CALT-68-2879

Naturalness and the Weak Gravity Conjecture

Clifford Cheung and Grant N. Remmen Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena, CA 91125∗ (Dated: July 31, 2014) The weak gravity conjecture (WGC) is an ultraviolet consistency condition asserting that an Abelian force requires a state of charge q and mass m with q > m/mPl. We generalize the WGC to product gauge groups and study its tension with the naturalness principle for a charged scalar coupled to gravity. Reconciling naturalness with the WGC either requires a Higgs phase or a low cutoff at Λ ∼ qmPl. If neither applies, one can construct simple models that forbid a natural electroweak scale and whose observation would rule out the naturalness principle.

PACS numbers: 11.10.Hi, 04.60.-m, 04.70.Dy, 11.15.-q

INTRODUCTION gauge symmetry is spontaneously broken or new degrees of freedom enter prematurely at the cutoff The naturalness principle asserts that operators not Λ ∼ qm . (2) protected by symmetry are unstable to quantum correc- Pl tions induced at the cutoff. As a tenet of effective field Ref. [3] conjectured Eq. (2) with the stronger interpre- theory, naturalness has provided a key motivation for tation that Λ signals the complete breakdown of four- new physics at the electroweak scale. However, the dis- dimensional quantum field theory. Supporting this claim covery of the Higgs boson [1,2] together with null results with compelling string theoretic examples, Ref. [3] fell from direct searches has led many to revisit naturalness short of a general argument. However, if one asserts the as a fundamental principle. Rather than amend natural- primacy of naturalness, then our logic provides a reason ness to fit the data, we instead explore its interplay with from quantum field theory for new states at Λ. established concepts in quantum field theory. To illustrate these ideas we present simple, concrete Our focus will be the weak gravity conjecture (WGC) extensions of the standard model (SM) in which a natu- [3], which states that a consistent theory of gravity cou- ral value of the electroweak scale—at the Planck scale— pled to an Abelian gauge theory must contain a state of is incompatible with Eq. (1) due to a new millicharged charge q and mass m satisfying1 force. These models offer the unique opportunity to test naturalness experimentally. Indeed, either naturalness q > m/mPl, (1) reigns, in which case Eq. (2) demands a low cutoff, or it fails. Absent additional ultralight states, a discovery i.e., gravity is the weakest force. While Eq. (1) is cer- of this millicharged force would then invalidate natural- tainly true of electromagnetism, Ref. [3] convincingly ar- ness and mandate an unnatural electroweak scale. In gued that it is a universal consistency condition of all particular, Eq. (1) would disallow a natural electroweak healthy quantum field theories. scale and the hierarchy problem would arise from as-yet- However, in theories with fundamental scalars Eq. (1) unknown ultraviolet dynamics. More generally, a fifth runs afoul of naturalness because it bounds a quadrat- force discovery of any kind would invalidate the inter- ically divergent mass by a logarithmically divergent pretation of Λ advocated in Ref. [3] as the cutoff of four- charge. For small charge, Eq. (1) forbids a natural spec- arXiv:1402.2287v2 [hep-ph] 30 Jul 2014 dimensional quantum field theory. If, as conjectured in trum in which scalars have masses near the cutoff. We Ref. [3], this breakdown is a universal feature of all string illustrate this contradiction with scalar quantum electro- compactifications, such an observation would also falsify dynamics (QED) coupled to general relativity, but this string theory. tension is a ubiquitous feature of any model with a hi- erarchy problem and a small charge. We also generalize Eq. (1) to the case of multiple forces and particles. EVIDENCE