IIPMU19-0062
Is Gravity the Weakest Force?
Satoshi Shirai and Masahito Yamazaki Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo Institutes for Advanced Study, University of Tokyo, Chiba 277-8583, Japan (Dated: April, 2019)It has recently been suggested that “gravity is the weakest force” in any theory with a suitable UVcompletion within quantum gravity. One formulation of this statement is the scalar weak gravityconjecture, which states that gravity is weaker than the force originating from scalar fields. We studythe scalar weak gravity conjecture in de Sitter space, and discuss its low-energy consequences in lightof the experimental searches for fifth forces and violations of the equivalence principle. We pointout that some versions of the scalar weak gravity conjecture forbid the existence of very light scalarparticles, such as the quintessence and axion-like particles. The absence of the quintessence fieldmeans that these versions of the scalar weak gravity conjecture are in phenomenological tension withthe recently-proposed de Sitter swampland conjecture and its refinements. Some other versions of thescalar weak gravity conjecture escape these constraints, and could have interesting phenomenologicalconsequences.
Introduction
Gravity has been studied extensively in physics for cen-turies. Yet despite its long history of research, gravity isstill one of the most enigmatic phenomena of Nature.The unification of gravity with all other forces of Nature,namely the quantum gravity, has been a notoriously dif-ficult subject, and gravity seems to be intrinsically dif-ferent from anything else.One of the rather peculiar features of gravity is that itis extremely weak. We learn this in kinder garden whenwe find that the electromagnetic force of a small magnetwins over the gravity of the whole earth. Recently thissimple observation, that “gravity is the weakest force”,has been promoted to a principle in the context of theswampland program [1, 2], a program to work out thelow-energy consequences of the existence of UV comple-tions with gravity.The most famous formulation of the weakness of thegravity is the weak gravity conjecture (WGC) [3], whichstates the following [58]: in a theory with a consistent UVcompletion in theories of quantum gravity, there shouldexist a particle with mass m and U (1) gauge charge q satisfying the inequality: √ q e ≥ mM Pl , (1)where M Pl (cid:39) . × GeV is the reduced Planck massand e is the gauge coupling constant for the U (1) gaugesymmetry. While this is still a conjecture, there has beengood arguments supporting this conjecture from the de-cay of non-extremal black holes [3], and there has recentlybeen many more supporting arguments (see e.g. Refs. [4–12]). [59]The inequality (1) means that the force mediated bythe gauge boson (spin 1 particle) is stronger by that bythe graviton (spin 2 particle), namely F gauge ≥ F gravity with F gauge = ( qe ) πr , F gravity = m πM r . (2)This raises a natural question: what happens if we havea spin 0 particle, namely a scalar? Is gravity still theweakest force, and if so, how should we articulate thiscondition?There has recently been several attempts towards an-swering this question, by formulating a set of “scalar ver-sions” of the WGC [13–17]. However, the contrast withthe case of the original weak gravity conjecture (whichwe call the gauge WGC or GWGC, to be distinguishedfrom the scalar WGC or SWGC), SWGCs have muchless evidence in general, and it is often not clear exactlywhich versions of the conjecture should be adopted.For this reason we work out low-energy consequencesof several versions of the WGC, and discuss observa-tional constraints on them. Such bottom-up constraintson swampland conjectures have been extremely useful insharpening our understanding of quantum gravity (seeRef. [18] for recent summary), and this Letter is not anexception—we will draw interesting conclusions, includ-ing some phenomenological tension with the de Sitterswampland conjectures. Scalar Weak Gravity Conjecture
