breaking as a quantum

Holger Gies, Franziska Synatschke and Andreas Wipf Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany

We explore supersymmetry breaking in the light of a rich fixed-point structure of two-dimensional supersymmetric Wess-Zumino models with one supercharge using the functional group (RG). We relate the dynamical breaking of supersymmetry to an RG relevant control param- eter of the superpotential which is a common relevant direction of all fixed points of the system. Supersymmetry breaking can thus be understood as a quantum phase transition analogously to similar transitions in correlated fermion systems. Supersymmetry gives rise to a new superscaling relation between the associated with the control parameter and the anomalous dimension of the field – a scaling relation which is not known in standard spin systems.

PACS numbers: 05.10.Cc,12.60.Jv,11.30.Qc

Supersymmetry is an important guiding principle for transition between the semi-metallic and the superfluid the construction of models beyond the standard model phase as discussed in the context of ultracold atoms [2] of particle , as it helps parameterizing the prob- or graphene [3]. As this transition occurs at zero temper- lem of a large hierarchy of scales, supports the unification ature, the critical value of the control parameter marks of gauge couplings, and facilitates an intricate combina- a quantum critical point of a quantum phase transition. tion of internal symmetries with the Poincar´egroup. On A standard approach to quantum phase transitions in the other hand, supersymmetry must be broken in na- strongly correlated fermion systems is Hertz-Millis theory ture in a specific way without violating these attractive [4, 5] where composite bosonic fields are introduced by features and other constraints imposed by the standard- a Hubbard-Stratonovich transformation. Subsequently, model precision data. In addition to a variety of break- the fermionic degrees of freedom are integrated out, leav- ing mechanisms often involving further hidden gauge and ing a purely bosonic description which is dealt with in a particle sectors, dynamical supersymmmetry breaking [1] local approximation. As this strategy is not always suffi- can play a decisive role in the formation of the low-energy cient, a treatment of fundamental fermionic and compos- standard model. Generically, symmetry breaking is re- ite bosonic degrees of freedom on equal footing beyond lated to collective condensation phenomena, often requir- Hertz-Millis theory has recently proved successful [6]. ing a nonperturbative understanding of the fluctuation- Now, supersymmetric systems by definition are com- induced dynamics. bined bosonic-fermionic systems with a high degree of From the viewpoint of , symmetry symmetry. A treatment of these degrees of freedom on breaking and phase transitions between ground states equal footing is even mandatory to preserve supersym- of a differently realized symmetry are often related to metry manifestly. Apart from this symmetry, we will fixed points of the (RG). In ad- show that supersymmetry breaking has many similari- dition, the quantitative properties of the system near a ties to a quantum phase transition in strongly correlated phase transition are determined by the RG flow in the fermion systems. Beyond these similarities, supersymme- fixed-point regime. As the understanding of symmetry try invokes additional structures which go beyond those breaking phenomena in quantum field theory and parti- known in statistical physics. cle physics has often profited from analogies to statistical Euclidean two-dimensional = 1 Wess-Zumino mod- N systems, this letter explores supersymmetry breaking in els can be defined by the Lagrange density in the off-shell the light of phase transitions and critical phenomena. formulation = kin + int with L L L As a simple example, we concentrate on Wess-Zumino 1 µ i i µ 1 2 models with = 1 supercharges in two dimensions in kin = ∂µφ∂ φ + ψ∂ψ/ ∂µψγ ψ F , (1) L 2 4 − 4 − 2 Euclidean spacetime.N Dynamical supersymmetry break- 1 ′′ ′ ing in this and many other supersymmetric systems is int = W (φ)ψγ∗ψ W (φ)F, (2) L 2 − governed by a control parameter of the superpotential which plays the role of a bosonic mass term. This is very with superpotential W (φ), bosonic field φ, fermionic reminiscent to phase transitions in strongly correlated fields ψ and ψ and auxiliary field F . The prime denotes fermionic systems: here, symmetry breaking is governed the derivative with respect to φ, and we use a chiral rep- by a composite bosonic order parameter, e.g., a super- resentation for the γ matrices. This model allows for fluid condensate, the mass term of which serves as the dynamical supersymmetry breaking if the highest power control parameter of the phase transition. For instance, of the bare superpotential is odd. the inverse interaction strength of attractively interact- We use the functional RG to calculate the effective ing electrons on a honeycomb lattice governs the phase action Γ of the theory. The functional RG can be formu- 2 lated as a flow equation for the effective average action Γk Let us start with the lowest-order derivative ex- [7]. This is a scale-dependent action functional