DO-TH-96/27

December 1996

Nonequilibrium dynamics:

preheating in the SU(2) Higgs model

Jürgen Baacke^{1}^{1}1
e-mail: , Katrin Heitmann
^{2}^{2}2e-mail: and
Carsten Pätzold
^{3}^{3}3e-mail:

Institut für Physik, Universität Dortmund

D - 44221 Dortmund , Germany

Abstract

The term ‘preheating’ has been introduced recently to denote the process in which energy is transferred from a classical inflaton field into fluctuating field (particle) degrees of freedom without generating yet a real thermal ensemble. The models considered up to now include, besides the inflaton field, scalar or fermionic fluctuations. On the other hand the typical ingredient of an inflationary scenario is a nonabelian spontaneously broken gauge theory. So the formalism should also be developed to include gauge field fluctuations excited by the inflaton or Higgs field. We have chosen here, as the simplest nonabelian example, the SU(2) Higgs model. We consider the model at temperature zero. From the technical point of view we generalize an analytical and numerical renormalized formalism developed by us recently to coupled channnel systems. We use the ’t Hooft-Feynman gauge and dimensional regularization. We present some numerical results but reserve a more exhaustive discussion of solutions within the paramter space of two couplings and the initial value of the Higgs field to a future publication.

## 1 Introduction

Nonequilibrium dynamics in quantum field theory within the closed time path (CTP) formalism has become recently a fastly developing area of research. Pioneering work by [1] has been followed by applications to inflationary cosmology [2, 3, 4] and to the hadronic phase transition, especially the possibility of formation of chiral condensates [5, 6]. With the increase of the experimental lower bound of the Higgs mass the electroweak phase transition may be second order and could then become a realistic field of application as well.

The typical numerical computations - or ‘experiments’ - in this new field have included up to now, in addition to the ‘Higgs’ of ‘inflaton’ field, scalar and fermionic fluctuations. The cosmological application has been prepared [7, 8, 9] by considering the nonequilibrium time development in a constantly curved space. Besides Friedmann - Robertson - Walker cosmology another typical ingredient of inflationary scenarios or of the cosmological electroweak phase transition are nonabelian spontaneously broken gauge theories. So the formalism should also be developed to include gauge field fluctuations. It is the aim of our present work to describe the analytical and computational tools for such applications.

The analytical part includes the formulation of the theory and renormalization. As a convenient gauge, used extensively in perturbative and nonperturbative calculations in the electroweak theory, we have chosen the ‘t Hooft-Feynman background gauge. We will not be able to discuss gauge invariance, especially since already the Ansatz for the classical Higgs field implies a choice of gauge. In the - closely related - formulation of the effective potential it has been proposed recently [10] to use the absolute value squared of the Higgs field as a gauge invariant order parameter, a choice that merits consideration also in the present context.

The renormalization conditions were chosen such the tree effective potential remain unchanged around the minimum corresponding to the broken Higgs phase. The renormalization has been based on dimensional regularization.

The numerical computation and the analytical one are both contained in a common scheme that we have proposed recently [11] for such nonequilibrium processes. The main characteristics of the method are: (i) a clean separation of the divergent and finite parts of the fluctuation integrals in close relation to CTP perturbation theory; (ii) analytic computation of the leading order contributions using standard covariant regularization schemes; (iii) numerical computation of the finite parts avoiding small differences of large numbers - the leading orders are not subtracted from the integrand but omitted from the outset. A fourth property has been mentioned in [11] but not yet used: the fact that the method can be extended easily to coupled channel systems. This application of the method will be demonstrated in this paper within the context of the SU(2) Higgs model.

The plan of this work is as follows: in section 2 we recall the basic definitions and relations; in section 3 we present the one-loop nonlinear relaxation equations; we prepare the regularization in section 4 by expanding the fluctuation modes in orders of the vertex function governed by the classical field and by deriving the large momentum behaviour of the first terms; regularization is then straightforward, the renormalization requires some algebra, both are presented in section 5; the numerical computation is discussed in section; we conclude in section 6 with a discussion of the numerical results and an outlook to more realistic and more general applications of the method.

