Do We Live in the Swampland?
aa r X i v : . [ h e p - t h ] O c t IPMU-18-0143
Do We Live in the Swampland?
Hitoshi Murayama , , , Masahito Yamazaki and Tsutomu T. Yanagida Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa, Chiba 277-8583, Japan Berkeley Center for Theoretical Physics and Department of Physics,University of California, Berkeley, CA 94720, USA Physics Division, Lawrence Berkeley National Laboratory,Berkeley, CA 94720, USA
Abstract
A low-energy effective theory is said to be in the swampland if it does not have anyconsistent UV completion inside a theory of quantum gravity. The natural questionis if the standard model of particle physics, possibly with some minimal extensions,are in the swampland or not. We discuss this question in view of the recent swamp-land conjectures. We prove a no-go theorem concerning the modification of the Higgssector. Moreover, we find that QCD axion is incompatible with the recent swamp-land conjectures, unless some sophisticated possibilities are considered. We discuss theimplications of this result for spontaneous breaking of CP symmetry. We commenton dynamical supersymmetry breaking as well as the issue of multi-valuedness of thepotential.
Introduction
String theory has been one of the most promising candidates for the theory of quantumgravity. While string theory has been very successful in a number of different directions,a fundamental question is if it has any direct experimental consequences in particle phe-nomenology. The effective field theory dogma suggests that quantum gravity at the Planckscale is irrelevant for a particle physicist, who often studies energy scales much lower thanthe Planck scale.However, there is growing evidence that a vast class of effective field theories, which aretotally consistent as low-energy effective theories, do not have consistent UV completionswith gravity included. In such cases, in the terminology of [1, 2], the low-energy effectivetheories are in the swampland as opposed to the landscape . Indeed, there are indicationsthat a significant portion of the low-energy effective field theories fall into the swampland.If this is indeed the case, it could be misleading to be confined to the effective field theoryframework: the constraints for the existence of suitable UV completion with gravity imposesimportant constraints for particle physics, which are hard to see otherwise.It is then natural to ask the following question: is the standard model of particle physics,possibly with some extensions, be in the swampland or not? In this paper we discuss thisquestion. Our main focus will be the QCD axion, which has long considered to be one ofthe most promising solutions to the strong CP problem [3, 4, 5, 6].To set the stage, we begin by summarizing the recent swampland conjecture of [7]. Wethen recall the quintessence explanation for the present-day vacuum energy [7, 8] (section3), and for the Higgs potential [9] (section 4). We then come to our main ingredient, theQCD axion. After pointing out the problem with QCD axion (section 5), we discuss somepossible loop holes (section 6). Our conclusion is that the QCD axion is excluded by aset of swampland conjectures [7, 2, 10], unless exotic scenarios are considered. We discussthe implications of this result for the strong CP problem (section 7). We comment ondynamical supersymmetry breaking and multi-valuedness of the potential (section 8). Wefinally comment on the modification of the swampland conjecture (section 9) The Appendix(Appendix A) contains some no-go result for the modification of the Higgs potential. Themathematical result there could be useful for an analysis of the conjecture in [7] in othercontexts. 1
Swampland Conjectures
Suppose that we have an effective field theory coupled with Einstein gravity, containing afinite number of scalar fields { φ i } . We then have the Lagrangian L = √− g " R + X i D µ φ i D µ φ i + V total ( { φ i } ) + . . . . (1)Here V total ( { φ i } ) is the potential for the scalar fields, and we added an index ‘total’ toemphasize that this is the full potential for all the scalar fields in the theory. Note that forthe scalar fields we have chosen a canonical kinetic term in the Einstein frame. If this is notthe case then a suitable re-parametrization of the fields is needed to bring the Lagrangianinto the form of (1).The question is when this theory has a well-defined UV completion inside a suitable theoryof quantum gravity, such as string theory. In other words, is the theory in the swampland,or in the landscape?In the literature several necessary conditions, for the effective theory to be in the land-scape, have been proposed. We call these the swampland conjectures. Over the years severalsuch conjectures have been proposed [1, 2, 10, 11, 7], see [12] for recent summary.One of the most recent of such swampland conjectures is the following remarkable con-jecture due to Obied et. al. [7]: Conjecture 1 (de Sitter derivative conjecture): The potential V total satisfies the followinginequality: M Pl ||∇ V total || > c ⋆ V total , (3)where c ⋆ is a O (1) constant and M Pl = 2 . × GeV is the reduced Planck scale. Recall that the size of the gradient is given by ||∇ V total || = sX i,j g ij conf ( ∂ φ i V total )( ∂ φ j V total ) , (2)where g ij conf is the inverse metric on the configuration space for the scalar fields. In practice, in the followingwe always have a canonical diagonal metric