Super-light baryphotons, Weak Gravity Conjecture and Exotic instantons in Neutron-Antineutron transitions

In companion papers \cite{Addazi:2015pia,Addazi:2016rgo}, we have discussed current bounds of a new super-light baryo-photon, associated to a $U(1)_{B-L}$ gauged, from neutron-antineutron current data, which are competitive with E\"otv\"os type experiments. Here, we discuss the implications of a possible baryo-photon detection in string theory and quantum gravity. The discovery of a very light gauge boson should imply the violation of the Weak Gravity Conjecture, carrying deep consequences in our understanding of holography, quantum gravity and black holes. On the other hand, we show how the detection of a baryo-photon would also exclude the generation of all $B{-}L$ violating operators from Exotic Stringy Instantons. We will disclaim the common statement in literature that neutron-antineutron may indirectly test at least a $300-1000\, \rm TeV$ scale. Searches of baryo-photons can provide indirect informations of the Planck (or String) scale (quantum black holes, holography and non-perturbative stringy effects). This strongly motivates new neutron-antineutron experiments with adjustable magnetic fields dedicated to the detection of super-light baryo-photons.


I. INTRODUCTION
As it is known, B and L are accidental symmetries of the Standard Model. Their conservation is in agreement with all current data. Some symmetry principles could be behind B, L accidental conservations. The simplest idea is to recover B, L number conservations as residual discrete symmetries of spontaneously broken global U (1) L , U (1) B , or a linear combination of the two (as U (1) B−L or U (1) B+L and so on). This class of models predicts the existence of new pseudo-goldstone bosons known in literature as Majorons [3,4] 1 .
An alternative way is to gauge B, L symmetries. However, as it is well known, U (1) B and U (1) L gauged are anomalous. The only way-out from anomalies is to consider a U (1) ζ(B−L) gauged, where ζ is an arbitrary constant which can be redefined in particle charges, i.e. U (1) B−L for convention. In particular, U (1) B−L gauge group may be spontaneously broken by a new Higgs singlet field (Higgs mechanism) or a Stueckelberg gauged axion (Stueckelberg mechanism). Usually, U (1) B−L is thought as a spontaneously broken gauge group at high scales, i.e. a new Z boson possibly testable at LHC or future colliders. On the other hand, from the point of view of quantum field theory consistency, a gauge U (1) B−L could also be massless. But certainly, this would be not phenomenologically healthy: it would be in contradiction with baryogenesis which necessary requests a violation of B − L 2 . If we desired to break B − L at a intermediate or high scale while a semi-massless gauge boson, we would introduce a very weakly coupled B − L boson with M 2 b ∼ α B−L v 2 B−L and α B−L << 1. Such a * 3209728351@qq.com 1 Majorons can also provide a good candidate of (warm) dark matter [5]. See also Refs. [6][7][8][9][10][11] For instance, this is not the case of U (1) B or U (1) L gauged which would be corrected by quadratically divergent contributions and they should be enormously fine-tuned from their mass scale to the Planck scale. However, the new Higgs field χ introduced to spontaneously break U (1) B−L can mix with the ordinary Higgs field as χ † χHH and this could introduce a new hierarchy problem. This is connected with the old hierarchy problem of the Higgs mass, which presently remains still unsolved. On the other hand, not all possible allowed gauge interactions in quantum field theories decoupled by gravity arXiv:1708.02956v1 [hep-ph] 9 Aug 2017 FIG. 2. a) The mixed disk amplitude coupling the physical RH (super)quark U with two instantonic zero modes τ and α. In (b) the Mixed disk amplitude dual picture in terms of intersecting D-branes. a) The mixed disk amplitude coupling the physical RH (super)quark D with two instantonic zero modes τ and β. In (d) the Mixed disk amplitude dual picture in terms of intersecting D-branes. seem to be compatible with quantum gravity. For instance, the Weak gravity conjecture (WGC) states that the weakest interaction is gravity and it excludes the presence of new very light U (1) bosons like U (1) B−L coupled to ordinary matter. This means that for each interactions, it must exist a particle satisfying where m, q are mass and U(1)-charge of the particle respectively [12]. At the present status, WGC is only based on heuristic arguments sustained by holography and absence of global symmetries in quantum gravity and string theory. In particular, L. Susskind suggested that, according to holography, Black Hole remnants should be impossible [13]. The WGC argument is the following: let us consider an hypothetical interaction of a particle with mass m and α < 1, whereα = α Y M /G N m 2 . In this case a black hole can have a charge from 0 toQ =α −1 (for examplẽ α ∼ 10 −10 , i.e.Q ∼ 10 10 ) and these charges cannot be radiated away as Hawking's radiation. This should imply a final remnant extremal BH with M = QM P l and Q in range (0,Q], contradicting Susskind's arguments. This seems to lead to the conclusion that WGC is sustained from the holographic principle.