## 2 Basic relations

The SU(2) Higgs model is defined by the Lagrangean density

(1) |

Here denotes a complex Higgs doublet. The covariant derivative is defined as

(2) |

and the corresponding field strength tensor is given by

(3) |

We write the Higgs potential in the form

(4) |

In the SU(2) Higgs model where is the mass of the Higgs field. In the case of unbroken symmetry - not considered here - the mass term would be defined by , where is the mass of the complex scalar doublet. We denote the ”classical” or ”inflaton” field as ; it is supposed to be constant in space and depends only on the time . We will treat here the fluctuations in one-loop order, generalizations to the expansion and to the Hartree approximation are straightforward; they are discussed e.g. in [12, 13]. We therefore decompose the Higgs field into the inflaton and the fluctuation parts as

(5) |

Here is the isoscalar Higgs field fluctuation, the isovector fluctuations , the “would-be Goldstone fields” will couple to the gauge fields. Since there is no classical gauge field, the gauge field reduces to

(6) |

in terms of the fluctuations . Furthermore we have to introduce a gauge fixing and Faddeev-Popov Lagrangean. It is convenient to use the ’t Hooft- Feynman gauge with the gauge condition

(7) |

and the gauge fixing term

(8) |

The corresponding Faddeev-Popov Lagrangean reads

(9) | |||||

The complete Lagrangean is then given by

(10) |

The propagators and vertices can be read off from the free Lagrangean

(11) | |||||

and the interaction Lagrangean

(13) | |||||

## 3 Equations of motion

The formalism of nonequilibrium dynamics in quantum field theory and the use of the tadpole method [14] have been presented or reviewed recently by various authors [15]. We give here just the one-loop equations of motion which are obtained from this formalism.

The basic graph from which the equation of motion of the inflaton field is derived is depicted in Fig. 1. The propagators are here the exact propagators in the inflaton background field. In contrast to the free propagators some of them involve coupled channels. We adapt the notation to this general case by using for the Green functions the notation . They will be defined below. The lower latin subscripts correspond to the different fluctuating fields and introduced in the previous section. For all one loop integrals the contributions of the space components of the gauge fields and the Faddeev-Popov fields will combine since they involve the same propagators (Green functions). We will therefore introduce the set of subscripts and for the isoscalar component of the Higgs field, the ‘transverse’ components of the gauge fields (i.e.the combination of their space components and the Faddev-Popov fields), the time components of the gauge fields and the isoscalar components of the Higgs field, respectively.

The vertices are realized by a matrix of vertex operators which has the following nonvanishing components

(14) | |||||

With these notations the differential equation for the inflaton field reads

(15) |

The propagators are the usual time ordered Green functions. We have omitted the spatial variables and since the Green functions are taken at and, due to translation invariance, are then independent of . These Green functions at can be written in terms of Fourier components as

(16) |

The Green functions for momentum are obtained in the usual way from the mode functions for the various fluctuations in the time dependent background field . We discuss briefly the different channels.

For the isoscalar part of the Higgs field is expressed as

(17) | |||||

where the mode functions satisfy the differential equation

(18) |

The mode frequency is defined by

(19) |

and the ‘frequency at t=0’ is given by

(20) |

The initial conditions are

(21) |

For we get

(22) |

For the transversal components of the gauge fields the mode functions satisfy

(23) |

where

(24) |

The Green function is then given by

(25) |

where again is defined as

(26) |

The Green functions for the time component of the gauge field and of the isovector part of the Higgs field satisfy the coupled system of differential equations

(27) |

Here we have introduced the metric taking into account the minus sign of the kinetic term and of the propagator of in the Feynman gauge and where the matrix is defined as

(28) |

The latin subscripts take the values and .

