U(1) mixing and the Weak Gravity Conjecture
aa r X i v : . [ h e p - ph ] J u l U(1) mixing and the Weak Gravity Conjecture
Karim Benakli ♠ , Carlo Branchina ♦ and Ga¨etan Lafforgue-Marmet ♣ Laboratoire de Physique Th´eorique et Hautes Energies (LPTHE),UMR 7589, Sorbonne Universit´e et CNRS, 4 place Jussieu, 75252 Paris Cedex 05,France.
Abstract
Tiny values for gauge couplings of dark photons allow to suppress their kinetic mixing with ordinaryphotons. We point out that the Weak Gravity Conjecture predicts consequently low ultraviolet cut-offs where new degrees of freedom might appear. In particular, a mixing angle of O (10 − ), requiredin order to fit the excess reported by XENON1T, corresponds to new physics below O (100) TeV,thus accessible at a Future Circular Collider. We show that possible realizations are provided bycompactifications with six large extra dimensions and a string scale of order O (100) TeV. ♠ [email protected] ♦ [email protected] ♣ [email protected] his short note aims to investigate some possible relations between U (1) mixing and the WeakGravity Conjecture (WGC) [1, 2] (for a review see e.g. [3]). In particular, the case of tiny mixing haswitnessed a recent surge of interest following the results announced by XENON1T [4]. The collabo-ration has reported an excess between 1 and 7 keV, close to the lower threshold of the experiment,with a peak around 2 − − ǫ ≃ O (10 − − − ) (1)which is in agreement with the upper bound limit given by XENON1T on ǫ , that also claims a 3 σ significance for a 2 . ǫ ≃ O (10 − ) , O (10 − ) or O (10 − ), with,respectively, an order O (GeV) massive dark photon in the first two cases and a massless one in thelast one have been advocated. We discuss below a possible origin of such mixing parameter, especiallyfor challenging tiny values, where we find that the WGC allows to hope for an accompanying signalat collider experiments.We focus on the sector of the low energy effective field theory describing the U (1) gauge groupsrepresentations and interactions. One of the two, U (1) v , is called visible as we have in mind hyper-charge or electromagnetism. Another, U (1) d , corresponds to an extra factor we call ”dark” U (1),having in mind an hidden sector. It is straightforward to generalize to cases with more abelian gaugegroups. The associated gauge fields and gauge fields strengths are denoted as A µ ( v ) , F µν ( v ) and A µ ( d ) , F µν ( d ) ,respectively. The corresponding two-derivative Lagrangian reads: L ⊃ − F µν ( v ) F ( v ) µν − F µν ( d ) F ( d ) µν − ǫ vd F µν ( v ) F ( d ) µν + g v J µ ( v ) A ( v ) µ + g d J µ ( d ) A ( d ) µ . (2)For massless visible and dark photons, this mixing in the two-derivative Lagrangian can be elimi-nated by performing the appropriate rotation. When the U (1) d gauge boson acquires a mass, througha Stueckelberg or Higgs mechanisms, the mixing has physical implications. The visible and darkphotons couple in the new basis to the currents J µv and J µd through: L ⊃ g d q − ǫ vd J µ ( d ) − ǫ vd g v q − ǫ vd J µ ( v ) A ( d ) µ + g v J µ ( v ) A ( v ) µ , (3)thus implying that the visible matter is charged under the dark gauge symmetry with charge ∼ ǫ vd g v .It is most natural to assume that the dark U (1) mass and mixing vanish in the fundamental theoryat the ultra-violet (UV) cut-off and are generated at lower energies. The mixing can be generated at ne loop by states with masses m i and charges ( q ( i ) v , q ( i ) d ) under ( U (1) v , U (1) d ). It is then given by: ǫ vd = g v g d π X i q ( i ) v q ( i ) d ln m i µ , (4)where µ is the renormalisation scale. In the case of the hyper-charge U (1) v ≡ U (1) Y we have g v = g ′ and q ( i ) v = Y ( i ) , while g v = g ′ cos θ w and q ( i ) v = q ( i ) em the electrical