Testing B-violating signatures from exotic instantons in future colliders

We discuss possible implications of exotic stringy instantons for baryon-violating signatures in future colliders. In particular, we discuss high-energy quark collisions and transitions. In principle, the process can be probed by high-luminosity electron-positron colliders. However, we find that an extremely high luminosity is needed in order to provide a (somewhat) stringent bound compared to the current data on NN → ππ,KK. On the other hand, (exotic) instanton-induced six-quark interactions can be tested in near future high-energy colliders beyond LHC, at energies around 20–100 TeV. The Super proton-proton Collider (SppC) would be capable of such measurement given the proposed energy level of 50–90 TeV. Comparison with other channels is made. In particular, we show the compatibility of our model with neutron-antineutron and NN → ππ,KK bounds.

In this paper, we will suggest that the exotic instantons can be tested in collider physics. As shown in [8] six quarks ∆B = 2 violating transitions can be generated by one exotic instanton in the context of a low scale string theory scenario with M S 10÷100 TeV. The model suggested in [8] is theoretically motivated by baryogenesis and the hierarchy problem of the Higgs mass, while embedded a theory of quantum gravity. As regards to baryogenesis, the tunneling probability of B − L violating processes can be enhanced in the thermal bath, as it happens for standard model electroweak sphalerons [14,15]. This implies a B − L first order phase transition which could explain the observed matter-antimatter asymmetry. On the other hand, a low string scale theory can alleviate the strong hierarchy problem of the Higgs mass of 17 orders, reducing the hierarchy to a small number (comparable to the Yukawa coupling of the electron). The possibility to test exotic instantons in collider may allow to the test low * andrea.addazi@infn.lngs.it † kxw198710@126.com ‡ khlopov@apc.in2p3.fr scale string theory in the fully non-perturbative regime. This may be insightfully important also to understand the issues from geometric moduli stabilization in string compactification. In fact, string instantons and string fluxes may generate non-perturbative effective potentials stabilizing the geometric moduli and allowing for a string vacuum safety.
In particular, we will show how exotic instantons can generate Λ −Λ transitions, and qq →qqqq qq →qqqq high-energy collisions. Electron-positron colliders with luminosity much higher than Belle and BaBar could be used to (indirectly) test such a scenario. Indeed, we find that an extremely high luminosity is needed, and even that look unrealizable in the near future. On the other hand, we argue how compared measured in neutronantineutron physics and in high energy colliders beyond LHC can provide tests for our model in the near future. In this sense, the high energy frontier is the preferred experimental direction of our model, with respect to the high luminosity one.
The letter is organized as following: In Sec. II we discuss our theoretical model, in Sec. III its phenomenology in electron-positron colliders, and in Sec. IV its phenomenology and parameter space in comparison with several other different possible channels before our conclusions.

II. THEORETICAL SIDE
B, L number conservations can be dynamically violated by non-perturbative quantum gravity effects known as exotic stringy instantons. Exotic stringy instantons arXiv:1705.03622v2 [hep-ph] 1 Jun 2017 are Euclidean D-branes (or E-branes), intersecting physical D-brane stacks. In our letter, we will consider IIA string-theories 1 . In this case, "exotic instantons" are E2-branes, wrapping different 3-cycles on a Calabi-Yau compactification with respect to physical D6-branes. A MSSM can be embedded in a quiver theory with three or more nodes. In the low energy limit, these quivers can produce gauge theories or eventually higher node extensions 2 . Let us remark that the basic fundamental elements are: i) D6-branes wrapping 3-cycles in the Calabi-Yau CY 3 ; ii) Ω-planes; iii) E2-instantons; iv) open strings 3 .
