Instantons in Four-Fermi Term Broken SUSY with General Potential. Agapitos Hatzinikitas University of Crete, Department of Applied Mathematics, L. Knosou-Ambelokipi, 709 Iraklio Crete, Greece, Email: ahatzini@tem.uoc.gr arxiv:hep-th/0300v 3 Apr 003 and Ioannis Smyrnakis University of Crete, Department of Applied Mathematics, L. Knosou-Ambelokipi, 709 Iraklio Crete, Greece, Email: smyrnaki@tem.uoc.gr Abstract It is shown how to solve the Euclidean equations of motion of a point particle in a general potential and in the presence of a four-fermi term. The classical action in this theory depends explicitly on a set of four fermionic collective coordinates. The corrections to the classical action dueto the presence of fermions are of topological nature in the sense that they depend only on the values of the fields at the boundary points τ ±. As an application, the Sine-Gordon model with a four-fermi term is solved explicitly and the corrections to the classical action are computed.
Introduction It is well-known that the equations of motion for a point particle in Euclidean space moving under the influence of a Minkowski potential possessing at least two degenerate minima, admit finite action solutions which are the instantons. Their existence is responsible for tunnelling processes in the Minkowski space. The symmetries of the action are reflected in the space of instanton solutions. Local deformations of these solutions in the directions determined by the symmetries give rise to zero modes in the semiclassical expansion. To avoid infinities in the path integral due to these zero modes one is forced to introduce collective coordinates [, ]. In the presence of rigid supersymmetry the above solutions are still instantons provided that the fermions vanish [3, ]. Applying the rigid supersymmetry transformation rules on these instantons one determines a more complete set of instantons where the fermions no longer vanish but instead they depend linearly on Grassmann collective coordinates. The number of Grassmann collective coordinates is equal to the number of the Fermi fields present. Since the new instantons are related to the previous ones by supersymmetry, the classical action remains the same. It is possible to break supersymmetry through the introduction of a four-fermi term [5]. The new equations of motion can be solved iteratively starting with the instanton solution in the supersymmetric case. The iterative process terminates due to the nature of Grassmann collective coordinates so in this way it is possible to obtain exact solutions to the new equations of motion. Since there is no symmetry involved in obtaining these solutions the instanton action will change. The correction is the integral of a total derivative term, so it depends only on the boundary values of the fields. It depends also on the Grassmann collective coordinates introduced by supersymmetry. Note that the fermionic fields become infinite as τ ± while it is possible to keep the bosonic field finite by appropriate choice of the integration constants. Nevertheless, despite this infinity, the action remains finite. As an explicit example we consider the quantum mechanical Sine-Gordon potential with a four-fermi term. The iterative solution of the equations of motion is demonstrated explicitly determining in this way the instantons. The integration constant that renders the bosonic field finite is determined. In this case the instanton action becomes S = 8m3 λ +ǫijkl mg ξ i ξ j ξ k ξ l where g, λ are coupling constants. Finally, we solve the equations of motion for a general potential in terms of the bosonic part of the instanton when the Fermi field is zero, which is used as a new variable instead of τ. It is interesting that in order to compute the action corrections it is not necessary to solve the nonlinear BPS equation. In fact it is possible to express completely both the instanton and the finite boson integration constant in terms of the new variable. The Quantum Mechanical Model We start by considering the following one dimensional quantum mechanical model in Euclidean space: S cl = [ ẋτ +U x ] dτ. The potential U x of the equivalent particle is asumed to have a number of degenerate minima.
