aa r X i v : . [ h e p - t h ] J u l Particle-like Q-balls
E. Ya. Nugaev a ∗ , M. N. Smolyakov b † a Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary prospect 7a, 117312, Moscow, Russia b Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,119991, Moscow, Russia
Abstract
Usually the charge and the energy of stable Q-balls vary in a wide range or are evenunbounded. In the present paper we study an interesting possibility that this range isparametrically small. In this case the spectra of stable Q-balls look similar to the one offree particles.
Among the variety of non-topological solitons (see [1, 2] for review) Q-balls [3, 4] and theirproperties were thoroughly examined, in particular, due to the interest encouraged by cosmology(see, for example, [5]). The main soliton characteristics, the energy E and the charge Q , arefunctions of the parameter ω (the standard Q-ball solution in a scalar field theory with global U (1) invariance has the form φ ( t, ~x ) = f ( | ~x | ) e iωt ), which results in the possibility of differentforms of E ( Q ) dependence for different scalar field potentials.Of course, the most interesting Q-ball solutions are stable solutions. In general, there arethree types of the Q-ball stability:1. The first type is the quantum mechanical stability, i.e., the stability with respect to decayinto free particles. If E ( Q ) < M Q for a Q-ball of charge Q , where M is the mass of afree particle in the theory under consideration (without loss of generality, from here onwe suppose that ω ≥ Q ≥
2. The second type is the stability against fission. Q-balls are stable against decay intoQ-balls with smaller charges if d E/dQ < ∗ e-mail : [email protected] † e-mail : [email protected] In the case of special interactions with fermions this kinematic consideration should be revised, see [6].
1. The third type is the classical stability, i.e., the stability with respect to small perturba-tions of the scalar field. The stability criterion proposed in [2, 8] implies that a Q-ball isclassically stable if dQdω < dEdQ = ω always holds for Q-balls, the latter leads to d EdQ = dωdQ . (1)Thus, the criterion of stability against fission and the criterion of classical stability coincide,i.e., classically stable Q-balls are stable against fission.Note that our definition of the absolute stability (at least in the absence of fermions) differsfrom the one of papers [9, 10], where the stability with respect to decay into free particles issupposed to be the strongest criterion, which Q-balls should satisfy, and such Q-balls are calledabsolutely stable in these papers. Our definition is different because, as we will see below, thestability with respect to decay into free particles does not imply the classical stability in thegeneral case.As it was noted above, the E ( Q ) dependencies may have rather different forms in modelswith different potentials. As the first example one can recall the model presented in the well-known paper [8]. The E ( Q ) dependence in this model consists of two branches, one of which(the lower one) is classically stable. Moreover, there exists Q S such that for Q > Q S theinequality E ( Q ) < M Q holds for the lower branch (see Fig. 3(a) in [8]). Thus, Q-balls with Q > Q S from the lower branch of the E ( Q ) dependence are absolutely stable. An analogousform of the E ( Q ) dependence is inherent to other models, see, for example, [7, 11, 12, 13].Another type of Q-balls is the one with only one branch. As an example one may considerthe model with | φ | potential studied in [14]. The E ( Q ) dependence in this model consistsof only one branch with d E/dQ >
0, and all Q-balls in such a model are even classicallyunstable (this was also shown explicitly in [14]).An interesting model with a logarithmic unbounded scalar field potential was proposed in[15] and thoroughly examined in [16]. The E ( Q ) dependence in this model also consists of twobranches, one of which is classically stable (again it is the lower branch). The charge of theQ-balls from the stable branch varies from 0 to Q max < ∞ . An analogous E ( Q ) behavior hasthe model with a simple polynomial potential discussed in [13].In all the examples presented above the spectra of stable Q-balls (if they exist) either haveno upper limit, or have an upper limit, but start from zero. In any case, the charge and the Surely, one can add positive terms to the potential for very large values of the field modulus without alteringthe physics at the scale of stable Q-balls. E ( Q )dependence. It consists of three branches, one of which, – the “lowest” branch, contains Q-ballswhich are classically stable. An important