High Q BPS Monopole Bags are Urchins
aa r X i v : . [ h e p - t h ] N ov High Q BPS Monopole Bags are Urchins
Jarah Evslin ∗ and Sven Bjarke Gudnason † TPCSF, Institute of High Energy Physics, CAS, P.O. Box 918-4, Beijing 100049,P.R. ChinaandRacah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Abstract
It has been known for 30 years that ’t Hooft-Polyakov monopoles of charge Q greaterthan one cannot be spherically symmetric. 5 years ago, Bolognesi conjectured that, atsome point in their moduli space, BPS monopoles can become approximately sphericallysymmetric in the high Q limit. In this note we determine the sense in which this conjectureis correct. We consider an SU (2) gauge theory with an adjoint scalar field, and numericallyfind configurations with Q units of magnetic charge and a mass which is roughly linearin Q , for example in the case Q = 81 we present a configuration whose energy exceedsthe BPS bound by about 54 percent. These approximate solutions are constructed bygluing together Q cones, each of which contains a single unit of magnetic charge. In eachcone, the energy is largest in the core, and so a constant energy density surface contains Q peaks and thus resembles a sea urchin. We comment on some relations between anon-BPS cousin of these solutions and the dark matter halos of dwarf spherical galaxies.October 9, 2018 ∗ jarah(at)ihep.ac.cn † gudnason(at)phys.huji.ac.il Introduction
Within the large moduli space of solutions of BPS monopoles [1, 2, 3, 4] of charge Q , itis plausible that there exists a high Q limiting sequence of monopoles that are sphericallysymmetric up to 1 /Q corrections. However even the most qualitative features of such solu-tions are, as yet, unknown. In this note we will solve an easier problem, we will explicitlyconstruct configurations in an SU (2) gauge theory with an adjoint scalar whose energyslightly exceeds the BPS bound. These configurations do not provide time-independentsolutions of the equations of motion , but the near saturation of the BPS bound supportsa conjecture that certain qualitative features of these solutions will be shared by some truesolutions. In particular, our approximate solutions become spherically symmetric in thehigh Q limit, and so the corresponding true solutions are also asymptotically sphericallysymmetric.We will construct these approximate solutions by decomposing space into Q identicalcones which extend from the origin, each of which asymptotically contains a unit ofmagnetic flux and is axially symmetric in a sense which will be made precise below. Ofcourse, the classification of regular polyhedra implies that no decomposition of space intoidentical cones exists for general Q . Therefore the cones may not fill all of space. We willprovide configurations in which the space between the cones consists of a vanishing gaugefield and a continuous Higgs field, which at high Q provide a contribution to the energywhich decreases with Q , and so is subdominant with respect to the total energy of theconfigurations, which is proportional to Q .Our problem is then reduced to the following steps. First, in Sec. 2 we will choosean Ansatz and boundary conditions for the configuration in a single cone. The finitenessof the energy fixes all of the functions in our Ansatz except for two angular functionswhich must be chosen. Different choices yield different energies, none of which satisfythe BPS bound but all of which satisfy the bound up to a factor which is Q -independentat large Q . This Q -independence, which is critical for the qualitative agreement with aBPS configuration, is achieved if we impose that the radial dependence of the solutionsatisfies an ODE in a single variable. Next in Sec. 3 we will explicitly solve this equationin the large and small radius limits, and discuss how these solutions grow together. Inthis way we are able to establish the qualitative profiles of these solutions, confirming the They do provide initial conditions for oscillatory time-dependent monopole solutions. /Q corrections are unknown.In Sec. 5 we describe the kinds of field theories in which we expect such monopoles to bethe only monopoles which survive a non-BPS deformation, and provide a very speculativeapplication of such theories.Since Bolognesi’s groundbreaking proposal [5], the field of monopole bags has beensteadily advancing with the understanding of various limiting behaviors [6] and of large Q limits of Nahm transforms of bag solutions [7]. Yesterday a paper by Manton appeared onthe arXiv with significant overlap with our results [8]. He found exactly BPS Bolognesi bagsolutions with an approximate spherical symmetry of the type considered here. This begsthe question: how we can justify presenting an approximate solution after exact solutionshave appeared? Our justification is that the physical application that we have in mind inSec. 