g ij conf = δ ij . O (1) constant c ⋆ depends on the setup. For example,four-dimensional compactifications of the eleven-dimensional supergravity suggests the value c ⋆ = 6 / √ ∼ . c ⋆ , and could easily accommodate the value c ⋆ ∼ − , for example.We will discuss the phenomenological consequences of this and other swampland con-jectures. Note that even if the Conjecture 1 in itself does not hold in full generality, ourconclusion still applies to some well-understood corners of string/M-theory vacua, as shownin the analysis of [13, 14, 7]. This is one of the reasons why the Conjecture 1 should be takenseriously. An immediate consequence of the conjecture (3) is that there is no de Sitter vacua, evenmetastable ones [15, 7] (cf. [16, 17, 18]): V total > , ∇ V total = 0 not allowed . (4)This in particular excludes the constant positive cosmological constant. We can instead con-sider a dynamical vacuum energy as generated by a scalar field Q , the so-called quintessencefield ([19, 20, 21], see [22] for review). This is an extremely light scalar field, and we can forexample choose the potential to be V Q ( Q ) = Λ Q e − c Q QM Pl , (5)where c Q is some O (1) constant. It was shown that this kind of potential can indeed beincorporated in supergravity, namely the effective field theory of string theory [23].In this potential the shift of the origin of the quintessence field can be absorbed in theredefinition of the scale Λ Q . We choose the present-day value of the quintessence field to be Q/M Pl ∼
0. To explain the current value of the cosmological constant, the energy scale Λ Q is chosen to be Λ Q ∼ O (1) meV. The quintessence is the only scalar field at this energyscale, and the condition (3) is satisfied easily if c Q ≥ c ⋆ . Other potentials for quintessence has been proposed in the literature. Our conclusion does not dependmuch on the precise form of the quintessence potential, as long as the quintessence potential satisfies theconstraint (3). The possible connection that Λ Q ∼ Λ /M Pl was explored in [24]. Higgs
In the minimal version of the standard model, the only fundamental scalar field is the Higgsparticle. At the electroweak (EW) scale Λ EW ∼ O (100) GeV, the only scalar fields in thetheory are the Higgs field H and the quintessence field Q introduced above.The potential for the Higgs field is V H = λ ( H − v ) , (6)where H is the absolute value of the complex Higgs field: in the following we always havethe symmetry of rotating the phase of the complex Higgs field, and the phase part of theHiggs field will not play any role. The total potential at the EW scale is then V total ( Q, H ) = V Q ( Q ) + V H ( H ) . (7)The Higgs potential (6) has (a) a local minimum at H = v and (b) a local maximum at H = 0.As already pointed out in [9], the latter (namely the local maximum (b)) is contra-dictory with the swampland conjecture (3). In the neighborhood of the point (b) we ob-tain ∂ H V total ( Q, H ) = ∂ H V H ( H ) ∼
0, and hence ||∇ V total ( Q, H ) || = ∂ Q V Q ( Q ) ∼ O (Λ Q ).By contrast the value of the potential is given by V total ( Q, H ) ∼ O (Λ ) and is positive V total ( Q, H ) >
0. We therefore obtain0 < M Pl ||∇ V total ( Q, H ) || V total ( Q, H ) ∼ O (cid:18) Λ Q Λ (cid:19) ∼ O (10 − ) , (8)which is in sharp contradiction with the Conjecture 1 (3). Note that in this analysis we didnot assume anything about the history of the Universe—the swampland conjecture appliesto any possible field values which can be theoretically considered in the effective field theory.One possibility to escape this contradiction is to modify the EW sector, and consider acoupling of the Higgs field to some other field. For example, we can introduce a real scalarfield S so that the potential is now given by V H,S = λ ( H − v ) + κ ( S − u )( H − w ) + m S + Λ S , (9) This is the general structure when analyzing the Conjecture 1 in (3)—when the potential is a sum ofcontributions from several different energy scales with large hierarchies in between, then the existence ofextremal values for the largest-energy-scale potential contradicts the conjecture (3). u, w, κ, m, Λ S , which are assumedto be in the electroweak scale. The potential (9) is the most general expression in H and S up to the quadratic order, up to a shift of the origin of S and H . Since we have manyfree parameters, one might hope that one can adjust the free parameters such that thereis no extremal values with positive potential values, perhaps at the cost of fine-tuning theparameters.It turns out, however, this modification does not work—one either finds a de Sitterextremum of the potential and thus violating the conjecture (3), or the EW vacua becomesunstable (the determinant of the Hessian about the EW vacua becomes negative). Thedetailed analysis for this is provided in Appendix A. Indeed, we can show that this conclusionholds for a much broader class of models than the particular model (9) (see a no-go theoremin Appendix A). While we did not completely exclude the possible EW modifications, webelieve that this is a strong evidence that the EW modification of the Higgs sector requiresmore sophisticated scenarios, to say the least.Instead of modifying the EW sector, we can take advantage of the quintessence field, asalready pointed out in [9]. Namely we can modify the Higgs potential to be V ′ H ( H ) = e − c H QM Pl λ ′ ( H − v ) . (10)We can then easily verify that the combined potential V total ( H, Q ) = V Q ( Q )+ V ′ H ( H ) does nothave any extremal values. We therefore no longer have any contradiction with the Conjecture1 in (3).In conclusion, by applying the Conjecture 1 to the Higgs potential we obtained somesupporting evidence for an existence of the quintessence field. This is independent from theargument from the previous section concerning the cosmological constant. In addition to the Higgs field, some extensions of the standard model could contain otherscalar fields, at energy scales lower than the EW scale.A good example is the QCD axion, which if present we will encounter at the QCD scale5