Let us begin by stating one version of the SWGC [13](we will later comment on variations of the conjecture).Suppose a charged particle has a mass of m . This particleis arbitrary and not necessarily an elementary particle aslong as it is a GWGC state, namely if it satisfies theinequality (1).Suppose that the mass m depends on a set of exactlymassless scalar fields ϕ i . This means that the GWGCstates have trilinear couplings with the massless fields ϕ i . To see this, suppose that the GWGC state is acomplex scalar φ . We then have a field-dependent mass a r X i v : . [ h e p - t h ] A p r term m ( ϕ ) | φ | in the Lagrangian L , and when expandedaround a VEV (vacuum expectation value) ϕ of the field ϕ as ϕ = ϕ + δϕ , we obtain a trilinear coupling L ⊃ ∂ ϕ m ( ϕ )( δϕ ) | φ | . (3)The case of a fermion is similar.Let us denote the metric for the kinetic term of ϕ i to be g ij (which can depend on ϕ themselves), with theLagrangian L kin = − g ij ∂ϕ i ∂ϕ j . Then the SWGC statesthat we have an inequality | ∂ ϕ m | ≡ (cid:88) i,j g ij ( ∂ ϕ i m )( ∂ ϕ j m ) ≥ m M . (4)The physical content of this inequality is that the totalforce mediated by the massless fields ϕ i is stronger thanthat by gravity, when we consider the 2 → F scalar ≥ F gravity with F scalar = | ∂ ϕ m | πr , F gravity = m πM r . (5)Note that the trilinear coupling as in (3) generates thescalar force as on the left hand of (4). Scalar Weak Gravity Conjecture in de SitterSpace
In this Letter, we wish to apply the SWGC to our Uni-verse, namely to a de Sitter space with a positive valueof the cosmological constant and a positive value of theHubble constant H . While swampland conjectures areoften formulated as theoretical constraints on possiblelow-energy physics and does not refer to the history ofthe Universe, our considerations in this Letter are morephysical, and our intention is to honestly formulate thestatement that “gravity is the weakest force” in our Uni-verse.In this context, we claim that we should allow the fields ϕ i to be not exactly massless, as long as they are nearlymassless. More quantitatively, we allow the mass compa-rable to the m ϕ i are of order of the value of the Hubbleconstant of the Universe, H : m ϕ i (cid:46) H , and on the lefthand side of (4) we sum over all such fields.There are several motivations for allowing such nearly-massless fields in the sum. First, when we consider 2 → ϕ i are practically massless, since theirCompton wavelength is comparable or larger than thecurrent horizon scale ∼ H − , where the scattering takesplace. Second, in the Universe with a positive value ofthe cosmological constant and one then expects that thefield φ will generically obtain a mass of the order H fromcurvature couplings.Unlike the supersymmetric case, it is generally hardto keep a scalar particle massless in the de Sitter space. Even when the fields ϕ i are massless classically, we ex-pect that the one-loop effect from the trilinear coupling(3) will generate a mass for ϕ , unless there is some sym-metry reasons. Note that we cannot forbid such a massterm by imposing an exact global symmetry, since globalsymmetry is not allowed in theories of quantum gravity(unless it is emergent in the IR) [19–21]. [60]When applied to the present-day Universe, the massesof the ϕ i are extremely small: m ϕ i (cid:46) H ∼ O (10 − ) eV . (6)As this discussion makes clear, we impose weakness ofgravity in at the IR, namely the horizon scale. Whilethere could be other fields with masses much larger thanthe nearly-massless fields ϕ i , they create only short-rangeforces at the horizon scale and hence do not affect ourargument, as in the case of the GWGC (1). Constraints on Very Light Scalars
In the Standard Model of particle physics, there areno nearly-massless scalar fields whose masses satisfy (6).Such particles, however, often arise in extensions of theStandard Model.One scenario for a nearly-massless scalar is an ultra-light axion-like particle (ALP). It has been argued thatstring theory motivates the presence of multiplet ALPsacross all energy scales [22], possibly including those sat-isfying (6). Another example is the quintessence field[23–25], a dynamical scalar field for the dark energy.Such a scalar field should be nearly massless and sat-isfy (6), to avoid rapid change of the size of the darkenergy. More generally, string theory has many moduli,and some of these fields could have flat directions whichare only broken by non-perturbative effects. Of course,these possibilities are not mutually exclusive, since partof the moduli could generate an ALP, which could playthe role of the quintessence, for example.Let us consider a minimal extension of the StandardModel where we only one very light scalar particle satis-fying (6), and let us denote this scalar by ϕ . We choose acanonical kinetic term for this scalar field, by a suitableredefinition of