which in- pansion, corresponding to a local-potential approx- terpolates between the classical action S = ( + ) imation for the superpotential. In this trun- Lkin Lint and the full quantum effective action Γ. TheR interpola- cation, the effective action is approximated by 1 ′′ ′ tion scale k denotes an infrared (IR) regulator scale, such Γ = dτ + W (φ)ψγ∗ψ W (φ)F with a k- k Lkin 2 k − k that no fluctuations are included for k Λ (with Λ be- dependentR superpotential Wk. Projecting the flow equa- → ing the UV cutoff), implying Γ → S. For k 0, all tion (3) onto this truncation leads to the flow equation k Λ → → fluctuations are taken into account such that Γ → Γ for the superpotential, k 0 → is the full quantum solution. The effective average action 2 ′′ can be determined from the Wetterich equation [7] k Wk (φ) ∂tWk(φ)= 2 ′′ 2 , (4) − 4π k + Wk (φ) −1 1 (2) k ∂tΓk = STr Γ + Rk ∂tRk , t = ln , (3) where we have used the simplifying regulator choice r = 2  k  Λ 1 h i 0 and r = ( k/p 1) θ 1 (p2/k2) . 2 | |− − which defines an RG flow trajectory in theory space The corresponding flow equation for the dimensionless (2) potential wk = Wk/k admits a variety of nontrivial fixed- with S serving as initial condition. Here, Γk = Z −→ ←−  ab point solutions: already a polynomial expansion for 2 δ δ antisymmetric superpotentials to order n, δΨa Γk δΨb where the indices a,b summarize field com- ponents, internal and Lorentz indices, as well as space- ′ ∂w time or momentum coordinates, i.e., ΨT = (φ, F, ψ, ψ¯). w k = λ(φ2 a2)+b φ4 +b φ6 + +b φ2n, (5) k ≡ ∂φ − 4 6 · · · 2n The regulator function Rk is introduced in the form of a quadratic contribution to the classical action, ∆Sk = reveal n independent solutions to the fixed-point equa- T Ψ RkΨ. Rk can thus be viewed as a momentum- tion ∂twk = 0, in addition to the Gaußian fixed point[20]. dependentR mass term, implementing the IR suppression The latter has infinitely many relevant directions in of modes below k. Different functional forms of Rk cor- agreement with perturbative power-counting in two di- respond to different RG schemes. Physical quantities do mensions. The number of relevant RG directions de- not depend on the regulator. creases for the other fixed points, culminating in a maxi- For a manifestly supersymmetric flow equation, we de- mally IR-stable fixed point with only one relevant direc- rive the regulator from a D term of a quadratic superfield tion. This relevant direction exactly corresponds to the operator, similarly to our construction for supersymmet- parameter a2 in the superpotential, and it is common ric quantum mechanics [8]. The most general supersym- to all other fixed points as well. As a2 is related to the metric cutoff action in the off-shell formulation in mo- bosonic mass term, we introduce a critical exponent νW mentum space is given by for this direction, such that a2 scales as a2 k−1/νW at a fixed point. This critical exponent thus corresponds∼ to 1 2 2 2 2 ∆Sk = d p(r1ψγ∗ψ 2r1φF +p r2φ +r2ψpψ/ r2F ). the negative inverse of the associated eigenvalue of the 2Z − − stability matrix. The exponent νW plays a similar role for the superpotential, as the exponent ν for the effective The regulator shape functions r = r (p2) determine 1,2 1,2 potential in Ising-type systems. At leading order in the the precise form of the Wilsonian momentum shell inte- derivative expansion, we obtain from Eq. (4): ν = 1. gration. The form of ∆S guarantees that the whole RG W LO k Fixed-point potentials and critical exponents at| next-to- trajectory remains in the hypersurface of supersymmet- leading order will be discussed below. ric action functionals even within approximative calcula- In order to clarify the role of this relevant direction tions; see [9, 10] for further supersymmetric RG studies. and its relation to supersymmetry breaking, we study In general, the full quantum effective action consists the phase diagram of a Gaußian Wess-Zumino model at of all possible operators which are compatible with the infinite volume. This is defined in terms of a quadratic symmetries of the theory. In this Letter, we use a super- perturbation of the Gaußian fixed point at the UV cut- covariant derivative expansion to systematically classify off Λ: W ′ = λ¯ (φ2 a¯2 ). The dimensionless control these operators. Truncating the expansion at a given or- Λ Λ Λ parameter δ and coupling− γ can be associated with der constitutes a consistent approximation scheme. In a variety of bosonic and fermionic systems, such expan- −1 ¯ 2 −1 ¯ δ := Λ λΛa¯Λ, γ := 2Λ λΛ, (6) sions have proven to capture the physics of phase transi- tion and critical phenomena quantitatively, see, e.g. [11]. where the scale is set by the UV cutoff Λ. In the follow- For our quantitative results presented below, we observe ing, we determine the region for the parameters δ, γ for a satisfactory convergence: leading-order results generi- which supersymmetry is dynamically broken in the IR. cally receive (1 10%) corrections at next-to-leading A criterion for supersymmetry breaking is provided by order, increasingO − up to 50% for observables in the a nonvanishing ground state energy, given by the min- ∼ 1 ′ 2 infinite-coupling limit. imal value of V (φ) = 2 (Wk→0(φ)) . As a true phase 3