5 is in a theory in which we expect the radial dependence to be very distorted, butunder a radial deformation the cone structure of our configurations is preserved. As described in the introduction, charge Q magnetic monopole configurations will beconstructed from Q identical cones extending from the origin together with an extrapo-lation in the region between the cones which yields a contribution to the energy which issubdominant at large Q . The crucial step in the construction is therefore to provide theconfiguration in a single cone. The first step is to determine the size of the cone and tofit it with coordinates.For simplicity let Q = n be a perfect square. This limitation will not be relevant forour analysis of the large Q regime. We will consider a cone whose core extends along thepositive z axis, and will let ψ be the azimuthal angletan ψ = yx . (2.1)If we define the radius in cylindrical coordinates to be ρ ≡ p x + y , (2.2)2hen the cone will be defined by the condition ρ ≤ σ zn , (2.3)where the cone is small enough that n non-overlapping copies fit inside of R . Theconstant σ parametrizes the size of the cone.The base of each cone at a radius r = p x + y + z is then a circle of radius ρ = r/ q n σ + 1 ≃ σr/n and so the area of the base of the cone is about πσ r /n = πσ r /Q which is σ / Q times the total area at that radius, implying that σ ≤ Q cones to fit inside of R . Here we are interested in the large Q limitand hence we can consider the size of the base of the cone to be much much smaller thanthe radius of curvature of the sphere at r . Then the limit, which is an upper bound onthe size of the cone, is given by the size of the circle which can fit inside of a hexagon .More precisely, in the large Q limit the upper bound on σ is σ max = 2 s A circle A hexagon = 2 r π √ ≃ . . (2.4)Clearly such a configuration cannot hope to saturate the BPS bound, instead a qual-itative agreement with a true solution leads to the requirement that it exceeds the BPSbound by a factor (of order unity) which tends to a constant in the high Q limit. We will consider an SU (2) gauge theory with a massless adjoint scalar Φ. Furthermore,we will make the crude approximation that the configuration of the Higgs and gauge fieldfactorizes into z and ρ/z -dependent functions, yielding the following axially-symmetricAnsatz Φ = h ( z ) h F ( η ) (cid:0) c t + s t (cid:1) + ǫ p − F ( η ) t i , (2.5)for the Higgs field and A = α ( z ) z (cid:0) cs [ J ( η ) − G ( η )] t + (cid:2) c G ( η ) + s J ( η ) (cid:3) t − sH ( η ) t (cid:1) ,A = α ( z ) z (cid:0) − (cid:2) c J ( η ) + s G ( η ) (cid:3) t − cs [ J ( η ) − G ( η )] t + cH ( η ) t (cid:1) ,A = α ( z ) nz I ( η ) (cid:0) s t − c t (cid:1) , (2.6) This hexagonal lattice approximation is exact for monopoles in AdS [9, 10]. SU (2) gauge field A i where we have defined the variables η ≡ nρz ∈ [0 , σ ] , ǫ ≡ sign( F ′ ( η )) , c ≡ cos ψ , s ≡ sin ψ , (2.7)and the functions F , G , H , I and J of η and also h and α of z .Such solutions are invariant under a rigid axial symmetry which simultaneously rotatesthe vectors x t y t ! → cos φ − sin φ sin φ cos φ ! x t y t ! , (2.8)and the potential A A ! → cos φ sin φ − sin φ cos φ ! A A ! , (2.9)by an arbitrary angle φ . Topological conditions
We define the charge of the magnetic monopole to be equal to the number of cones Q = n in which the Abelian magnetic field Tr( F Φ) asymptotically pointing in the outward radialdirection integrates, over the base of each cone, to a single Dirac quantum 4 π . Theconfiguration then will have a finite energy only if, for all sufficiently high z , the value ofΦ on the cone’s base is in the correct topological sector. More precisely, we will demandthat, at large z , Φ be constant on the boundary ρ = σz/n of the cone. Therefore the valueof Φ on the base of the cone defines a map from a 2-sphere S , which is the ( z = ∞ , η ≤ σ )two-disc D with its boundary identified with a point, to the space S of values of Φ ofconstant norm | Φ | = v . The topological condition is then that this map S → S be ofdegree one.We will satisfy this condition by imposing that F be a continuous function such that0 ≤ F ( η ) ≤ , F (0) = 0 , F ( σ ) = 0 , (2.10)and F ( η ) = 1 for some intermediate value η of its argument. We will assume thatthe function F has only one maximum on the interval [0 , σ ]. One function obeying thiscondition is F ( η ) = sin (cid:16) πησ (cid:17) . (2.11)4e will see shortly that the analysis of the region between the cones is simplified if onerequires the derivative of F to be zero at η = σ which can be achieved via the deformation F ( η ) = sin (cid:16) πησ (cid:17) (cid:18) − η κ σ κ (cid:19) , (2.12)where κ ≫ F needs to be rescaled so that itsmaximal value is equal to one. It is difficult to calculate the resulting normalization of F analytically, but for κ ≫ High z asymptotics The topological condition on the Higgs field is necessary but not sufficient for the finiteenergy of the configuration. The total energy is the sum of the integrals of the gauge fieldstrength energy density | F ij | and the kinetic energy density of the Higgs field | D i Φ | . Solong as the local energy density is finite, a divergence may only arise from the noncompactregion z → ∞ , where we will demand that h ( ∞ ) = gv , α ( ∞ ) = n , (2.13)and so their derivatives tend to zero. The former condition is an arbitrary choice, howeverin the non-BPS generalizations discussed in Sec. 5 it minimizes the potential for the Higgsfield. As will be clear from the analysis that follows, any different choice of limiting valuefor α will lead to a rescaled value of the finite energy conditions on the functions G , H , I and J . The combination of the rescaling of α with that of the functions leads toprecisely the same form of the connection, and so this reflects a simple redundancy in theparametrization of our Ansatz. Higgs field kinetic energy