the field if needed.Let us apply the SWGC to our setup. First, we havethe choice of GWGC states, but there is practically littleconstraint from this condition. As stated in introduc-tion, gravity is extremely weak compared to other forcesand hence all charged states in the Standard Model areGWGC states.We obtain a constraint on a single function m ( ϕ ) fromthe SWGC: 2 | ∂ ϕ m | ≥ (cid:18) mM Pl (cid:19) . (7)This inequality (7) represents the presence of the fifthforce, which acts stronger than gravity in any chargedstates, e.g., the electron and the proton.Such a scenario is highly constrained in observationalconstraints on fifth force searches (see e.g., Refs. [26–29]). First, one generically expects that the coupling ofthe field ϕ (as in eq. (3)) to other GWGC states are non-universal. This causes the violation of the weak equiva-lence principle, which has been checked to be very highaccuracy (of order 10 − ). The constraint is amelioratedwhen we somehow manage to couple the light field ϕ universally to the matters as the gravity. It is still thecase, however, that ϕ modifies the gravitational poten-tial, leading to sizable deviations of the gravitational po-tential in the parametrized post-Newtonian expansion.The observational constraint from this reads | ∂ φ m | < O (10 − ) (cid:18) mM Pl (cid:19) . (8)This is in immediate tension with the SWGC (7). [61]The intuitive reason for our findings is clear—SWGCstates that the fifth force originating from the scalarforce should be stronger than gravity, which is alreadyexcluded from fifth-force searches.Let us add that if φ has a potential and the VEV of φ has a time dependence, the masses of particles are time-dependent, and such time-variations are also stronglyconstrained by cosmological observations.Summarizing, we have concluded that the SWGC asformulated above is in tension with the existence of avery light scalar. The case of multiple such scalars issimilar. Tension with de Sitter Swampland Conjecture
The results we have presented above has an interest-ing consequence: the SWGC is in phenomenological ten-sion with another swampland conjecture, the de Sitterswampland conjecture [30].Let us quickly summarize the de Sitter swampland con-jecture. After the initial proposal [30], some bottom-up constraints of the conjecture has been pointed out inRefs. [31–35], leading to several proposals for refinements[33, 36–41]. In particular, Refs. [38, 39] proposed that ascalar potential V for a low-energy effective field theoryshould satisfy the inequality M Pl |∇ V | > c V or M min( ∇∇ V ) ≤ − c (cid:48) V. (9)Here c and c (cid:48) are O (1) positive constants (a version with c (cid:48) = 0 was proposed in Ref. [33]).Irrespective of the details of the refinements, all ofthese conjectures imply that the (meta)stable de Sitterspace ( ∇ V = 0 , V >
0) are excluded (see also Refs. [42–47] for related discussion). This gives a rather strongmotivation for quintessence models as an explanation ofthe dark energy. This, as we have seen, contradicts theSWGC.This in itself does not prove the inconsistency betweenSWGC and the (refined) de Sitter swampland conjec- tures, since one can try to explain dark energy by mod-ify the gravity instead of incorporating extra quintessencematter fields. However, many modifications of the grav-ity, such as scalar-tensor theories, have light scalar fieldsand tend to have issues similar to the quintessence fields.Moreover, it is a non-trivial question to see if such mod-ifications of gravity can really be embedded into theoriesof quantum gravity, such as string theory.We therefore come to the conclusion that the versionof the SWGC discussed above is in tension with the (re-fined) de Sitter swampland conjecture. This is a non-trivial result. Indeed, both conjectures are partly mo-tivated by the same conjecture, namely the swamplanddistance conjecture [2, 48], at least in asymptotic regionsof the parameter space. Moreover it was suggested inRef. [39] that there are close analogies between the twoconjectures, when the scalar potential is given by themass itself. More broadly, consistency checks betweendifferent conjectures has been one of the primary guide-lines in the swampland program. It is fair to say thatneither conjectures have solid evidence, and are open forpossible modifications and refinements. Since we alreadymentioned on refinements on the de Sitter swamplandconjectures, let us next discuss variations of the SWGC.
