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In each example, the scale f can be determined un- in imposing the entropy bound on such states. ambiguously. For example, in gravity it is given by the Planck mass f = MP which sets the strength of the Finally, in appendix, we provide an explicit example graviton-graviton interaction. On the other hand, for illustrating how the formulation (4) eliminates an ap- topological solitons such as ’t Hooft-Polyakov monopole parent violation of the Bekenstein bound (1) by a fully the same scale is set by the vacuum expectation value of consistent unitary theory. the Higgs field, f = v. At the same time, in case of a baryon/skyrmion f is set by the pion decay constant fπ. We thus encounter the following universal pattern at the II. SCALAR NO-SCALE SOLITON IN 5D saturation point, As first example we consider a scale-invariant soliton 1 max. entropy = = area . (7) in 5D theory without gauge redundancy. The simplest coupling Lagrangian which leads to such a soliton has the following form,

1 1 2 4 L = ∂ σ∂µσ + g σ (8) For all isolated solitons existing in Lorentzian space- σ 2 µ 4 times the bound (4) fully agrees with the standard 2 Bekenstein bound (1). However, the form (4) allows Here σ is a real scalar field. The parameter g > 0 is a for a consistent generalization of the concept of the five-dimensional coupling constant that has the dimen- 2 1 entropy bound to Euclidean field configurations such as sionality [g ] = [mass]− . The canonical dimensionality 3 instantons that describe virtual processes rather than of the scalar field is [σ] = [mass] 2 . The non-perturbative states. To such entities the standard formulation of solutions of 4D version of the above theory are very Bekenstein bound (1) cannot be applied directly since well-known [10]. Due to an attractive self-coupling their energy is not defined. the energy of the theory (8) is unbounded from below. Despite this, it represents an useful prototype model for We proceed in the following steps. First, we generalize testing our ideas. We shall later move to a gauge theory the analysis of [4] to solitons in 5D theory. We consider that does not suffer from such a problem. examples of scale-independent solitons in theories with global and gauge symmetries. The forms of these solitons The dimensionless four-point quantum coupling at en- are identical to instantons in corresponding 4D theories. ergy scale E is given by We show that 5D solitons saturate both bounds, (1) α g2E. (9) and (4), simultaneously and together with unitar- ≡ ity. We check explicitly that at the saturation point the entropy is equal to the area, in full accordance to (6). Due to a dimensionful gage coupling the theory violates perturbative unitarity above the following cutoff scale, Next, using the formulation of the bound (4) and the 1 Λ . (10) connection between the 5D soliton and 4D instanton, we ∼ g2 generalize the concept of the Bekenstein entropy bound to an instanton. Note, we are not talking about the en- That is, above the scale Λ the dimensionless coupling tropy for a gas of instantons but rather the entropy of a α becomes strong. The above theory has a time- single isolated instanton of fixed size and position. independent spherically symmetric solution of the follow- We assign the instanton a micro-state entropy given ing form, by the entropy of its counterpart soliton in a theory with one dimension higher. Assigning an entropy to an √8 R σ = 2 2 , (11) instanton may sound unusual since the instanton does g x + R not describe a “real” object but rather a tunnelling where x is the space four-vector. The form of the solution process. Nevertheless, using the connection between an is identical to an instanton that appears in 4D version of instanton and a soliton in one dimension higher gives a the theory [10] where it describes a tunneling process. well-defined physical meaning to this assignment. The mass of the soliton is given by