Open strings are attached to D6 and E2-branes. Let us also recall that open (un)oriented strings attached to two intersecting D6-brane stacks will reproduce (MS)SM matter fields as lower energy excitations, in the limit α s → 0. The number of intersections among the stacks will correspond to the number of generations. As an example, the three generations of lepton superfields L comes from open strings attached to U (2) or Sp(2) stack and a U (1) stack, with these stacks intersecting three times each other. The hypercharge U (1) Y is reconstructed as a massless linear combinations of the U (1)'s in the model. For example for where On the other hand, two linear independent combinations of U (1)'s orthogonal to (1) will be anomalous 4 . After this short introduction, let us consider, the presence of an E2-brane intersecting two times the U (3)-stack, two times the U (1)-stack, four time theÛ (1)-stack (image of U (1) with respect to Ω). For α s → 0, this construction generate , effective Lagrangian terms among ordinary superfields U c , D c and fermionic moduli (also called modulini): where τ i modulini live at U (3)-E2 intersections, α modulini at U (1) − E2 and β atÛ (1) − E2. Here, we consider 1 For classic papers on open string theories see [19][20][21][22][23]. In open string theories, calculations of scattering amplitudes are much simpler than F-theories or heterotic string theories. 2 See [16][17][18][25][26][27][28] for (incomplete) litterature on intersecting Dbranes models. 3 Before the orientifold projection, in order to restore the correct balance of arrows one has also to introduce flavor branes for some of these models. See [24] for a discussion of Unoriented quivers with Flavour branes. 4 In string theory, anomalous U (1)s can be cured through the generalized Green-Schwarz mechanism. The two anomalous vector bosons Z , Z get a mass of the order of the String scale, through Stückelberg mechanism. Peculiarly, they have to interact with Z, γ through generalized Chern-Simon (GCS) terms in order to have anomalies' cancellations. See [29,31] for an extensive discussion on these aspects.
an E2-instanton with a Chan-Paton factor O(1). As prescribed by instanton calculation, we will integrate out modulini at the D6-E2 intersections, and we will obtain where Y (1) f6 is the flavor matrix determined by the couplings k (1,2) derived from mixed disk amplitudes. We can assume these as free parameters, parametrizing our ignorance about the particular geometry of the E2-instanton considered. A Superpotential (3) can generate diagrams like the one in Fig.2. In particular, n −n and Λ −Λ transitions are generated by the two effective operators 112112 . More details and analysis of explicit quivers were extensively discussed in [8]. These operators correspond to the generation of an effective Majorana mass for the neutron and for the Λ baryon.
e −S E2 is the effective action of E2-brane wrapping 3cycles on the CY 3 . It is related to the string coupling as where V E2 is the volume wrapped by the E2-brane and the imaginary part consists of a sum all over Ramond-Ramond axions. For l > l S , Exotic instantons have to respect the universal string theory bound on nonperturbative effects: interpreted as a bound from instanton-antinstanton virtual pair diagram, i.e as a bound on their partition function. This bound is very important for the considerations in the following. In fact, g s << 1 will suppress E2-transitions. On the other hand, scenari in which g s = 0.25 ÷ 1 suggest a coupling strong as e −10 ÷ e −2 5×10 −5 ÷1.1×10 −1 , with a small volume V E2 This result is peculiarly different with respect to non-perturbative classical configurations in field theories, like sphalerons, usually suppressed as e −1/g 2 Y M . In particular, for electroweak gauge instantons, the suppression factor can be proportional to e −10 4 or so. A so high string scale as the one desired in our case has to be consistently compatible with Yang-Mills coupling and M P l /M S ratio. Let us remind that, by dimensional reduction to 4d, where V 3 is the volume wrapped by D6-branes (stacks) corresponding to YM bosons; V 6 is the total internal volume. The hierarchy among YM couplings and a high string coupling is understood as a volume suppression hierarchy. Let us note that generically the volume of three-cycles are not directly related to the total internal volume, even if in simple compactifications like isotropic toroidal ones they are related as V 3 ∼ √ V 6 . This allows more variability among String scale and Planck scale hierarchies.