We will now impose that the total energy in each cone is finite. There are two contributionsto the energy, one from the magnetic field F ij F ij and another from the kinetic term of theHiggs field | D i Φ | . As both contributions are positive definite, we must demand that theyare finite separately. We now begin by imposing the finiteness of the Higgs field kineticenergy.The finiteness of the Higgs field kinetic energy places non-trivial conditions on thefunctions in our Ansatz. At high z the Higgs field tends to a constant limit, and so itscovariant derivative is at most of order O (1 /z ) and the energy density | D i Φ | is therefore atmost of order O (1 /z ). The finiteness of the energy of the configuration is then equivalent5o the vanishing of the order O (1 /z ) and O (1 /z ) terms of | D i Φ | for each spatial index i and each gauge direction. As there are three spatial directions and three gauge directions,this consists of 9 directions, of which we will see 3 are independent.The axial symmetry (2.8) and (2.9) implies that the energy density is axially symmet-ric. Therefore it suffices to calculate the energy at ψ = 0, corresponding to x = ρ and y = 0. This is merely to simplify the expressions in the following. The three contributionsto the kinetic energy are then the squares of the quantities D i Φ = ∂ i Φ + i [ A i , Φ], D x Φ | y =0 = ǫhz (cid:0) n | F ′ | − αG √ − F (cid:1) (cid:18) t − ǫF √ − F t (cid:19) ,D y Φ | y =0 = hFz (cid:18) nη − α (cid:18) ǫJ √ − F F + H (cid:19)(cid:19) t , (2.14) D z Φ | y =0 = h ′ (cid:0) F t + ǫ √ − F t (cid:1) + ǫhz (cid:16) − η | F ′ | + αn √ − F I (cid:17) (cid:18) t − ǫF √ − F t (cid:19) . The terms of order O (1 /z ) and O (1 /z ) in the kinetic energy will only arise if there areterms of order O (1 /z ) in D i Φ. Therefore if we assume that h − gv and α − n go to zeroat least as quickly as 1 /z , then we may drop the first term in the expression for D z Φ andapproximate α by n everywhere in (2.14) without affecting the divergent terms.The O (1 /z ) terms in each of the three D i Φ can then be seen to be proportional to acombination of the functions F , G , H , I and J . By setting these combinations to zero,we eliminate the divergence. These three conditions can be solved to yield, for example, G , H and I as functions of F and J , we find respectively G = | F ′ |√ − F , H = 1 η − ǫJ √ − F F , I = η | F ′ |√ − F . (2.15)Recall that F ( σ ) = 0 therefore H is only finite if J ( σ ) = 0 as well, and in fact the ratioof the two needs to tend to a constant as z tends to σ . F (0) = 0 as well, and so thefiniteness of H (0) requires that the divergences in both terms cancel. Here ǫ = 1 and solim η → JF = lim η → η + O (1) . (2.16)Substituting these relations into (2.14) the J dependence cancels and so we can findexact expressions for the covariant derivatives as functionals of F alone D x Φ | y =0 = hF ′ z ( n − α ) (cid:18) t − ǫF √ − F t (cid:19) ,D y Φ | y =0 = hFzη ( n − α ) t , (2.17) D z Φ | y =0 = h ′ (cid:0) F t + ǫ √ − F t (cid:1) − ηn D x Φ | y =0 . z → hz < ∞ , lim η → Fη < ∞ , (2.18)and imply in particular h (0) = α (0) = 0 , (2.19)as they must be in order for Φ and A i to be well defined at the origin. A finite energydensity at the maximum of F also requires that F ′ vanish as quickly as √ − F .The total Higgs kinetic energy density is found by adding the squares of the compo-nents in Eq. (2.17) ρ kin = 1 g X i =1 Tr | D i Φ | = 12 g ( h ′ ) + h g z ( n − α ) (cid:18) F η + ( F ′ ) − F (cid:18) η n (cid:19)(cid:19) . (2.20)As the covariant derivatives were independent of J , so is the total kinetic energy density. Magnetic field kinetic energy
As was the case for the Higgs kinetic term, the axial symmetry of the energy means that,for the purposes of calculating energy, it suffices to consider the gauge field strength F ij = ∂ i A j − ∂ j A i + i [ A i , A j ] , (2.21)at y = 0. Again this contribution to the total energy will be finite if the energy densityis at most of order O (1 /z ), which means that the field strength itself must be at mostof order O (1 /z ). Therefore it will again suffice to impose that terms of order O (1 /z )vanish, and that the coefficient of the order O (1 /z ) terms is finite.These conditions are only nontrivial in the case of F , which is F | y =0 = αz (cid:18)(cid:20) n (cid:18) − J ′ + G − Jη (cid:19) − αGH (cid:21) t + (cid:20) n (cid:18) H ′ + Hη (cid:19) − αGJ (cid:21) t (cid:19) . (2.22)A divergence at the tip of the cone can be avoided iflim z → αz < ∞ . (2.23)The expression (2.22) is then, at any constant η , of order O (1 /z ) and so the energy isonly divergent if the field strength itself diverges. To avoid such a divergence as η tendsto 0 we will impose that the two quotients by η are finitelim η → G − Jη < ∞ , lim η → Hη = lim η → (cid:18) η − J √ − F ηF (cid:19) < ∞ , (2.24)7here we have used the expression for H in Eq. (2.15). From here we learn that G (0) = J (0), H (0) = 0 and also this yields a refinement of the boundary condition (2.16)lim η → JF = lim η → η + O ( η ) . (2.25)If we now expand F as F ∼ c j η j + O (cid:0) η j +1 (cid:1) , (2.26)where j = 1 , , , . . . and j represents the first non-zero term in the expansion thenEq. (2.25) implies J ∼ c j η j − + O (cid:0) η j +1 (cid:1) . (2.27)This is sufficient for rendering H/η finite at small η . Finally, G ∼ jc j η j − + · · · togetherwith Eq. (2.27) implies that the first constraint of Eq. (2.24) is trivially satisfied for j > j = 1 due to the matching of the coefficients of G and J .The other components of the field strength can be evaluated easily. Using the factthat I = ηG from Eq. (2.15) one finds the y component of the magnetic field F | y =0 = − α ′ z G t , (2.28)which yields a finite energy contribution at large z since α ′ is at most of order O (1 /z ).A divergence is avoided at small z if in addition to (2.23) one imposeslim z → α ′ z < ∞ . (2.29)In fact this condition follows from (2.23) if α is differentiable at η = 0.The