Variations of the Scalar Weak Gravity Conjecture
Let us discuss variations of the SWGC—when stat-ing the inequality (6) there are many choices. Somevariations, which impose stronger condition than above,clearly does not affect the argument above. For exam-ple, we can impose the inequality for arbitrary states, asopposed to only for GWGC particles. We can also re-place the inequality ( ≥ ) by a strict inequality ( > ) or anapproximate inequality ( (cid:38) ), without affecting our argu-ment.One plausible modification of the SWGC is to considerthe following: instead of imposing the inequality (4) forany state (or any GWGC state), we can require that there exists at least one particle (the SWGC particle)satisfying the SWGC condition (4). This extends the“weak version” of the SWGC in Ref. [13] to the de Sitterspace, and represents most directly the spirit that thegravity is the weaker than the other forces. [62]Note we still assume that there exists at least one par-ticle satisfying the GWGC condition (1), but this particlecan in general be different from the particle satisfying theSWGC. We can impose a stronger constraint that bothinequalities are satisfied by the same particle.In these versions of the SWGC, if the mass m of theSWGC particle satisfying the inequality (4) is extremelylarge (e.g. near the cutoff scale of theory), then it wouldlikely be difficult to constrain the existence of such par-ticles observationally. One possible exception is the casewhere such a particle is stable and is a dark matter can-didate.In addition to the SWGC particle, we discussed verylight scalar fields ϕ i . One should note that morally speak-ing the conjecture asks for the existence of a very lightscalar particle. While we assumed the existence of sucha particle in the discussion above in the formulation ofthe SWGC, the idea that gravity is the weakest force atleast at the horizon scale seems to require such a particle–without such a light scalar particle the scalar force decaysin the IR. While this light scalar field does not need tocouple directly to the Standard Model particles, it willnecessarily couple to them through gravitational interac-tions. Strong Scalar Weak Gravity Conjecture
We have to this point formulated SWGC in the IR. Onemight however try to consider a “mixed UV/IR version”of the SWGC. In this situation, many states, includingmassive states, begin to contribute to the forces, and apriori we need to take into account all of them, and onemight wonder if there is any hope for a simple statement,while keeping the statement applicable to any low-energyeffective theory with UV completions.Despite these potential obstacles, recently the authorsof Ref. [17] boldly conjectured a version of the SWGC,which we call the strong scalar WGC (SSWGC) (see also[49]). [63]The SSWGC states that the scalar potential V , in alow-energy theory with UV completion with gravity, sat-isfies an inequality χ ≡ V (cid:48)(cid:48)(cid:48) ) − V (cid:48)(cid:48) V (cid:48)(cid:48)(cid:48)(cid:48) ≥ ( V (cid:48)(cid:48) ) M . (10)If we write m = V (cid:48)(cid:48) , then this inequality can be writ-ten as 2( ∂ φ m ) − m ( ∂ φ m ) ≥ ( m ) M . (11)which looks similar to (4).There are, however, crucial differences between (4) and(10). First, as already stated the conjecture (11) is meantto be a “mixed UV/IR statement” [17], while we haveabove been using the versions formulated in the IR. Sec-ond, (11) has a term involving the fourth derivative ofthe potential, which did not appear in (4). Third, thenew proposal (10) does not refer to existence of nearly-massless or massless fields ( ϕ i in our previous notation),and refers only to the scalar field φ , which can be takenarbitrary.In SSWGC the self-interaction of the scalar field, with-out any help from other scalars, is claimed to win overgravity. We find a tower of massless states when theequality in (10), which states are proposed to play therole of the nearly-massless states ( ϕ i in the notation ofthis Letter). In this situation, since we have a towerof very light scalar fields in the theory, we expect thatour conclusions presented above from the fifth-force con-straints seems to apply essentially the same manner to this new conjecture. However, such a tower often ariseswhen the field range is of order the Planck scale, wherethe effective field theory breaks down according to thedistance conjecture [2, 48]. For this reason SSWGC couldpossibly escape our