QCD ∼ O (100) MeV. Since axions are abundant in string theory compactifications [25, 26],it might be natural to imagine that we have the QCD axion in the string/M theory landscape.The QCD axion, which we denote by a , couples to the QCD gauge field as a dynamical θ -angle: L axion = 132 π af a e µνρσ Tr F µν F ρσ , (11)where f a is the axion decay constant.Perturbatively we have a shift symmetry for the axion a → a + (const.), which is brokenonly by the non-perturbative effects: V axion ( a ) = Λ (cid:20) − cos (cid:18) af a (cid:19)(cid:21) . (12)This potential forces the axion to be at the origin, and hence the QCD axion provides anelegant solution to the strong CP problem [3, 4, 5, 6].Let us consider the QCD scale where only the quintessence and the axion are present, sothat the total potential is given by V total ( Q, a ) = V Q ( Q ) + V axion ( a ) . (13)There is a problem with the potential (13) at the field value a = πf a , which is a localmaximum for the potential V axion ( a ). We can now apply the similar logic as before: whilewe have ||∇ V total || ∼ | ∂ Q V Q | ∼ O (Λ Q ), the value of the potential is given by V total ∼ V a ∼ O (Λ ) >
0, leading to the ratio0 < M Pl ||∇ V total || V total ∼ Λ Q Λ ∼ − , (14)in contradiction with the Conjecture 1 in (3).As a cautionary remark, the cosine potential in (12) is obtained by the one-instantonapproximation, and does not quite match the actual form of the potential, as was suggestedby the chiral Lagrangian analysis long ago [27, 28] and studied more in detail by recentworks (e.g. [29, 30]). However, while the detailed form of the potential is different from (12),the crucial fact that we have a local maximum at values a ∼ O ( f a ) stays the same, and we6till have a contradiction mentioned above. While this is a rather simple argument, thispotentially invalidates the QCD axion and hence one should try to find loopholes to theargument, as we will do below.
One can think of several possible loopholes in the no-go discussion for the QCD axion in theprevious section. Let us discuss these in turn.
The first possible loophole is to make the value of the decay constant f a to be large, so thatwe have f a & O ( M Pl ) . (15)We can then appeal to the following swampland conjecture by Ooguri and Vafa [2] (seealso [31, 32, 33, 34] for recent discussions): Conjecture 2 (field range conjecture): The range ∆ φ traversed by scalar fields in field spaceis bounded as ∆ φ . O ( M Pl ); at the field range ∆ φ ∼ O ( M Pl ) we inevitably encounter aninfinite tower of nearly massless particles, thus invalidating the effective field theory.When we assume the inequality (15), Conjecture 2 means that we have moved away theproblematic value a = πf a beyond the regions of the validity of the effective field theory,thus removing the immediate contradiction with the Conjecture 1.However, this is in sharp tension with yet another swampland conjecture, namely theweak gravity conjecture by Arkani-Hamed et. al. [10] (see also [35, 36, 37, 38, 39]). Oneparticular consequence of the weak gravity conjecture is an existence of the upper bound onthe decay constant [10]: See however the discussion in section 8.2. onjecture 3 (weak gravity conjecture): There is an upper limit on the decay constant f a . M Pl S inst , (16)where S inst is the value of the instanton action.For QCD axions we have S inst ∼ , to obtain f a . − M Pl . (17)The two results (15) and (17) are clearly contradictory. The option of making the decayconstant large is therefore eliminated. The next is to appeal to higher-dimensional operators. In the discussion of the potential(12) the shift symmetry of the axion is only an approximately symmetry which is brokenby non-perturbative effects. The shift symmetry can instead by broken by quantum gravityeffects represented as higher-dimensional operators in the Lagrangian.For example, suppose that the U (1) Peccei-Quinn symmetry is broken to a discrete Z n subgroup, by the effect of the higher-dimensional operator. We then expect the followingextra contribution to the axion potential δV axion ( a ) = λ f na M n − cos (cid:18) n ( a + a ) f a (cid:19) , (18)where n is an integer (such that this term is a dimension n -operator), and λ and a are thecontinuous parameters.One should notice, however, that the potential as well as the original potential (12) isstill periodic in a with period 2 πf a . This immediately implies that there is still a maximumof the axion potential somewhere in the region a ∈ [ − πf a , πf a ], again in contradiction withthe Conjecture 1. While there are attempts to evade the weak gravity constraints by N-flation [40, 41, 42] or alignment[43], the O (10 ) gap between the two constraints (15) and (17) makes is rather difficult to fill in the gap.See [44, 45, 46, 47] for related discussion.