Next, using the form (4) we impose the Bekenstein 8π2 Msol = , (12) bound on an instanton in a self-consistent way. We 3g2 then discover that the instanton entropy saturates the bound and assumes the form of the area when the and is independent of the localization radius R. Cor- theory saturates the bound on unitarity. This happens respondingly the moduli space of the soliton consists when the ’t Hooft coupling assumes a critical value of translation and dilatation zero modes that can be order-one. This gives us confidence in meaningfulness of regarded as Goldstone bosons of corresponding broken generalizing the concept of entropy to virtual states and symmetries. These modes are not sufficient for endowing 3 the soliton with a large entropy that can be increased Consequently, there emerge N 1 Goldstone modes parametrically. In order to achieve such an entropy, we localized within the soliton. − need to increase the number of localized zero modes. We shall accomplish this by assuming that σ-field transforms The situation that we got is fully analogous to the as a large irreducible representation of some internal “fla- example of a ’t Hooft-Polyakov monopole constructed in vor” symmetry group. For example, σα, α = 1, 2, ..., N [4]. There too, a global SO(N)-symmetry was sponta- can form an N-dimensional representation of a global neously broken within the monopole core. Therefore, the SO(N)-symmetry. Correspondingly, the Lagrangian now counting of the micro-state degeneracy goes in the same takes the form, way. Of course, classically, the degeneracy of the moduli space is infinite, since every point on it counts as a 1 µ 1 2 2 Lσ = ∂µσα∂ σα + g (σασα) . (13) different Goldstone vacuum of the soliton. This is also a 2 4 manifestation of the fact that in ~ = 0 limit the entropy Note, with the enlargement of symmetry the unitarity is infinite, as it should be. Of course, in quantum theory bound becomes, the entropy of a localized soliton becomes finite.

1 1 The micro-state degeneracy can be deduced from the N . 2 . (14) α ≡ (Eg ) effective Hamiltonian describing the vacuum structure of the soliton. This can be written in the following simple Correspondingly, the cutoff Λ is no longer defined by (10) form [4], but instead by the following relation, N −1 Λ ˆ † N . (15) H = X aˆαaˆα Nσ , (19) 2 − ∼ g α=1 ! X The soliton solution (11) goes through almost un- where X is a Lagrange multiplier which enforces the con- changed. However, it becomes highly degenerate: straint (17) and the parameter Nσ will be determined below. All the non-zero frequency modes have been ex- √8 R a α cluded as they do not contribute into the ground-state σα = x2 2 , (16) g + R √Nσ structure but only into the excited states. Therefore, the † where aα are arbitrary subject to the following con- operatorsa ˆβ, aˆα represent the creation and annihilation straint, operators of the zero frequency moduli. They satisfy the † usual commutation relations [ˆaα, aˆβ] = δαβ. Obviously, N 2 2 the quantities aα entering in (17) and (16) must be under- aα = Nσ . (17) =1 stood as the expectation values of the number operators α † X nˆα aˆ aˆα. The micro-state degeneracy is then given by ≡ α The mass of the soliton is given by the same expression all possible distributions of these numbers subject to the (12). In classical theory the value of the parameter Nσ constraint (17). The total number of such states is given is irrelevant. However, in quantum theory it acquires an by the binomial coefficient important meaning, as we shall explain. Nσ + N 1 nst = − . (20) The above degeneracy can be understood in two equiv- Nσ   alent languages. The first is the language of a mod- uli space. Since the soliton became embedded into the In order to evaluate this expression, we need to determine SO(N)-space, its moduli space got increased. Indeed, an the value of the parameter Nσ which of course must de- arbitrary orthogonal SO(N)-transformation, pend on R. For fixing it, notice that we can describe the soliton as the bound-state of Nσ quanta of the field σ σα Oαβσβ , (18) and treat it in Hartree approximation essentially follow- → ing the method applied by Witten to baryons/skyrmions that acts on the soliton non-trivially, gives again a in [6]. 1. Indeed, the bound-state has the size R and is valid solution with exactly the same mass Msol and size stabilized due to an attractive interaction measured by R. Thus, the soliton’s internal moduli space becomes the coupling g2. This means that the positive kinetic en- SO(N)/SO(N 1) which has a topology of an SN−1- ergy of each quantum, which scales as 1/R, is exactly − sphere. Thus, we have N 1 moduli parameterizing the balanced by the negative potential energy∼ of attraction location of the soliton on this− N 1-dimensional sphere. − An alternative language for describing the soliton degeneracy is of Goldstone modes. The non-zero 1 Here we use the bound-state view for a simple estimate, oth- expectation value of the σ-field breaks spontaneously erwise not relying on it. More detailed steps in understanding the global SO(N)-symmetry down to SO(N 1). corpuscular structure of solitons/instantons were taken in [11]. − 4

