A Weak Gravity Conjecture for Scalar Field Theories

1,2,3 1,3 1,3 1,3 Miao Li ,∗ Wei Song ,† Yushu Song ,‡ and Tower Wang § 1 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China 2 Interdisciplinary Center of Theoretical Studies, Academia Sinica, Beijing 100080, China 3 Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: February 1, 2008) We put forward a conjecture for a class of scalar field theories analogous to the recently proposed weak gravity conjecture [4] for U(1) gauge field theories. Taking gravity into account, we find an upper bound on the gravity interaction strength, expressed in terms of scalar coupling parameters. This conjecture is supported by some two-dimensional models and noncommutative field theories.

PACS numbers: 04.60.-m, 04.60.Kz

Scalars, such as Higgs and inflaton, play significant whether this is also true in higher dimensions, but we roles in both particle physics and cosmology. Recently, feel that a consistent theory in higher dimensions should Arkani-Hamed et.al. conjectured an upper bound on the result in a consistent theory upon dimensional reduction, strength of gravity relative to gauge forces in quantum thus we lift this conjecture to higher dimensions, in par- gravity [4]. Their conjecture enriches the criteria of con- ticular, to four dimensions. sistent effective field theory proposed in [2, 3]. These When the scalar is coupled to two-dimensional dilaton criteria help to constrain the string landscape [1] in the gravity, the action is generally taken to be vast vacua of string theory. After a careful investigation, 1 2 2φ 1 2 we find that a similar bound exists in a class of scalar S = d x√e γ − [ R + 4( φ) 2π − 8G ∇ field theories. In this class of theories at least, “gravity Z is the weakest” even in the appearance of scalars. 1 2 1 2 2
( ϕ) µ ϕ Vc(ϕ, λ, µ)] (1) −2 ∇ ∓ 2 − We will study scalar field theories with soliton solu- tions. we will focus our attention on a special class of In this notation, the potential of scalar ϕ is V = Vc 1 2 2 ± such theories. In these theories, the coefficient of the 2 µ ϕ . In most cases, after a redefinition of λ, the higher 1 2 2 mass term is 2 µ with µ > 0, and there are higher order coupling terms Vc can be rewritten as order terms controlled± by a coupling constant λ in ad- 2 dition to µ. So the theories are described by only two µ α Vc(ϕ, λ, µ) = V ′c(g ϕ) (2) parameters: the mass parameter µ and the coupling con- g2α stant λ. When the scalar is coupled to gravity, we find √λ an interesting constraint on µ and λ. in which g = µ , and the value of α depends on the form of V . As we will show later, for a given V , the value of The idea is to study a scalar system coupled to two- c c α can be worked out explicitly. dimensional dilaton gravity, which is assumed to be di- We rescale the coordinates and the scalar to make them mensional reduction of some consistent quantum gravity dimensionless system in higher dimensions. In two dimensions, the sys- α tem contains solitons in the weak gravity limit (plus some t′ = µt, x′ = µx, ϕ′ = g ϕ (3) other massive particles in a sector not considered here). These solitons appear as asymptotic scattering states so the action (1) becomes there is a well-defined S-matrix. As one increases the gravity coupling, these solitons disappear, thus there are 1 2 2φ 1 2 S = d x′√e γ − [ R′ + 4( ′φ) no more massive particles and the S-matrix ceases to ex- 2π − 8G ∇ Z ist. We assume that in this case the quantum system be- 1 1 2 1 2
