Prepared for submission to JHEP Intercommutation of U(1) global cosmic strings Guy D. Moore Institut f¨urKernphysik, Technische Universit¨atDarmstadt Schlossgartenstraße 2, D-64289 Darmstadt, Germany E-mail: [email protected] Abstract: Global strings (those which couple to Goldstone modes) may play a role in cosmology. In particular, if the QCD axion exists, axionic strings may control the efficiency of axionic dark matter abundance. The string network dynamics depend on the string intercommutation efficiency (whether strings re-connect when they cross). We point out that the velocity and angle in a collision between global strings \renormalize" between the network scale and the microscopic scale, and that this plays a significant role in their intercommutation dynamics. We also point out a subtlety in treating intercommutation of very nearly antiparallel strings numerically. We find that the global strings of a O(2)- breaking scalar theory do intercommute for all physically relevant angles and velocities. Keywords: axions, dark matter, cosmic strings, global strings arXiv:1604.02356v1 [hep-ph] 8 Apr 2016 Contents 1 Introduction1 2 Global string review3 3 Renormalization of string angle and velocity5 4 Microscopic study of intercommutation9 5 Discussion and Conclusions 12 1 Introduction fsec:introg Cosmic strings [1,2] are hypothetical extended solitonic excitations which may play a sig- nificant role in cosmology. Their original motivation, for structure formation [3{5], appears in conflict with modern microwave sky data [6]. But cosmic strings may be important in other contexts. In particular, if the QCD axion [7,8] exists, the axion field may contain a string network in the early Universe [9] which may dominate axion production and play a central role in the axion as a dark matter candidate. String defects can occur whenever the vacuum spontaneously breaks a symmetry, say, breaking G down to H 2 G, such that the quotient group (vacuum manifold) G=H has nontrivial π1 homotopy. The simplest example is the complete breaking of an SO(2) or U(1) symmetry. Generally, strings are divided into two sorts; \local" strings, which typically occur when G is a gauge group and which do not couple to any massless fields [1], and \global" strings, which typically occur when G is a global (non-gauged) symmetry, in which case the strings couple to the associated massless Goldstone bosons [10]. There is a rich literature studying local strings and their networks [11{13]. This includes a rather careful study of when crossing strings intercommute with each other [14{17]. Global strings, such as the axionic string networks alluded to above, are harder to study. The strings interact with each other via massless fields, which change the network's dynamics, both by allowing strings to radiate energy efficiently [18{21], and by communi- cating inter-string forces. To simulate such a network faithfully requires a hybrid algorithm, which treats the string cores via a Nambu-Goto action and treats the Goldstone field with lattice methods [22], together with some string-field interaction. We have recently pre- sented such an algorithm [23]. However, to implement it we must know when strings which cross each other will inter-commute, and when they will simply pass through each other (see Figure1). This problem has been previously addressed via microscopic simulations [24]. But we will argue here that the physics is more complicated due to the long range inter-string interactions, and the issue of string inter-commutation deserves some more study. { 1 { B A B A Pass Through? A B Intercommute? B A A B A B fig:intercommute Figure 1. Left: two strings approach, with A descending and B rising out of the page. The arrows indicate the \sense" of the string. There are two possibilities: they pass through each other (top right), or they intercommute (bottom right). Which occurs is determined by the microphysics at the point where they touch. Cosmic string networks involve two disparate scales. There is a microscopic scale −1 mh , set by the inverse mass of the (Higgs) radial excitation of the symmetry-breaking field. And there is a macroscopic scale, the average inter-string separation L, which is typically of order the Hubble scale H−1. For axions around the time where the dark matter density is established, these scales differ by a factor of ∼ 1030 [25]. The energy in a −1 local cosmic string is carried within a few mh of the string's core. But for a global string, −1 the energy is distributed logarithmically over all scales between mh and L. As global strings approach each other through these intermediate scales, this energy distribution is responsible for