2 2 2 t= 0.0 by the non-universality of the quadratic initial conditions W'Λ(φ)=Λ(0.5γφ -δ ) δ γ t= - 0.6 near the Gaußian fixed point, implying an RG scheme de- ] =0.03, =0.2

2 1.5 t= - 3.0 Λ t= - 10 pendence of the critical value). [ -3 The mass of the lowest bosonic excitation is given by 1 )*10

φ the curvature of the full renormalized effective potential ( 2 k at the minimum. In the supersymmetric phase, fermionic

W' 0.5 and bosonic masses are identical and nonzero. In the bro- 0 ken phase, the lowest fermionic excitation is a Goldstino -0.8 -0.6 -0.4 -0.2 0 φ 0.2 0.4 0.6 0.8 with vanishing mass. The latter holds also at finite k 0.26 in the supersymmetry-broken regime. As the flow in the 0.24 unbroken SuSy 0.22 IR is attracted by the maximally IR-stable fixed point, 0.2 the bosonic mass in the fixed-point regime is governed by 0.18 the only relevant direction a2. As the minimum of the 0.16 δ 0.14 effective potential in the broken regime is necessarily at 0.12 10 vanishing field, the bosonic mass yields 0.1 broken SuSy φ φ8 − 1 0.08 4 2 2 2 2 2 νW φ mk =2k λ a k , (9) 0.06 φ2 | |∼ 0.04 0 2 4 6 8 10 where the latter proportionality holds in the vicinity of γ/2 the IR fixed point where λ const. and a2 is governed FIG. 1: Top: Typical flow of the superpotential in the by the critical exponent ν .→ For ν > 1/2, the bosonic supersymmetry broken phase. Bottom: Phase diagram in W W the coupling–control-parameter plane for the Gaußian Wess- mass scales to zero upon attraction by the maximally IR- Zumino model. The lowest set of data points corresponds to stable fixed point. We conclude that the broken phase the lowest-order result (8). Higher orders converge rapidly. remains massless in both bosonic and fermionic degrees of freedom. On the other hand, the limit k 0 is an idealized IR transition only occurs in the infinite-volume limit [12], limit. Any real experiment as well→ as any lattice simu- the ground state energy can only be identified at k 0. lation will involve an IR cutoff scale k characterizing → m Four snapshots of a typical flow of the superpotential in the measurement, e.g., the scale of momentum trans- the broken phase are depicted in Fig. 1 (top) for δ =0.03 fer, detector size or lattice volume. Any measurement and γ =0.2. is therefore not sensitive to k 0 but to k km > 0. For a simple estimate of the phase diagram, we trun- We conclude that a measurement→ of the bosonic→ mass in cate the superpotential at order φ2 in Eq. (5). From the broken phase will give a nonzero answer proportional Eq. (4), we obtain the flow of the dimensionless couplings to the measurement scale, whereas the goldstino will be truly massless. The bosonic mass in both phases as a 2 2 3 2 1 6λ a 6λ function of the control parameter δ is depicted in Fig. 2 ∂ta = , ∂tλ = λ + , (7) 2π − π − π (top) for a coupling of γ =0.2. which can be solved analytically. The phase transition At next-to-leading order in the derivative expansion, curve in the (γ,δ) plane is given by the initial