was calculated to one − n0 n1 loop [17,18]. The one loop RG flow was also calculated for a four- In terms of U, the Lagrangian (1) becomes parameter generalization of the squashed sphere target space [19] which in another limit reduces to the integrable 1 μ † κ μν 3 3 L ∂ U ∂μU − η IμIν; sausage model [20]. This four-parameter model and related ¼ 4λ2 Trð Þ 2λ2 ð5Þ target spaces were further considered in [13]. where Besides this four-parameter model, another generaliza- tion of the squashed sphere model is the Yang-Baxter 1 model [21] which is also integrable [22]. The Yang-Baxter 3 ≡ − †∂ τ Iμ 2 Trð iU μU 3Þ; model with a WZNW term was considered [23], and the classical flatness condition for the currents was shown to μν with τ3 the third Pauli matrix. Moreover, η is the persist to one loop. Two-loop renormalizability of such spacetime (or world sheet, depending on the interpretation) models were recently addressed in [24,25]. metric. In 2D Minkowski space ημν ¼ diagf1; −1g while in Other issues in higher-loop calculations in sigma models Euclidean ημν ¼ diagf1; 1g. with two or more coupling constants addressed in the The key point is that the first term is just the Lagrangian literature are crucial for what follows. For instance, of SUð2Þ principal chiral model (PCM), which makes sense Metsaev and Tseytlin [26] pointed out that the β functions since the target space S3 of the Oð4Þ model is equivalent to in such theories depend on a renormalization scheme the Lie group SUð2Þ. choice starting with two-loop order. This generally speak- The principal chiral model can also be modified by ing applies to our model as well as to the generic sigma a topological term: the Wess-Zumino-Novikov-Witten models with metric gab and two-form gauge field hab, see (WZNW) term [7–9]. In this paper we will investigate Sec. II C.
025016-2 SIGMA MODEL ON A SQUASHED SPHERE WITH A WESS- … PHYS. REV. D 103, 025016 (2021)
In our calculation we follow the pattern described in Sec. III C a particular RG trajectory is found that is argued detail in [27] (see also references therein). The background to be valid to all orders in perturbation theory. And finally, field method is used combined with dimensional regulari- in Sec. III D, nontrivial UV fixed points of the model are zation. This is insufficient to unambiguously determine found at two loop order. The possible nonperturbative two-loop terms. To eliminate the ambiguities we apply two existence of these fixed points is further discussed in the different consistency conditions which had been suggested conclusion, Sec. IV. in the past. The first constraint stems from string theory (to be referred to below as an integrability condition). II. INTRODUCING THE WZNW TERM Perturbative in α0 string calculations [26] are related to ours through the integrability condition. This will be A. Introducing the model discussed in Sec. III. We will begin by recalling the Lagrangian of the SUðNÞ The integrability requirement for the σ model β functions PCM, is insufficient to constrain them unambiguously. The second constraint comes from analysis of expansion in 1 μ † L ∂ U ∂μU ; the vicinity of the conformal point. For instance, the exact PCM ¼ 4λ2 Trð Þ ð6Þ nonperturbative result for the derivative of the β function at the critical point is known from [28]. This aspect is where U is a N × N matrix in SUðNÞ. This has SUðNÞL × incorporated in Sec. III A while in Sec. III B we carry SUðNÞR global symmetry, out some additional checks. As a result, at two loops we → −1 fully determine the β functions of the model. Three-loop UðxÞ VLUðxÞVR : analysis carried out in [27] leaves a few constants unde- termined. We briefly discuss this issue in Sec. II C, Up to a normalization factor, the Noether currents corre- deferring its solution for the