+ ( ′ϕ′) ϕ′ V ′c(ϕ′) ] (4) comes inconsistent, because without the S-matrix, only g2α −2 ∇ ∓ 2 − correlation functions exist which are not observables in   a gravity theory. We therefore conjecture that the quan- We stress that rescaling λ in (2) by a numeric factor tum theory exists only in the weak gravity region. In fact, will rescale the expression of V ′c also by a numeric factor. in some exact S-matrices, such as the Sine-Gordon model To remove this uncertainty, when writing (2), one should [11, 12] and the chiral field model [13], only solitons ap- define λ appropriately to get a reasonable V ′c whose nu- pear in spectra. It is a matter subject to debate that meric coefficients are of order 1. 2

From (4), it is clear that the gravitational strength rel- with a,b =0, 1 in two dimensions. ative to the strength of scalar interaction is controlled by We are interested in static solutions, in which the dila- G 2α g2α . If g G, the effect of gravity is just a small ton φ, the scalar field ϕ and the metric γab are indepen- correction to≫ the original scalar field theory, so one can dent of t. treat the weak gravity perturbatively, expanding in terms For any two-dimensional static metric √G 2α of α . On the other hand, if g G, the correction g ds2 γ dt2 γ dtdx γ dx2 would be large enough to destroy the≪ original solutions = 00 +2 01 + 11 2 and make the theory inconsistent. Hence quantum grav- γ01 2 γ01 γ00γ11 2 = γ00[ (dt + dx) + − 2 dx ](11) ity provides a criterion of consistent scalar field theories, − − γ00 γ00 parametrically By redefining coordinates g2α > G (5) ∼ γ γ2 γ γ ˜ 01 01 00 11 This constraint can be lifted to higher dimensions: dt = dt + dx, dx˜ = − 2 dx (12) γ00 s γ00 λ α > ( 2 ) G (6) one can always switch to the conformal gaugeγ ˜ab = µ ∼ 2w e ηab while keeping the solutions static. Henceforce, In D-dimensional spacetime, the Newton constant G has we will work in conformal gauge and omit the tilde for 2 D the dimensions of mass − while the coupling constant brevity. 2 D−2 λ has the dimensions of mass − α . The equations of motion (10) now reduce to In the special case dφ d2φ 1 dϕ 1 d2w 2 2 2 2w 1 1 1 1 µ 2 2( ) 2 2 + ( ) + e V = 2 (13) V = µ2ϕ2 + λϕn + ( )µ2( ) n−2 (7) dx − dx 4 dx 2 − dx −2 n 2 − n λ 2 dφ 2 d φ 1 dϕ 2 1 2w dw dφ 2 2( ) 2 2 + ( ) + e V = 2 (14) with an even number n> 2, we have α = n 2 , therefore dx − dx 4 dx 2 − dx dx the relation (6) becomes − dφ d2φ 1 dϕ 1 2( )2 2 + ( )2 + e2wV dx − dx2 4 dx 2 λ 2 ( ) n−2 > G (8) dw dφ 1 dϕ d2φ µ2 =2 + ( )2 2 (15) ∼ dx dx 2 dx − dx2 In four dimensions, setting n = 4 leads to the constraint dϕ dφ d2ϕ dV µ 2w < M . 2 2 + e = 0 (16) √λ Pl dx dx − dx dϕ In∼ the examples studied below, when gravity decou- ples, solitons are in the spectrum. Disappearance soli- We have written them in forms that the left hand sides tons in the strong gravity singals breakdown of unitarity: of (13), (14) and (15) are the same. Equating the right when we tune up gravity interaction strength gradually, hand sides of (13) and (14), we find a simple relation the rank of S-matrix will have a jump at some point in dw 2 the parameter space. Indeed, we do not even know if = Ce φ (17) dx the S-matrix exists in the strong gravity region. So for the parameter region violating (6), there is no consistent Combining it with (14) and (15), it follows that quantum gravity theory. We suspect that the same kind dφ d2φ dϕ of relation (6) remains true in a larger class of theories 2φ 1 2 4Ce =2 2 ( ) (18) without solitons, although we do not have evidence. dx dx − 2 dx A kink is the most familiar soliton in two-dimensional Using (14) and (15) to eliminate dw , we obtain spacetime. The action of the coupled system reads dx 2 dφ 2 d φ 2w 1 2 2φ 2 1 2 4( ) 2 + e V = 0 (19) S = d x√ γe− [R + 4( φ) ( ϕ) V ] (9) dx dx2 2π − ∇ − 2 ∇ − − Z 2w the equations of motion are Combined with (16) to eliminate e , it gives 1 dV d2φ dφ d2ϕ dϕ dφ R +4 2φ 4( φ)2 ( ϕ)2 V = 0 2 [ 2( )2]= V ( 2 ) (20) ∇ − ∇ − 2 ∇ − dϕ dx2 − dx dx2 − dx dx ab 2 dV 2γ aϕ bφ ϕ + = 0 We will restrict our attention to a kink potential V (ϕ) ∇ ∇ − ∇ dϕ with α = 1 in (2). Based on the work of ’t Hooft [6], we 1 aϕ bϕ +2 a bφ assume −2∇ ∇ ∇ ∇ 2 2 1 2 1 1 +γab[2( φ) 2 φ + ( ϕ) + V ] = 0 (10) ϕ = f 1(x) + subleading terms (21) ∇ − ∇ 4 ∇ 2 g − 3