8n fact, the location of the maximum of the potential stays close to the value a ∼ πf a .To see this, note that the combined axion potential V ′ axion ( a ) = V axion ( a ) + δV axion ( a ) nearthe origin is given by V ′ axion ( a ) = λ f na M n − sin a naf a + Λ f a a + O ( a ) + . . . , (19)and hence the minimum of the axion potential V ′ axion ( a ) is no longer at the origin a = 0, andrather at a non-zero value a = a ⋆ , where a ⋆ is given by a ⋆ ∼ f a λ f na M n − Λ f a = f a λ f na M n − Λ . (20)For a solution of the CP problem (the small effective theta angle | ¯ θ | = | θ +arg det( m ) | < − for quark mass matrix m ), one needs a ⋆ f a ∼ λ f na M n − Λ ∼ δV axion ( a ) V axion ( a ) < − . (21)The contribution from the higher-dimensional operator (18) is therefore much smaller thanthe original potential (12) (cf. [48]). Another possible loophole is to consider the coupling to the quintessence. This might benatural possibility to consider, since the similar solutions works for the Higgs potential, asexplained before around (10).There is a big difference for the QCD axions, however. The potential for the axion (12)is determined by the non-perturbative instanton effects, and there is no option of modifyingthe potential (12), say by coupling the axion directly to quintessence—one would then breakthe shift symmetry of the axion, and hence the axion will no longer provide a solution to thestrong CP problem.One can still try to couple the quintessence field to the kinetic term of the axion. Thiskeeps the shift symmetry of the axion, and hence the potential (12). The total Lagrangian9s now L total = f (cid:18) QM Pl (cid:19) ∂ µ a∂ µ a + V axion ( a ) + ∂ µ Q∂ µ Q + V Q ( Q ) , (22)for some function f ( Q/M Pl ). Since this is no longer has the canonical kinetic term, oneshould do the field redefinition. We can choose the transformation a → Z dQ r f (cid:16) QM Pl (cid:17) := g ( Q ) , Q → a , (23)so that the Lagrangian afterwards is L total = ∂ µ Q∂ µ Q + V axion ( g ( Q )) + ∂ µ a∂ µ a + V Q ( a ) . (24)We can exchange the label of Q and a , to bring the Lagrangian into the more familiar form: L total = ∂ µ a∂ µ a + V axion ( g ( a )) + ∂ µ Q∂ µ Q + V Q ( Q ) . (25)This computation shows that for the analysis of the Conjecture 1, the only practical effectof the function f ( Q/M Pl ) is the replacement of the argument a of V axion by g ( a ). Despitethis change, the potential V axion ( g ( a )) still has maximum at a = a max with g ( a max ) = πf a ,and hence we run into the same contradiction with the Conjecture 1 as before.The only potential caveat for this is to appeal to the loophole of section 6.1. Supposethat the function g ( a ) is chosen such that a max & O (10 ) πf a , (26)such that the condition a max & M Pl can be imposed without contradicting the constraintsfrom the weak gravity conjecture (17): f a . O (10 − ) M Pl . (27)If f a saturates the bound (27) (where the constraint should be the least severe), we need g ( M Pl ) ∼ O (10 − ) M Pl . (28)This scenario is not impossible. For example, we can choose f (cid:18) QM Pl (cid:19) = e c QA QM Pl , (29)10o that we have g ( Q ) M Pl = 1 c QA (cid:16) − e − c QA QM Pl (cid:17) , (30)Then (28) can be satisfied for c QA ∼ .The interaction (22) for the function (29) includes a linear coupling L total ⊃ c QA QM Pl ∂ µ a∂ µ a . (31)When we have a large coefficient c QA ∼ , this violates the Born unitarity of the Q + a → Q + a scattering amplitude before arriving at the Planck scale.Other than coupling the quintessence to the kinetic term of the axion, yet another possi-bility then is to keep the form of the potential (12), and make the parameter Λ QCD dependenton the quintessence: V axion ( Q, a ) = Λ
QCD ( Q ) (cid:20) − cos (cid:18) af a (cid:19)(cid:21) . (32)This can indeed be realized by coupling the quintessence Q to the kinetic term for the gluons: L kin . = (cid:18) λ QF F QM Pl (cid:19) g Tr F µν F µν . (33)This is equivalent to making the gauge coupling constant Q -dependent:1 g → g ( Q ) := (cid:18) λ QF F QM Pl (cid:19) g , (34)which leads to the Q -dependence of the QCD scale Λ QCD after transmutation:Λ
QCD ( Q ) = Λ exp (cid:18) − πb (cid:18) g ( Q ) − i θ π (cid:19)(cid:19) → Λ exp (cid:18) − πb (cid:18)(cid:18) λ QF F QM Pl (cid:19) g − i θ π (cid:19)(cid:19) ∼ exp (cid:18) − c ′ Q QM Pl (cid:19) , (35)where c ′ Q = 8 πλ QF F b g . (36)and b is the coefficient of the one-loop beta function. The constraint from the Conjecture1 in (3) is then satisfied by choosing c ′ Q ≥ c ⋆ .11he coupling (33) causes a serious problem, however. Once the quintessence couples tothe gluons, then the quintessence couples to the nucleons through the gluon loops, so thatwe generate an effective interaction L QNN ∼ λ QNN Λ QCD M Pl QN N , (37)where N here stands for nucleons. Since we expect the coefficient λ QNN to be of ∼ O (1),the coefficient is λ QNN Λ QCD /M Pl ∼ O (10 − ) and this is in tension with the equivalence-principle constraints on fifth-force between the nucleons: (Yukawa) < O (10 − ) [49]. This isin contrast with the case of the Higgs particle, where the similar coupling (10) between theHiggs and the quintessence is less constrained due to suppression of the loop diagrams byYukawa couplings and electroweak couplings [9].While this eliminates the coupling (33) between the quintessence and the gluon, one cantry to save the loophole by coming up with a more complicated, if exotic, scenario. Oneidea is to use the mirrored copy of the QCD [50, 51, 52, 53]. Here we have two copies of theQCD, our original QCD and its mirror image. There is no direct coupling between the twocopies of QCD. We assume that the quintessence field couples only to the mirror QCD as in(33), but not to the original QCD. One then obtains the potential V axion ( Q, a ) = (cid:0) Λ ′ QCD ( Q ) + Λ (cid:1) (cid:20) − cos (cid:18) af a (cid:19)(cid:21) , (38)where the mirror QCD scale Λ ′ QCD ( Q ) comes from the mirror QCD (see (35))Λ ′ QCD ( Q ) = Λ ′ QCD4 exp (cid:18) − c ′ Q QM Pl (cid:19) , (39)and another scale Λ from the original QCD. The potential (38) satisfies the constraints from Conjecture 1 in (3). Indeed, the totalpotential is now given by V total ( Q, a ) = V Q ( Q ) + V axion ( Q, a ) , (40) In the potential (38) we need to make sure that the phases of the two cosine functions from the twocopies of QCD match. One expects that this is possible by imposing the mirror symmetry between the twocopies of QCD at the value Q = 0. V Q is the quintessence potential (5). The derivatives of the axion potential are com-puted to be M Pl ∂ a V axion ( Q, a ) = M Pl f a (cid:0) Λ ′ QCD ( Q ) + Λ (cid:1) sin (cid:18) af a (cid:19) ,M Pl ∂ Q V axion ( Q, a ) = c ′ Q Λ ′ QCD ( Q ) (cid:20) − cos (cid:18) af a (cid:19)(cid:21) . (41)We find that the problematic point a = πf a no longer extremizes the potential. We stillhave a = 0 as an extremal point, but this is of course the minimum V axion ∼ V Q , which satisfies the Conjecture1 as discussed in section 3.There is no constraint from the long-range force since the quintessence does not coupleto the original copy of the QCD. Suppose that we have managed to evade the constraints on the QCD axion, so that theConjecture 1 in (3) is satisfied for the total potential V total ( Q, a ) = V Q,a ( Q, a ) := V Q ( Q ) + V axion ( Q, a ) at the QCD scale. Namely, we have M Pl q ( ∂ Q V Q,a ( Q, a )) + ( ∂ a V Q,a ( Q, a )) ∼ V Q,a ( Q, a ) . O (Λ QCD4 ) . (42)for all possible values of Q and a .Since we now have the QCD axion, we should re-do the analysis of the Higgs potentialin section 4 at the EW scale. Let us start with the standard Higgs potential (6) (withno coupling to the quintessence field), so that the total potential at the EW scale (7) nowincludes the axion: V total ( Q, a, H ) = V Q ( Q ) + V axion ( Q, a ) + V H ( H ) , (43)where V H is the standard Higgs potential (6).Let us study the neighborhood of the local maximum H = 0 of the Higgs potential, where V H ( H ) ∼
0. We then have, using (42), M Pl ||∇ V total ( Q, a, H ) || = M Pl q ( ∂ Q V Q,a ( Q, a )) + ( ∂ a V Q,a ( Q, a )) . O (Λ QCD4 ) . (44)13his implies 0 < M Pl ||∇ V total ( Q, a, H ) || V total ( Q, a, H ) . O (cid:18) Λ Λ (cid:19) ∼ O (10 − ) . (45)This is still in contradiction with Conjecture 1 in (3). We can eliminate this problem by thecoupling of the quintessence to the Higgs potential (10), as in section 4.There seems to be a possible loophole in this argument. In the discussion above (e.g.in (42)) we have implicitly assumed that the QCD scale Λ QCD is the only scale relevant forthe QCD axion. This is not the case when we have mirror copies of QCD as in (38), wherewe also have the mirror QCD scale Λ ′ QCD . This scale can taken to be Λ ′ QCD ∼ Λ EW [53], inwhich case other ratio in (45) will be replaced by an O (1) constant. Namely, for H = 0 and a ≁ < M Pl ||∇ V total ( Q, a, H ) || V total ( Q, a, H ) ∼ O (cid:18) Λ ′ QCD4 Λ (cid:19) ∼ O (1) . (46)There is another problem in the neighborhood of the special locus H = a = 0, however.In this special case both the axion potential V axion and the Higgs potential V H are extremized,and the norm of the gradient of the potential is given by the quintessence potential V Q , sothat ||∇ V total ( Q, a, H ) || ∼ O (Λ Q ). By contrast the value of the potential is dominated bythe Higgs contribution, so that we have 0 < V total ( Q, a, H ) ∼ O (Λ ). We therefore find0 < M Pl ||∇ V total ( Q, a, H ) || V total ( Q, a, H ) ∼ O (cid:18) Λ Q Λ (cid:19) ∼ O (10 − ) , (47)which is again in contradiction with the Conjecture 1. Having discussed the possible loopholes in the previous section, we now arrived at one of ourmain conclusions. Let us assume the recent swampland conjecture (Conjecture 1 in (3)) aswell as the two more swampland conjectures (Conjecture 2 and Conjecture 3 in section 6.1),and of course impose observational constraints. Then in effective field theories admittinga consistent UV completion inside theories of quantum gravity, almost all of the existingscenarios for the QCD axion are ruled out. Our conclusion applies only to the QCD axions, and does not necessarily exclude more general non-QCDaxions. and it would be desirable to come to a definiteconclusion in the near future. Regardless of the outcome, let us emphasize again that theConjecture 1 is known to hold in some corners of string/M-theory vacua.Let us for now suppose that the swampland conjectures are true. Then we sill need tosolve the strong CP problems. There are several options. • One still uses the QCD axion. As mentioned already this requires some sophisticatedmodel building, such as the possibility discussed towards the end of section 6.3. • There has been a proposed solution of the QCD by making the up quark (nearly)massless [65]. This option seems to be disfavored by lattice gauge theories [66], whichsuggests non-zero up quark mass with high statistical significance; see [67, 68] for recentdiscussion. • Another possibility is that the CP symmetry is an exact symmetry of the Lagrangian(so that the bare value of the theta angle is θ = 0), and that the CP symmetry isspontaneously broken. Such a scenario was consider before, see e.g. [69, 70, 71, 72]. Inview of the results of this paper, it would be interesting to study if any of these modelscan be properly embedded into string theory. • Of course there could be other solutions of the strong CP problem, not traditionallydiscussed in the literature. See the recent paper [77] for such an attempt. The literature is too large to be summarized here. See [54, 54, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]for a sample of recent references which discuss the construction of de Sitter vacua in string theory. Perturbative analysis of Calabi-Yau compactifications show that the CP is either unbroken, or brokenby the vacuum expectation value of the CP-odd moduli [73, 74]. Even non-perturbatively it believed thatCP is a gauge theory in string theory and can be broken only spontaneously [75, 76].