observational constraints, despite thefact that the conjecture applies to any scalar.One obvious question is of course if the SSWGC is true,at least in some corners of the landscape of quantumgravity. While we leave this question for future work,let us consider a simplest setup where we have a com-pactification of the eleven-dimensional supergravity on aCalabi-Yau four-fold. This is considered to be the classi-cal limit of M-theory compactifications, and we assumethat all the corrections, stringy and α (cid:48) , can be safely ne-glected.The bosonic part of the action of the eleven-dimensional supergravity reads S = 12 κ (cid:90) d x (cid:112) − g (cid:18) R − | G | (cid:19) , (12)where R is the curvature and G is the four-form fieldstrength. Let us compactify this theory on R × (7-manifold), and denote the overall modulus of the com-pactification manifold by ρ . As explained in [30], the ef-fective potential for the overall modulus ρ takes the form V eff = V R e − √ ρM Pl + V G e − √ ρM Pl . (13)The two terms represents the contributions from the cur-vature and the four-form field strength in (12), and wehence have V G ≥
0. Interestingly, this seems to satisfy(10), irrespective of the sign of V R .Another question, which is natural for this Letter, isif the SSWGC holds for the Standard Model of particlephysics, e.g. the Higgs field. [64]For the Standard Model Hversioniggs, the effective po-tential of the Higgs boson h , is approximately given by V ( h ) = λ eff ( h )4 h , (14)for h much greater than the electroweak scale. The SS-WGC criteria (10) for the Higgs boson is approximatedas, χ ( h ) (cid:39) h (cid:2) λ eff ( h ) + β eff ( h )) + 23 β ( h ) (cid:3) , (15)where β eff ( h ) ≡ dλ eff /d log( h ) and we neglect higherderivatives on β eff , as these values are higher-loop sup-pressed. Unless both λ and β eff are zero, the value of χ is positive and greater than O ( h / (16 π ) ). There-fore, in the Standard Model, the value of χ is greaterthan the Planck-suppressed combination V (cid:48)(cid:48) /M . InFig. 1, we show the χ/h − V (cid:48)(cid:48) /M as a function ofthe Higgs VEV h . The red band in the figure shows2 σ uncertainty from the experimental and theory errors.To calculate the effective potential, we use the proce-dure of Refs. [50, 51] and take the physical parametersas top mass m t = 173 . ± . m h = 125 . ± .
16 GeV [52]. We add another error ± . − − − − − − ( χ − V / M P l ) / h h [GeV] σ bandcentral value FIG. 1: The plot of ( χ − ( V (cid:48)(cid:48) ) /M ) /h for the StandardModel Higgs effective potential. The SSWGC states that thisquantity is non-negative. Note that we use the inequality (10) from version 2of Ref. [17]. In version 1 of Ref. [17], the criteria is χ/V (cid:48)(cid:48) ≥ /M , so that the direction of this inequal-ity is opposite from that in version 2 when V (cid:48)(cid:48) <
0. Thecriteria in version 1 is violated by the potential (13), if ρ > M Pl log( − V G / V R ) / (cid:112) / V R < h (cid:38) GeV.As discussed before, the some versions of the SWGCare in tension with the existence of very light scalars.Since the Higgs or meson scalar field mediates forcesmuch larger than the gravity, this tension may be relaxedin the SSWGC condition (10), if we suitably generalizethe conjecture to multi-field cases. Of course, the fun-damental question remains how to better motivate theSSWGC in itself.In conclusion, SWGC is an attempt to articulate pre-cise our intuition that gravity is the weakest force in Na-ture. Theoretically, however, it is not clear which versionof the conjecture should hold. We have seen that obser-vational constraints on fifth-force searches provide usefulguidelines in formulating the SWGC. Our considerationshighlights the fascinating link between theoretical studiesof “high-scale” quantum gravity and experimental stud-ies of yet unknown “low-scale” physics.
Acknowledgements
We would like to thank Ryo Saito for useful discussionsand for comments on the manuscript. MY would like tothank Perimeter institute for hospitality where part ofthis work was performed. This research was supported in part by WPI Research Center Initiative, MEXT, Japan(SS, MY) and by the JSPS Grant-in-Aid for ScientificResearch 17H02878 (SS), 18K13535 (SS), 19H04609 (SS),17KK0087 (MY), 19K03820 (MY), and 19H00689 (MY). [1] C. Vafa, (2005), arXiv:hep-th/0509212 [hep-th] .[2] H. Ooguri and C. Vafa, Nucl. Phys.
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