√λ where we have used the notation of g = µ , while f 1(x) It is the same as the kink equation in [6] after a simple is some function independent of g. − calculation Remember that in the notation of (4), the potential dϕ = √2V (29) and its derivative with respect to ϕ can be written in the dx form We conclude that in weak gravity region the kink solution µ2 µ2 survives when this scalar field is coupled to gravity. V (ϕ, λ, µ) = V (gϕ)= V (ϕ ) g2 ′ g2 ′ ′ The equation of motion of the scalar field (28) reduce to the equations of motion for scalar field theory in flat dV µ2 dV dϕ µ2 dV = ′ ′ = ′ (22) spacetime. So if there are 2-soliton solutions in the pure dϕ g2 dϕ dϕ g dϕ ′ ′ scalar field theory, there should be when the scalar field theory is coupled to gravity. On the other hand, from In the leading order ϕ′ f 1, therefore ∼ − (27) we see that even there is no soliton, there may be µ2 linear dilaton, and it does no harm to the one soliton V = V ′(f 1) + subleading terms g2 − soluton. With the soliton, the second derivative of the dilaton becomes positive in some localized region, and dV µ2 dV = ′ + subleading terms (23) zero elsewhere. So asymptotically the soliton will induce dϕ g df 1 − another linear dilaton. But the effect is only a shift of the Substitute the leading order terms of (21) and (23) into coefficient of the linear dilaton in pure gravity, and hence (20), We find it will do harm to two soliton solution. Thus, if S matrix exists in the original pure scalar field theory, it will exist 2 2 dV ′ d φ dφ 2 1 d f 1 df 1 dφ when the scalar field theory is coupled to dilaton gravity. 2 [ 2 2( ) ]= 2 V ′(f 1)( −2 2 − ) In the strong gravity region, we take the limit g 1 df 1 dx − dx g − dx − dx dx ≪ − (24) and expand φ and ϕ in g In order to be consistent with this equation, φ has to take 1 ϕ = f 1 + f0 + ... the form g − 1 1 1 φ = Φ 2 + subleading terms (25) φ = 2 Φ 2 + Φ 1 + ... (30) g2 − g − g − Plugging (30) into (18) and (20) leads to the following In the strong gravity limit g 1, the value of φ in lowest order equations (25) will generate a divergent value≪ of w in (17), unless 2 C = 0. So we set C = 0 from now on in order to avoid d Φ 2 1 df 1 2 − = ( − ) the essential singularity. dx2 4 dx We shall use the perturbative expansion to study equa- dΦ 2 1 V df 1 − = − (31) tions (18) and (20) with C = 0 in the weak gravity limit dx 2 dV dx df−1 and thestrong gravity limit respectively. In the weak gravity region, we assume g 1 and ex- We will analyze the equations in two particular models, 1 ≫ i.e. the ϕ4 model and the Sine-Gordon model. pand φ and ϕ in g 2 1. V (ϕ)= λ (ϕ2 µ )2 1 1 4 − λ ϕ = f 1 + f 2 + ... Substituting this potential into the equations (31), g − g2 − we have the solution 1 1 φ = Φ 2 + Φ 3 + ... (26) 8C1x+C2 g2 − g3 − f 1 = e− − 1 16C1x+2C2 Φ 2 = e− + C1x + C3 (32) Substituting (26) into (18) and (20), we have the follow- − 16 ing lowest order equations From (32), it can be seen easily that f 1 does not 2 − d Φ 2 1 df 1 2 satisfy the kink condition, so the kink solution − = ( − ) dx2 4 dx breaks down in this case. 2 2 d Φ 2 1 V d f 1 − = − (27) 4 dx2 2 dV dx2 2. V (ϕ)= µ [1 cos( √λ ϕ)] df−1 λ − µ Inserting this potential into the equations (31), we So we have have the solution 2 V d f 1 1 df 1 2 f 1 = 2 arctan sinh(D1x + D2) − = ( − ) (28) − dV dx2 2 dx df−1 Φ 2 = ln cosh(D1x + D2)+ D3 (33) − 4