future. sponding to these symmetries are In this paper our focus is on the two-loop RG flow, which † † leads to new features not seen in the one loop case. With the JL;μ ¼ i∂μUU ;JR;μ ¼ −iU ∂μU: ð7Þ second loop included, the RG flow drastically changes, in particular due to the emergence a second separatrix and one Of course, once additional terms are added to the action, the or more new nontrivial fixed points in the UV. In general, a Noether currents JL, JR will need to be modified, but these tentative existence of an extra fixed point from balancing original combinations of U matrices will still be useful. The one- and two-loop terms (with not necessarily small higher- Noether current under right isospin transformations here order corrections) cannot not proven. As a counterexample will also be referred to as I, and it may be expanded in we could refer to symmetric space models with negative terms of the Lie algebra basis τa, curvature (but without the Wess-Zumino-Novikov-Witten a † term) [29]. Higher-loop corrections are small if the param- Iμ ¼ Iμτa ≡ −iU ∂μU: eter k ≫ 1. Since the phenomenon we detected1 disappears at k ≥ 9 we can only hope that at k smaller than 9, but not The Lie algebra basis is normalized as too small, (say, k ¼ 7, Fig. 2), the extra fixed points survive. To complete the proof nonperturbative methods TrðτaτbÞ¼2δab: are needed. The paper is organized as follows. In Sec. II we will It will be convenient to reexpress the Lagrangian in terms of review briefly the squashed sphere model in the formu- these currents. lation (5) close to the SUð2Þ PCM, and we will explain how 1 X to add a WZNW term. The discussion closely follows that a;μ a L I Iμ: PCM ¼ 2λ2 ð8Þ of [30]. The general results on renormalizing a sigma model a with a WZNW term are reviewed briefly in Sec. II C. The RG equations of the squashed sphere sigma model Of course we could have equally well expanded in terms of with a WZNW term are found and discussed in Sec. III. JL instead of JR ¼ I, but as in [31] we will explicitly break Section III A discusses the ordinary regime in which the the SUðNÞL × SUðNÞR symmetry down to SUðNÞL × 1 1 model flows to the CP sigma model coupled to a massless Uð ÞR by adding extra terms depending on JR. fermion in the UV. Section III B begins the discussion of As far as eventually adding a WZNW term is concerned, the two loop results by comparing the loop expansion of the the interesting case is when N ¼ 2, in which case the beta functions to the expansion about the WZNW CFT. In squashed sphere sigma model Lagrangian is just [31] 1 1The additional fixed points show up at 2 025016-3 DANIEL SCHUBRING and MIKHAIL SHIFMAN PHYS. REV. D 103, 025016 (2021) where the upper index taking values 1,2,3 refers to the usual the manifold multiple times. In any case, as usual the Pauli matrix basis of the Lie algebra. When κ 0 this is the ambiguity in signed volume will be some integer multiple ¼ pffiffiffiffiffiffiffiffiffiffiffi ordinary SUð2Þ PCM, which is equivalent to the sigma of the total volume of the target space, 2π2λ−3 1 − κ. This 3 model on the sphere S . When κ ¼ 1, the Uð1Þ global ambiguity will be harmless in the path integral if we symmetry becomes a gauge symmetry, and the model is normalize it to be some integer multiple k of 2πi. So the equivalent to the sigma model on CP1. WZNW action is just Z B. Adding a WZNW term 2πik pffiffiffi S ¼ pffiffiffiffiffiffiffiffiffiffiffi dx3 g; WZNW 2π2λ−3 1 − κ In two spacetime dimensions, the only finite action field Z configurations are those which have a unique limit as the k 3ϵλμν spacetime argument goes to infinity, and thus these field ¼ 12π dx TrðIλIμIνÞ: ð12Þ configurations map out a two-dimensional surface in the squashed sphere target space homeomorphic to S3. Since Note that all dependence