Euclidean solutions of Yang-Mills theory coupled to a massive dilaton

The Euclidean version of Yang-Mills theory coupled to a massive dilaton is investigated. Our analytical and numerical results imply existence of infinite number of branches of globally regular, spherically symmetric, dyonic type solutions for any values of dilaton mass $m$. Solutions on different branches are labelled by the number of nodes of gauge field amplitude $W$. They have finite reduced action and provide new saddle points in the Euclidean path integral.


I. INTRODUCTION
Motivated mainly by the Bartnik-McKinnon discovery [1] of globally regular, static, spherically symmetric solutions of the Einstein-Yang-Mills (EYM) equations and similar results obtained in the EYM and YM theory coupled to a dilaton field [2,3,4,5,6,7] the Euclidean version of the EYM and YM theory coupled to a dilaton were investigated recently [8], [9]. While in the four dimensional space-time with Lorentzian signature electric part of non-abelian SU(2) gauge field should necessarily vanish for asymptotically flat solutions [10,11] situation is changed in Euclidean sector: electric field here plays role similar to the Higgs field of the Lorentzian sector [12] and there are nontrivial dyonic type solutions [8], [9].
Dilaton field in these investigations was considered to be massless. On the other hand there are very strong experimental bounds on dilatonic coupling constant for the massless dilaton case and it is thought that at some stage dilaton should obtain potential possibly with stable or metastable minimum. Since close to the minimum potential can be well approximated by the dilaton mass term it is of certain interest to investigate Euclidean solutions in the YM theory coupled to a massive dilaton, which is exactly the aim of present Letter.
The rest of the Letter is organized as follows: in next Section we formulate our model, derive equations of motion and asymptotic behaviour of finite action solutions. In Sect. III we discuss some special solutions of our system. In Sect. IV we present our numerical results. Sect. V contains concluding remarks.

II. THE MODEL
Starting point of our investigation is the Euclidean version of Yang-Mills theory coupled to a dilaton field φ, which is defined by the Euclidean action where F µν is the non-Abelian gauge field strength, V (φ) is dilaton field potential and γ and g denote the dilatonic and gauge coupling constants, respectively. The simplest choice of a dilaton potential would be to have just a mass term: where m is dilaton mass.

A. The Ansatz and equations
In this study we will restrict ourselves with the SU(2) gauge group and will be interested in spherically symmetric solutions. Following Witten [13,14] the general spherically symmetric SU(2) Yang-Mills field can be parameterized as follows: with r = x 2 i being the radial coordinate, i, j = 1, 2, 3 are spatial indices and a, b = 1, 2, 3 group indices.
The reduced action S red is defined as follows: with τ = x 4 being Euclidean time. Without loss of generality we can put A 1 = A 2 = 0. Then the reduced action reads: with where and the prime denotes the derivative with respect to r. The equations of motion which follow from the reduced action (5) read: The rescaling removes the gauge and dilatonic coupling constants g and γ, respectively, from the equations of motion. We thus set without loosing generality γ = g = 1 in the following.

B. Asymptotic behaviour
Close to r = 0 we find a 3-parameter family of solutions regular at the origin, which can be parameterized in terms of b, u 1 and φ 0 : The behaviour of the solutions at infinity is more involved. In leading order we have: So, at infinity solutions are characterised by three free parameters U, Q e and C.
Note that in the massless case the shift of dilaton field φ → φ +φ for any finite valueφ can be compensated by a rescaling of r and u as follows: This symmetry is obviously broken by dilaton mass term and for finite action solutions we have to choose normalisation φ ∞ ≡ φ(r = ∞) = 0.