From the potential in this case, we find that this the energy is solution connects two maximum points rather than 1 minimum, so there is no kink solution in this case. E = d2z (∂ϕ)2 + θV (ϕ) . (37) 2 In action (9) we have set 8G = 1. By recovering G, Z   one finds that the expansion parameter is not g but g . √G When θV is large, the potential energy dominates and This accounts for the name “weak/strong gravity”. we can find an approximate solitonic solution by solving dV The calculation above tells us that kink solutions ap- the equation dφ = 0. pear in weak gravity limit g 1 but disappear in √G In our notations, the scalar potential is V (ϕ) = g ≫ 1 2 2 1 3 strong gravity limit 1. In the absence of gravity, 2 µ ϕ + 3 λϕ . The value of the potential at the sta- √G 2 ≪ dV µ the existence of soliton is dictated by unitarity. Natu- tionary point = 0, namely ϕcr = , determines dϕ − λ rally we expect that unitarity will still require solitons the soliton energy completely [8] g when gravity is switched on. Therefore, when √ 1 G ≪ 6 the breakdown of kink solutions implies that we cannot π θµ E =2πθV (ϕ )= (38) take the strong gravity limit smoothly. In other words, cr 3 λ2 there must be a lower bound on the value of g , which √G In three-dimensional spacetime, a massive pointlike soli- is determined by the critical point of the equations of ton will generate a deficit angle in the metric. The re- motion. Parametrically, the critical value is of order 1. quirement that the deficit angle be less than 2π implies This leads to λ > 8πGE < 2π (39) ( 2 ) G (34) µ ∼ Observing (38) and (39), we obtain which agrees with the conjecture (6). It would be useful g to work out the exact value of at the critical point. 6 √G 1 θµ > (40) All the discussions above are on the classical level. G λ2 One may wonder whether the conclusion is reliable on ∼ the quantum level. To check this, we should consider Actually, for a general potential (not necessarily of some quantum corrections such as higher derivative cor- polynomial form, see [7]) with a noncommutative soliton rections. When we consider the original action with the whose energy is positive correction term (α R)n, then the equations of motion ′ 2 2 (13), (14) and (15) will deserve the correspondent cor- µ α 1 2 2 µ 1 2 V = V ′ (g ϕ) µ ϕ = [V ′ (ϕ′) ϕ′ ] g2α c λ α c rection terms, proportional to some positive power of R. ± 2 ( µ2 ) ± 2 Note that our discussion below (12) shows that the we (41) can always take the conformal gauge and set the field there is a similar relation static in 2 dimensional spacetime. So we will always have 2 2 2ω d ω 1 θµ R = 2e− 2 . So fortunately these terms will vanish > E (42) − dx G λ α dω ∼ ∼ ( µ2 ) under the classical solution (17) with C = 0, e.g, dx = 0. So we can conclude that our results must be preserved at the quantum level. On the other hand, the condition for the existence of Another piece of evidence comes from noncommuta- soliton solution in the noncommutative theory is tive solitons. In noncommutative field theories, when 1 θ . (43) the noncommutativity parameter θ is sufficiently large ≥ µ2 (but finite), there are stable soliton solutions in some three-dimensional scalar theories [7, 8]. Based on a cubic Combine (42) and (43), a constraint with the same form potential, in [9], Huang and She found a relation involv- of (6) can be obtained. Using (43) on the right hand side ing the noncommutative parameter. In the following, we of (42), we get will briefly repeat part of their work. Consider a scalar field theory in three-dimensional non- 1 θµ2 1 > , (44) commutative spacetime, whose Euclidean action is G λ α λ α ∼ ( µ2 ) ≥ ( µ2 )