on the parameters λ and κ has the target space is three dimensional it is meaningful to canceled, so the WZNW term for the squashed sphere is consider the three-dimensional volume enclosed by the 3 field configuration in the target space. A term proportional exactly the same as for the unit sphere S . to this volume is exactly the WZNW term we will add to the Adding the WZNW action to the original action of the action. PCM (6), the Noether currents are modified. Using the This can be calculated by integrating over the volume same normalization as in (7), they become form on the squashed sphere, thus we will need an 2 expression for the determinant of the metric. In terms of † λ † J μ i∂μUU − k ϵμν∂νUU ; an orthonormal basis on the target space expressed in terms L; ¼ 2π a of the vielbeins eμ, it is easy to show that the determinant of λ2 − †∂ − ϵ †∂ the metric is JR;μ ¼ iU μU k 2π μνU νU: ð13Þ pffiffiffi λμν 1 2 3 g ¼ ϵ e eμeν: ð10Þ λ At a special value of the coupling constant, which we will λ2 The coordinates on the target space are not yet fixed, but call k, they can be chosen to be compatible with the spacetime coordinates. The two-dimensional spacetime coordinates 2 2π λ ¼ ; ð14Þ can be thought of as defining a coordinate system on the k k image of the field configuration UðxÞ in the target space. And if we introduce an arbitrary third coordinate, the image these currents reduce to one independent component in of UðxÞ can be continued throughout the bulk of the target holomorphic coordinates, z ¼ x0 þ ix1, space. This choice of coordinates is useful because the a quantity Iμ can be interpreted as the projection of the † † JL;z¯ ¼ 2i∂z¯UU ;JR;z ¼ −2iU ∂zU; ð15Þ coordinate vector ∂μ onto the left-invariant vector field corresponding to the Lie algebra element τa. And since the components of the target space metric can be read off the and the Noether current conservation law implies that JR;z sigma model Lagrangian, our Lagrangian (9) is telling us only depends on z, and JL;z¯ only depends on z¯. These that the left-invariant vector fields τa are orthogonal. In currents form a Kac-Moody algebra of level k and this is of a other words, up to a constant normalization, the currents Iμ course just the WZNW conformal fixed point first found can be identified with the vielbeins. in [9]. ffiffiffiffiffiffiffiffiffiffiffi Since we will be modifying the Lagrangian by terms of p μ ∝ pffiffiffi 1 − κ λμν 1 2 3 the form I Iμ IzIz¯ we will need to consider not only Iz, g ¼ ϵ IλIμIν; λ3 which is one component of a conserved current, but also Iz¯ pffiffiffiffiffiffiffiffiffiffiffi − 1 − κ which is no longer conserved and thus can have an i ϵλμν ¼ 3 TrðIλIμIνÞ: ð11Þ anomalous dimension. The scaling dimension of this 12 λ operator was calculated by Knizhnik and Zamolodchikov [28]. It takes the form 2Δ1 1, where, in the case of group Now we will integrate this over the interior of the field þ SU 2 , configuration, but since the target space is homeomorphic ð Þ to S3 there is some ambiguity in which side is considered the interior and which is the exterior. We could even allow 2 Δ1 ¼ : ð16Þ for the “interior” of the field configuration to wrap around 2 þ k 025016-4 SIGMA MODEL ON A SQUASHED SPHERE WITH A WESS- … PHYS. REV. D 103, 025016 (2021) C. Renormalizing WZNW models This will simplify some of the formulas for the β functions 2 The problem of renormalizing sigma models with a slightly. To two loops the beta function is given as [34] WZNW term has been considered by a number of authors ∂ 1 1 [32–34]. The key step is to rewrite the WZNW term in the μ g Rˆ c Rˆ Rˆ cdf ∂μ ab ¼ 2π abc þ 8π2 acdf b action as a two-dimensional integral on the same footing as the ordinary sigma action and then apply the same back- 1 − Rˆ Sd Sfgh; ð21Þ ground field method which works to