inter-string interactions, which apply both torques, and attraction or repulsion, to the strings. The angle and velocity with which the strings approach therefore evolves logarithmically with scale. We illustrate this idea in Figure2. Here we study this evolution, and its impact on the physics of string intercommutation. We concentrate on strings arising from a complex scalar (the O(2) model or relativistic 3D xy model); the details and results could be quite different for strings with other more complicated symmetry breaking patterns. In the next section we review the physics of global cosmic strings, and we explain why long range inter-string interactions can be important. Section3 treats the inter-string forces explicitly and finds how the angle and velocity of string approach changes with scale. This is combined with a new study of microscopic string intercommutation in Section4. We end with a discussion and conclusion, but we will give the main findings here. At the macroscopic scale strings may approach each other at a range of angles and relative { 2 { Zoom Zoom Zoom In In In etc. fig:scales Figure 2. Two string just before crossing. They twist towards larger crossing angle; zooming in, we see that at short distances they twist more; zooming in, they twist still more, and so on. Even if the twist is slight, over many orders of magnitude in scale it can become very large. velocities. But accounting for inter-string forces, at the microscopic scale of 100=m, the strings are essentially always almost antiparallel (the attractive channel) and approaching with a highly relativistic velocity v ' 0:9. We show in contradiction to previous results [24] that such strings always intercommute. Therefore in treating O(2)-model global strings, one should assume that intercommutation always occurs. 2 Global string review fsec:reviewg Here we review global strings (see also [2, 10, 26]). Consider a complex scalar field ' with U(1) symmetry (or equivalently, two real scalars ' , ' with an O(2) symmetry; p r i 1 ' = ('r + i'i)= 2). The most general renormalizable Lagrangian is λ 2 − L = @ '∗@µ' + 2'∗' − f 2 : (2.1) fLagrangiang µ 8 a p 2 iθA If fa > 0 then the classical vacuum 2' = fae spontaneously breaks the U(1) symmetry. The excitation spectrum about this vacuum contains a radial \Higgs" excitation with mass 2 2 mh = λfa and angular excitations, which are massless Goldstone modes. We are interested in solitonic solutions corresponding to an extremely large number of quanta, so a classical description of the field and its dynamics are sufficient. Now θA is only defined modulo 2π, and there are configurations where it cannot be defined in a continuous and single-valued way. Specifically, a topological string solution is where there is a 1-dimensional oriented (linelike) locus of points, called a cosmic string, where j'j = 0, and such that θA varies by 2π as one circles the string with positive sense. Locally the string is nearly straight and we can use it as the z-axis in polar (z; r; φ) co- ordinates. The string solution is '(z; r; φ) = fah(r) exp(i(φ − φ0)), with h(r) chosen to minimize the energy per unit length (string tension) Z T = r dr dφ jr'j2 + V ('∗') Z h2 m2 = πf 2 r dr (@ h)2 + + h (h2 − 1)2 ; (2.2) fE_hg a r r2 8 1We use the [−; +; +; +] or mostly-positive metric { 3 { which is minimized when h(r) obeys 2 2 2 1 h mh 2 @ h + @rh + − h(h − 1) = 0 ; h(0) = 0; lim h(r) = 1 : (2.3) fh_of_rg r r r2 2 r!1 2 R 2 2 2 The most important term in Eq. (2.2) is the πfa r dr h =r term, arising from jrφ'j . All other terms only contribute appreciably for mhr ∼ 1 and fall off quickly at large r. But −1 this term receives equal contributions from all logarithmic scales larger than mh : Z rmax 2 2 2 T ' r dr πfa (1=r ) = πfa ln(mhrmax) + O(1) ; (2.4) fTensiong −1 mh where rmax is a long-distance cutoff, which would physically be provided by the curvature scale of the string or the distance to the next string. This logarithmic energy scaling will be essential to our arguments. The other essential feature of the strings is the way they interact with their environ- ment. Suppose that a string exists in an environment where θA also varies uniformly, for instance because of the far field of another string. Outside the string core the solution is described by θA alone, and its equation of motion is linear so we can superpose solutions. In this case our Ansatz for the field becomes iφ iθA;ext '(x) = fah(r)e e ; θA;ext ' xiriθA;ext ; (2.5) fAnsatzg where we have approximated the external field by the first term in its Taylor series. Near the string core the equation of motion is not linear, so riθA;ext will cause an acceleration in the string.