15t is too early to tell which of these possibilities are realized in Nature. Regardless of theresult, it is tantalizing that the insights from the quantum gravity are now intimately tiedwith the phenomenological search for the solutions of the strong CP problem.
In this paper we discussed the implications of the conjecture at the energy scales for thequintessence, QCD axion and the Higgs. We can try to go further to higher energy scales.While the analysis there depends on the details of the physics beyond the standard model,one ingredient one might wish to include is the supersymmetry, which we hope to be brokendynamically [78].For some models of the dynamical supersymmetry breaking, it is subtle to understandwhether or not the Conjecture 1 excludes the model. For example, in the Polonyi model, theconclusion depends crucially on the behavior of the K¨ahler potential when the Polonyi fieldtakes an O (1) M Pl value. This is also the case when the Polonyi model arises dynamically,as in the case of the IYIT model [79, 80], see [81].The conclusion is much more clear-cut for other cases. For example, in the models ofmetastable supersymmetry breaking (such at the ISS model [82]), the supersymmetry isbroken at a metastable de Sitter vacuum, where the field value is parametrically smallerthan the Planck scale and hence the physics is still calculable. The existence of such vacuaimmediately contradicts the Conjecture 1 in (3). This is an important consequence of theConjecture 1—such metastable supersymmetry breaking is known to dramatically simplifythe supersymmetric model building [83, 84], but these are excluded by the swampland con-jecture. 16 .2 Multi-Valuedness of the Potential and Inflation Let here us comment on one subtlety concerning the Conjecture 1. In the formulation ofthe conjecture it is implicitly assumed that the potential is single-valued. However, thereare situations where the potential is multi-valued as a function of the field value, say φ . Inother words, we have several stable as well as metastable branches labeled by 1 , , . . . withdifferent potentials V ( φ ) , V ( φ ) , . . . , and we will have transitions between the branches. Inthis case, the energy-minimizing potential, which corresponds to the stable branch, is givenby V min ( φ ) = min n V n ( φ ) . (48)In this situation, we can consider two different possibilities in interpreting Conjecture 1:1. We impose the constraint (3) for each branch, namely for each function V n ( φ ).2. We impose the constraint (3) for the minimum-energy potential V min ( φ ).Our proposal is that we should choose the latter option.This has important consequences regarding the Conjecture 1. While one expects that thepotential V n ( φ ) to be a continuous function of the field φ , the energy-minimizing potential V min ( φ ) is in general a discontinuous function of φ , when the minimal branch changes froma branch n to another branch n ′ . This means the some mathematical results assumingcontinuity of the function, such as the no-go theorem of Appendix A.3, does not apply tothe potential V min ( φ ).Moreover, an inconsistency with the Conjecture 1 often happens when we have a localmaximum of a smooth function, which in our case is V n ( φ ). But when we have a localmaximum in a branch n , then one might expect another branch where the value of thepotential is smaller, so that the branch n is not chosen for the energy-minimizing potential(48). One therefore expects that having the multi-branch structure will help to amelioratethe constraints from Conjecture 1.An excellent example for such multi-branch structure is provided by an axion a coupledwith pure SU ( N ) Yang-Mills theory. In the large N limit it was argued by Witten [28, 85]17hat we have an infinitely many branches labeled by an integer n , and the potential is givenby V n ( a ) = Λ a f a ( a − nπf a ) , V min ( φ ) = min n ∈ Z Λ a f a ( a − nπf a ) . (49)Note that the potential on each branch does not have the expected 2 πf a periodicity; thisperiodicity is restored only after gathering together all the branches. The potential (49) hasdiscontinuities at a = (2 Z + 1) πf a , where the sign of the derivative of the potential differsbetween the left and the right. It is believed that the multi-branch structure is preserved evenfor a finite value of N , where we expect O ( N ) branches of vacua ([28], see [86, 87] for recentdiscussion). While the potential is quadratic near the origin, the potential eventually isbounded by the dynamical scale O (1)Λ a , and we expect a plateau near the values a ∼ N πf a .The existence of such multi-branch structure was also observed in supersymmetric QCD[88]. For the case of (non-supersymmetric) QCD, this was analyzed via the chiral Lagrangianin [27, 28] (see also [89]), and we do have multiple branches for some quark masses. We do nothave such branch structures for realistic values of the quark masses, however. The discussionof section 5 is therefore not affected.Let us finally comment on the relevance of this remark for inflation. Instead of QCDaxions we can choose the axion above to be the inflaton. The multi-branch structure men-tioned above gives a field-theory realization [95, 96, 97, 98, 99] of the axion monodromyinflation, originally discussed in string theory [100, 101].It has recently been pointed out that an inflation model based on this multi-branchstructure [102] is in perfect agreement with the current observational constraints . Theinflaton rolls down the potential for a single branch ( V n in the previous notation), since wecan argue that the transition between different branches are irrelevant for the time scales ofinflation [96, 102]. Since the model assumes the slow-roll condition, the current bounds forthe scalar-to-tensor ratio is in mild tension with the current Planck constraints ( c ⋆ ∼ . See e.g. [90, 91, 92, 56, 93, 94] for recent discussion of the swampland conjectures in the context ofinflation. The potential in [102] was inspired by the holographic computation of [96], and is different from thecosine potential used for natural inflation [103]. Note that the deviation from the cosine potential for pureYang-Mills is now firmly established by lattice gauge theory results [104], see also [105].