On the other hand, the suppression factor e −S E2 can also be compensated by coefficients in k (1) f of mixed disk amplitudes. These coefficients can be higher than one and their combinations can give rise to an appreciable enhancement of Y (1) flavor components 5 .

A. Exotic instantons in high energy collisions
In this section, we will consider two quarks high energy collisions induced by exotic instantons. For s << Λ 2 , the scattering amplitude is just reduced to a contact six quark interaction with q c = u c , d c , ....t c RH quarks and Λ −3 M −3 S e −S E2 . A less suppressed channel is q c q c →q cqcqcqc . In the low energy limit, its amplitude is The corresponding quarks-squarks cross section can be evaluated by integrating the squared of the amplitude all over the 4-dimensional phase space dΦ 4 . We define the Mandelstam variables s 12 = (p 1 + p 2 ) 2 , s 34 = (p 3 + p 4 ) 2 , where p 1,2,3,4 are squark momenta (final states) and s = E 2 CM . One finds that where 5 However, let us mention that recent trends in non-perturbative string theory and string phenomenology suggest that for distances of l < l S 3-cycle volume factor can be collapsed on the CY 3 singularity [32,33]. In this regime, saddle point approximation cannot be trusted anymore so that our previous estimates are not more valid in this case. In this case an enhancement of exotic instanton processes is expected.
with C 4 a combinatorial factor depending on the number of equal particles in the final state, and dΩ the integral all over scattering angles . (13) The notationφ ij indicates variables evaluated with respect to the rest frame of q ij . P 2 is a complicated polynomia of s, s 12 , s 34 and subleading logaritimic functions of expression later: with m 1,2,3,4 squark masses in the final state. For s → Λ 2 , cross sections rapidly grows up.s = Λ 2 corresponds to the bound of the unitarity break-down. In other word, our effective cross-section would badly breaks the Froissart bound.
For s Λ 2 , the contact interaction approximation looses validity: scatterings are probing the fully non-perturbative regime: an exotic instanton is a fully non-perturbative configuration. Resonant production at the s Λ 2 corresponds to an infinite series of stringy amplitudes with six open-strings' insertions ∞ g=0 A g (z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ), where g is the genus (loops). This is a technical problem in the high energy limit common to all non-perturbative (euclidean) configurations, like QCD instantons, sphalerons and so on. However, let us note that in our model Λ = e +S E2 M S > M S . As a consequence, an infinite tower of stringy higher spins states are excited at Λ and they will unitarize the S-matrix. This is a generic conclusion coming from unitarization arguments of string theory amplitudes, also connecting to the CPT symmetry in string theory. We consider the problem in the fixed angle kinematical regime, which is also particularly suitable with the experimental set-up of colliders. We argue that fixed angle scattering amplitudes of (six) open strings are expected to exponentially fall down with energy as FIG. 1. We show the two quark four squarks cross section σ(y) (flavor independent part) with respect to the normalized s-variable y = s/Λ 2 and neglecting squark masses. For y < 1, σ ∼ y 2 ; for y 1 the amplitude will saturate the unitarity bound of string theory. For y > 1 and fixing the scattering angle, the amplitude falls off exponentially with the production of the non-perturbative configuration -as displayed in this figure. while N-loops amplitudes behaves as 6 [34] A N −loops In Fig.1, we show the qualitative universal part (flavor and combinatory independent) of two quark four squark cross section, assuming squark masses smaller than Λscale. For E CM Λ the process lies into the fully nonperturbative stringy regime. From the physical point of view, we expect that, with the growing of CM energy, the E2-brane starts to oscillate and its dynamics is described by oscillations of modulini fields. The number of modulini is a topological invariant of the exotic instanton solution, associated with the invariance of intersections with physical D-branes. At the non-perturbative scale, open strings associated to modulini have to reggeize. Loop-corrections to tree-level mixed-disk amplitudes at fixed angle are expected to add exponentially suppressed contributions in the form of Eq. (15). We conjecture that this is the main contribution to the unitarization of the scattering amplitude for fixed angle kinematical regime and s >> Λ 2 .