III. SPECIAL SOLUTIONS
There are few special solutions of our system, which play an important role. The simplest is the vacuum solution with Note here that W (0) = 1 and W (0) = −1 are gauge equivalent and in the Eq. (12) we have chosen positive sign as in most of the investigations on the subject. In massless case the equations of motion have elegant solution [6,9,12], for which the gauge field amplitudes are simply with U an arbitrary constant and dilaton field φ = φ (0) ab has logarithmic behaviour In massive case this abelian solution is modified as follows: gauge amplitudes W (r) and u(r) are the same as in Eq. (20) and dilaton field φ = φ (m) ab now satisfy massive equation [15] which unfortunately cannot be integrated analytically. Solution of this equation φ (m) ab plays an important role, while as we discuss below in some limit solution of full system will be approaching this abelian solution.
In the regime, when dilaton decouples, φ ≡ 0, one finds the BPS monopole solution. Note that in flat space-time there are no radial excitations of BPS monopole [16] in contrast to case when gravity is taken into account [17,18].

IV. NUMERICAL RESULTS
We integrated the equations of motions (8)-(10) using the differential equations solver COLSYS [19]. To construct solutions we used the following boundary conditions: at the origin and at infinity. From numerically found solutions with fixed U and n, the number of nodes of gauge field amplitude W , one can extract the values of b, u 1 and other parameters which enter in the asymptotic behaviour Eqs. (12)- (14) and Eqs. (15)- (17). Examples of solutions with zero, one and two nodes of the gauge field amplitude W (r) are shown on Fig. 1, where we plot profiles of W (r), u(r) and φ(r) for U = 1.0 and m 2 = 10.0. Typically dilaton field grows monotonically from some (negative) value φ 0 at r = 0 to φ ∞ = 0 at r = ∞. Electric component of the YM field, u(r), also grows monotonically from u(0) = 0 to some u(∞) = U and gauge amplitude W (r) starts from its vacuum value W (0) = 1 and for the solution with n nodes oscillates n times and then tends to zero according to Eq. (15).
In order to interpret our numerical results, we find it convenient to analyze the Yang-Mills part of the action density, i.e. the piece L ym , Eq. (7), appearing in the full action density Eq. (6). There are two factors which can make L ym large in the neighbourhood of the core of the soliton: (i) the case U ≫ 1 , (ii) when many nodes appear, the function W ′ and/or the ratio (1 − W 2 )/r can get large and lead to important contribution to the action density.
The consequence of these features on the pattern of solution is observed already in the massless case [9]. For fixed n and increasing the parameter U, we observe a decreasing of the value φ 0 . The factor exp(2φ) then get very small and decreases considerably the contribution to the action of L ym in the region of the core of the soliton (i.e. for r ∼ 0).
In the case of a massive dilaton, the situation is more complex because, increasing too much the value of the parameter m also leads to a large term in the action density and there appears to be a competition between the Yang-Mills part and the dilaton part.
Note that in problem under consideration we have three different scales: YMD scale, l ymd = γ g , 'monopole' scale, l mon = 1 U and scale associated with the dilaton mass, l dilaton = 1 m . With the rescaling (11) we effectively put l ymd = 1, so we still have two scales to vary. It turns out that the large m 2 behaviour of nodeless solutions and solutions with the nodes are different. So we consider these cases separately.