2 3 from the rest, which scales as g Nσ. We thus conclude, Since, σ has dimensionality of [mass] 2 , it is clear that the ∼ relevant scale in the problem is R Nσ . (21) ∼ g2 1 f = 2 . (27) (gR) 3 Now, comparing the above expression to (14), it is clear that whenever N saturates the unitarity bound at energy This is the correct scale that must be used for measur- 1/R, we have ing the surface area of the soliton. Expressing then (25) through (27) it is clear that the entropy at the saturation unitarity limit Nσ N. (22) point can be written as the surface area in units of f: → ∼ 3 Or equivalently, this equality takes place whenever the Ssol (Rf) . (28) soliton size becomes equal to the unitarity cutoff scale ∼ R =1/Λ. In this case, from (20) using Stirling’s approx- Note, the surface area of a soliton in 5D is a three- imation we get that the micro-state entropy of the soliton dimensional surface. So, (28) is in full accordance with in the unitarity limit becomes equal to, (6).

Ssol = ln(nst) N. (23) In summary, we observe that an instanton-like soliton ∼ in 5D exhibits exactly the same tendency as was observed We are now ready to test our claims. The first task earlier for ’t Hooft-Polyakov monopoles and baryons in is to see explicitly that the above entropy saturates [4]; the saturation of the Bekenstein bound (1) is equiva- the Bekenstein bound (1). Next, we wish to see that lent to the saturation of the bound (4) and both are satu- this saturation is in full agreement with (4). Thirdly, rated together with the unitarity bound (15). The corre- we must show that at the saturation point the entropy sponding entropy exhibits the area-law (28) in agreement assumes the form of the area according to (6). with the general relation (6).

Taking into account the expression for the soliton mass (12), the Bekenstein bound (1) reads, III. NO-SCALE GAUGE MONOPOLE