One could think that a heuristic argument may be not enough satisfiable and the conjecture should be tested with high precision. To test WGC should be crucially important for our understanding of quantum gravity, holography and black holes. For instance, a violation of WGC would imply that some fundamental aspect in our understanding of black holes and quantum gravity is still missing. In particular, it is commonly retained that holography is a crucial feature of black hole physics and a violation of WGC could lead to revisit such a concept itself.
However, the detection of super-light baryo-photon can lead to rethink semiclassical non-perturbative solutions in string theory. In particular, exotic D-brane instantons can generate B−L violating operators and their implications in particle physics were recently discussed in our papers [24][25][26][27][28][29][30][31][32][33][34]. As is known, B−L violating exotic instantons have necessary to be synchronized with a Stueckelberg mechanism for U (1) B−L , sending the B − L gauge boson mass to a large scale. So that, a super-light baryophoton is in tension with exotic instanton effects, which should be suppressed by non-perturbative stringy corrections beyond the semiclassical approximation. So that, the detection of a baryo-photon implies a prohibition of exotic instanton effects from the spontaneous symmetry breaking scale v B−L up to the String scale! In this letter, we suggest to test both the weak gravity conjecture and non-perturbative stringy effects from neutron-antineutron oscillations data. The neutronantineutron transition was not observed and last limits were obtained in by Baldo-Coelin et al. [14]. From these data, Z. Berezhiani, Y. Kamyshkov and the author of this paper have recently discussed limits to a new bary-photons coupled to the (anti)neutron from neutron-antineutron experiments [1,2]. The possibility to improve current neutron-antineutron limits was discussed in Ref. [15]. However, authors of Ref. [15] 3 have emphasized aspects of neutron-antineutron experiments as a test for the effective ∆B = 2 Majorana mass operator (udd) 2 /M 5 , in order to indirectly test M 1000 TeV scale. So that, it was suggested to search n −n transitions with very suppressed external magnetic field (B < 10 −4 Gauss). But according to our papers [1,2], a neutron-antineutron transition should be suppressed by the presence of an external baryo-photon background field. For example, for a baryo-photon background field with scale 10 −11 ÷10 −13 eV on the Earth surface, neutron-antineutron transitions would be enhanced in strong magnetic field conditions B ∼ 1 ÷ 10 Gauss rather than suppressed ones. In this letter, we suggest that the search of a baryo-photon can provide a test for the Planck (and String) scale physics, even if M P l , M s >> 1000 TeV. In fact, according to our considerations above, the detection of a baryo-photon in neutron-antineutron should violate Weak Gravity Conjecture as well as should be a test for exotic D-brane in-stantons. In other words, a detection of a baryo-photon would lead to re-discuss the same basic principles of quantum gravity and string theory, such as holography, stringy instantons, black hole remnants and so on. In this sense, searches for bary-photons in neutron-antineutron experiments can indirectly test quantum gravity.

II. BARYO-PHOTON
The Baryo-photon model is based on Standard Model gauge group extension with an extra B − L gauge symmetry: The B − L baryo-photon gauge coupling with neutron, proton and lepton currents is where b µ is the baryo-photon associated with U (1) B−L .