18t is worth pointing out that the setup of [102], together with the proposed implemen-tation of the Conjecture 1, eliminates the problem of the plateau of the inflaton potential.In many of the inflationary models today the inflaton potential has a plateau region wherethe potential is nearly flat. This is clearly a dangerous region for the Conjecture 1. Sucha plateau, however, does not appear in the energy-minimized potential V min in (48). Themulti-valued structure of the potential has traditionally been invoked for increasing the fieldrange traversed by the inflaton. What we are finding here is that it has a different virtue,namely the consistency with the swampland conjecture of (3). In view of the phenomenological constraints discussed in this paper, one of the most naturalpossibilities is to weaken the swampland conjecture (3).One plausible possibility is to modify the conjecture (3) to be in the following form: M Pl ||∇ V total || > c ⋆ V total , whenever Hessian( V total ) > . (50)This should be compared with (4). Namely, we allow for the point V total > , ∇ V total = 0as long as the Hessian has at least one non-positive eigenvalue. This restriction seemsnatural since the point is unstable if the Hessian has a negative eigenvalue. This proposalimmediately removes the problem with the QCD axion and the Higgs discussed in this paper. Acknowledgements
We would like to thank C.-I. Chiang, H. Fukuda, K. Hamaguchi, A. Hebecker, M. Ibe, S. Mat-sumoto, T. Moroi, H. Ooguri, S. Sethi, S. Shirai and C. Vafa for discussions. This researchwas supported in part by World Premier International Research Center Initiative, MEXT,Japan. H. M. was supported in part by U.S. DOE under Contract DE-AC02-05CH11231,NSF under grants PHY1316783 and PHY-1638509, JSPS Grant-in-Aid for Scientific Re-search (C) No. 26400241 and 17K05409, and MEXT Grant-in-Aid for Scientific Research on See e.g. [54, 56] for other modification of the conjecture (3).
A Analysis of the Higgs Potential
As mentioned in the main text, one possible way to escape the constraint from the Conjecture1 in (3) is to extend the EW sector, so that we have a potential involving multiple fields. Inthis Appendix we discuss some difficulties in this approach.
A.1 The Potential of (9)
Let us start with the potential of (9), where we included a real field S in addition to theHiggs field H .By extremizing the potential ( ∂ H V H,S = ∂ S V H,S = 0), one finds two different solutions.The first solution, which we call solution (a), is what should be the EW vacuum, correspond-ing to the value H = v in the original Higgs potential (6): H = κ w − κm u − λm v κ − λm , S (a) = κ u + 2 κλv − κλw κ − λm . (51)Another solution, which we call solution (b), corresponds to the local maximum H = 0 ofthe original Higgs potential (6): H = 0 , S (b) = w κm . (52)There are several conditions to be imposed. First, since we wish to keep the EW vacuum(namely solution (a)), we need H a ) = κ w − κm u − λm v κ − λm ≥ . (53)20econd, we should have zero energy at the solution (a); if this is not the case we havenon-zero constant cosmological constant at lower energy scales, and we spoil the quintessencediscussion in section 3. This requires us to choose the constant Λ S to beΛ S = − κ (cid:16) m u ( κu + 4 λ ( v − w )) + 2 κλ ( v − w ) (cid:17) κ − λm ) . (54)Third, we impose the condition that the solution (a) is at least a local minimum. Thisin particular implies that the determinant of the Hessian at (a) is positive, leading to theconstraint m (cid:0) κu + 2 λv (cid:1) − κ w > . (55)Finally, for the consistency with the conjecture (3) we require that the value of thepotential is non-positive at the solution (b). This gives V ( b ) = ( κ w − m ( κu + 2 λv )) λm − κ m < , (56)namely κ w − m (cid:0) κu + 2 λv (cid:1) = 0 or λ < κ m . (57)The three conditions (53), (55), (57) are mutually incompatible. We therefore concludethat the potential (9) does not serve our purposes. A.2 General Possibilities: a No-Go Theorem
While the discussion of the previous subsection was restricted to a particular potential (9),the lesson is actually more general.Suppose that we have a set of scalar fields ~S such that the total potential, involving theHiggs field, is given by V H,S ( H, ~S ) = V H ( H ) + . . . , (58)21here . . . represents the terms involving the field ~S . We assume that V H,S is a continuousand differentiable function of the arguments H and ~S .In general we find multiple solutions to the extremal condition: ∂ H V H,S = ∂ ~S V H,S = 0 . (59)In general there are many other solutions to (59), and Conjecture 1 in (3) could beviolated at any of these points. We therefor impose the condition(A) The potential is non-positive at all the solutions of (59).Moreover, we wish to have the EW vacuum ( H = v in the original Higgs potential (6)).This motivates us to impose(B) There exists a solution of (59), which is a local minimum for the potential V H,S . We moreover assume that there are no flat directions around the solution,and the value of the potential vanishes at the solution: V H,S = 0.Let us further assume that(C) There exists at least one solution to (59) other than the EW vacuum solutiondiscussed in (A).Namely we exclude the possibility that (A) is the only extremal value of the potential in theconfiguration space.It turns out that it is not possible to satisfy all the constraints (A), (B), (C). This is ourno-go theorem. 