The equation of motion is ẍ UxU x = 0. The instanton solution of this equation satisfies the BPS equation ẋ in +Ux in = 0 3 subject to the conditions x in ± = C ± where UC ± = 0. The action is invariant under time translations which implies that if x in τ is a solution to the BPS equation then x in τ τ 0 is also a solution. This means that Z 0 τ τ 0 = ẋ in τ τ 0 = d dτ 0 x in τ τ 0 is a zero mode of the operator corresponding to the quadratic variation of the action around x in. It satisfies the equation Z 0 τ τ 0 +U x in Z 0 τ τ 0 = 0. 5 Note that the normalization of Z 0 defined as ẋ in is just the absolute value of the classical action. This is so because S cl = ẋ in dτ = Z 0dτ 6 where we use the BPS equation. So it is reasonable to define the normalised zero mode as follows Z 0 = S cl / Z0. 7 It is possible to introduce fermions into this model by adding the terms S f = [ ψ T i ψ i +ψi T σ ψ i U ] dτ. 8 0 i where σ =, ψ i 0 i are two component Majorana fermions and the fermionic index is a colour index ranging in i =,,. Each fermion is related to a boson through rigid N = supersymmetry realised by the transformations δx = ǫ T σ ψ; δψ = σ ẋǫ Uǫ. 9 The spinor ǫ can be expanded in terms of the eigenstates ψ ± = ±i of σ as follows and then it can be proved that if ǫ = +σ ǫ then ǫ = ξ +ψ + +ξ ψ 0 δx = 0; δψ = ξ + ψ + Z 0 τ. 3
Starting from the configuration x = x in, ψ = 0 and integrating over the above supersymmetry transformations we arrive at the instanton given by x = x in and ψ i = ξ i Z 0 τ τ 0 ψ +. To each colour index we associate the same bosonic zero mode but different Grasmannian collective coordinates ξ i. Following [5] we add to the action a four Fermi term which breaks supersymmetry where σ = 0 0 S f = g. The new field equations are now ǫ ijkl ψ T i σ ψ j ψ T kσ ψ l dτ 3 ẍ UU = ψt i σ ψ i U ψ i +σ ψ i U = gǫ ijkl σ ψ j ψ T k σ ψ l. 5 Expanding ψ i w.r.t the GCC one has ψ i = ψ i + ψ 3 i. Note that all the other terms in the expansion vanish. Making the ansatz and plugging it into the fermionic field equation 5 one gets ψ 3 i = ατǫ ijkl ξ j ξ k ξ l ψ, 6 α αu = gz 3 0. 7 Since the solution of the homogeneous equation is Z 0 it is natural to set This leads to the equation ατ = Z 0 yτ. 8 ẏτ = gz 0. 9 Similarly we can expand xτ = x in τ+x τ and plug this into the bosonic field equation which gives ẍ x UU +U x in = ǫξ Z 0 ατu x in. 0 The solution of the homogeneous equation is Z 0 τ so it is natural to set x = ǫξ Z 0 τβτ where βτ satisfies d dτ Z 0 β = yτz 0 τu x in and ǫξ = ǫ ijkl ξ i ξ j ξ k ξ l. It is interesting to compute the corrections to the classical action due to ψ 3 i and x. The two Fermi term gives S f = [ ψ T i ψ 3 i +ψ T3 i ψ i +ψ T i σ ψ 3 i +ψ T3 i σ ψ i U x in ] = [ T3 ψ i ψ i +σ ψ i U +ψ T 3 i ψ i +σ ψ 3 i U ] = ǫξ = ẏdτ ǫξ y y
and the four Fermi term S f = g ǫ ijkl ψ T i σ ψ j ψ T k σ ψ l = ǫξ y y. 3 One easily checks that S f = S f. The bosonic correction gives S bc = [ẋin ẋ +x Ux in U x in ] d = dτ x ẋ in dτ = ǫξ S cl lim τ Z 0 τβτ lim τ Z 0 τβτ. It is worth noting that if the bosonic field is finite as τ = ± then lim τ βτz 0 τ is finite and since lim τ Z 0 τ = 0 the bosonic correction vanishes. 3 The Sine-Gordon model For the Sine-Gordon model the action is Here Scl SG = [ẋ + m λ λ cos m x] dτ. 5 Ux = m λ sin x. 6 λ m The BPS equation takes the form ẋ in + m λ sin λ m x in = 0 7 and can be easily solved to give x in τ = ± m λ tan e mτ τ 0. 8 The minus sign corresponds to the instanton solution and the plus sign to the anti-instanton. In what follows we are going to work with the instanton. The zero mode corresponding to the instanton is and the classical action becomes Z 0 τ = dx in dτ = m λ coshmτ τ 0 9 So S cl = m Z 0 = Z 0dτ = 8m3 λ 30 coshmτ τ 0. 3 5