charge for U (1) v ≡ U (1) em .In order to generate such a small mixing as the one required by XENON1T, we either requirethe dark photon coupling to be appropriately small, a cancellation in the one-loop logarithms, orappeal to higher order non-renormalisable operators. The cancellation can be partial, for instancebetween particles with (order one) charges ( q ( i ) v , q ( i ) d ) and ( q ( j ) v , q ( j ) d = − q ( i ) d ) and masses m i and m j with m j = m i + ∆ m ij . For ∆ m ij ≪ m i , we have an approximation: ǫ vd ∼ g v g d π ∆ m ij m i . (5)For complete cancellation, this one loop contribution is replaced by higher loop ones. However,gravitational loops are expected to show up at some order and lead to a lower bound. It was shownin [16] that this is expected at six loop order giving rise to an ǫ vd & O (10 − ) for a bona fide fourdimensional theory. We shall discuss below the first alternative of a tiny dark sector coupling.We start by considering an abelian gauge symmetry U (1) with gauge coupling g . The WeakGravity Conjecture requires the presence of at least one state with a mass: m ≤ g q M P (6)where we use use natural units ~ = c = 1 and M P = √ πG ∼ . × GeV is the reduced Planckmass. Obviously this is satisfied by all the Standard Model states for the hypercharge/electromagneticgauge symmetry. This bound was generalised to the case of multiple U (1)s by the replacement q → P i q i in (6) [17]. In theories with supersymmetry, extra-dimensions can furnish a set of BPSstates (Kaluza-Klein modes or solitonic objects as branes) that saturate this bound. Their masseswill receive corrections after supersymmetry breaking but some should still satisfy the bound. TheSwampland Lamppost Principle (SLP) [18] implies that all the sets of charges are present in thetheory. These states will then contribute to generating a mixing between the U (1)s. For two U (1)s,the masses of the charged particles can be expressed as m = c p ( g v q v ) + ( g d q d ) , where c < i and j for the visible anddark charge of the particle respectively, as would be for quantized charges forming a lattice in chargespace. Eq. (4) then becomes in this scheme ǫ vd = g v g d π X i,j q i q j ln c ( i,j ) (cid:2) ( g v q i ) + ( g d q j ) (cid:3) µ ! (7)Though the number of states is infinite, we include in the loop only states below the cut-off. If aparticle with charge ( q i , q j ) is in the spectrum, there are also particles with charge ( q i , − q j ), ( − q i , q j )and ( − q i , − q j ) giving ǫ vd ≃ g v g d π X i,j q i q j ln c ( i,j ) c ( − i, − j ) c ( − i,j ) c ( i, − j ) ! (8) hich for small number of states, as a result of the diverse cancellation between different contributions,is expected to remain small.The most relevant facet of the WGC for this work is the prediction of an ultraviolet cut-off scale forthe effective field theory at Λ UV . gM P l . This was dubbed as the magnetic weak gravity conjecturein [2] and, in the weak coupling limit g →
0, it predicts the absence of global symmetries in quantumgravity [19–21]. For electromagnetic or hypercharge gauge coupling, the cut-off scale set by the WGCremains close to the Planck scale.We generalize here, as done by [22], this requirement to the case with multiple U (1) gauge groupsby requiring that none of the gauge symmetry factors should turn into a continuous global symmetryby taking the corresponding coupling to vanish. This implies that a tiny value of the dark photon gaugecoupling, introduced to make the mixing tiny, require a UV cut-off at most of order Λ UV . g d M P . Thisis sensibly lower than M P and could have important consequences in phenomenology and cosmology.Starting from (4), we identify the visible photon with the SM photon, i.e. g v = e ∼ .