22 .3 Proof of the No-Go Theorem
Let us give a proof of this no-go theorem. Let us denote the EW vacuum of (B) as P B = ( H (B) , S (B) ), and another Anti-de Sitter vacuum of (C) as P C = ( H (C) , S (C) ). Let usassume we have a total of D fields and the configuration space of ( H, ~S ) is D -dimensional.Let choose a set of path L ˆ n starting from P B into P C , so that (1) L ˆ n points in thedirection ˆ n ∈ S D − in the neighborhood of P B and then reaches P C and (2) there are nomutual intersections of L ˆ n with different ˆ n ∈ S D − , so that L ˆ n with ˆ n ∈ S D − foliates thewhole ( H, ~S )-plane. See Figure 1 for the case of D = 2. We can think of the combination( P, ˆ n ) with P ∈ L ˆ n , ˆ n ∈ S D − as providing a coordinate chart in the configuration space.Figure 1: We can foliate the D -dimensional configuration space by a set of L ˆ n with ˆ n ∈ S D − starting with P B and ending at P C . Here we show the case of D = 2, where ˆ n ∈ S is apoint of the circle, namely specifies the direction in the neighborhood of the point P B .Let us fix ˆ n ∈ S D − and consider the function V H,S along the line L ˆ n , starting with thepoint P B . Since P B was the EW vacuum we start with V H,S = 0, and by assumption (B)the potential grows into positive values as we gradually move along L ˆ n . Since we know (byassumption (C) and (A)) that the potential should reach negative values by the time weget to the point P C , and since the potential is the continuous function of the arguments,we quickly conclude that there should be at least one local maximum along the path L ˆ n . Ifthere are multiple such local maximums, we take the one closest to the point P B , and wecall this point P ˆ n . Obviously we find V ( P ˆ n ) > We thank Kyoji Saito for suggesting some refinement on this proof. The possible error in the followingproof, however, should be attributed solely to the authors. Strictly speaking L ˆ n for some particular value ˆ n ∞ of ˆ n runs off to infinity, one might worry that forthis ˆ n = ˆ n ∞ the path L ˆ n ∞ has infinite length and the local maximum we mentioned here might be locatedat infinity. When this happens, we can replace the minimum in (60) by a maximum to apply the same V ( P ˆ n ) as we change ˆ n ∈ S D − . Since S D − is a compact space, there is necessarily a point ˆ n ∗ ∈ S D − which attains the minimum: V ( P ˆ n ∗ ) = min ˆ n ∈ S D − V ( P ˆ n ) . (60)Since V ( P ˆ n ) > n ∈ S D − , we in particular find that V ( P ˆ n ∗ ) > . (61)Moreover, we find P ˆ n ∗ is a extremal point of the potential: ∇ V ( P ˆ n ∗ ) > . (62)Indeed, the derivative of the potential vanishes along the path L ˆ n ∗ from the definition of P ˆ n ∗ ,and vanishes along the direction of the sphere S D − thanks to the definition (60); since thederivative of the potential vanishes in all the linearly-independent directions, the derivativeshould vanish in all the directions. The result (61) and (62) are in contradiction with ourassumption. This concludes our proof.Our result excludes many of the possible EW modifications of the Higgs potential. Forexample, if we have a polynomial potential V H,~S for complex scalars ~S , then we generi-cally expect many extremal points (thus satisfying (C)), so that we can conclude withoutany explicit computations that the potential does not satisfy our criterion. Note that thequintessence modification in (10) solves the problem by violating the condition (C).While we discussed this result in the context of the Higgs potential, our mathematicalno-go theorem can be used in other contexts, e.g. the discussion of the moduli stabilizationin the the swampland conjecture (see [60] and version 2 of [9] for one-parameter version ofour discussion). References [1] C. Vafa, “The String landscape and the swampland” , hep-th/0509212 .[2] H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and the Swampland” , Nucl. Phys. B766, 21 (2007) , hep-th/0605264 . argument, to arrive at (61) and (62).
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