Upon introducing the two and four Fermi terms given in the previous section, we get that the fermionic field is ψ i = ψ i + ψ 3 i where ψ i is given by and ψ 3 i is given by 6. The function yτ is the solution of the equation ẏτ = gz0 τ = gm cosh mτ τ 0 3 This can be solved by setting z = tanhmτ τ 0. With this change of variable the solution is written as yz = gm a+z 3 z3 33 and thus ψ 3 i = g m z a+z 3 z3 ǫ ijkl ξ j ξ k ξ l ψ. 3 To determine the form of x we solve the bosonic field equation. In terms of the new variable z this equation becomes This gives dβ dz = g m λ m The solution to this equation is βz = g m λ [ A m + 5 d dβ dz dz Z 0τ = λ yτ. 35 m A z + αz z + z z z. 36 z z A z ln 6 The contribution to the two and four Fermi terms is z + z + z + α z +B] 37 The bosonic correction is S f +S f = S f = ǫξ gm. 38 S bc = ǫξ gm A+ 5. 39 If the bosonic field is bounded at infinity then S bc vanishes and this implies that A = 5. In this case the full action is S tot = 8m3 λ +ǫξgm. 0 6
Generalisation It is worth mentioning that it is possible to solve exactly the equations 9, if we change variables from τ to x in. This is so because ẋ in = S cl Z 0. Applying this change of variable to 9 we get dy = g dx in S cl Z3 0 = g x S cl U3 in where S cl = C + C Ux in dx in. This admits the solution y = gm α+ g U 3 x S cl in dx in. The integration constant α has been chosen to be dimensionless. Similarly equation becomes d dβ U 3 x in = S cl yx in U x in. 3 dx in dx in Integrating once this equation and using we get dβ = S cl yx in U x in dx in U 3 x in g S cl 3 Uxin +gm S cl à U 3 x in. Integrating again we get βx in = S cl 3 gm S U cl α+g U 3 x in dx in + x in g S cl 3 Ux in dx in + gm S cl à U 3 x in dx in +g S cl B. 5 Recall now that as τ ±, x in C ± where UC ± = 0. Demanding that the bosonic field correctionis finitewe getthat lim xin C ± βx in Ux in isfinite, so lim xin C ± βx in U x in = 0. This translates into the conditions gm S cl à lim x in C + gm S cl à lim x in C U x in U x in à = m S cl xin x 0 U 3 s ds xin x 0 U 3 s ds g S cl 3 g S cl 3 C+ x 0 U 3 sds = αgm S cl 6 C x 0 U 3 sds = αgm S cl 7 where the value of α is determined by the choice of x 0. Subtracting the two equations we arrive at C+ C U 3 sds limxin C + lim xin C U x in x in x 0 U 3 s ds. 8 These formulae have been checked in the cases of the double well and the Sine-Gordon potentials. For the double well potential the results agree with those of [5] provided that we redefine our integration constants appropriately. In the case of the Sine-Gordon model the results agree with those of the previous section under the following identification of the integration constants à = 6 A ; α = α ; B = B. 9 7
5 Conclusion We have determined the corrections to the supersymmetric instanton, for a point particle in a general potential that admits at least two degenerate minima, due to the presence of a supesymmetry breaking four-fermi term. Starting from the instanton solution of the supersymmetric case and applying an iterative procedure we obtain exact solution of the new equations of motion. There is no symmetry involved in obtaining these solutions so the classical action will receive corrections. If we demand that thebosonic field remains finiteas τ ± then onlythe two- and four-fermion terms contribute corrections to the classical action. The fermionic fields diverge as τ ± nevertheless their corrections to the action remain finite. In the case of the Sine-Gordon potential the classical action gets modified by the contribution ǫ ijkl mg ξ i ξ j ξ k ξ l where the ξ i are fermionic collective coordinates. Finally, by a suitable change of variables, we determine the corrections to the classical action for a general potential and we compute the integration constant A that makes the bosonic field finite when τ ± in terms of the potential only. References [] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 98. [] M. A. Shifman, Particle Physics and Field Theory, World Scientific, Lecture Notes in Physics -Vol.6 Singapore, 999. [3] P. Di Vecchia and S. Ferrara, Classical Solutions in Two-Dimensional Supersymmetric Field Theories, Nucl. Phys. B30 977 93. [] E. Witten and D. Olive, Supersymmetry Algebras that Include Topological Charges, Phys. Lett. B78 978 97. [5] S. Vandoren and P. van Nieuwenhuizen, New Instantons in the Double-Well Potential, Phys. Lett. B99 00 80. 8