With an exact U (1) B−L , the neutron-antineutron transition is forbidden, otherwise a gauge symmetry is unlikely violated. So that, U (1) B−L has to be spontaneously broken and this can be synchronized with the generation of a effective Majorana mass for the neutron. For example, one can introduce effective operators like χ M 6 (udd)(udd), where χ is a Higgs scalar field with charge Q B−L = −2 and getting a vacuum expectation value χ = v χ . In this case, a n −n transition is generated with an effective suppression scale M nn = (M 6 /v χ ) 1/5 and consequently a Majorana mass term δm nn = Λ 6 QCD /M nn . An example of UV completion of such an operator was suggested in Refs. [2,16] as a see-saw mechanism for the neutron. Alternatively, it is also possible that the generation of the effective Majorana mass term for the neutron is totally disconnected by the spontaneous symmetry breaking mechanism and it happens after the spontaneous breaking. Then, in full generality, one can also consider the more complicated case in which U (1) B−L is spontaneously broken by a combination of scalars χ, where Q i , Q χ are B-L charges of the scalars while M b is the mass of the baryophoton. As a consequence, the baryophoton mediates a spin-indpendent force among SM particles with baryon and lepton charges: where V G n is the gravitational potential, q A = Q A m n /(M A ) andα = α B−L /α G and α G = G N m 2 n . If aα << 1 gauge boson was detected, WGC would be violated.
Yukawa radius larger than Earth's radius λ > R Earth → α B−L < 10 −49 ,α < 1.7 × 10 −11 (7) The Earth induces a gravitational energy for the neutron at its radius V E Earth = −Gm n M Earth /R Earth 0.66 eV, while the Sun V G Sun = −Gm n M Sun /AU 10 eV, while the Galaxy V G Galaxy 1 keV. The total energy potential contribution from baryo-photon on a (anti)neutron in laboratory frame is The effective interaction enters in the oscillation probability as where ∆ ± = V ∓ Ω B , Ω B = |µ n · B| 6 · 10 −12 (B/1G)eV (Zeeman energy shift induced by the external magnetic field), ± corresponds to two polarizations states, δm nn is the effective Majorana mass term. In Fig. 3, we report various exclusion plots for (λ, α B−L ) parameter space compared with Eötvös-like experiments. As one can see, for λ > 10 9 cm, which is comparable with the Earth radius, for v B−L > 1 GeV the parameter space is very constrained. On the other hand, v B−L < 1 meV is not possible in a minimal model: it would imply a spontaneous breaking of U (1) B−L only in very late Universe (1 ÷ 10 Gyrs) which is clearly excluded by baryogenesis. However, we suggest that this scenario could have a subtle way-out: it is possible that B −L was broken in the early Universe because of thermal bath induced expectation values to one (or more) scalar Higgs, allowing Baryogenesis, and restored later. For instance, our idea is inspired by various old models of high temperature symmetry breaking suggested in Refs. [17][18][19][20][21][22]. An interesting possibility could be a phase transition mechanism from a electroweak conserving and B − L broken vacua (G SM ) to an electroweak breaking and B − L preserving vacua G = (SU (3) c × U (1) em × U (1) B−L ). In this case, CP-violating scatterings of primordial plasma to expanding Bubbles associated to the broken-restored phase G can generate a Baryon-asymmetry as in standard electroweak baryogenesis (See [23] for a review).
Among the landscape of parameters, we would like to point out the attention on v B−L 1 keV allowing for λ 10 16 cm a ∆V = |V n − Vn| 10 −11 eV which would correspond to a magnetic field of 5 Gauss (10 times the Earth magnetic field or so) coupled to the neutron magnetic moment. As a consequence, a so strong background would completely suppress a n −n transition searched in condition of |B| < 10 −4 Gauss as suggested in Ref. [15]. On the contrary in this case a neutronantineutron transition should be searched in resonant condition |µ n · B| ∆V . Roughly speaking, neutronantineutron experiments seriously risk to not detect any new physics with the wrong magnetic field set-up.