R Smax 2 . (24) We shall now discuss a gauge soliton in 5D. We first ∼ g consider a simplest model that contains such a soliton. 2 This is a theory with a gauged SO(3) symmetry with a Notice, this is exactly the bound (4) with α = g E eval- a triplet of gauge fields Aµ where a = 1, 2, 3 is an SO(3)- uated at the scale E =1/R. Thus, the two bounds fully index. Sine we are in five space-time dimensions, the agree. They become saturated when (24) becomes equal indexes µ,ν take values 0, 1, 2, 3, 4. The Lagrangian has to (23). This happens when the following form: R 1 N . (25) L = F a F µνa (29) ∼ g2 −4 µν a a a abc b c where F ∂µA ∂ν A + gǫ A A . The parameter As it was already said, by taking into account (14), the µν ≡ ν − µ µ ν expression (25) means that N saturates the unitarity g is a five-dimensional gauge coupling constant that − 1 bound and at the same time R = Λ−1. Indeed, for has a dimensionality [g] = [mass] 2 . The canoni- −1 cal dimensionality of five-dimensional gauge field is R = Λ the equation (25) takes the form (14). Thus, 3 a 2 we see that the soliton saturates the Bekenstein bound [Aµ] = [mass] . The dimensionless four-point coupling (1) (and simultaneously (4)) precisely when its size at energy scale E is given by the same expression R becomes equal to the cutoff scale Λ−1 fixed by the (9) as in the previous example. The same is true unitarity bound (14). about the cutoff scale Λ above which the coupling α becomes strong and violates unitarity. It is given by (10). Let us now investigate whether at the saturation point the entropy takes the form of the area (6). For this, we The above theory admits a time-independent localized monopole-like soliton solution with the topological charge first need to identify the scale f. This scale is determined 2 g 4 0µναβ a a by the order parameter that breaks the SO(N)-symmetry given by Q = 32π2 d xǫ Fµν Fαβ . This soliton spontaneously. The latter is given by the maximal value is identical to an instanton solution in four-dimensional that the field σ reaches in the core of the soliton and is Euclidean space [9].R Therefore, when lifted to a five- equal to dimensional space-time with the Lorentzian signature, it describes a localized monopole-like soliton. These √8 5D solitons and their relation to 4D instantons are well σmax = σ(0) = . (26) gR known and were studied previously in various contexts, 5 in particular, in the context of large extra dimensions we are correlating the resulting entropy bound with the [13]. unitarity and the area of the system. For definiteness, we shall focus on the case Q = 1. The solution has the well-known form: Indeed, the moduli space can be understood as the de- 2 xν generacy of the soliton vacuum due to spontaneous break- Aa = ηa , (30) ing of SU(N) global symmetry by the monopole. This µ g µν x2 + R2 2 is fully analogous to the breaking of global SO(N) sym- a 4 a R metry by a soliton considered in the previous example. Fµν = ηµν , − g (x2 + R2)2 Consequently, up to order-one factors, the counting of a the entropy is very similar. Thus, the micro-state degen- where ηµν are ’t Hooft’s parameters that shall not be dis- played explicitly. As in the case of the soliton in previous eracy is again described by the effective Hamiltonian of the form (33) with the number of zero modes now being example, the mass of the gauge monopole is independent of R and is given by the similar expression, of order 4N 4N 8π2 ˆ † H = X aˆαaˆα Nmon , (33) Mmon = 2 . (31) − g α=1 ! X The zero mode spectrum of the above monopole con- and the parameter Nmon given by (21). However, we sists of a standard set of translation, dilatation and orien- would like to be more precise. We shall make a guess and tation moduli, eight in total. These are not sufficient for choose Nmon to be a dimensionless quantity constructed delivering a micro-state entropy that we could increase in out of the size of the soliton R and the topological in- a controllable way. For achieving the latter goal, we must variant increase the number of bosonic and/or fermionic gapless 2 4 0µναβ a a 32π modes localized within the monopole as it was done in Nmon = R d xǫ F F = R . (34) µν αβ g2 [4]. Z The number of resulting micro-states at large-N is given by the expression analogous to (20), IV. BOSONIC ZERO MODES Nmon +4N nst . (35) ≃ Nmon In this case we need to enlarge the monopole mod-   uli space by embedding the SU(2) gauge symmetry as a Interestingly, this qualitatively matches what would be subgroup into a larger SU(N) symmetry group. This em- the pre-factor measure for the contribution of 4D instan- bedding does not change the expression for the monopole ton of size R into the vacuum-vacuum transition proba- mass (31). However, the unitarity constraint becomes bility. In particular, for weak ’t Hooft coupling the equa- (14) with the unitarity cutoff scale Λ given by (15). In tion (35) can be written as order to fix the numerical coefficients, we shall define a five-dimensional analog of the ’t Hooft coupling in the 2 4N 1 8π R λ following way, nst e t . (36) ∼ (N!)4 g2 2   g E The visual similarity with the pre-factor in a standard λt N 2 . (32) ≡ 8π instanton transition probability [12, 19] is clear. The