final component of the magnetic field is F | y =0 = (cid:20) αηz (cid:18) G − Jη − αGHn − J ′ (cid:19) + α ′ Jz (cid:21) t − (cid:20) αηz (cid:18) − Hη + αJ Gn − H ′ (cid:19) + α ′ Hz (cid:21) t . (2.30)Again the finiteness of this contribution to the energy is guaranteed by the conditionEq. (2.23). To choose a consistent configuration outside of the cones, one must first determine theconfiguration on the boundaries of the cones. We have already imposed the boundarycondition F ( σ ) = 0 and we have seen that the finiteness of H ( σ ) implies that J ( σ ) =8 as well. In general the shape of the region between the cones is quite complicated.However, we are only interested in an approximately BPS configuration, which we defineas a series of configurations at various values of Q such that at large Q the energy isasymptotically proportional to Q . We will see in this subsection that it is thereforesufficient to consider a configuration in which the gauge field vanishes in the region betweenthe cones, considerably simplifying our analysis.The vanishing of the gauge field in the region between the cones means that it alsovanishes on the boundaries of the cones G ( σ ) = H ( σ ) = I ( σ ) = 0 . (2.31)According to Eq. (2.15), for G ( σ ) and I ( σ ) to vanish it is sufficient to fix F ′ ( σ ) = 0 , (2.32)which is the reason for the modification proposed in Eq. (2.12). The vanishing of H ( σ ) isonly slightly more complicated. By (2.15), now with ǫ = − − σ = lim η → σ J ( η ) p − F ( σ ) F ( η ) . (2.33)Since F ′ ( σ ) vanishes, so must J ′ ( σ ) and so taking yet another derivative of the numeratorand denominator J ′ ( σ ) = 0 , J ′′ ( σ ) = − F ′′ ( σ ) σ . (2.34)With these choices the gauge field vanishes on the boundary and the Higgs field isequal to Φ = − h ( z ) t = − h (cid:16) r/ p σ /n (cid:17) t . (2.35)These fields may then be easily extended to the region outside of the cones, simply byasserting that A i always vanishes and Φ always obeys (2.35). This choice certainly doesnot minimize the energy, but as we will now argue, it yields a contribution to the energywhich at large Q is increasingly subdominant.Notice that the energy density between the cones is simply | ∂ i Φ | which depends onlyupon r . Thus it is equal to a Q -dependent constant multiplied by the area of a sphere ofradius 1 which is not within a cone. This area tends to a constant at large Q . In fact, atlarge Q the maximal number of cones that can be packed into a finite volume approachesthe maximum packing of their circular cross-sections on a plane, which is given by theratio of the area of a unit hexagon to a unit circle 2 √ /π . The area at unit radius outsideof a cone is then 8 √ − π and hence, integrating over the radius E out = (cid:0) √ − π (cid:1) g Z dr r Tr | ∂ i Φ | = (cid:0) √ − π (cid:1) g Z dr r ( h ′ ( r )) , (2.36)9ields the energy outside of the cones.Therefore at large Q the geometric factor asymptotes to a Q -independent constant.Also at large Q the σ/n in the argument in Eq. (2.35) becomes negligible. Thereforethe only source of Q -dependence is in h itself, which interpolates between 0 and the Q -independent constant gv . We will argue in Sec. 3 that this interpolation occurs over aregion of size of order O ( √ Q ). Therefore one expects that the derivatives will decrease,and in particular the derivative squared energy density will scale as 1 /Q . However therange of integration scales as √ Q , and so one expects the total contribution to the energybetween the cones to decease as 1 / √ Q , becoming ever subdominant as compared withthe BPS energy which is proportional to Q itself. At an arbitrary point in space, the covariant derivatives of Φ are D Φ = (cid:0) c X + s X (cid:1) t + cs ( X − X ) t − cX t , (2.37) D Φ = cs ( X − X ) t + (cid:0) s X + c X (cid:1) t − sX t , (2.38) D Φ = (cid:16) Z − ηn X (cid:17) (cid:0) ct + st (cid:1) + (cid:16) Z + ηn X (cid:17) t , (2.39)where we have defined the following functions X ≡ hF ′ z ( n − α ) , X ≡ hFzη ( n − α ) , X ≡ hF Gz ( n − α ) , (2.40) Z ≡ h ′ F , Z ≡ h ′ ǫ √ − F , (2.41)and the field strength components are F = nη U (cid:0) ct + st (cid:1) + nη U t , (2.42) F = (cid:2) c ( W J + U ) + s W G (cid:3) t + cs ( W ( J − G ) + U ) t + c ( U − W H ) t , (2.43) F = cs ( W ( J − G ) + U ) t + (cid:2) c W G + s ( W J + U ) (cid:3) t + s ( U − W H ) t , (2.44)where we have defined U ≡ αz h G − J − η (cid:16) J ′ + αn GH (cid:17)i , U ≡ αz h H + ηH ′ − αn ηGJ i , W ≡ α ′ z . (2.45)10he BPS equations read Z − ηn X − nη U = 0 , Z + ηn X − nη U = 0 , (2.46) X = W J + U , X = W G , X = W H − U . while the energy density for BPS-saturated configurations reads g H cone , BPS = ǫ ijk ∂ i Tr ( F jk Φ) = ǫ ijk Tr ( F ij D k Φ)= nη ( U Z + U Z ) + W ( J X + GX + HX ) . (2.47)The only function of η which is not given in terms of the function F is J .We will in the following formally integrate the above boundary term, obtaining thecorrect magnetic charge. This procedure imposes some criteria on J . Using the relations(2.15) we obtain the topological contribution to the energy density g H cone , BPS = αh ′ z η (cid:20) ( n − α ) ǫF F ′ √ − F + nηJ F ′ F − n ( J + ηJ ′ ) F (cid:21) + α ′ hz η n − α ) ǫF F ′ √ − F . (2.48)The total energy of the cone is given by E cone = 2 πn Z dz Z σ dη z η H cone , (2.49)which tells us that it will be convenient to have the energy density of the form η d ( ··· ) dη .After this integration, the BPS part of the energy depends only on the boundary dataand so by Stokes’ theorem is of the form z d ( ··· ) dz . Rewriting Eq. (2.48) as discussed g H cone , BPS = αh ′ z η (cid:20) ( n − α ) ∂ η (cid:16) − ǫ √ − F (cid:17) − n∂ η (cid:18) ηJF (cid:19)(cid:21) + α ′ hz η n − α ) ∂ η (cid:16) − ǫ √ − F (cid:17) , (2.50)we see from the first and third terms that − ǫ √ − F | σ = 2. Hence, in order to obtain atotal derivative in z , we need to impose − ηJF (cid:12)(cid:12)(cid:12)(cid:12) σ = 2 . (2.51)This can be done consistently with the boundary condition (2.34) in many ways, amongwhich we will choose J = Fη ǫ √ − F . (2.52)11arrying out first the η integration, we obtain Z σ dη η H cone , BPS = 2 αh ′ g z (2 n − α ) + 2 α ′ hg z n − α ) = 2 g z ∂ z [(2 n − α ) hα ] , (2.53)which can readily be integrated E cone , BPS = 4 πg n [(2 n − α ) hα ] ∞ = 4 πvg . (2.54)This is exactly one Dirac quantum contained in a single cone.We will however see that we can only approximately satisfy the above BPS equationsand hence we will need to calculate the total energy density H cone = 12 g (cid:20) X + X + X + (cid:16) Z − ηn X (cid:17) + (cid:16) Z + ηn X (cid:17) (2.55)+ (cid:18) n η (cid:19) (cid:0) U + U (cid:1) + 2 W ( J U − HU ) + W (cid:0) G + H + J (cid:1) (cid:21) , for our approximate solutions. The first five terms are the kinetic energy contributionsgiven in Eq. (2.20).The total energy of the monopole is then E monopole = QE cone + E out . (2.56)Ideally the BPS equations could all be satisfied, in which case E cone = E cone , BPS , however,as we will see this does not turn out to be the case for our configuration. Hence theenergy of the cone is given by Eq. (2.49) with H cone given by Eq. (2.55). This energy willnecessarily exceed the BPS bound (2.54). The conditions in Secs. 2.3 and 2.4 guarantee that the energy of these configurationsis finite. However, a qualitative agreement with true BPS solutions is only possible ifthe total energy is, at large Q , proportional to Q . Such a scaling constrains the radialdependence of the Higgs field and gauge fields. There are many different choices ofradial dependence which yield the correct scaling . We will choose one of the simplest,we will impose that the Bogomol’nyi equations be satisfied near the cores of the cones.More precisely, we will expand the solution as a power series in η/n and will apply the Indeed two such choices were described yesterday in Ref. [8]. F as follows F ( η ) = aη − bη + O ( η ) , (2.57)and choose J = Fη ǫ √ − F , (2.58)as was suggested in Eq. (2.52), we obtain at leading order in η/n the following ODEs h ( z ) = α ′ ( z ) n − α ( z ) , (2.59) h ′ ( z ) = a α ( z ) z (2 n − α ( z )) , (2.60)which can be combined into a single equation for αα ′′ ( z ) n − α ( z ) + (cid:18) α ′ ( z ) n − α ( z ) (cid:19) − a α ( z ) z (2 n − α ( z )) = 0 . (2.61)With the radial differential equation at hand we are now ready to write down all of theangular profile functions in the next subsection. Summarizing the construction of this section, given the functions F and J one maydetermine all of the angular functions in the Ansatz (2.5,2.6). The radial functions onthe other hand are determined by the Bogomol’nyi equations (2.59) and (2.60) at leadingorder in η/n .For instance, if F is given by Eq. (2.12) and J by Eq. (2.52), then the other angularfunctions are given by G ( η ) ≃ (cid:12)(cid:12)(cid:12) πσ cos (cid:0) πησ (cid:1) (cid:0) − η κ σ κ (cid:1) − κη κ − σ κ sin (cid:0) πησ (cid:1)(cid:12)(cid:12)(cid:12)q − sin (cid:0) πησ (cid:1) (cid:0) − η κ σ κ (cid:1) , I ( η ) = η G ( η ) , H ( η ) = F ( η ) η , (2.62)whose η/n expansion contains (2.61) with a = πσ . (2.63)13s we have mentioned, we will not be able to satisfy the BPS equations everywhere(or equivalently to all orders in the η/n expansion) due to the fact that true BPS solutionsdo not exactly factorize. Hence, in order to measure the excess energy with respect to atrue BPS configuration we need to calculate the total energy density as follows g H cone = 12 h ′ ( z ) + (cid:18) σ n P ( η, σ ) + Q ( η, σ ) (cid:19) h ( z ) ( n − α ( z )) z (2.64)+ Q ( η, σ ) α ′ ( z ) z + (cid:18) σ R ( η, σ ) + 1 n S ( η, σ ) (cid:19) α ( z ) (2 n − α ( z )) z , which should be integrated over the volume of the cone as in Eq. (2.49) while the totalenergy is given by Eq. (2.56). All the above functions P , Q , R and S are defined suchthat their integral R σ dη ηX = const is independent of σ , where X = P , Q , R , S .By evaluating the integrals over the angular functions numerically we can write thetotal energy for the cone E cone ≃ πg n Z dz z (cid:20) σ h ′ ( z ) + (cid:18) . σ n + 3 . (cid:19) h ( z ) ( n − α ( z )) z (2.65)+ 3 . α ′ ( z ) z + (cid:18) . σ + 1 . n (cid:19) α ( z ) (2 n − α ( z )) z (cid:21) . One learns from this expression that there is a competition in energy which is determinedby the value of σ . Only the first and the last term are dependent on σ at large n andthe first wants σ to be small while the last one prefers a large value of σ . It so happensthat the last term wins the competition due to the fact that h varies only over a fairlysmall fraction of the integration range while the last term is roughly n σ z which is largefor large n . Hence a minimization of energy leads to a value of σ which is as large aspossible, i.e. σ ∼ . As we have described, the monopole is characterized by the configuration in a singlecone. This consists of two parts, an angular function F which needs to be chosen as wellas radial functions α , h . The factorizing Ansatz (2.5,2.6) implies that the radial functionsare independent of the choice of F . In particular the function α may be determined usingthe second order ODE (2.61) and then h may be determined from α using Eq. (2.59).14 .1 Small z asymptotic behavior At small z , near the center of the monopole, the gauge and Higgs fields approach zero.These two conditions imply the boundary conditions α (0) = α ′ (0) = 0 . (3.1)In the small z region, α, α ′ ≪ n . Therefore (2.61) may be approximated by α ′′ ( z ) − a n α ( z ) z = 0 . (3.2)This is a homogeneous, linear, ODE with the solution α ( z ) = k z e , e = 12 + r