FOR THE WGC As we will show, reconciling naturalness with Eq. (1) requires a revision of the original theory: either the Let us summarize the justification for the WGC [3]. Consider a U(1) gauge theory with charged species la- beled by i, each representing a particle (anti-particle) of charge qi (−qi) and mass mi. We define dimensionless ∗ cliff[email protected]; [email protected] charge-to-mass ratios, 1 We define the Planck mass, mPl, such that Eq. (1) is saturated for an extremal black hole. zi = qimPl/mi, (3) 2 so Eq. (1) implies that there exists some particle i with Here a and b are dimensionless numerical coefficients. 2 zi > 1. The authors of Ref. [3] offered theoretical evi- We assume that δm is positive so that the theory re- dence in support of Eq. (1). They presented many ex- mains in the Coulomb phase. In a natural theory, the amples from field theory and string theory, all satisfying physical mass of φ cannot be parametrically smaller than Eq. (1). Further, they argued that Eq. (1) reconciles the its radiative corrections. Equivalently, the counterterm inherent inconsistency of exact global symmetries with for the scalar mass should not introduce a delicate can- the naïvely innocuous q → 0 limit of a gauge theory. cellation. This is formally equivalent to requiring that This limit yields an exact global symmetry; however, the coefficients a and b take on O(1) values. such charges are not conserved by quantum gravity [4,5] Let us set the physical mass squared for φ to its natural because, in accordance with no-hair theorems [6], a sta- value, δm2, which the WGC forbids from exceeding its tionary black hole is fully characterized by its mass, spin, charge in Planck units. The charge-to-mass ratio of φ is and charge. Of course, examples and consistency with no-hair the- 4πmPl 1 z = p , (6) orems only provide circumstantial evidence for Eq. (1). Λ a + bλ/q2 Importantly, Ref. [3] also argues for Eq. (1) via reductio 2 ad absurdum, drawing only on general relativity, conser- where the WGC implies that z > 1. If q  λ, then vation of charge and energy, and minimal assumptions 4πmPl about the ultimate theory of quantum gravity. Consider Λ < √ , (7) a a black hole of charge Q and mass M decaying solely to particles of species i, which can occur via Hawking which is the reasonable requirement that the cutoff not radiation or Schwinger pair production [7,8]. By charge exceed the Planck scale. conservation, Q/qi particles are produced. Conservation Turning to the opposite hierarchy, q2  λ, which is of energy dictates that the total rest mass of the final also radiatively stable, we find that the WGC implies state, miQ/qi, be less than M. In terms of the black r 2 hole charge-to-mass ratio, Z = Q mPl/M, this implies q Λ < 4πmPl . (8) zi > Z. An extremal black hole corresponds to Z = 1 bλ and is stable unless some state i exists for which zi > 1. If Eq. (1) fails, the spectrum contains a large number of As q2/λ → 0, a sensible cutoff requires b → 0, indicating stable black hole remnants, in tension with holographic mandatory fine-tuning in order to satisfy the WGC. We bounds [9, 10] and afflicted with various quantum grav- are left with a remarkable conclusion: scalar QED with itational and thermodynamic pathologies [11, 12]. q2  λ and natural masses fails Eq. (1) and is thus inconsistent with a quantum theory of gravity. We have not traded a mass scale hierarchy problem THE LIMITS OF NATURALNESS for an equivalent hierarchy problem of couplings. Small charges are radiatively stable and thus technically nat- 2 The WGC is straightforward at tree-level, but radia- ural. In principle, q  λ is no worse than the small tive corrections introduce subtleties. In fermionic QED, electron Yukawa coupling. q and m run with renormalization scale, as does their ra- To reconcile naturalness with Eq. (1), one alternative tio, naïvely making q/m