feature of this branch is that there exist both a lowerbound on the charge Q min and an upper bound Q max such that Q-balls with Q min < Q < Q max are classically stable. Such an E ( Q ) dependence arises in the models with piecewise parabolicpotentials examined in [7, 12] (these scalar field potentials were originally proposed in [3]), andin the model with a polynomial potential discussed in [13]. Below we will focus on examinationof such an E ( Q ) dependence with three branches.In order to find out whether it is possible that the range of charges, where the absolutelystable Q-balls exist, can be made small, it is better to have an analytically solvable model.The models discussed in [7, 12] are analytically solvable (the model of [7] provides a verysimple analytic Q-ball solution, which is very useful for examining perturbations above the Q-ball solution explicitly), but the scalar field potentials utilized in these models contain breaks,which is rather unphysical and demands an additional regularization of the potentials. Belowwe will propose a model with a continuous and differentiable potential, admitting the existenceof a simple analytic Q-ball solution and providing the E ( Q ) dependence with three branches,one of which corresponds to classically stable Q-balls. We will calculate Q S and Q max in thismodel and answer the question posed above.We consider the globally U (1) invariant scalar field theory with a piecewise potential of theform V ( φ ∗ φ ) = M φ ∗ φ θ ( v − φ ∗ φ ) (2)+ (cid:16) m φ ∗ φ + 2 v ( M − m ) p φ ∗ φ − v ( M − m ) (cid:17) θ ( φ ∗ φ − v ) , where M > θ is the Heaviside step function with the convention θ (0) = . The form of thisscalar field potential for different values of the dimensionless parameter mM is presented in Fig. 1.We will be looking for a solution to the corresponding equation of motion of the standard form Φ * Φ v1V H Φ * Φ L M v Φ * Φ v1V H Φ * Φ L M v Φ * Φ v1V H Φ * Φ L M v Figure 1: The forms of the scalar field potential described by Eq. (2): m < | m | /M = 2 (leftplot); m = 0 (middle plot); m > m/M = 0 . = f ( r, ω ) e iωt , where r = | ~x | . Without loss of generality, we suppose that f ( r, ω ) >
0. Themonotonic solution for f such that dfdr (cid:12)(cid:12) r =0 = 0 and f | r →∞ = 0 can be easily found and has theform f ( r, ω ) = v ( M − m )( ω − m ) − v ( M − ω )( ω − m ) Rr sin( √ ω − m r )sin( √ ω − m R ) , r < R, (3) f ( r, ω ) = v Rr e −√ M − ω r e −√ M − ω R , r ≥ R, (4)where the matching radius R is such that f ( R, ω ) = v . For r < R we have f ( r, ω ) > v , whereasfor r > R we have f ( r, ω ) < v . It is evident that if m >
0, then
M > ω > m ; if m = 0, then M > ω >
0; otherwise
M > ω ≥ f ( r, ω ) and of its first derivative leads to the following equation for R = R ( ω ): (cid:18) M − m ω − m + √ M − ω R (cid:19) tan( √ ω − m R ) = M − ω √ ω − m R. (5)This equation can be easily solved numerically for a given ω . Note that equation (5) is validonly for Q-ball solutions without nodes. For such a solution and for a given ω one should takethe first (smallest) root of (5) satisfying the condition R ( ω ) > π √ ω − m .The Q-ball charge and energy can also be easily calculated and have the form Q = 2 ω ∞ Z f d x = (6)4 πωv (cid:20) R ( ω − m ) (cid:18)
23 ( M − m ) + ( M − ω ) + ( ω − m )( M − ω ) (cid:19) + R √ M − ω ( ω − m ) (cid:0) M − m + ω ) (cid:1) + 5 R ( M − m )( ω − m ) + R √ M − ω (cid:21) ,E = ωQ + 4 πv M − m ω − m (cid:20) R M − ω ) + R √ M − ω + R (cid:21) , (7)where we have used Eq. (5) in the derivation.Now let us examine the E ( Q ) dependence for different values of the model parameters, i.e.,for m > m = 0 and m <
0. It is not difficult to show that the charge (6) and the energy(7) can be represented as Q = 4 πv M ˜ Q, E = 4 πv M ˜ E, (8)where ˜ Q and ˜ E depend only on ωM and mM and do not depend on v . So, below we will notspecify the values of v and M while examining the main properties of Q-balls in our model:4he E ( Q ) dependencies can be examined by considering the dimensionless quantities ˜ Q and ˜ E for different choices of mM . Such a simplification is possible only because of the simple form ofthe scalar field potential, which appears to be very useful for calculations.The corresponding plots are presented in Figs. 2 and 3. We see that the E ( Q ) diagrams forthe cases m > m = 0 resemble those in the models discussed in [8, 11, 12, 13]. All fourcases, presented in Figs. 2 and 3, also exist in the model discussed in [7].
200 250 300 350 Q M Π v M Π v
150 200 250 300 350 Q M Π v M Π v Figure 2: E(Q) for m > mM = 0 . m = 0 (right plot). The dashed linecorresponds to free scalar particles of mass M .