find the renormaliza- ð2πÞ2 adfb gh tion of the sigma model without a WZNW term (an early ˆ example of which is found in [35]). The WZNW term in the where Rabcd is the Riemann curvature tensor R modified by action (12) may be rewritten as S in such a way that it has an interpretation as a Riemann Z curvature for a target space metric with torsion [32], i 3 ϵλμν∂ ϕa∂ ϕb∂ ϕc SWZNW ¼ dx Sabc λ μ ν ; ð17Þ ˆ f 3 Rabcd ≡ Rabcd − SfabS dc: ð22Þ where ϕa are the fields in the sigma model mapping to The last term in (21) involved some ambiguities in continuing the Levi-Civita tensor in dimensional regulari- coordinates in the target space, and Sabc is proportional to the volume form on the target space ω , zation which were fixed by matching the beta function to abc the dimension of the operator perturbing the conformal fixed point [33]. λ3 kffiffiffiffiffiffiffiffiffiffiffi For the ordinary (κ ¼ 0) SUð2Þ WZNW model, Sabc ¼ p ωabc: ð18Þ 2π 1 − k 2 Rabcd ¼ λ ðgadgbc − gacgbdÞ; Of course since S is a three form on a three-dimensional manifold, it is a closed form. And even on higher dimen- and given that S is proportional to the Levi-Civita tensor the f sional target spaces, the WZNW term may be written as an contraction SfabS dc produces an identical structure. So the integral over a closed form. This means that locally (but not full Riemann curvature with torsion Rˆ is globally), we may write S as the exterior derivative of a ˆ 2 2 two-form gauge field h, Rabcd ¼ λ ð1 − η Þðgadgbc − gacgbdÞ; ∂ where we have defined the useful parameter3 Sabc ¼ ½ahbc ; kλ2 λ2 and apply Stokes’s theorem to rewrite the action as η ≡ : 2π ¼ λ2 ð23Þ Z k i 2 μν a b S ¼ dx h ϵ ∂μϕ ∂νϕ ; ð19Þ When we are at the point η ¼ 1, the modified Riemann WZNW 3 ab ˆ curvature Rabcd ¼ 0 and thus the β function vanishes. And λ2 λ2 this is of course just the point where ¼ k which was which just looks like an antisymmetric version of the introduced above in the context of the current algebra. ordinary sigma model action, Z III. RG FLOW OF THE SQUASHED SPHERE WITH 1 − 2 ημν∂ ϕa∂ ϕb WZNW TERM Sσ ¼ 2 dx gab μ ν ; ð20Þ It is now a simple matter to calculate the β function when μν κ ≠ 0. Now the Riemann tensor will no longer take a form where g is the target space metric and η is defined after proportional to ðg g − g g Þ, but it can still be calcu- Eq. (5). Note that in our conventions all λ and κ dependence ad bc ac bd lated from the structure coefficients of the group as in, e.g., is absorbed into the definitions of g and h. [4,31]. Taking the left invariant vector fields corresponding The problem of finding the renormalization of a sigma model with general g and h has been solved up to three 2 loops [27], although there are still some ambiguities left to The expression for the derivative of the β function in (21) be cleared up, as we will discuss later. In our case, S is generally speaking must involve symmetrization in a, b on the right-hand side. However, in the case under consideration in proportional to the volume form, so its covariant derivative which the antisymmetric tensor hab does not run (corresponding vanishes, and it also satisfies the identity to the WZNW term) this is irrelevant, as discussed above Eq. (21). 3 η h 0 The parameter in (23) and below (without indices) is not to S½ab Sc gh ¼ : be confused with the spacetime metric ημν. 025016-5 DANIEL SCHUBRING and MIKHAIL SHIFMAN PHYS. REV. D 103, 025016 (2021) to the standard Pauli matrix basis for the Lie algebra as a Also recall that the antisymmetric S tensor is proportional basis for the tangent space, the metric is to the volume form ω, 1 1 − κ g11 ¼ g22 ¼ 2 ;g33 ¼ 2 ; ð24Þ λη λ λ ffiffiffiffiffiffiffiffiffiffiffi Sabc ¼ p ωabc: ð26Þ 1 − k and the Riemann tensor Rabcd ¼ R½ab ½cd is 1 1 2 Applying the general two-loop RG equations (21), we find R1212 ¼− ð1þ3κÞ;R1313 ¼R2323 ¼− ð1−κÞ : ð25Þ λ2 λ2 beta functions for λ and κ, d 1 1 1 η 1 þ κ 1 μ ¼ ð1 þ κÞ − η2 þ 1 þ 2κ þ 5κ2 − 4 η2 þ 3 η4 ; dμ λ2 π 1 − κ πk 1 − κ ð1 − κÞ2 d 1 − κ 1 η 1 μ ¼ ½ð1 − κÞ2 − η2 þ ð1 − κÞ3 − 4ð1 − κÞη2 þ 3 η4 : ð27Þ dμ λ2 π πk ð1 − κÞ When κ ¼ 0 this agrees with the usual two-loop RG perturbation theory, but we will have more to say on this κ ≠ 0 equation for the WZNW model [33,34]. When but later. pffiffiffiffiffiffiffiffiffiffiffi k ¼ 0, which implies η ¼ 0 and η=k ¼ λ2=2π, then the RG Actually, as shown in [17,23], the separatrix η ¼ 1 − κ equations agree with the two-loop equations first found is exactly the condition needed for the SUð2Þ currents at in [4]. nonzero κ to satisfy the flatness condition of [16] without modification by a topological current. The condition η pffiffiffiffiffiffiffiffiffiffiffi ¼ A. Below the first separatrix 1 − κ solving the RG equations to one loop can be seen as a specific case of the preservation of the classical integra- Let us first consider the one-loop behavior. The RG flow bility condition for Yang-Baxter models with a WZNW is plotted in Fig. 1. Everythingffiffiffiffiffiffiffiffiffiffiffi flows to the WZNW CFT in p term to one loop as found in [23]. Note that the exact the IR. The curve η ¼ 1 − κ solves the RG equations, and pffiffiffiffiffiffiffiffiffiffiffi η 1 − κ is in fact a separatrix determining two classes of UV condition ¼ is no longer preserved by the RG behavior. Below the separatrix every trajectory flows in the equations to two loops, although there is still a separatrix UV to the asymptotically free fixed point related to CP1. that approximately satisfies this for largepkffiffiffiffiffiffiffiffiffiffiffi. η ≪ 1 − κ And above the separatrix the coupling constant η and thus Far below the separatrix, for , the RG λ2 appears to diverge in the UV. This divergence is quite equations reduce to the RG equations for the squashed possibly an artifact of taking only a finite order in sphere sigma model without a WZNW term. There is an RG invariant in this case first found in [4], ffiffiffi pκ λ2 1.5 K ¼ 1 − κ : As pointed out in [31], this value K is the exponent of 1.0 power law behavior of two-point functions of the unit vector n field in the UV. This makes sense considering the representation of the squashed sphere model in terms of a Stueckelberg field ϕ discussed earlier, see Eq. (3). In the 0.5 UV, where κ ≈ 1 and λ ≈ 0, the correlation function is being dominated by a phase factor expðiϕÞ, where ϕ is essentially a massless free field. In terms of the parameter α in (4), this has two-point function, 0.0 0.0 0.2 0.4 0.6 0.8 1.0 − ϕ ϕ 0 −α − K he i ðzÞei ð Þi ∝ jzj ≈ jzj 2π ; FIG. 1. RG flow for the squashed sphere sigma model with a which is exactly the value of the exponent found in [31]. ϕ WZNW term to onep loop.ffiffiffiffiffiffiffiffiffiffiffi The flow is pointing towards the IR. Note in passing that we had been considering in (3) as The separatrix η ¼ 1 − κ is plotted in red. a Stueckelberg field coupled to the gauge field A,butwe 025016-6 SIGMA MODEL ON A SQUASHED SPHERE WITH A WESS- … PHYS. REV. D 103, 025016 (2021) can fix the gauge by making one of the components of n B. Testing the loop expansion real, and then integrate out A. In doing so, the sigma model The RG equations (27) are based on a loop expansion of ϕ Lagrangian takes a form where has the interpretation as the action. Our model has three parameters, λ2 (or equiv- 1 the coordinate along the Uð Þ fibers of the squashed sphere alently η), κ, and the discrete parameter k. So it perhaps is target space in a coordinate system closely related to the not clear at first which small parameter we are expanding Fubini-Study coordinates (detailed in the appendix of [31], in. But recall that the