We will return to phenomenological implications in the next sections. 6 The result found by Gross-Mende in [34] is valid for four open string amplitude. Their result is expected to be qualitatively the same as for six points amplitudes in 2 → 4 fixed angle scatterings.

III. PHENOMENOLOGY OF Λ −Λ TRANSITIONS
Now let us discuss the implications of Λ − Λ oscillations on the phenomenological side 7 . So far that no baryon number violating process has been observed in the Standard Model (SM), thus any possible signal would be definitely exciting and efforts along this direction are deserved. New physics effects can be also probed at hadronhadron colliders. As discussed in Ref. [40], BES detector with huge J/ψ data sample has been proposed to measure the possible Λ − Λ oscillation by studying the quantum coherent ΛΛ states. Following the treatment of D 0 − D 0 mixing [41]. To avoid the complications due to mixing, the charged B was used as an example in Ref. [42]. The eigenstates emerging from the oscillations can be written as with the normalization condition |p 2 | + |q| 2 = 1. The subscript "H" ("L") means the heavy (light) mass eigenstates. Here p and q can be parametrized as √ 1 + z/ √ 2 and √ 1 − z/ √ 2, respectively, where z involves the oscillation mass δm ΛΛ appearing in the off-diagonal elements of the effective Hamiltonian used to describe the time evolution of the Λ − Λ system. For details, see Ref. [40] or the general consideration in [41]. One can define the following mass and width: 7 Of course, in the standard model analysis, these baryon-violating effects need not to be considered, e.g., see the CP violation of Λc → Λπ decay [35] and also the conventional NN scattering [36][37][38][39].
from which the mixing parameters are defined as For the free Λ case without external magnetic fields, ∆m = 2δm ΛΛ . For simplicity, we will confine ourselves to this case. The influence of external magnetic field is discussed in Ref. [40]. The dominant decay mode of Λ is Λ → pπ − with branching ratio 64% [43], and correspondingly Λ → pπ + assuming CP invariance. In the case that Λ−Λ oscillation happens, the process Λ → Λ → pπ − will be possible. This phenomenon can be probed by counting the event number N for J/ψ → ΛΛ → (pπ − )(pπ − ) -wrong-sign decay-while the right-sign decay would be J/ψ → ΛΛ → (pπ − )(pπ + ). Time-integrated decay rate for the wrong-sign decay relative to the right sign is found to be A discussion of time-dependent observables is also presented in [40], and the BES-III detector is capable to access this time information. Assuming y Λ = 0 -it is indeed a quite small quantity -one can get an estimate of the mixing mass parameter δm ΛΛ from Eq. (19) as According to the designed luminosity of BEPC-II in Beijing [44], 10 × 10 9 J/ψ and 3 × 10 9 ψ data samples will be collected by the runnings per year. If finally no signal of wrong-sign decay can be detected, one should put an upper limit R ≤ 4 × 10 −7 and correspondingly δm ΛΛ ≤ 10 −15 MeV at the 90% of confidence level (C.L.) [40], inferring from the knowledge of interval estimate for very rare signal. Currently, a sample of 1.31 × 10 9 J/ψ events has been collected [45], then the aforementioned upper limit will be increased by a factor of 10/1.31 ≈ 2.8, i.e. δm ΛΛ ≤ 3 × 10 −15 MeV. Consequently, the oscillation time will be bounded by the upper limit as 2.5 × 10 −7 s at 90% confidence level. This would be the first search in the experiment. See also [40] for more details on this analysis. Note that Υ(4S) also has the same quantum numbers as J/ψ, i.e., I G (J P C ) = 0 − (1 −− ), and can also decay to coherent ΛΛ states. Belle and BaBar detectors could be used to probe this process as well. Currently, there are 772×10 6 Υ(4S) data available in Belle [46] and 471×10 6 for BaBar [47]. Taking into account the fact that the non-BB decay mode only constitutes less than 4% (at 95% confidence level) of the total decay rate, much less ΛΛ events are