Obviously, for strong λt the theory violates perturbative underlying physical reason for this connection shall unitarity. become more transparent below, after we extend the notion of entropy to instantons. The above embedding results into the appearance of additional bosonic zero modes. These modes The entropy of the monopole/instanton saturates the parameterize the orientation of the monopole’s SU(2)- Bekenstein bound when the ’t Hooft coupling becomes subgroup within SU(N). They correspond to global order one. In order to make this more precise, let us SU(N)-transformations that act non-trivially on the evaluate the monopole entropy for λt = 1. First, tak- monopole. Therefore, the orientation moduli space is ing into account (32) and (46) we see that in this limit 32π2 SU(N)/SU(N 2) U(1). However, the U(1)-factor Nmon =4N = R g2 . Next, plugging this in (35) and us- is only partially− residing× within the stability group of ing Stirling’s approximation, we find that the monopole the monopole. At large N this moduli-space has dimen- entropy for λt = 1 takes the following form sionality 4N. This space is identical to a standard 2 ∼ 64π moduli space of an instanton of SU(N) gauge theory Smon = ln(nst) 2 R ln(2) (37) in 4D. The novelty here is that we are interpreting this ≃ g ≃ degeneracy in terms of the Goldstone phenomenon and 4π 1.3 MmonR , giving it a meaning of the micro-state entropy. Next, ≃ 3   6 where Mmon is given by (31). The last expression in the via coupling the gauge field to a large number of fermion brackets is nothing but a 5D Bekenstein bound on an species. Let us assume that fermions form an N- entropy of an object of mass Mmon and the size R, dimensional representation of a global flavor symmetry group. Due to existence of N fermion species the 4π ∼ Smax = MmonR. (38) unitarity bound is given by (14) as in the previous case. 3 Correspondingly, the cutoff scale Λ is determined by (15). At the same time, of course, this expression is equal to As it is well-known, in the instanton/monopole the Bekenstein-Hawking entropy of a would-be static background fermions give rise to localized zero modes. black hole with the same parameters but in 5D theory A detailed construction of fermion zero modes in the with gravity. instanton backgrounds can be found in several excellent reviews [12]. For us, the important fact is that the We thus observe that the monopole/instanton entropy number of zero modes scales as N and can be made saturates the Bekenstein bound when ’t Hooft coupling arbitrarily large by increasing the∼ number of flavors. is close to one. Note, since the sensitivity to the value of λ is exponential, the pre-factor 1.3 simply indicates that t For any given R, the existence of N fermionic zero the saturation takes place when λt is very close to one. N modes creates nst = 2 degenerate micro-states in the Equivalently, we can write, monopole spectrum. Correspondingly, the monopole ac- quires a micro-state entropy give by, Smon(λt 1) = Smax . (39) ≃ Smon = ln(nst) N. (42) Simultaneously, the entropy acquires a form of the ∼ area measured in units of the scale (28) with the scale f From here, it is obvious that all the effects observed in the given by (27). Notice, the scale f has a very well defined previous examples repeat themselves. This can be easily physical meaning, as it represents an order parameter seen by taking into account (14) and (15) and comparing of spontaneous breaking of the global SU(N)-symmetry (42) with (24). Thus again the monopole saturates the by the monopole. Equivalently, it sets the interaction entropy bounds (1) and (4) together with unitarity. The strength of the orientation moduli. In this sense, there area form of the entropy (28) at the saturation point also is no ambiguity in defining the physical meaning of remains intact. the scale f. Thus, the area form of the entropy at the saturation point is unambiguous. VI. INSTANTON IN 4D It is interesting to map the entropy of a monopole on an entropy of a wold-be black hole of the same mass and the size. For this, we write the monopole entropy in form We shall now move to 4D and consider an instanton in- of the black hole entropy by explicitly separating 1/4: stead of a soliton. It is given by exactly the same solution (30) but “downgraded” to 4D Euclidean space. There- fore, we have to take into account the change of dimen- Smon = A , (40) 4G sionalities of the parameters. We shall denote them by the same symbols as in 5D. However, we must remember 2 3 a where 2π R is the monopole surface area in 5D, that g now is dimensionless and Aµ has a dimensionality and theA parameter ≡ G has the following form, of mass. Correspondingly, we define the four-dimensional analog of the ’t Hooft coupling as 3 2 2 3 G g R f − . (41) ≡ 64π ≡ g2 λt N 2 . (43) As said above, G has a very transparent physical meaning ≡ 8π as it sets the coupling strength of orientation moduli that Note, in 4D we wish to keep the theory asymptotically- represent the Goldstone bosons of broken global symme- free. So we must assume that the fermion content is try. In this sense it is fully analogous to the Newton’s chosen appropriately. In the present case, for simplicity, constant that sets the interaction strength among gravi- we shall ignore the fermion flavors. tons. Correspondingly, the scale f plays the role of the Planck mass. We shall now undertake the following two steps. First, we shall assign entropy to an instanton. Secondly, we shall try to understand what is the upper bound on V. FERMION ZERO MODES this entropy and what is the physical meaning of its saturation. An alternative possibility for increasing the micro- state entropy is to populate the 5D monopole by fermion We assign the entropy to the instanton by a direct gen- zero modes, as it was done in [4]. This is accomplished eralization of the entropy of its 5D counterpart soliton. 7