values for the fields acquire a wave function renormalization Zk, which a2 → changes sign. This condition yields which is a multiplicative factor of the kinetic term in k 0 2 | Eq. (1), kin Zk kin. The flow of Zk gives rise to an L → L 2 arcsin(α) π anomalous dimension, η = ∂t ln Z . Convergence of the δ = , α2 =1 . (8) − k √24π α − 3γ derivative expansion requires η . (1). From the polynomially expandedO superpotential flow Beyond this simple estimate, the phase transition curve at next-to-leading order we can derive the following flow can, of course, be identified by a full numerical integra- of the renormalized parameter a2: tion of the flow equation. Already upon inclusion of 2 higher orders of Eq. (5), we observe a rapid convergence 2 1 η η 2 at ∂tat = 1 1 at ∂tλt. (10) of the phase boundary; see Fig. 1 (bottom). There exists 2π  − 4  −  − 2  − λt a critical value for δcr characterizing the phase transition At any fixed point with ∂tλ 0, we can read off that in the strong coupling limit γ . Our best estimate a2 again denotes a relevant direction→ with scaling a2 → ∞ −1/νW ∼ from a numerical higher-order solution is δcr 0.263; k , where (the lowest-order result from Eq. (8) is π/96 ≃0.181). ≃ 2 We conclude that supersymmetry canp never be broken ν = . (11) W 2 η dynamically above this critical value. This agrees qual- − itatively with earlier results in the literature [12–15] (a This remarkable superscaling relation connects the su- quantitative comparison, e.g., with the lattice is inflicted perpotential exponent νW with the anomalous dimension 4

defines a distinct asymptotically safe UV completion of # nodes η νW the Wess-Zumino model, i.e., RG trajectories emanating 0 0.4386 1.2809 from different fixed points correspond to different theo- 1 0.20 1.11 ries. Our findings are reminiscent of those in 2D scalar 2 0.12 1.06 field theories [16, 17], where the fixed points of the effec- tive potential can be related to conformal field theories TABLE I: Critical exponents of the first fixed points. [18]. The connection between the present supersymmet-

ric models at their fixed points and conformal field theo- 6 ries remains an interesting question. t= - 9 5 t= -10 broken SuSy unbroken SuSy To summarize, we have constructed a manifestly su- t= -11 t= -12 persymmetric flow equation for 2D = 1 Wess-Zumino 4 ] N Λ ] t= -13 models. These models have nontrivial fixed-point super- Λ

[ 3 k

/25 [ potentials which can be classified by their relevant direc- k m

2 m tions. All fixed points share a relevant direction in the

1 form of a bosonic mass term. At next-to-leading order, the associated critical exponent νW and the anomalous 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 dimension η satisfy a superscaling relation which has no δ 5 analogue in Ising-type spin systems. η=0.4386 η The fixed point of this relevant direction corresponds 4 =0.20 η=0.12 to the critical point separating the supersymmetric from 3 the broken phase. The initial condition for this rele- )