action (9) implies see also [36]). The coupling between ϕ and the CP1 degrees of freedom involves an extra factor of λ2 and so Z ϕ κ → 1 λ2 → 0 k 2 1 2 2 2 1 − κ 3 2 the field decouples in the , limit. S ¼ 4πη dx ½ðI Þ þðI Þ þð ÞðI Þ This means that the squashed sphere model at κ ≈ 1 is Z 1 not a small perturbation on CP alone, the additional k 3 λμν þ dx ϵ TrðIλIμIνÞ: degree of freedom given by ϕ is very important to the 12π behavior of the correlation functions. This can also be seen by considering the Zamolodchikov c theorem [37]. The Since k−1 multiplies each term in the same position as ℏ,it central charge at the WZNW fixed point is should be treated as the loop expansion parameter, and indeed each term of the RG equations (27) has a definite −1 3k power of k . At large k successive loop contributions are c ¼ ; suppressed. k 2 −1 þ As a consistency check this expansion in k can be compared with a perturbative expansion about the WZNW and, since the central charge monotonically decreases along fixed point action S0, in a calculation similar to the κ ¼ 0 the RG flow and we are free to take k as large as we want, case [33,34]. If the CFT action S0 is perturbed by operators i there must be a central charge of at least three in the UV. O each of which has a well-defined scaling dimension Δi, This emphasizes the point that the theory near κ ¼ 1 can 1 not be just the CP sigma model, which has a central charge i S ¼ S0 þ g O ; of 2 at its asymptotically free UV fixed point. In order for i the flow to be consistent we need the extra degree of μ μ0 freedom given by the ϕ field. then under a RG coarse graining from to , to lowest This is in contrast to the situation with the Haldane order in the small parameter gi the action will transform to 1 θ π conjecture [10,11]. There we are considering CP ð ¼ Þ Δ flowing to only the level k ¼ 1 WZNW model, which has μ0 i μ0 Oi ≈ 1 Δ Oi central charge c ¼ 1. So, as pointed out in [12],itis S0 þ gi μ S0 þ gi þ i log μ : perfectly consistent for the UV behavior to be given by the asymptotically free fixed point of the CP1 model alone. So to the lowest order the β function for g is just, Returning now to our discussion of K and the correlation i functions, note that even with the WZNW term, the β Δ O 2 quantity K has an unambiguous definition below the gi ¼ gi i þ ðg Þ: ð28Þ separatrix through the limit of λ2=ð1 − κÞ in the UV. Correlation functions of n now have power law behavior In our case the total action is in both the UVand IR. As just mentioned, the UV behavior K Z is determined by the value of , and the IR behavior is 1 − η 1 given by the scaling dimension of n field in the WZNW k 2 1 2 2 2 S ¼ S0 þ 2π η dx 2 ½ðI Þ þðI Þ CFT. Since the field n is just components of U, which is a Z primary field in the WZNW CFT, it has scaling dimension k 1 − κ − η 1 þ dx2 ðI3Þ2: 2Δ1=2, where Δ1=2 was calculated in [28], 2π η 2 3=4 So the coefficients of the operators perturbing the fixed Δ1 2 ¼ : = k þ 2 point action are k 1 − η k 1 − κ − η Furthermore, as in the squashed sphere model without a ≡ ≡ gη 2π η ;gκ 2π η : ð29Þ WZNW term, trajectories with small values of K pass near the κ ¼ 0, η ¼ 0 fixed point and so display crossover behavior associated to the UV behavior of the SUð2Þ PCM. Expanding the beta functions (27) in these parameters, 025016-7 DANIEL SCHUBRING and MIKHAIL SHIFMAN PHYS. REV. D 103, 025016 (2021) 1 4 8 1 − κ X 2π X X d 2 d n n μ ¼ − gη þ Oðg Þ; μ ¼ gκ ia þ gη ði þ jÞa dμ λ2 k k2 dμ λ2 knþ1 i;j i;j n i;j i;j 1 − κ 4 8 d 2 O 2 ¼ − gκ þ Oðg Þ: ð30Þ þ ðg Þ: dμ λ2 k k2 But as in (28) there should be no first order gη term. So 2Δ 1 And indeed, using the dimension 1 þ Rof the operator Iz¯ besides (31), there is also the condition 2 a 2 in (16), the dimension of the operators dx ðI Þ is X ði þ jÞan ¼ 0: ð32Þ 4 