For the Green functions of this system we make the Ansatz [16, 17]

(29) | |||||

Here the mode equations

(30) |

have a fundamental system of two independent solutions labelled by the superscript which takes again the values and . We choose as a basis the two independent solutions characterized by the initial conditions

(31) |

where

(32) |

with

(33) | |||||

(34) |

By a simple extension of the derivation given in [17] one can show that the coefficients are then related to the Wronskians

(35) | |||||

via

(36) |

The Wronskians can be computed from the initial conditions for the ; we obtain for the coefficients

We use the notation for the single channel fields and as well, implying that and and extend the metric to these components via corresponding to the ordering . Collecting the various expressions for the Green functions we define the (yet unrenormalized) fluctuation integral

(37) |

The equation of motion for is then

(38) |

In order to obtain the energy we first obtain the Hamiltonian from the Lagrangean (11), insert the field expansion in terms of the mode functions and the corresponding annihilation and creation operators and take the expectation value in the initial state. For the subsystem we expand the fields as

(39) |

With our initial conditions (3) for the mode functions are at the annihilation and creation operators of the field and those of the field . So averaging in the initial vacuum state of the system we have in the Feynman gauge . Therefore, the metric enters twice: once in order to take into account the sign of the kinetic terms of the field components (latin subscripts) in the Hamiltonian and a second time due to the averaging of the field operators (greek indices) in the initial state. The unrenormalized total energy of the system consisting of the inflaton field and the fluctuations is therefore given by

where the summation is over all fields and
.
is the isospin degeneracy, i.e.
for and for ^{4}^{4}4The
vertex operators are defined such to include such degeneracy factors.
The vertex operators and the potentials are
obviously related by
a functional derivative via
..
The frequency is defined by

(41) |

Using the equations of motion for and the mode functions it can be checked easily that the energy is conserved. In both the fluctuation integral and the energy we have already taken into account the cancellation between the transversal gauge modes and the Faddeev-Popov ghosts so that the latter ones do not appear any more.

## 4 Perturbative expansion

In order to prepare the renormalized version of the equations given in the previous section we introduce a suitable expansion of the mode functions . For a single channel this expansion has been presented in [11]. We extend here the discussion to coupled channel systems. Adding the term on both sides of the mode equations (30) they take the form

(42) |

with

for the coupled system and

(44) | |||||

(45) |

for the single component systems. We assume that to vanish at . Including the initial conditions (3) the mode functions satisfy the equivaltent integral equation

(46) |

with

(47) |

We separate into the trivial part corresponding to the case and a function which represents the reaction to the potential by making the Ansatz

(48) |

satisfies then the integral equation

(49) |

and an equivalent differential equation

(50) |

with the initial conditions .

We expand now
with respect to orders in
by writing

(51) | |||||

(52) |

where is of n’th order in and is the sum over all orders beginning with the n’th one. The are obtained by iterating the integral equation (49) or the differential equation (50). The function is identical to the function itself which is obtained by solving (50), the function can be obtained as

(53) |

or by solving the inhomogeneous differential equation

(54) |

Note that in this way one avoids the computation of via the small difference . This feature is especially important if deeper subtractions are required as in the case of fermion fields.

The order on the potentials will determine the behaviour of the functions at large momentum. We will give here the relevant leading terms for and . We have

(55) |

Integrating by parts we obtain

(56) | |||||

For we need to know only that the leading behaviour is

(57) | |||||

The leading terms of and in this expansion in powers of are the same as for and respectively.

## 5 Renormalization

Using the expansion in orders of the potential the fluctuation term (37) occuring in the equation of motion (38) can be written as

(58) |

To zeroth order in the functions vanish and we have

(59) |

These correspond to the tadpole graphs depicted in Fig. 2; they are removed by including into the Lagrangean a mass counterterm for the Higgs field.

To first order in the potentials we find

(60) | |||

The first part, the sum over the diagonal terms proportional to , corresponds to the graphs of Fig. 3a and their divergent parts are removed by the coupling constant renormalization; the second term corresponds to the graph of Fig. 3b and its divergent part is removed by a wave function renormalization counter term.

The sum of all contributions of order higher than in the potential is finite. It is given by