Namely, the entropy of the instanton will be counted as Similarly to the 5D monopole, at the saturation point the log of the number of micro-states (46) due to the ex- the entropy takes the form of the area. istence of zero mode bosons and/or fermions, exactly as it was counted for the 5D monopole. So we shall assign Smon = A (48) to an instanton the following number of “micro-states”, 4G 2 Ninst +4N where 4πR is the area and the parameter G has nst , (44) the form,A ≡ ≃ Ninst   g2R2 where taking into account the dimensionality of the cou- G f −2 . (49) pling constant we take, ≡ 64π ln(2) ≡ 2 As in 5D case, the parameter G has a very well defined 4 µναβ a a 32π Ninst = d xǫ F F = . (45) µν αβ g2 physical meaning as it sets the coupling strength of the Z Goldstone modes. In this respect it is fully analogous to From the perspective of Lorentzian space, (44) counts Newton’s constant in gravity. the number of micro-processes that cost the same Eu- 8π2 Notice, we get the area law as for a localized object clidean action Iinst = g2 . For weak ’t Hooft coupling the “degeneracy” (44) can be approximated as, in 4D Minkowski space-time. The physical meaning of this is easy to understand. An instanton describes a 2 4N tunnelling process. We can connect this process to 5D 1 8π λ nst e t . (46) monopole discussed in the previous section. For this, we ∼ (N!)4 g2   can imagine that the 4D Minkowski gauge theory resides 2 on a 4D slice of the 5D gauge theory. Now, the 5D theory The coupling g is evaluated at the scale R. This houses monopoles. These monopoles can tunnel through matches the expected measure for the contribution the 4D surface. This “passing-by” virtual monopole is of a given size instanton into the vacuum-vacuum 2 “seen” by a 4D observer as instanton. transition probability [12, 19]. This is remarkable since In such a picture, an each passing-through event can be the correlation between the degeneracy of actual states attributed a particular location in 4D space. Of course, and their contribution into the virtual processes is by 4D Poincare invariance we effectively integrate over all rather non-trivial. The interpretation of the transition possible locations and sizes whenever we compute instan- probabilities in terms of instanton entropy can be highly ton contribution into the physical observables. However, instructive. Namely, the question that we would like to this is not important for the present discussion since we ask is: are interested in the entropy of an instanton with fixed size and location. Then, the area of a sphere surrounding What is the physical significance of the violation of each event is a two-dimensional surface. the entropy bound by an instanton?

In order to answer this question, we first need to solve VII. BLACK HOLES the following dilemma. Since we are in Euclidean space, there exists no notion of energy. So the Bekenstein en- tropy bound (1) is not well-defined and cannot be ap- Although in the present paper we focus on non- plied to an instanton directly. However, the bound (4) gravitational theories, we must comment on an obvious is well-defined and can be applied. This bound is satu- connection with the black hole entropy (2). As it was rated when the ’t Hooft coupling (43) becomes order-one. already noticed in [8] the entropy of a black hole of size R can be written in the form (6) where α must be un- Indeed, taking λt =1 we get derstood as the gravitational coupling at the scale 1/R: 64π2 Sinst = ln(nst) 2 ln(2) , (47) 1 ≃ g αgr = d−1 . (50) (RMP ) where g2 has to be evaluated at the energy scale 1/R. This fact suggests that the remarkable similarity between solitons/instantons/baryons/skyrmions on one hand and Equivalently, we can say that an instanton of a given size R saturates the entropy bound (4) when the counter- part monopole in 5D saturates the ordinary Bekenstein bound as described by (37). The fact that the entropy 2 In the present discussion the embedding of 4D theory into 5D saturation takes place when ’t Hooft coupling becomes is just a mental exercise for understanding the connection, with order one, cannot be a simple coincidence and must be no real physical meaning implied for the fifth dimension. The revealing some deep underlying physics. embedding however can be given a direct physical meaning in specific constructions such as [13] or [18]. 8 black holes on the other exhibited at the saturation Another novelty is that we have generalized the point lies in the fact that all these seemingly-different concept of the entropy bound to an instanton of a fixed entities are maximally packed composite objects [8]. size and location. For this, we first needed to assign the Such objects consist of maximal occupation number of entropy to an instanton. We did this in two equivalent quanta compatible with the strength of the coupling ways. First, we assign to an instanton an entropy that 1/α. It is then natural that the entropy capacity of would be carried by its counterpart soliton in a space ∼such objects fully saturates the unitarity bound of the with one dimension higher. An alternative way is to system for a given α. use the formulation (4) which for solitons and baryons fully agrees with Bekenstein bound (1). However, since It is also clear why the Planck mass plays the role the bound (4) does not include any dependence on the analogous to symmetry breaking parameter. Notice, mass, we can directly apply it to an instanton. We then the canonically normalized graviton field reaches value observe that the instanton saturates the bound (4) when MP near the black hole horizon as seen by an external its counterpart soliton in a theory one dimension higher observer∼ in Schwarzschild coordinates. This is analogous saturates the Bekenstein bound (1). Both saturations to the field σ reaching the values set by the scale f. The are synchronized with the saturations of unitarity in same is true about the baryon that can be viewed as the respective theories and take the forms of the respective skyrmion soliton [14, 15]. There too the maximal value areas. The saturation of the entropy bound takes place of the order parameter is set by the pion decay constant when the ’t Hooft coupling reaches the critical value fπ. order one.