14 + 2 a n . (3.3)At large n , corresponding to large Q , this implies α ( z ) ≃ k z √ an . (3.4)The function h can then be found from the small z limit of Eq. (2.59) h ( z ) ≃ α ′ ( z ) n = √ ak z √ an − . (3.5) z asymptotic behavior Far from the monopole, a finite energy solution requires that α tends to n . It will proveconvenient to define the function β ( z ) ≡ n − α ( z ) , (3.6)which tends to 0 at large z . Eq. (2.61) may be re-expressed in terms of β as − β ′′ ( z ) β ( z ) + (cid:18) β ′ ( z ) β ( z ) (cid:19) + a z (cid:0) β ( z ) − n (cid:1) = 0 . (3.7)We will now assume that β may be expanded at large z such that the leading term is β ( z ) = k z e e − mz (3.8)where m >
0. Substituting this into (3.7) one finds e − a n z + · · · = 0 , (3.9)15here the ellipsis denote exponentially suppressed terms, which may be canceled by sub-dominant corrections to (3.8). Therefore we demand only that the first term vanishes,fixing e = a n .Again the Higgs field profile function h is easily found from (2.59) h ( z ) = − β ′ ( z ) β ( z ) = m − a n z . (3.10)BPS monopoles are characterized by the boundary condition that h tends asymptoticallyto the vacuum expectation value gv (recall that we have rescaled the coupling into thescalar field). Therefore m is determined entirely by this boundary condition m = gv . (3.11)Physically, m is the mass of the W -bosons, and the exponential decay in Eq. (3.8) is justthat of a massive field. Our final asymptotic form for the function β ( z ) is hence β ( z ) ≃ k z a n e − gvz . (3.12) We have studied two regimes: one, at small z , where α ≪ n and another, at large z , where β = n − α ≪ n . Clearly it is important to determine at what value z = z the solution of(2.61) interpolates between these two regimes. Neither approximation of z leads to a well-controlled expansion in the intermediate regime. However one may arrive at a qualitativeunderstanding of the solution by imagining that both expansions are roughly correct, inthe sense that will be described below, at z . This is potentially a dangerous assumption,but numerically we have verified that the results of this subsection are indeed correct.We will define z to be the midpoint of the function αα ( z ) = β ( z ) = n . (3.13)Therefore the matching of the two limits yields α ′ ( z ) α ( z ) = − β ′ ( z ) β ( z ) . (3.14)This relation is exact. But the consequences may be approximated by substituting theasymptotic behaviors (3.4) of α at small z and (3.12) of β at large z which gives anestimate for z ∼ angv (cid:16) √ an (cid:17) ∼ a n gv , (3.15)16here in the rightmost expression we have approximated n ≫
1. From this expression itis clear that the 1 /z tail of the Higgs field gives the monopole its size ∝ n .The value of z in Eq. (3.15) is the approximate size of the monopole, the value of theradius at which the W -bosons begin to fall exponentially. At charge 1, corresponding to n = 1, one recovers the fact that the monopole size is the inverse W -boson mass. Howevermore generally it reveals, as was conjectured in Ref. [5], that the radius of the monopoleis proportional to the charge Q = n .One may also determine the thickness w of the boundary region, the distance overwhich α tends from a value near 0 to a value near n . This depends on how quickly α changes at the boundary z α ′ ( z ) = √ an α ( z ) z ≃ gv √ a (3.16)valid for n ≫
1. The distance over which α changes by n units is then approximately w ∼ nα ′ ( z ) ∼ √ angv . (3.17)Therefore the width of the boundary of the monopole is proportional to n = √ Q , whilethe radius of the monopole is proportional to n = Q . Thus at large n the walls ofthe monopole are much thinner than its radius, confirming a conjectured description ofthese solutions as monopole bags in Ref. [5]. However one should note that, given thelarge moduli space of solutions, it seems possible that there are other sequences of BPSsolutions with approximate spherical symmetry in the large Q limit but with differentqualitative radial profiles, and so the bag description may not apply to them.The boundary of the monopole is not only relatively thin, with a width w much smallerthan z at large n , but also it is very sharp, as β exponentially decays at large z as seenfrom Eq. (3.12). As we have mentioned, this is due to the fact that the gauge field ismassive. On the other hand, the massless Higgs field decays only as O (1 /z ), as seenin Eq. (3.10). Therefore the Higgs field does not exhibit such a sharp transition at theboundary of the monopole. In this section we will provide a few numerical solutions to the ODE (2.61) illustratingthe radial profile functions for the monopole. We have used a shooting method to find thesolution using Eq. (3.4) as the initial condition and hence k as the shooting parameter.17 solution with Q = 81 , n = gv = 9 , σ = 1 . πvn /g . α h Figure 1: Left panel: the profile function α . Right panel: the profile function h both asfunctions of z for Q = 81 , n = gv = 9 , σ = 1 . , z = 25 . a = π/σ . E ne r g y den s i t y η Energy densityz
Figure 2: Left panel: the radial energy density R σ dη ηg H cone as function of z for themonopole with Q = 81 , n = vg = 9 , σ = 1 .