4 8 i;j −3 i;j 2Δ1 ¼ ¼ − þ Oðk Þ: 2 þ k k k2 This condition can also be derived by considering the 1=λ2 So the loop expansion to two loops agrees with conformal beta function at κ ¼ 0 and demanding that the dimension 2 perturbation theory to first order. agrees with the ð1 − κÞ=λ beta function at gη ¼ 0,as in (30). C. An exact RG trajectory The point is now that a simple way to satisfy this second η 1 − κ condition (32),isifi þ j is a constant for all nonvanishing Notice that the straight line ¼ satisfies the RG an , in which case it simply reduces to the first condition equations (27) to two loops. The meaning of this may be i;j (31). And, as mentioned above, this is exactly what is clarified by considering the parameters perturbing the needed for η ¼ 1 − κ to solve the RG equations. So while WZNW fixed point (29).Ifgκ is set to zero at one scale, 3 2 this is not a proof, it seems rather plausible that η ¼ 1 − κ, it remains zero under the RG flow. No ðI Þ perturbation on or equivalently gκ ¼ 0, is preserved under the RG flow to the fixed point is generated through renormalization. all orders. In fact, we conjecture that this will hold not just to two loops, but to all orders in perturbation theory. The reason that η ¼ 1 − κ satisfies the RG equations is that the β D. Additional fixed points 2 function for ð1 − κÞ=λ (or equivalently, for gκ) is of the In Fig. 2, the RG flow is plotted to two loops. The form (27) behavior below the first separatrix, whichpffiffiffiffiffiffiffiffiffiffiffi is now a slight deformation of the one loop result η ¼ 1 − κ, is quali- d 1 − κ 1 1 tatively the same as for one loop, but the behavior above the μ ¼ ½a0 ð1 − κÞ2 þ a0 η2 þ ½a1 ð1 − κÞ3 dμ λ2 k0 2;0 0;2 k1 3;1 first separatrix is dramatically different. There is now a second separatrix, and between the first and second a1 1 − κ η3 a1 1 − κ −1η5 : þ 1;3ð Þ þ −1;5ð Þ separatrices the trajectories flow to the WZNW CFT in the IR, but flow to one or more new nontrivial fixed points n Here aj;k is just notation for the numerical coefficient of the in the UV, plotted in red in Fig. 2. term ð1 − κÞjηk in the (n − 1)-loop correction to the beta Where the second separatrix meets κ ¼ 0 there is an IR function. Since κ ¼ 0, η ¼ 1 is a fixed point, we must have unstable fixed point. Setting κ ¼ 0 in the RG Eqs. (27), and the condition dividing out the factor of ð1 − η2Þ, we see that the unstable X fixed point is at the value η ¼ η0, which is the real root of n 0 ai;j ¼ : ð31Þ the cubic equation i;j 3 3η0 − η0 − k ¼ 0: ð33Þ So, given this condition, the reason why η ¼ð1 − κÞ solves the RG equations is because at each loop order n, the sum When k is in the range 2 Expanding to first order in gη, gκ we have And we can also solve for the RG equation of κ alone, 025016-8 SIGMA MODEL ON A SQUASHED SPHERE WITH A WESS- … PHYS. REV. D 103, 025016 (2021) 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 2. RG flow for the squashed sphere sigma model with a WZNW term to two loops. On the left is the level k ¼ 7 and on the right is k ¼ 9. The green dot is the ordinary WZNW CFT, and the red dots are additional fixed points. The two fixed points with κ > 0 disappear for levels k ≥ 9. dκ 4η 2η unstable fixed point scales as k−2=3, naïvely it appears to be μ ¼ − κ ð1 − κÞþ ðð1 − κ2Þ − 2η2Þ : dμ k k in the weak-coupling regime when k is large. There are actually three-loop results available that can β κ 0 4 The fixed points ðκ0; η0Þ are roots of the bracketed poly- illuminate the issue [27]. The function at ¼ is nomials on the rhs of these two equations, and these can be κ 1 1 η further manipulated so that 0 is given by the two real roots μ d 1 − η2 1 − η2 1 − 3η2 2 ¼ ð Þþ ð Þð Þ of a single quartic equation, and then η0 can be easily found dμ λ π πk from the value of κ0, η2 1 − η2 8 1 − η2 