As already pointed out in [4], yet another similarity in It would be interesting to understand if this criticality black hole case is the existence of the species length-scale is somehow related with possible phase transitions in 1 −1 −2 Λ = N d /MP [17]. This is the size of a smallest black large-N QCD, for example, of the type suggested by hole which saturates the bounds (1) and (4) on informa- Gross and Witten [20] and by Wadia [21], or with other tion storage capacity. At the same time, at the same non-analiticities at large-N [22]. scale, the gravitational interaction saturates unitarity. One lesson we are learning is that the solitons and instantons are no less holographic [16] than black holes. VIII. DISCUSSIONS Our observations indicate that saturation of the entropy bound and its area-form go well beyond gravity and are defined by fundamental aspects of quantum field theory In the present paper we gained an additional support such as unitarity and asymptotic freedom. for the results obtained in [4] and made some further steps. First, we continue to observe that manifestations of the Bekenstein entropy bound that are usually attributed to gravity are universal. They are shared by ACKNOWLEDGEMENTS non-perturbative objects in generic consistent theories regardless of renormalizability and/or the presence of gravity. We observe that the saturation of the entropy We thank Cesar Gomez, Andrei Kovtun, Oto Sakhe- bound and its area form are linked with the saturation lashvili, Goran Senjanovic, Nico Wintergerts and Sebas- of unitarity. tian Zell for discussions. This work was supported in part by the Humboldt Foundation under Humboldt Pro- For soliton- and instanton-like non-perturbative fessorship Award, by the Deutsche Forschungsgemein- objects the entropy bound can be formulated in the schaft (DFG, German Research Foundation) under Ger- form (4) in which the mass of the object does not enter many’s Excellence Strategy - EXC-2111 - 390814868, and explicitly. Instead, the bound is set by the inverse of Germany’s Excellence Strategy under Excellence Cluster the coupling constant. This form makes the connection Origins. between the saturation of the entropy bound and unitarity more transparent. Through the relation (6) it also gives a quantum field theoretic explanation to the IX. APPENDIX: COUPLING AND ENTROPY area-form of the entropy at the saturation point. BOUND

In appendix we highlight the importance of the cou- Here we would like to discuss the importance of the pling constant in formulation of the entropy bound (4). coupling for formulation of the entropy bound. This Namely, we provide an explicit example of manifestly should provide an additional justification for writing unitary theory in which the Bekenstein bound in its the bound in the form (4). In order to do this, it is original formulation (1) is seemingly violated, however the simplest to give an explicit example of a consistent the formulation (4) restores the consistency. field theory that naively violates the entropy bound formulated in the original Bekenstein form (1) but 9 the violation is avoided after we take into account the form (1) cannot capture this difference explicitly, since formulation (4). it is independent of the strength of the coupling.