3 1 ij S = d x√ γ γ ∂iϕ∂j ϕ + V (ϕ) . (35) which is just the the form of (6). This is a hint that our − 2 Z   conjecture (6) is universal. Written in terms of the canonically commuting noncom- One may notice that in U(1) gauge field theories the mutative coordinates weak gravity conjecture has a sharp form [4]: there are x1 + ix2 x1 ix2 always particles with M < Q. However, it is difficult z = , z = − , (36) √θ √θ to similarly sharpen our∼ conjecture (6). The particle 5 charge Q is naturally defined for U(1) gauge field the- † Electronic address: [email protected] ories but not for scalar field theories. Fortunately, it has ‡ Electronic address: [email protected] § been shown that in [4] their conjecture can be expressed Electronic address: [email protected] < g [1] L. Susskind, “The Anthropic Landscape of String The- in different forms, one of which is Tel √ . This form is ∼ G ory,” hep-th/0302219. similar to conjecture (6) for scalar field theories. Essen- [2] M. R. Douglas, “Is the number of string vacua finite?” tially, in both U(1) gauge field theories and scalar field talk at the Strings 2005 Conference, theories, the weak gravity conjecture is a statement about http://www.fields.utoronto.ca/audio/05-06/strings/douglas/ effective field theories. So it can be used to identify the [3] C. Vafa, “The String Landscape and the Swampland,” swampland, a series of effective field theories which are hep-th/0509212. consistent semiclassically yet inconsistent in a quantum [4] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, “The String Landscape, Black Holes and Gravity as the Weak- gravity theory. est Force,” hep-th/0601001. We have so far been dealing with real scalar field the- [5] M. Li, W. Song and T. Wang, “Some Low Dimensional ories. The generalization to theories of complex scalars Evidence for the Weak Gravity Conjecture,” JHEP 0603, and scalar multiplets is not straightforward. However, we 094 (2006), hep-th/0601137. have some hints. In [5], we have offered some evidence [6] G. ’t Hooft, “Under the Spell of the Gauge Principle,” supporting the weak gravity conjecture of gauge field the- World Scientific. ories. To avoid possible confusion, let us use e to replace [7] R. Gopakumar, S. Minwalla and A. Strominger , 2 0005 2 2 µ “Noncommutative Solitons,” JHEP , 020 (2000), g in [5]. By noticing mW =2e λ , one can easily obtain hep-th/0003160. the relation (6) for three-dimensional examples studied [8] M. R. Douglas and N. A. Nekrasov, “Noncommuta- in [5]. tive Field Theory,” Rev. Mod. Phys. 73, 977 (2001), In [10], the weak gravity conjecture in [4] has been gen- hep-th/0106048. [9] Q. G. Huang and J. H. She, “Weak Gravity Conjec- eralized from flat spacetime to dS/AdS spacetime, lead- 0612 ing a bound on cosmological constant. It is also interest- ture for Noncommutative Field Theory,” JHEP , ing to study scalar theory in a dS background. 014 (2006), hep-th/0611211. [10] Q. G. Huang, M. Li and W. Song, “Bound on the U(1) Acknowledgments. We would like to thank Qing- gauge coupling in the asymptotically dS and AdS back- Guo Huang, Jian-Huang She and Peng Zhang for useful ground,” JHEP 0610, 059 (2006), hep-th/0603127. discussions. This work was supported by grants from [11] V. E. Korepin, L. D. Faddeev, “Quantization Of Soli- CNSF. tons,” Teor. Mat. Fiz. 25, 147 (1975). [12] A.B. Zamolodchikov, “Exact Two Particle S Matrix Of Quantum Sine-Gordon Solitons,” Commun. Math. Phys. 55, 183 (1977). [13] P. Wiegmann, “Exact factorized S-matrix of the chiral 142 ∗ Electronic address: [email protected] field in two dimensions,” Phys. Lett. B , 173 (1984).