9. Right panel: the energy density g H cone ofthe cone.In order to check the size of the monopole estimated in Eq. (3.15), we have numericallycalculated the value of z such that α ( z ) = n/ n . 18 g v z Figure 3: The approximate monopole size gvz which is predicted to be πσ n (cid:0) √ πσ n (cid:1) and shown as the red line (we have set σ = 1 . % E ne r g y / BPS
Figure 4: The total excess energy of the monopole compared to the BPS-saturated energypercentage as a function of n = √ Q . It is seen that the energy excess is roughly constantat large n around 54%. 19 Future directions
First of all we should provide a word of caution due to the fact that the BPS solutionsdo not share the factorization property of our configurations. For this reason we havecalculated the radial profiles in terms of angular expansion parameters expanded aroundthe origin. This means that our monopole configurations are only approximately BPS,and in particular do not provide time-independent solutions of the equations of motion.However we do believe they capture some key features of those true BPS monopoleswhich are spherically symmetric. As a support to this claim we found numerically thatat high Q the total energies of our configurations exceed that of true BPS solutions by a Q -independent factor of roughly 54%.As we have mentioned, it may seem as though we are studying a particularly diffi-cult and uninteresting part of the moduli space of solutions. We would however like toconjecture that in a certain class of theories it is the most interesting part: Conjecture 1
The approximately spherically symmetric BPS monopoles are the onlyones which survive the strong non-BPS deformation described below. They all reduce tothe same non-BPS configuration.
Our deep interest lies in non-BPS monopoles in which a Higgs potential is includedfor the scalar Φ. We are interested in these monopoles because of a series of perhapscoincidental facts relating Q = 1 non-BPS monopoles with the dark matter halos of themany minimal size dwarf spheroidal galaxies which have recently been discovered in ourlocal group, for example by the Sloan Digital Sky Survey. Some of the most strikingsimilarities, which in fact are shared by other dark matter dominated galaxies such aslarger dwarf and low surface brightness spiral galaxies, are as follows:1) Dark matter halos, like topological solitons, have a minimum mass. For solitons thiscorresponds to the charge Q = 1. The lightest satellites of the Milky Way have massesof about 10 M ⊙ within 1,000 light years of their center [11] and between 10 and 10 M ⊙ within 2,000 light years [12]. This observed minimum dark matter mass leads to the dwarfgalaxy problem: For particle models of dark matter, like WIMPs, consistency with thisminimum mass is a problem because simulations generally suggest 4 to 400 times moredwarf galaxies in our local group than have been observed, most of which should be muchlighter than the minimum observed mass. In fact, only one dwarf galaxy (ComaBer [13])has been observed with a mass near 10 M ⊙ , and it is very elongated and irregular andappears to be in the process of being ripped apart by tidal forces.20he existence of a topological charge, equal to one, for these small galaxies not onlyexplains the fact that no small dwarf galaxies have been seen, but also the related factthat smaller dark matter bodies cannot exist. In this way one avoids the fatal gravitationallensing constraints faced by other MACHO dark matter models. Indeed the upper limitof the range of radii of dark matter candidates excluded by gravitational lensing is manyorders of magnitude smaller than these 1000 light year solutions, and so these monopolesare too large to be excluded by the lensing bounds.2) At least in cases in which there is enough visible matter to determine the density profile,dark matter dominated galaxies and non-BPS monopoles have cores with relatively con-stant densities, with intermediate regions with 1 /r densities and external regions with afaster radial fall-off. For Q = 1 non-BPS monopoles this 1 /r intermediate region densityprofile is inevitable, unlike the the higher Q BPS 1 /r density profile in the Bolognesigalaxy bags of Ref. [8] in which there are many inequivalent choices of r -dependence, suchas solutions which the author called planets, etc.