We note that apparent violations of the Bekenstein On the other hand, the bound formulated as (4) does bound in gravitational context and attempts of resolving capture the difference. Indeed, in the absence of any such violations where discussed previously, e.g., in mediator quantum field, the inter-monopole coupling [23],[24]. The formulation (4) can be useful in this strength αij vanishes. So written in the form (4) respect also, as it avoids violation of the bound in any 1 unitary theory. Smax < (51) αij Consider any unitary theory that admits a large num- the bound is maintained for arbitrary large N and ber of non-interacting localized solitons. For example, consistency is restored. The advantage of the bound (4) let us focus on a set of N-copies of SO(3) gauge theories j is that it accounts for the fact that in order to read-out in 4D, where j = 1, 2, ...N is the label of the group. the information and/or to maintain the bound-state the Assume that each copy is Higgsed by an accompanying aj coupling must be non-zero. triplet scalar field Φj , where aj = 1, 2, 3 is the gauge index of the respective SO(3)j . In this way, each SO(3)j Let us now create a measuring device. The role of it is Higgsed down to an U(1)j -subgroup. As a result, we can be played by a field that couples to all the Higgs obtain a ’t Hooft-Polyakov magnetic monopole residing triplets. For example, a gauge-singlet scalar χ and/or in each theory. For simplicity, let us assume that the a Majorana gauge-singlet fermion ψ with the following parameters of the theories such as gauge couplings couplings, ej = e and masses are equal and the theories do not talk to each other. That is, the Higgs vacuum expectation 2 2 aj aj aj ¯aj gj χ (Φj Φj )+˜gj Φj ψj ψ , (52) values are all equal Φj Φj = v and so are the gauge | | h i j boson masses mj = ev = mv. X would do the job. Notice, in order to be connected via Then, we get the following situation. The masses a gauge-singlet fermion ψ, an each j-sector must also 2 aj of the monopole species Mj = mv/e as well as contain a gauge-triplet fermion ψ . There is no need their sizes Rj = 1/mv are all equal to each other, for this in the scalar case. Mj = Mmon , Rj = R. Since the theories are decou- pled, the monopoles from different sectors experience For simplicity, we can assume all the gj andg ˜j to be no interaction and they can be placed at arbitrary of the same order gj g˜j g. Then, unitarity puts the distances from each other. In particular, we can place following bound on this∼ coupling-strength∼ them right on top of one another and create a stack 2 2 of N monopoles. The size of the stack is R but the g g N . 1 . (53) j ∼ mass scales as M = NMmon. Thus, the maximal j X entropy permitted by the Bekenstein bound (1) scales linearly with N: Smax = N(MR). On the other Now, the coupling to the connectors induces the coupling 2 hand the naive entropy scales as Smon N due to among the different monopole species already at one-loop the number of moduli that parameterize∼ the relative level due to the exchanges by χ and ψ. This can be seen positions. Their number is growing as N 2. So, it from the following part of one-loop effective Coleman- appears that by taking N sufficiently large∼ the entropy of Weinberg potential, the monopole stack can violate the Bekenstein bound (1). 2 2 1 4 Mχ 4 Mψ V = 2 Mχ ln 2 Mψ ln 2 (54) However, the issue is more subtle. Although the 64π µ ! − µ !! entropy formally grows unbounded with N, the infor- 2 2 aj aj 2 2 aj aj mation content stored in such micro-states is “sterile” where Mχ j gj Φj Φj ,Mψ j g˜j Φj Φj and µ and therefore meaningless. Indeed, in order to read-out is a renormalization≡ scale. ≡ the quantum information stored in a state of displace- P P ment moduli, a device must exist that can measure This potential induces the coupling among the i and j 2 2 2 2 4 such displacements. However, in quantum field theory monopole species of the strength αij = gi gj g˜i g˜j g . there exist no external devices. All the “devices” are By a suitable choice of parameters this interaction− can∼ be manufactured out of the quantum fields. Thus, in made attractive, repulsive or neutral. Thus, the effective order to measure the quantum information stored in the inter-monopole coupling by the unitarity bound (53) 2 monopole displacements, an agent is required in form scales as αij . 1/N . So the entropy bound formulated of a quantum field that interacts with all the monopole as (4) is not violated by the stack of monopoles as long species. If such an agent is absent, the information is as the system is unitary. unreadable. The Bekenstein bound formulated in the 10

Finally, we notice that at the saturation point e2 = Thus, although increasing N increases the entropy of the g2 =1/N the area law (6) is satisfied. Indeed, the scale stack of monopoles without increasing its size R, never- f is given by f 2 = Nv2 and we have, theless the area measured in units of the order parameter f increases wth N in such a way that it always matches 1 2 2 Smax = = N = (Rf) . (55) the entropy at the saturation point. α

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