20 40 60 80 100 120 Q M Π v M Π v M Π v M Π v Figure 3: E(Q) for m < | m | M = 1 (left plot) and | m | M = 5 (right plot). The dashed linecorresponds to free scalar particles of mass M .5t should be noted that, though ω is bounded from above, ω < M , in the limit ω → M the charge and the energy tend to infinity in all four cases, presented in Figs. 2 and 3. Thishappens because the factor √ M − ω in the exponent of (4) tends to zero for ω → M , whereas R ( ω ) | ω → M → π √ M − m ; so the scalar field falls off not exponentially, but as r in this limit. Thelatter leads to infinite charge and energy of the Q-ball for ω → M . Due to the large size of theQ-ball core, such Q-balls were called “Q-clouds” in [11].As it was noted above, we will be interested in the last case m <
0. As it can be seen fromFig. 3, there are two phases: the first phase contains three branches on the E ( Q ) diagram,whereas the other phase contains only one branch (the latter case is similar to the one ofthe model with | φ | potential studied in [14]). The transition between the phases occurs at | m | M ≈ . E = M Q line corresponding to free particles, which means that therange of charges of absolutely stable Q-balls is rather large. Note that the part of the upperclassically unstable branch (which starts from Q = 0) on the left plot in Fig. 3 also lies underthe E = M Q line corresponding to free particles, which means that the stability with respectto decay into free particles indeed does not imply the classical stability in the general case.We would like to note that the existence of a locally maximal charge in the phase with threebranches (see Fig. 3) seems to be a consequence of the turnover of the scalar field potential.We think that this is a rather general property, which is inherent to other models of Q-balls.Although we can not prove it in a rigorous way, we do not know exceptions from this rule.Meanwhile, the opposite is not correct — the existence of the turnover of the scalar fieldpotential does not guarantee the existence of a locally maximal charge, which is confirmed by theexistence of the phase without maximal charge for | m | M > .
775 in our case and by the examplesof other models (see, for example, [14]). We also stress that the maximal value of f ( r, ω ) (whichis simply f (0 , ω )) of the Q-ball with locally maximal charge is not connected with the pointof the maximum of the scalar field potential for m <
0. Indeed, the scalar field potential ismaximal at f Vmax = v (cid:16) M | m | (cid:17) ; whereas the value of f (0 , ω ) decreases monotonically (thiscan be checked numerically) from f (0 ,
0) to f (0 , M ) = 2 v . So, if | m | M ≥ f (0 , ω ) > v ≥ f Vmax : the maximum of the absolute value of theQ-ball scalar field is larger than the point of the maximum of the scalar field potential for any0 ≤ ω < M , i.e., for any Q-ball. An interesting observation in the opposite case | m | M < ω → M lie on the unstable branch, whereas f (0 , M ) = 2 v < f Vmax in thiscase. These examples demonstrate that there is no (at least obvious) connection between themaximal absolute value of the Q-ball scalar field, the point of the maximum of the scalar fieldpotential and the Q-ball stability.Now let us check what happens when we change the parameter ˜ m = | m | M . The result ispresented in Fig. 4. We see that the larger ˜ m is, the smaller the “triangle” in the corresponding E ( Q ) diagram is. Moreover, the larger ˜ m is, the smaller part of this “triangle” turns out to lie6
00 400 600 800 1000 1200 Q M Π v M Π v H a L
30 35 40 45 Q M Π v M Π v H b L
24 26 28 30 32 Q M Π v M Π v H c L
18 20 22 24 26 28 Q M Π v M Π v H d L Figure 4: E(Q) for m < | m | M = 0 . | m | M = 1 . | m | M = 1 . | m | M = 1 . M .under the E = M Q line corresponding to free particles. For ˜ m = ˜ m x ≈ . m > ˜ m x all the classically stableQ-balls are quantum mechanically unstable. For ˜ m & .
775 the “triangle” disappears and thereis no classically stable branch in the E ( Q ) dependence at all.The observations presented above indicate that there exist such values of the parameters thatabsolutely stable Q-balls can exist only in a very narrow range of charges (and, consequently,energies). As an example, let us take ˜ m = 1 . Q = QM πv and ˜ E = EM πv , corresponding to the dots on the plot, are the following:˜ Q S ≈ . E S ≈ . Q max ≈ . E max ≈ . Q = ˜ Q max − ˜ Q S ≈ . , ∆ ˜ E = ˜ E max − ˜ E S ≈ . . These ranges are much smaller than the absolute values of the charges and the energies respec-tively.The closer (from below) ˜ m to ˜ m x ≈ . Q and ∆ ˜ E are. For ∆ ˜ Q ≪ ˜ Q max max M Π v M Π v Figure 5: E(Q) for m < | m | M = 1 . M .the E ( Q ) dependence of the absolutely stable Q-balls is similar to the one in the limiting case˜ m → ˜ m x : E = M Q x , where ˜ Q x = ˜ Q max | ˜ m = ˜ m x ≈ . E ( Q ) dependence of freeparticles at rest! The only difference is that the charge of free particles Q p = 1, whereas for Q-balls we have in the limiting case Q x ≈ πv M . Of course, analogous anti-Q-balls (i.e., Q-ballswith ω < Q <
0) also exist and possess the same properties.We think that the existence of such particle-like Q-balls should be inherent not only to themodel presented above, but to other models providing an E ( Q ) dependence with three branches(namely, to models with scalar field potential admitting the existence of a true vacuum at φ ∗ φ > cknowledgements The authors are grateful to D. Levkov, M. Libanov and I. Volobuev for discussions and tothe unknown referee for useful comments. The work was supported by RFBR grant 14-02-31384. The work of E.Y.N. was supported in part by grant NS-2835.2014.2 of the Presidentof Russian Federation and by RFBR grant 13-02-01127a. The work of M.N.S. was supportedin part by grant NS-3042.2014.2 of the President of Russian Federation and by RFBR grant12-02-93108-CNRSL-a.
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