3, and thelogarithm to be O (1 − ǫ vd ∼ g v g d π ∼ O (10 − − − ) g d ⇒ g d ∼ O (10 − ) ǫ vd (9)Per se, the WGC does not provide information on the new physics required at Λ UV . g d M P l . Asimple possibility is that the U (1) d becomes part of some non-abelian gauge group SU (2) D with fieldstrength F µν ( D ) broken by a vacuum expectation value (v.e.v) of h Σ i = v ≃ Λ UV /g d of a field in theadjoint representation. One could then induce a contribution ǫ NRvd to the kinetic mixing through theeffective non-renormalizable operator (see e.g. [23]): c NR M P T r h Σ F µν ( D ) i F ( v ) µν ⇒ ǫ NRvd ≃ c NR vM P (10)where c NR is a constant. For this contribution to remain sub-leading, we require: c NR v . ǫ vd M P ⇒ c NR v . − g d M P , (11)which for ǫ vd ∼ − gives c NR v . TeV.Kinetic mixing might also arise from D-terms in supersymmetric theories through effective oper-ators [24, 25]: D Λ D F µν ( d ) F ( v ) µν (12)that are expected to be very small. For example, they can be suppressed by the value of the ratio SMHiggs v.e.v over the scale Λ D for hypercharge D -term or through powers of the dark sector couplingfor the dark U (1) D-term.In the following we will use (9) to compute g d from ǫ . A value of ǫ vd ∼ − as in (1) wouldrequire g d ∼ O (10 − − − ). The WGC implies then that the theory has a UV cut-off:Λ UV . g d M P ∼ O (10 − )TeV . (13)Therefore, new physics must appear below energies of order O (100) TeV. Such physics could beaccessible at future experiments at collider, such as the 100 TeV Future Circular Collider (FCC).Following the SLP, such a scenario is consistent with quantum gravity only if it could arise froma string theory model. We will discuss now one possible venue for realizing this UV completion in a tring theory. We do not attempt an explicit string model building which is beyond the scope of thiswork. We contemplate the possibility that a hierarchy g d ≪ g v is obtained through the suppression of g d by the volume of the internal compactified space. More precisely, we consider a scenario where westart from ten-dimensional type IIB string theory compactified on a six-dimensional space of volume V ≡ (2 πR ) . The four-dimensional reduced Planck mass M P is related to the string scale mass M s and string coupling g s through (multiplied by 2 for type I strings): M P = R M s πg s M s (14)The visible U (1) v is taken to live on a D5-brane wrapping a small two-dimensional cycle of approximatestring size of volume (2 πr ) & π M − s . Then, the visible coupling reads: g v = 2 πg s r M s ≃ πg s (15)The dark U (1) d is instead chosen to live on a D9-brane wrapping the whole six-dimensional compactspace and its gauge coupling is given by: g d = 2 πg s R M s (16)Then, we get: ǫ vd ∼ g v g d π ∼ g s πR M s ⇒ ǫ vd ∼ √ π M s M P ∼ − M s M P (17)thus ǫ vd ∼ − ⇒ M s ∼ O (100)TeV (18)This is merely two orders of magnitude above the proposals of TeV strings for solving the hierarchyproblem [26–35]. Note that our analysis is similar to the analysis performed in [25]. However there isa notable difference in that we have considered D5-D9 branes instead of D3-D7, leading to differentresults, and in particular allowing smaller values of the mixing. The Dp-D(p-4) set-up is enforced bysupersymmetry, but, in the case of low string scale, we could have taken instead, without change in ourresults, a non-supersymmetric configuration of D3-D9 branes, our world being non-supersymmetric atleast up to TeV energy scales. However, one should keep in mind that some of the non-supersymmetricconfigurations tend to fall in the Swampland [36].The above scenario implies the appearance of large extra-dimensions at a scale of order:1 R = (cid:18) M s √ πM P α Y M (cid:19) / M s (19)where we have identified the tree-value of the SM gauge couplings as α Y M = g s /
2. Taking anapproximate value for α Y M ∼ /
