# Tunnelling of pulsating strings in deformed Minkowski spacetime

## Abstract

Using the WKB approximation we analyse the tunnelling of a pulsating string in deformed Minkowski spacetime.

## Keywords

Gauge Theory Fundamental Form Minkowski Spacetime Classical Stability Tunnelling Amplitude## 1 Introduction

There is a huge body of evidence that the AdS/CFT correspondence holds true for strings in \(AdS_5 \times S^5\) and \(\mathcal{N}=4\) super Yang–Mills theory in four dimensions. For instance, the energy of spinning and rotating strings matches the anomalous dimension of operators in the gauge theory in the range where they can be compared. Integrability on both sides of the correspondence also provides further support for the correspondence [1]. It is also important to test the correspondence in situations with less supersymmetry where the gauge theories have deformed potentials leading to marginally deformed \(\mathcal{N}=2\) or \(\mathcal{N}=1\) supersymmetric gauge theories [2, 3]. The gravitational dual of such theories have a deformed five-sphere characterised by a real parameter \(\gamma \) and the dilaton and some RR and NS-NS fields are also present [4]. In this situation both sides of the correspondence also have integrable structures [5]. Spinning and rotating strings have also been considered in such a deformed context and they confirm the correspondence whenever they can be compared [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

There is a class of string configurations, pulsating strings, which has not received much attention since its dual operator is not completely understood. They have been analysed in \(AdS_5 \times S^5\) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], \(AdS_4 \times CP^3\) [28, 29] and other backgrounds [30, 31, 32, 33, 34, 35, 36, 37], and more recently they have been studied in the deformed case as well [38]. Since the string presents a periodic motion its dynamics can be characterised by its oscillation number. It is not one of the string charges but it is quite useful to parametrise its behaviour [23, 38, 39]. At the quantum level it is an adiabatic invariant so it provides information as regards the semi-classical regime for higher values of the oscillation number. In [38] we analysed pulsating strings in deformed Minkowski spacetime and in deformed \(AdS_5 \times S^5\) for a small deformation. We have found the classical energy in terms of the oscillation number in the high and low energy limits. For high energy we performed the quantisation of the highly excited string states to second order in perturbation theory and found that the oscillation number has to be even. In the low energy case we found a new term, proportional to \(\gamma \), which is not present in the classical case.

In order to analyse the classical dynamics of the pulsating string we introduced an effective potential which captures all relevant information as regards the deformed background. When the string pulsates on the deformed five-sphere its effective potential grows smoothly as one of the angles increase from zero to \(\pi /2\) and the oscillation number can be expressed in terms of complete elliptic integrals [38]. In the case of deformed Minkowski spacetime the string pulsates along the radial direction and the effective potential starts growing from the origin until it reaches a maximum value of \(m^2/(2\gamma )\) at \(r^2=1/\gamma \) and then goes back to zero far away from the origin (see Fig. 1). It is clear that at low energies or small deformation the string has a periodic motion that can be quantised perturbatively as done in [38]. However, since the potential has a maximum, it is possible for the string to tunnel through the potential barrier and the computation of the transition rate for such a process is the main goal of this paper.

A non-perturbative phenomenon like tunnelling may be studied semi-classically using the WKB approximation whenever the amplitude or the phase of the wave function is taken to be slowly changing. The WKB method has been applied in several situation involving strings [40, 41, 42, 43, 44] and here it will be used to analyse the behaviour of a pulsating string in deformed Minkowski spacetime.

This paper is organised as follows. In Sect. 2 the pulsating string in deformed ten-dimensional Minkowski spacetime will be briefly described. In Sect. 3 we will use the WKB technique to calculate the transition rate for the pulsating string to tunnel through the potential. In Sect. 4 we will analyse the classical stability of the pulsating string and show that for small deformation it is stable. We then present some conclusions in the last section.

## 2 Pulsating strings in deformed Minkowski spacetime

*m*is the string winding number. This ansatz is compatible with the classical equations of motion for the pulsating string and is suitable for the classical stability analysis of Sect. 4. It corresponds to a string dynamically equivalent to the simplest and well-known pulsating strings studied in [19, 38]. Using (2) in (1) we get

The equation of motion can be integrated in terms of elliptic functions and the energy can be found in terms of the oscillation number \(\mathcal{N} = \oint \Pi \mathrm{d}r/2\pi \). For more details see [38].

## 3 String tunnelling

*r*so that we can take for \(\Pi ^2\) the radial part of the Laplacian

*d*arbitrary since we want to consider the general situation. For the full ten-dimensional case \(d=6\). Then the Schrodinger equation reads

*A*,

*B*,

*C*,

*D*and

*F*are constants and \(p(r) =\sqrt{E^2 - V^2(r)}\).

*p*(

*r*) vanishes at these points. To avoid this problem we will consider solutions of (10) around these two points. To this end we introduce coordinates \(x = c_i(r - R_i)\), \(i=1,2\), where \(R_i\) stands for \(R_1\) or \(R_2\), and we find that (10) reduces to

*Ai*(

*x*) and

*Bi*(

*x*) are the two linearly independent Airy functions.