expected. Otherwise assuming that most of Υ(4S) decays into ΛΛ, the event number can increase by one or two orders lager more than BES. Although the above upper limit for Λ −Λ oscillation could be accessed from the beginning in the experiment, we require an extremely higher luminosity to get a comparable bound to the n −n case. One can find this point from Eq. (20)the oscillation mass is proportional to the square root of the luminosity. n −n oscillation time is constrained to be 10 7 ÷10 8 smaller than the Λ−Λ oscillation time, and thus the luminosity should be increased by order 10 15 , which seems unrealizable in the near future. In other words, extremely huge data samples would be needed to pin down a more stringent bound on such new-physics (NP) signal. However, we will show that the required energy to generate such NP phenomenon (e.g., in exotic instanton model below) can be accessed in the near future. Recently, the Circular Electron Positron Collider (CEPC) has been proposed in China, which arouses great interests in the community [48]. According to the design agenda, the electron-positron collider will be converted into a proton-proton collider, with an unprecedented center-ofmass energy of 50-90 TeV at the second phase. This is the project of super proton-proton collider (SppC) [49]. At such a high energy, the new physics beyond LHC discussed in the present paper would be accessible. Future measurements are expected to give valuable hints on this research line.

IV. SPACE OF PARAMETERS, n −n AND PROTON-PROTON COLLIDERS
An operator O ΛΛ generates not only Λ −Λ transition but also N N → KK transition. As discussed in the previous section δm ΛΛ can be constrained up to 10 −6 eV by next generation of experiments. However, N N → KK is just constraining O ΛΛ up to δm ΛΛ < 10 −21 eV. δm ΛΛ is related to a New Physics scale by so that M ΛΛ 100 TeV scale. On the other hand, O nn is actually constrained by δm nn < 10 −23 eV, corresponding to M nn > 300 TeV [50,51]. The next generations of experiments will test 8 M nn 1000 TeV [52]. In our model, Such a large hierarchy range is naturally understood by couplings arising from mixed disk amplitudes. For example the number 10 −18 is understood as As a consequence, a 100 TeV-scale test of Λ −Λ is motivated from the theoretical side, independently from n ↔n limits. A 100 TeV scale scenario for Λ −Λ can be naturally obtained with e +S E2 1 (small 3-cycles wrapped by the E2-brane), M S M SU SY 100 TeV. In this case, a final test-bed for this model can be provided by future proton-proton 100 TeV colliders beyond LHC. In fact a uds →ūds transition will be directly tested, mediated by an exotic instanton in collision. However, a more intriguing scenario can be opened in the case of 10 TeV < M SUSY M S < 100 TeV. In this case, planned proton-proton colliders can test other flavor amplitudes like ud →csss,csbb that are less constrained by n −n or N N → ππ, KK processes. In particular, the predicted experimental processes for a high energy proton-proton collider are pp → 4q. This leads to several different channels. This is the case of pp → 4q + 4χ 0 leading to four jets and missing transverse energy pp → 4j + E M.T or pp → 3j +t+E M.T (with standard top decays) and so on. Other interesting channels are coming from stops productions and successive decays liket As discussed in subsection A, cross sections of these processes have a peculiar behavior not common to gauge models because of their exponential decrease up to Λ for s >> Λ 2 .

V. CONCLUSIONS
In this paper, we have discussed phenomenological implications of a new class of instantons known as exotic instantons. They can generate ∆B = 2 violating transitions as n ↔n, Λ ↔Λ and ∆B = 2 high energy collisions like qq →qqqq,qqqq -in hypercharge preserving combinations. We have explored the possibility to detect exotic instantons in future colliders, in comparison with present low energy limit channels like n ↔n, N N → ππ, KK.