ambiguous; however, as Ref. [3] is to argue that the original theory—scalar QED with notes, the appropriate scale to evaluate q/m is the phys- q2  λ—is impossible. For example, this would be true ical mass of the particle. This is the mass scale that is if nature does not permit fundamental scalars or if a hi- relevant to the kinematics of extremal black hole decay, erarchy among couplings is somehow strictly forbidden. which provides the justification for the WGC. However, there are far less drastic options, elaborated However, the radiative stability question becomes below, if one modifies the original scalar QED theory. more interesting in scalar QED: i) Radiative corrections induce the Higgs phase. 1 λ It is possible that quantum effects generate a tachyon for L = − F 2 + |D φ|2 − m2|φ|2 − |φ|4, (4) 4 µν µ 4 φ, Higgsing the theory. Charge becomes ill-defined; the charge and mass eigenbases need not commute, leaving where D = ∂ + iqA is the gauge covariant derivative. µ µ µ q/m ambiguous. Further, the WGC is not justified in As for any effective field theory, we assume an ultraviolet the Higgs phase. The original argument for the WGC cutoff, Λ, above which new physics enters. Since φ is a [3] relied on stable extremal black holes. However, no- fundamental scalar, its mass is radiatively unstable and hair theorems imply that there are no stationary black corrected by m2 → m2 + δm2 where hole solutions supporting classical hair from a massive Λ2 photon [13], independent of the size of the black hole δm2 = (aq2 + bλ). (5) 16π2 relative to the Higgs scale. If a black hole accretes a 3 massive-U(1)-charged particle, it briefly supports an as- consistent with WGC inconsistent with WGC sociated electric field, but after a time of order the pho- ~z ton Compton wavelength, it balds [14] when the gauge 1 field is radiated away to infinity or through the horizon. ~z1 ~z ii) New physics enters below the Planck scale. The 2 ~z simplest way to reconcile the WGC with naturalness is 2 for the effective field theory to break down at a cutoff ~z2 ~z2 defined by Eq. (8). There could be new light states regu- ~z lating quadratic divergences of φ, effectively lowering Λ. 1 This option resolves the contradiction tautologically by ~z1 eliminating the hierarchy problem altogether. However, Figure 1. Vectors representing charge-to-mass ratios for two a more interesting alternative occurs when the new states species charged under two Abelian gauge symmetries. When do not couple to φ. The quadratic divergence of φ is ro- the convex hull defined by these vectors contains the unit bust and m is large. If one of these new states satisfies ball, then extremal black holes can decay to particles and Eq. (1), then φ is irrelevant: the WGC and naturalness the condition of the WGC is satisfied. are reconciled. Thus, asserting naturalness offers Eq. (8) as a more precise version of the low cutoff conjecture of ~ Ref. [3] stated in Eq. (2). hole of charge Q, mass M, and charge-to-mass ratio Z~ = Q~ mPl/M decaying to a final state comprised of ni particles of species i. Charge and energy conservation ~ P P MORE FORCES, MORE PARTICLES imply Q = i ni~qi and M > i nimi. If σi = nimi/M is the species i fraction of the total final state mass, then ~ P P ~ Extending our results to various charged species of Z = i σi~zi and 1 > i σi; decay requires that Z be different spins, the WGC implies that at least one state a subunitary weighted average of ~zi. This criterion has in the spectrum must satisfy Eq. (1) after taking into a geometric interpretation in charge space. Draw the account radiative corrections. Naturalness is violated in vectors ±~zi corresponding to the charge-to-mass ratio of parameter regions with a hierarchy between charges and each fundamental particle in the spectrum. A weighted couplings that generate quadratic divergences (quartic average of ~zi defines the convex hull