1 − η4 þ 2 ð Þ½ þð Þq2 þð Þq4 : 2 2 2πk 4ð1 − κ0Þð1 þ 3κ0Þð1 þ 5κ0Þ − k ¼ 0; ð34Þ ð36Þ k η The numerical parameters q2 and q4 in (36) are scheme 0 ¼ 10κ 2 : ð35Þ 0 þ dependent; in particular, they depend on the dimensional regularization of the Levi-Civita symbol. The authors of This quartic equation also indicates an upper bound on k for [27] were unable to fix them by first-order perturbation ≥ 9 which these two extra fixed points exist. At k , these theory about the WZNW CFT. So they are somewhat two fixed points disappear as in the right side of Fig. 2, and −1 questionable and below we will use the values suggested in since the loop expansion is essentially an expansion in k , [27] only for the purpose of orientation, it is not clear whether these two fixed points are an artifact of the loop expansion or not. 10 5 − − q2 ¼ 3 ;q4 ¼ 3 : ð37Þ IV. DISCUSSION OF RESULTS With the above remark in mind, we will nevertheless have a In fact it is not clear whether the new unstable fixed point look on whether the unstable fixed point (33) survives after κ 0 2 at ¼ which solves (33) is an artifact or not. Of course at including the three-loop terms. Dividing out the 1 − η κ 0 2 ¼ this is just the ordinary well-known SUð Þ WZNW factor associated with the WZNW fixed point, and keeping model, so it is perhaps surprising that there would be an only the highest order terms in η at each loop, we have the additional fixed point. But this fixed point does already fixed points as solutions to the equation appear in the two-loop RG equations published long ago [33,34]. q4 − η6 − 3η3 þ k ¼ 0: Note that the loop expansion may be expected to be more 2k accurate at high k, and according to (33) the η coordinate of 1=3 this fixed point scales as η0 ∼ k at large k. Although η 4 η As was mentioned in Sec. I, RG scheme dependence increases with k, the parameter actually multiplies the ambiguity at three loops has not yet been resolved (for a related action in the combination 2πη=k ¼ λ2. And since λ2 of the discussion in the pure metric case see, e.g., the Appendix in [24]). 025016-9 DANIEL SCHUBRING and MIKHAIL SHIFMAN PHYS. REV. D 103, 025016 (2021) The unstable fixed point survives for large k as long as might be useful to go beyond perturbation theory in k−1, q4 > −9=2, which is satisfied if the value given in [27] is and compare to conformal perturbation theory about the correct. What is more, given q4 < 0, there is a second fixed WZNW CFT at higher orders in gη, gκ. point which is stable in the IR and first appears at To summarize, in this paper we studied a two-dimen- three loops. sional field theory which is a deformation of the SUð2Þ Although these three-loop results are only for κ ¼ 0, let WZNW model that explicitly breaks one of the two SUð2Þ us consider what this might imply about the κ > 0 behavior. global symmetries in the UV. The RG flow is partitioned by As seen in Fig. 2, at the order of two loops, all RG separatrices. Below the first separatrix, all trajectories flow trajectories above the second separatrix flow to strong in the UV to the weak-coupling fixed point of the CP1 coupling in the IR. Now given there is an IR stable fixed sigma model coupled to a massless fermion or Stueckelberg point that appears at three loops, it is very tempting to field. Above the first separatrix, to lowest order all conjecture that the new stable fixed point at three loops lies trajectories appear to flow to a Landau pole in the UV, at the end of a new third separatrix, above which all but as higher loops are added these trajectories appear to trajectories flow to strong coupling in the UV. If the pattern instead flow to new “asymptotically safe” nontrivial fixed continues, at each loop order there could be a new points. 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