We summarize our main conclusions as follow: i) contrary to other non-perturbative solutions like electroweak gauge instantons, exotic instantons can induce effective operator with a high coupling. As a consequence, their effects can be seen in low energy observables as well as in high energy colliders.
ii) A neutron-antineutron transition can be generated by exotic instantons and it can provide an indirect testbed for a class of models mentioned in this paper 9 . iii) Our model predicts Λ−Λ transitions. The possibility to test these after in future high luminosity electronpositron colliders seems very far from our present technological possibility, if compared to the actual related limits from N N → KK transitions. iv) ∆B = 2 exotic instantons can be reached in the next generation of high energy proton-proton colliders beyond LHC, well compatible with neutron-antineutron limits, N N → ππ, KK and so on. We stressed how crosssection running with Center of Mass energy cannot be reproduced by any quantum field theory model. In fact an exponential softening of the cross-section cannot be reproduced from any other UV completion of the six quark effective operator in context quantum field theories. This is a feature distinguishing our string theory model from other quantum field theory ones.
v) Cosmological impact of exotic instantons revealed in [8,10] provides additional constraints on their parameters. Differently from the case of electroweak gauge instantons, due to the enhancement of the effect of exotic instantons in high energy collisions, a direct quantitative relationship between cosmological and physical consequences of the model considered is possible.
The class of models suggested here strongly motivates two directions for future experimental physics: neutronantineutron experiments and high-energy colliders beyond LHC. Eventually, a detection of exotic instantonmediated processes can motivate the construction of technologically challenging high luminosity collider in order to detect a Λ −Λ transition or new rare physics experiments searching for N N → KK. These measurements could constrain the exotic instantons' geometry and their intersections (with ordinary D-branes) and their wrapped 3-cycles on the CY 3 . Future beyond LHC (such as the proposed CEPC+SppC) might render us new exciting surprises in higher-energy and higher-luminosity frontiers.

Acknowledgments
We would like to thank M. Bianchi In our paper, we have considered an instanton with a rigid O(1) symmetry in context of IIA superstring theory. This instanton has the universal zero mode structure dx 4 d 2 θ which yields a new term to the holomorphic F-terms in the effective supergravity action. The computation of these effects can be done from the Conformal Field Theory prospective. In particular, one can compute the supepotential of an M-point correlator in a string instantonic background Φ a1b1 ...Φ a M b M E -where B denoted the instantonic background and Φ are generic physical fields in the bi-fundamental representations of two generic N and M stacks of D6-branes The correlator is related to the effective supergravity quantities in the action as follows [64,65]: where K is the non-holomorphic Khäler potential, K is the Khäler metric and Y is the holomorphic superpotential coupling. The following generic formula for the computation of the correlation function in semiclassical approximation is [64,65]: , whereΦ a k b k is the chain-product of all the the vertex operators at fixed value of k Φ a k b k = Φ a k x k,1 · Φ x k,1 x k,2 · ... · Φ x k,n−1 ,x k,n · Φ x k,n ,b k while Φ a1b1 [x 1 ] λa 1 ,λ b 1 a CFT disk correlator forΦ and for the charged zero modes λ a1 ,λ b1 inserted in the boundary of the mixed disk amplitudes. C ED * b , C EO * denote the sum all over the annulus diagrams with and without one cross-cap.
From Eqs.(22)-(23), the holomorphic superpotential coupling can be entirely re-expressed in terms of holomorphic couplings in the CFT amplitudes: where S 0 corresponds to the vacuum disk amplitude for the E2-brane: with V E2 is volume of the 3-cycles wrapped by the E2-instanton. This formula generalizes the prescription given in Section II, in the particular case of brane intersections described above. In particular, the instanton can reproduce the six quark operator if and only if with the number of branes intersections considered above.