spanned by the vec- ~ couplings, Yukawa couplings). tors, delineating the space of Z that is unstable to de- The story becomes more interesting for product gauge cay. Any state outside the convex hull is stable. Since QN extremal black holes correspond to |Z~| = 1, the general- symmetries. Consider a gauge group a=1 U(1)a and ized WGC requires that the convex hull spanned by ±~zi particles i with charges qia and masses mi. We rep- contain the unit ball. resent the charges, ~qi = qia, and charge-to-mass ra- Consider a model of two Abelian factors and two tios, ~zi = qiamPl/mi, as vectors of SO(N), the sym- metry transforming the N photons among each other. charged states. The left and right panels of Fig.1 repre- If present, photon kinetic mixing can be removed by a sent two possible choices for the charge-to-mass ratios of general linear transformation on the photons, which is the particles. Black holes of all possible charges are rep- equivalent to redefining charge vectors of states in the resented by the unit disc. The left panel of Fig.1 depicts theory. a theory that is consistent with the WGC: the unit disc To generalize the WGC for multi-charged particles, is contained in the convex hull. Extremal black holes, Eq. (1) is inadequate and requires upgrading to a con- the boundary of this disc, can decay. However, the right panel of Fig.1 depicts a theory that violates the WGC: straint on ~qi and mi. Ref. [3] briefly alluded to this scenario, but detailed analysis will reveal quantitative there are regions of the unit disc not within the con- differences between the WGC as applied to a single vex hull, corresponding to stable black hole remnants. U(1) versus many. By symmetry, the proper general- Remarkably, this theory fails the WGC despite the fact ized WGC must be SO(N) invariant. Naïvely, the WGC that |~z1| > 1 and |~z2| > 1. Simple geometry shows that the WGC imposes the more stringent constraint: could require at least one species i with |~zi| > 1. How- ever, this is insufficient—it guarantees the existence of 2 2 2 (~z − 1)(~z − 1) > (1 + |~z1 · ~z2|) . (9) one particle of large total charge, but preserves stability 1 2 for orthogonally-charged extremal black holes. A stricter For example, given orthogonal charges of equal magni-√ alternative is that for each U(1) there exists a species i tude, |~z1| = |~z2| = z and ~z1 ⊥ ~z2, Eq. (9) implies z > 2, charged under that U(1) with |~zi| > 1. Curiously, this is manifestly stronger than the z > 1 condition required for still actually weaker than the true generalized WGC. theories with a single U(1). Note that the WGC places To determine the proper generalized WGC, we re- constraints on ~z1 and ~z2 that are not mathematically in- visit black hole decay kinematics. Consider a black dependent. Were a particular value of ~z1 experimentally 4

2 2 −1 observed, this would fix a bound ~z2 > (1 − 1/~z1 ) . THE HIERARCHY PROBLEM A similar analysis can be applied for N Abelian factors and N charged states. Suppose each particle is charged We have presented explicit models in which natural- under a single U(1), with equal magnitude charge-to- ness contradicts Eq. (1). We now construct theories in mass ratios, so zia = δiaz for some z. The convex hull which natural values of the electroweak scale—at the defined by ±~zi is an N-dimensional cross-polytope of cutoff—are similarly incompatible. In these models, circumradius z. The largest√ ball contained in the cross- strict adherence to naturalness implies either a Higgs polytope has radius z/ N. Requiring that the radius√ phase or a parametrically low cutoff given by Eq. (2). of this ball be greater than unity then implies z > N, The obvious path is to relate the electroweak scale to parametrically stronger than the condition required for the mass m of a particle that carries a tiny charge q. The a single Abelian factor. SM gauge couplings are