III. EXOTIC INSTANTONS
The possible detection of a so light bary-photons would also rule-out B-L violating exotic instantons. In Refs. [31], we have shown how the intersection of E2-branes, wrapping different 3-cycles on CY 3 , with D6-brane stacks can generate new non-perturbative neutron-antineutron operators. For instance, the effective lagrangian is where τ, α, β are chiral fermionic zero modes (or modulini) associated to the Exotic instanton, while U, D are RH up and down quarks. In Fig.2, we report the mixed disk amplitudes generating the effective lagrangian Eq.(10) from string theory. Integrating over the modulini space, where Y is a 3 × 6 flavor matrix, combination of c (1), (2) couplings. The same lagrangian Eq.(10) can be considered with a one-half reduce number of modulini families, providing a trilinear ∆B = 1 term This operator can generate a Neutron-Antineutron transition mediated by a gluino exchange connecting quarksquark reduction currents. There are several different exotic instanton solution which cannot preserve U (1) B−L even if not directly connected to n −n transitions. For example exotic instantons with an effective lagrangian that integrating on the modulini space generates a Majorana mass matrix for the RH neutrino As is well known, such an operator can generate a Majorana mass for the LH neutrino from a see-saw type I mechanism. Alternatively, a Weinberg superpotential W = e −S E2 HLHL/M S can be directly generated by However, the generation of these superpotential is incompatible with a B-L light baryphotons. In fact, the generation of n −n is necessary synchronized with a Stueckelberg mechanism of U (1) B−L . In fact, all the e −S E2 factors have a structure e −S E2 = e −VΠ/gs+i r crar (16) where V Π is the volume of Π-cycles wrapped by a E-brane on the internal CY ; a r are RR axions and c r are E-brane couplings to them, g s is the string-coupling constant associated to the vacuum expectation value of the dilaton field (g s = e φ ). The Eq. (17) is not invariant under RR axion shifts, i.e. under U (1) B−L in our case: where I is the umber of intersection among the E-brane M and the background D-brane M , N A is the number of A D-brane stacks and Λ is an axion shift constant. 4 . and as is known this is exactly compensated by the shift factor of the superpotential combinations. The shift is associated to a Stueckelberg mechanism for B-L. As a consequence, the associated B-L boson gets a huge mass, typically of the order of the string scale or so. So that, we can argue that the observation of a very light baryo-photon would have strong implications for string phenomenology. In fact, this could imply that a non-perturbative protection mechanism would suppress all possible B-L exotic instantons for many orders magnitude from the string scale to the low scale of B-L spontaneous symmetry breaking. For instance, effects of RR and NS-NS fluxes wrapped by Euclidean D-branes could strongly suppress the mixed disk amplitudes associated to exotic instantons.

IV. CONCLUSIONS AND REMARKS
In this letter, we discussed possible implications of the detection of a super-light bary-photon coupled to (anti)neutrons in quantum gravity and string theory. Current available measures of n −n experiments impose unexpectedly stringent bounds to the baryo-photon mass and coupling constant. We have discussed how the detection of a super-weak baryo-photon may rule out the Weak Gravity Conjecture as well as the generation of non-perturbative (B−L)-violating operators from exotic D-brane instantons. It is commonly retained that neutron-antineutron experiments would indirectly test at least 1000 TeV scale physics in next generation of experiments [15]. However, we want to disclaim such a statement. In fact, following our arguments, neutronantineutron experiments could indirectly test the Weak Gravity Conjecture with very high precision.
We also have stressed how the detection of the superweak baryo-photon may rule out the presence of B − L violating Exotic Stringy Instantons up to the String scale! In fact, Exotic Instantons must necessary be associated with a Stueckelberg mechanism, providing a large mass to the B − L gauge field. In other words, a so light baryo-photon should be sequestered by Exotic Instantons, generating, for example, a mass term for the neutrino, or other R-parity violating operators.