A. Zero-node solutions
The result of the competition between the Yang-Mills field and the massive dilaton is illustrated by Fig. 2, where the value of φ 0 is plotted as function of m 2 for several values of U for lowest branch of solutions with no nodes.
For small values of U, we see that increasing the mass of the dilaton has the effect to increase the value φ 0 monotonically. As a consequence, for large enough dilaton masses, the function φ(r) becomes uniformly small and the corresponding Yang-Mills field converges to the BPS monopole, suitably rescaled, since U = 1. For U > 1 the scenario is different, as suggested by Fig. 2. Indeed, increasing m, we observe that the value of φ 0 becomes more negative in the region of the origin, so that the large values of L ym are lowered by the dilaton factor. While keeping m increasing, we observe that the value φ 0 reaches a minimum for some m = m cr and then increases to reach zero. We would like to point out that our numerical analysis becomes rather involved for m 2 > 1000. The occurrence of a minimal value for φ 0 can be related to the balance between the Yang-Mills and dilaton pieces of the action density.
The behaviour of the solutions in the large U limit and with m > 0 fixed is illustrated by Fig. 3 in the case n = 0 and m 2 = 10. Here we reported the Yang-Mills action density and the dilaton function φ(r) for three values of U, namely U = 2, 10, 100. We see in particular that the function L ym reaches its maximum closer to the origin while increasing U defining the radius of the soliton core. Outside this core the dilaton field approaches the profile φ ab . Interestingly, our numerical analysis strongly suggest that for large U, the relation holds, for both cases m = 0 and m = 0. In the massless case [9] this relation leads to the existence of critical valueÛ cr when we transform to solutions to an alternative gauge, saŷ φ,Û where the dilation shift symmetry Eq. (18) is fixed by demandingφ(0) = 0. The critical valueÛ cr is then determined by exp(F ) where F is the constant above. Our result strongly suggest that the constant F is independent on m; at least it was checked with a good level of accuracy for 0 ≤ m 2 ≤ 100. This remarkable numerical result shows that the critical value are independent on the mass and simplifies considerably the analysis of the domain of the solutions. It is illustrated on Fig. 4 where the value φ 0 is plotted as a function of the parameter δ ≡ φ 0 + ln(U) for m 2 = 0, 10, 100. We have for this case F ≈ −1.08.
The values of Euclidean action A = S red and parameter Q e are plotted as function of ln(U) for m 2 = 0, 10, 100 for zero nodes solutions on Fig. (5).

B. Solutions with the nodes
In m 2 > 0 case the node solutions constructed in [9] get smoothly deformed by the dilaton mass parameter. Keeping U fixed and increasing the mass, we observe that the value of the dilaton function at the origin becomes more and more negative. This is due to the fact that, the position of the node, say r 0 , of the function W , is very close to the origin, so that the terms W ′ and (1 − W 2 ) 2 /r 2 are quite large in this region and favorize negative values of φ(r) for r ∈ [0, r 0 ]. This is illustrated by Fig. 6 where the values φ 0 are reported as function of m 2 for U = 0.1 and U = 1.0 (for larger U the pattern is the same). Note that this behaviour of φ 0 of Fig. 6 is different from the zero-node case Fig. 2 and this difference is explained by absence of radial excitations of BPS monopole in flat space-time [16].
The large U-limit of one node solutions was also investigated for several values of m and is illustrated on Fig. 7. We see in particular on this figure that the property (25) also hold for one-node solution. Here we find F ≈ −4.70.
The values of Euclidean action A = S red and parameter Q e are plotted as function of ln(U) for m 2 = 0, 10, 100 for one nodes solutions on Fig. (8).

V. CONCLUDING REMARKS
In the present study we investigated O(3)−symmetric solutions of four dimensional Euclidean YMD theory and found branches of dyonic type solutions for any values of dilaton mass m. They can also be viewed as static (vortex type) solutions of 4 + 1 dimensional theory with Lorentzian signature. Note that in pure Euclidean YM theory solutions with higher, O(4) symmetry are well known instantons. Simple scaling arguments show that there should not be any finite action O(4) symmetric solutions in combined YM-dilaton theory.
The electric component of YM field in Euclidean theory plays role similar to the Higgs field and correspondingly present model is very similar to the static, spherically symmetric sector of the YM-Higgs model investigated by Forgács and Gyürüsi (FG) [15]. The main difference with the FG model is in the way the dilaton field couples to the the Higgs part of the lagrangian.
It was shown [20] that static spherically symmetric solutions of YM dilaton theory found in [5,6,7] have sphaleronic nature similar to their EYM "relatives" [21,22,23], namely they have unstable mode(s), half-integer topological charge and there are fermionic zero modes in the background of these solutions. Interpretation of Euclidean solutions discussed in the present Letter depend in particular on number of negative modes in spectrum of small perturbations about them. We plan to address this topic in further work.
Another interesting topic for further study is the investigation of Euclidean generalisations of axially symmetric [24] and platonic [25] solutions of YMD theory.