O(1), so we require an additional The WGC constraint grows at large N for fixed phys- U(1) gauge symmetry beyond the SM. It is tempting to ical Planck scale mPl. However, the presence of N ad- charge the Higgs, but this will spontaneously break the ditional species generally renormalizes the strength of U(1), invalidating the applicability of the WGC. gravity [15–17] as δm2 ∼ NΛ2/16π2. If corrections en- Pl √ However, we can charge the SM fermions under a very hance mPl by a factor of N, all factors of N encoun- weakly gauged unbroken U(1)B−L symmetry. Current tered in our earlier analyses cancel. That is, in a theory −24 limits on U(1)B−L require q . 10 [18, 19] and will with fixed Lagrangian parameters and cutoff, the limit likely be improved by several orders of magnitude by as- from the WGC is N-independent at large N. A simi- trophysical [20], lunar ranging [21], and satellite-based lar phenomenon was discussed in Ref. [3] for N Abelian [22–24] tests of apparent equivalence principle violation. factors Higgsed to a U(1) subgroup. The large-N limit N To cancel anomalies we introduce a right-handed neu- introduces a Z2 symmetry, which is subject to the large- trino νR that combines with the left-handed neutrino νL N species bounds considered in Ref. [15]. to form a U(1) preserving Dirac mass term of the The multi-charge generalized WGC has implications B−L N form mν ν¯LνR + h.c., where mν ∼ yν v is controlled by for naturalness. Consider a U(1) gauge theory with the electroweak symmetry breaking scale. The particle scalars φi of charges ~qi and masses mi, with the largest charge-to-mass ratio is the lightest neu- 1 X 2 X 2 2 2 λi 4 trino. Assuming its mass is of order the neutrino mass L = − Fµνa + |Dµφi| − mi |φi| − |φi| , (10) 4 4 scale, mν 0.1 eV [25, 26], we fix the charge to a tech- a i . nically natural albeit tiny value: q ∼ m /m ∼ 10−29. P ν Pl where Dµφi = (∂µ + i a qiaAµa)φi. Radiative correc- For this value of q, Eq. (1) is just marginally satisfied by 2 2 2 tions send mi → mi + δmi , where the lightest neutrino. While such a charge is permitted Λ2 in quantum field theory, it may be difficult to engineer δm2 = (a ~q 2 + b λ ) (11) i 16π2 i i i i in string theory if q arises from a string coupling con- stant requiring dilaton stabilization at large field values. and a and b are O(1) ultraviolet-sensitive coefficients. i i Similar issues arise in theories of large extra dimensions The charge-to-mass ratio vector for φ is i and it is a detailed question of string moduli stabilization 4πmPl ~qi 1 whether this is possible. In any case, at fixed Yukawa ~z = . (12) i Λ |~q | p 2 coupling y , were the electroweak scale any higher than i ai + biλi/~qi ν its measured value, Eq. (1) would fail. In this model, A necessary albeit insufficient condition for the WGC is regions of parameter space favored by naturalness—and that, for each U(1), there is a state i charged under that an electroweak scale at the cutoff—are inconsistent with Abelian factor such that |~z | > 1. This implies i Eq. (1). Strictly speaking, this logic hinges on the ab-  1 sence of additional U(1) charged states lighter than  √ , ~q 2  λ B−L  i i the neutrino. Depending on the cosmological history,  ai  however, such particles may be constrained experimen- Λ < 4πmPl × . (13) s tally by primordial nucleosynthesis.  ~q 2  i , ~q 2  λ  b λ i i Our model offers a direct experimental test of natu- i i ralness by virtue of a very specific prediction: a new 2 As for the single Abelian case, ~qi  λi corresponds gauge boson very weakly coupled to the SM. As dis- to the reasonable requirement of a sub-Planckian cutoff, cussed earlier, the assumption of naturalness mandates 2 while ~qi  λi implies tension with naturalness. How- either a Higgs phase or a low cutoff. The discovery of ever, the most stringent requirement of the WGC—that a fifth force would rule out the former, while current the convex hull spanned by ±~zi contain the unit ball— sensitivities would for the latter imply Λ . keV from √places a stronger limit than Eq. (13) by a factor of order Eq. (2). In the absence of such ultralight states, the ob- −29 N for fixed mPl. servation of a U(1)B−L gauge boson at q ∼ 10 would 5 then simultaneously falsify the naturalness principle and [9] G. ’t Hooft (1993), arXiv:gr-qc/9310026 [gr-qc]. suggest an ultraviolet-dependent reason for the why the [10] R. Bousso, Rev.Mod.Phys. 74, 825 (2002), arXiv:hep- weak scale takes an unnatural value. Moreover, given th/0203101 [hep-th]. present sensitivities, a fifth force discovery of any kind [11] L. Susskind (1995), arXiv:hep-th/9501106 [hep-th]. [12] S.B. Giddings, Phys.Rev. D46, 1347 (1992), arXiv:hep- would falsify string theory to the extent to which it pre- th/9203059 [hep-th]. dicts the strong interpretation of Λ as the scale at which [13] S.L. Adler and R.B. Pearson, Phys.Rev. D18, 2798 four-dimensional quantum field theory breaks down [3]. (1978). This mechanism can generally be incorporated into [14] J. Preskill, Phys.Scripta T36, 258 (1991). any theory where the electroweak scale sources the mass [15] G. Dvali, Fortsch.Phys. 58, 528 (2010), arXiv:0706.2050 of a U(1) millicharged state, e.g., dark matter [27] [hep-th]. charged under an unbroken U(1) dark force. For weak [16] N. Arkani-Hamed, S. Dimopoulos, and S. Kachru (2005), −16 arXiv:hep-th/0501082 [hep-th]. scale dark matter, a charge of q ∼ 10 is sufficient to [17] S. Dimopoulos, S. Kachru, J. McGreevy, and J.G. saturate Eq. (1). Wacker, JCAP 0808, 003 (2008), arXiv:hep-th/0507205 [hep-th]. Acknowledgments: We thank Nima Arkani-Hamed, Sean [18] T.A. Wagner, S. Schlamminger, J.H. Gundlach, and Carroll, Meimei Dong, Stefan Leichenauer, Jesse Thaler, E.G. Adelberger, Class.Quant.Grav. 29, 184002 (2012), and Mark Wise for useful discussions and comments. We arXiv:1207.2442 [gr-qc]. also thank the referees for helpful observations on this [19] S. Schlamminger, K.-Y. Choi, T.A. Wagner, J.H. Gund- work. C.C. is supported by a DOE Early Career Award lach, and E.G. Adelberger, Phys.Rev.Lett. 100, 041101 under Grant No. DE-SC0010255. G.N.R. is supported (2008), arXiv:0712.0607 [gr-qc]. by a Hertz Graduate Fellowship and a NSF Graduate [20] S.M. Ransom, I.H. Stairs, A.M. Archibald, J.W.T. Hes- sels, D.L. Kaplan, et al., Nature 505, 520 (2014), Research Fellowship under Grant No. DGE-1144469. arXiv:1401.0535 [astro-ph.SR]. [21] T.W. Murphy, Jr., E.G. Adelberger, J.B.R. Battat, C.D. Hoyle, N.H. Johnson, et al., Class.Quant.Grav. 29, 184005 (2012). [22] J. Overduin, F. Everitt, P. Worden, and J. Mester, [1] G. Aad et al. (ATLAS Collaboration), Phys.Lett. B716, Class.Quant.Grav. 29, 184012 (2012), arXiv:1401.4784 1 (2012), arXiv:1207.7214 [hep-ex]. [gr-qc]. [2] S. Chatrchyan et al. (CMS Collaboration), Phys.Lett. [23] A.M. Nobili, M. Shao, R. Pegna, G. Zavattini, S.G. Tu- B716, 30 (2012), arXiv:1207.7235 [hep-ex]. ryshev, et al., Class.Quant.Grav. 29, 184011 (2012). [3] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, [24] P. Touboul, G. Metris, V. Lebat, and A. Robert, JHEP 0706, 060 (2007), arXiv:hep-th/0601001 [hep-th]. Class.Quant.Grav. 29, 184010 (2012). [4] J.D. Bekenstein, Phys.Rev. D5, 1239 (1972). [25] E. Giusarma, R. de Putter, S. Ho, and O. Mena, [5] J.D. Bekenstein, Phys.Rev.Lett. 28, 452 (1972). Phys.Rev. D88, 063515 (2013), arXiv:1306.5544 [astro- [6] S.W. Hawking and G.F.R. Ellis, The Large Scale Struc- ph.CO]. ture of Space-Time (Cambridge University Press, Cam- [26] R. de Putter et al., Astrophys.J. 761, 12 (2012), bridge, 1973). arXiv:1201.1909 [astro-ph.CO]. [7] S.R. Coleman, J. Preskill, and F. Wilczek, Nucl.Phys. [27] C. Cheung and Y. Nomura, Phys.Rev. D86, 015004 B378, 175 (1992), arXiv:hep-th/9201059 [hep-th]. (2012), arXiv:1112.3043 [hep-ph]. [8] G.W. Gibbons, Commun.Math.Phys. 44, 245 (1975).