25, we get1 R ∼ O (10) GeV (20)Though these values of the compactification energy scale might seem low, they are not experimentallyexcluded. Gauge bosons propagate in these extra dimensions, in addition to the gravitons. However,in contrast to the case in [27–29,37–39], these are Kaluza-Klein excitations of the dark U (1) with tiny ouplings. It is the production of a huge number of them that will compensate the coupling strongsuppression. They can be observed as missing energy at collider experiments in particular at a 100TeV collider.We can express the string mass scale and the compactification radius as a fonction of g d and M P : M s ∼ √ g s g d M P and 1 R ∼ g d (8 π ) M P . g d M P (21)The most stringent bound on ǫ today is ǫ ∼ O (10 − ) (see e.g. [40]), with a mass for the dark photonaround the keV. For this case, one obtains for the string mass scale M s ∼ GeV, and R ∼ . − ǫ cannot be obtained through our simple large extra-dimension setup as theywill conflict with the current experimental limit on the string scale. For different values of the darkphoton mass, the constraints on ǫ are weaker, and consequently the extra-dimension and string massscales are set at higher energies. Taking the three values mentioned above we can have for ǫ ∼ − , ǫ ∼ − and ǫ ∼ − , respectively, a string mass scale and an inverse compactification radius oforder M s ∼ GeV and R ∼ GeV, M s ∼ GeV and R ∼ GeV, and finally M s ∼ GeV and R ∼ GeV. The intermediate scale ∼ GeV has diverse motivations [34, 41]. Italso corresponds to the energy where the SM quartic Higgs coupling vanish, thus a scale where newdegrees of freedom might be expected. Though we cannot proceed to the same string embedding aswe have done above, for kinetic mixing as small as ǫ ∼ − the WGC requires new physics aroundthe scale Λ ∼ −
10 MeV, that could then in turn be constrained by the Big Bang Nucleosynthesis.In our way to generate the large hierarchy between the two couplings g v and g d , we have assumedthe existence of small cycle with a size of order of the string scale inside six large compact dimensions.This is not the case in the simplest toroidal compactifications and requires some warping. Thus, theKK excitations of the dark photon are not expected in general to exhibit the same spectrum as inthe simplest case. However, assuming that the rough behaviour of the density of states goes with theenergy as E /M s , a sizable value of the effective coupling between SM states and the dark photonsis reached only at energies of the order of M s .In most phenomenological applications, the dark U (1) is massive. The WGC in [2] concernsmassless U (1) gauge bosons. For instance, in the case of a massive U (1), the charge is not conservedand there is no problem of remnants as charged black holes decay. However, one may argue that ifthe weak gravity states masses m W GC are much bigger than the dark photon mass m γ d , the massivecase is a (Higgs) phase of the same theory and remains in the landscape. Moreover, the comparisonof gravity and gauge forces should be done at energies of order m W GC and makes sense in the region m γ d ≪ m W GC . Finally, we have explicitly illustrated the WGC prediction for the UV cut-off of thetheory by a type IIB string scenario that we do not expect to break down because of an infraredHiggsing of the U (1). In fact, [22] have argued, through the explicit investigations of the propertiesof the WGC charge lattice, that the bounds used here on the mass, combination of charges ratios andultraviolet cut-off of the theory remain true.To conclude, we would like to stress that the main aim of this work is not to add to the plethoraof XENON1T analysis and interpretation [42–94], but to point out the amusing coincidence that theobservation of kinetic mixing between ordinary and dark photon would suggest new physics at scalesthat should be probed by a future collider. cknowledgments K.B. thanks Ignatios Antoniadis and Marco Cirelli for useful discussions. We acknowledge the supportof the Agence Nationale de Recherche under grant ANR-15-CE31-0002 “HiggsAutomator”.
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