*x*|. Around \(R_1\) we find that in the WKB solution \(\intop _r^{R_1} p(r) \mathrm{d}r = 2/3 \hbar (-x)^{3/2}\) for \(x<0\), while for \(x>0\), \(\intop _r^{R_1} p(r) \mathrm{d}r = -2/3 \hbar x^{3/2}\). The Airy functions go like \(e^{\pm 2/3 x^{3/2}}\) for \(x<0\) and \(\cos (2/3 (-x)^{3/2} - \pi /4)\) and \(\sin (2/3 (-x)^{3/2} - \pi /4)\) for \(x>0\). Similar expressions hold for the solutions around \(R_2\). Matching the solutions we find that around \(R_1\) we have

*d*has gone away and only \(\mathcal{P}\) depends on the dimension through the potential \(V^2(r)\).

To find the tunnelling probability we have to consider the probability current \(i( \Psi ^* \mathbf {\nabla } \Psi - \Psi \mathbf {\nabla } \Psi ^*)\) in the deformed Minkowski spacetime (1). Taking only the radial component and integrating it with the proper measure we find that the square root factors in (11) precisely cancel the measure factors so that in the region \(r<R_1\) it gives \(|B|^2 - |A|^2\). Unitarity is then respected since (17)–(24) imply that \(|B|^2 - |A|^2 = |F|^2\). This means that the tunnelling amplitude \(T=|F/A|^2\) is given by (25).

*r*as \(\tilde{r}= \sqrt{\gamma } r\) so that for \(\gamma <<1\) we have

## 4 Classical stability

As show in the previous section a pulsating string can tunnel through the potential barrier and this naturally raises questions about its classical stability. It is well known that spinning strings in anti-de Sitter spaces are classically unstable for large spin [45]. Pulsating strings, on the other side, have better stability properties than spinning strings, as shown in [46]. In the following we will analyse the stability properties of pulsating strings in deformed Minkowski spacetime. We will apply the technique developed by Larsen and Frolov [47] and we will show that when the deformation is small the classical pulsating string is stable.

*t*, \(\phi _1\) and \(\phi _2\).

*r*is the unperturbed solution to (39) which is a periodic function of \(\tau \). Then, by the Sturm theorem, \(F_3\) oscillates for large \(\tau \) so that the perturbation \(\delta X_3\) is stable. We can handle \(F_2\) and \(F_8\) by expanding in \(\gamma \) as \(F_2=U_2+\gamma V_2\) and \(F_8=U_8+\gamma V_8\) to get

*r*cannot be reached so that there is no resonance. The same result holds for \(V_8\) so the perturbations \(\delta X_2\) and \(\delta X_8\) are also stable. For arbitrary values of \(\gamma \), (55) still shows that \(\delta X_3\) is oscillatory but (56) and (57) could not be decoupled.

*m*, the mode

*n*of the perturbation and the stringy energy \(\kappa \), are fixed. In these cases the perturbation amplitude modulates and is stable.

*r*vanishes and could cast doubts about the stability. In fact, when plotting \(U_2\) we get warnings that there is some problem at \(\tau =\pi /(2m)\). The same sort of problem appears in (64) and (65) or in (68) and (69). Our previous arguments, however, show that there is nothing special in those points. Since (66) is an ordinary differential equation we can solve it. Changing variables to \(y = \sec ^2 (m\tau )\) and taking \(U_2(y)= y^{\frac{n}{2m} }G(y)\) we find that

*G*(

*y*) satisfies the hypergeometrical differential equation so that

*m*and

*n*, showing that it is well behaved everywhere. Notice also the amplitude of \(U_2\) modulates so that the \(\gamma \) independent part of the perturbation of the mode \(F_2\) is also stable. The \(\gamma \) independent part of \(F_8\), that is, \(U_8\), is also stable since it satisfies (67). Then the homogeneous solutions for \(V_2\) and \(V_8\) in (68) and (69) are stable and the particular solutions, which involve \(U_8\) and \(U_2\), respectively, are also stable unless there is some resonance frequency (Fig. 7). But as discussed previously, this cannot happen so that the full solutions for \(V_2\) and \(V_8\) are also stable.

## 5 Conclusions

We have applied the WKB method to compute the tunnelling amplitude for an oscillating string in deformed Minkowski spacetime. As expected it is proportional to the string energy and vanishes when the deformation goes to zero. We have also shown that for small deformation the classical pulsating string is stable. It is well known that pulsating strings in \(AdS_5\times S^5\) are dual to operators composed of non-holomorphic products of scalar fields [20, 23, 48, 49], but the theory corresponding to the deformed Minkowski spacetime is not known. Since the string tunnelling represents an instability of the system it would be very interesting to see what happens on the other side of the correspondence.

## Notes

### Acknowledgments

Sergio Giardino is supported by CNPq Grant 206383/2014-2 and thanks Prof. Paulo Vargas Moniz and the Center for Mathematics and Applications of the Beira Interior University for hospitality. The work of Victor Rivelles is supported by FAPESP Grant 2014/18634-9.

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