χ 2 vant direction defines a control parameter, such that su- u( persymmetry breaking can be understood as a quantum 1 phase transition. For the Gaußian Wess-Zumino model, 0 we observe that the control parameter stays finite even at -1 arbitrarily large coupling in accord with a general argu- 0 0.5 1 χ 1.5 2 2.5 ment by Witten. In the broken phase, our superscaling FIG. 2: Top: Scale dependence of the renormalized bosonic relation predicts that the measured bosonic mass is pro- mass. Bottom: Fixed-point superpotentials at next-to- portional to the momentum scale set by the detector. ′′ 2 leading order derivative expansion, χ = Zkφ, u = wk /Zk . Helpful discussions with G. Bergner, C. Wozar, T. Fis- chbacher and T. Kaestner are gratefully acknowledged. This work has been supported by the Studienstiftung η. Recall that in Ising-like systems the thermodynamic des deutschen Volkes, and the DFG under GRK 1523, main exponents (i.e., α, β, γ and δ) are related among Wi 777/10-1, FOR 723 and Gi 328/5-1. each other by scaling relations, and can be deduced from the correlation exponents ν and η by hyperscaling rela- tions. Beyond that, there is no general relation between ν and η. The superscaling relation (11) thus represents B188 a special feature of the present supersymmetric model [1] E. Witten, Nucl. Phys. , 513 (1981). [2] E. Zhao and A. Paramekanti, Phys. Rev. Lett. 97, 230404 and is an exact relation to next-to-leading order in the (2006). supercovariant derivative expansion. [3] B. Uchoa and A. H. C. Neto, Phys. Rev. Lett. 98, 146801 Finally, we determine the possible nontrivial fixed- (2007). point superpotentials. Solving the coupled fixed-point [4] J. A. Hertz, Phys. Rev. B 14, 1165 (1976). equations for the dimensionless superpotential as well as [5] A. J. Millis, Phys. Rev. B 48, 7183 (1993). the anomalous dimension self-consistently, we find a dis- [6] P. Strack, S. Takei, and W. Metzner (2009), crete set of fixed-point superpotentials with an increasing arXiv:0905.3894 [cond-mat.str-el]. [7] C. Wetterich, Phys. Lett. B301, 90 (1993). number of nodes, see Fig. 2 (bottom). The number of RG [8] F. Synatschke, G. Bergner, H. Gies, and A. Wipf, JHEP relevant directions at the fixed points in addition to the 03, 028 (2009). a2 direction is equal to the number of nodes. The su- [9] H. Sonoda and K. Ulker (2008), arXiv:0804.1072. perpotential with no nodes is the next-to-leading order [10] O. J. Rosten (2008), arXiv:0808.2150. analogue of the maximally IR-stable fixed point discussed [11] B. Delamotte (2007), arXiv:cond-mat/0702365. B202 above. Our quantitative next-to-leading order estimates [12] E. Witten, Nucl. Phys. , 253 (1982). et al. D70 of the critical exponent ν and the anomalous dimension [13] M. Beccaria Phys. Rev. , 035011 (2004). W [14] M. Beccaria, M. Campostrini, and A. Feo, Phys. Rev. η for the fixed-point superpotentials of Fig. 2 (bottom) D69, 095010 (2004). are summarized in Tab. I. [15] J. Ranft and A. Schiller, Phys. Lett. B 138, 166 (1984). Each fixed point constitutes a universality class and [16] T. R. Morris, Phys. Lett. B345, 139 (1995). 5

[17] R. Neves, Y. Kubyshin, and R. Potting (1998), hep- [20] Beyond the polynomial expansion, we find fixed-point so- th/9811151. lutions corresponding to oscillating sine-Gordon-type po- [18] A. Zamolodchikov, Yad. Fiz. 44, 529 (1986). tentials, bouncing potentials as well as potentials which [19] F. Synatschke, H. Gies, and A. Wipf (2009), confine the field to a compact target space; see [19] for a arXiv:0907.4229 [hep-th]. detailed discussion.