3) The cores of these non-BPS solutions in many cases naturally contain black holes [14,15, 16, 17]. In the case of the galaxies, models often suggest that there has not beenenough time to form the supermassive black holes known to inhabit most galactic cores, inaddition there are even some claims of supermassive black holes without luminous galactichosts. These problems are both naturally explained if the black hole is an integral partof the monopole solution, as it is in many models. The gradual consumption of stars,gas and dark matter particles is, in this scenario, no longer the main mechanism drivingsupermassive black hole formation.4) The simplest model in which non-BPS ’t Hooft-Polyakov monopoles exist is a Georgi-Glashow model with a simple Abelian Higgs quartic potential. In this case, a Q = 1non-Abelian monopole with the radius r and mass M of the smallest dwarf galaxies arisesif the value of the Higgs VEV is about v ∼ p ~ c M/r ∼ GeV. This number onlychanges by about a factor of 2 depending on whether the luminous region is within the coreor the intermediate radius regime. Had v been above the Planck scale, gravity would havedominated over the Georgi-Glashow interactions and the whole solution would have beena hairy black hole instead of a dwarf galaxy. Had it been smaller than about 100 eV, thesemonopoles could not have formed in time for dark matter to have played its crucial role inthe oscillations of primordial plasma which reproduces the oscillation spectrum observed Here we are considering a new sector, these gauge symmetries have no obvious relation with standardmodel or GUT symmetries and we do not specify the charges of standard model particles under this newgauge symmetry.
21n the CMB. Given that the two physical inputs in this calculation are of galactic scales,the fact that the output is a particle physics scale in this relatively narrow acceptablewindow is for us miraculous. If one naively uses the rotation curves of slightly largerdwarves one may similarly conclude that the Higgs coupling is of order λ ∼ − , hadit entered v with a different power, even a fourth root, the relation with dwarf galaxieswould have been ruined.5) Similarly the 1,000 light year scale radii of these solutions imply that they form when theuniverse is about 1,000 years old [18]. Again, this is in time to help increase the intensityof fluctuations in the primordial plasma as is required by observations of the CMB. Haddwarf galactic radii been larger by a factor of 100, they would have formed too late andan inconsistency would have arisen.6) While the monopole core excluding gravity has a constant density, and with gravity mayhost a black hole, the core itself is nonsingular. More precisely it avoids the cusp prob-lem of the Λ CDM model, in which many simulations predict galactic mass distributionprofiles, such as the historic Navarro, Frenk and White profile [19], with density cusps intheir cores, in stark contrast with observations.While these similarities are very strong, there is a serious problem with galaxy sizednon-BPS Bolognesi bags as a dark matter candidate. Non-BPS monopoles repel, andas v is less than the Planck scale this repulsion would dominate over gravity and allgalaxies would be minimal dwarves and would repel one another. There is a similarproblem of course for visible matter, which is mostly made of protons which also repel.In the case of protons, the solution is quite complicated. First of all there are electronswhich screen the interactions between protons. While electrons have antiparticles whichhave the same charge as protons, for reasons which have not yet been quantitativelyexplained by any model, there was a primordial excess of negatively charges electronsand positively charged protons. They did not annihilate each other because they carrydifferent conserved charges. They can combine, forming hydrogen bound states or viainverse beta decay they can even merge beyond recognition into neutrons. However, dueto the choice of parameters in the standard model, the later possibility is kinematicallydisfavored in the conditions that have existed in most of the universe since baryogenesis.We would like to propose that repulsion between non-BPS monopoles is avoided in asimilar manner. Additional conserved charges are easily introduced in a Georgi-Glashowmodel by including charged fermions, which via the Jackiw-Rebbi mechanism provide anadditional charge for each kind of monopole. If one adds two species of fermions, then22here are two kinds of charge, which can play a role analogous to baryon number andlepton number. Monopoles of different charges can have very different masses, in fact in N = 1 supersymmetric models some flavors are usually massless while some are massive.One can then demand that the dark matter halos are made of very massive magneticallypositively charged monopoles which carry one kind of flavor charge, and that the screeningis caused by light negative monopoles with the other flavor charge. This eliminates theproblem of galaxies repelling another.But one still needs to worry about the stability of Q > SU (2)dynamics will dominate over the attractive scalar dynamics at large distances, leadingto a net repulsion. The gravitational interaction in general is insufficient to counter thisrepulsion, as v is much less than the Planck mass. However at large distances one expectsthe screening to play a role. Unfortunately the exact role played by this screening ishighly model dependent and is not clear whether there exist any models which screenthis repulsion sufficiently to allow it to be dominated by gravitational attraction, withoutadding any new interactions (playing the role of strong interactions in the proton analogydescribed above). Strong constraints on models also arise from the fact that one does notwant the light monopoles to combine with the heavy monopoles, analogously to inversebeta decay, as the resulting bound state may not share the attractive features of themassive monopoles described above.Therefore the model-independent predictions for Q > Q = 0 and dwarf galaxies at Q = 1, there must also be a gap between Q = 1 and Q = 2. This prediction is much more general than the monopole dark matterproposal discussed in this section, but extends to any topological soliton dark mattercandidate which solves the dwarf galaxy problem by identifying minimal spherical dwarfgalaxies with Q = 1 solitons. The masses of these galaxies are at best known at the 100percent level, and so with current data such a gap cannot be verified. However one mayhope that radio surveys of gas in our galactic neighborhood such as that which will beperformed by the FAST telescope starting in 5 years will be able to test this claim.Another model independent prediction is that, while the charge Q is determined bythe flat part of the galactic rotation curve, the radius of the core must be proportionalto √ Q and, even more surprisingly, the outer radius of the region with the flat rotationcurves must at large Q be nearly Q -independent. This counter-intuitive prediction mayalready rule out these models, as it requires spatial extents of dwarf galaxy dark matter23alos to extend far beyond their most distant stars, but it is necessary for the convexity ofthe galactic mass as a function of Q , which in turn is necessary to prevent these galaxiesfrom exploding. Acknowledgments
We are eternally grateful to Stefano Bolognesi and Malcolm Fairbairn for many useful andenlightening discussions. JE is supported by the Chinese Academy of Sciences Fellowshipfor Young International Scientists grant number 2010Y2JA01. SBG is supported by theGolda Meir Foundation Fund.
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