Testing B-violating signatures from Exotic Instantons in future colliders

TL;DR

This paper explores how exotic stringy instantons could lead to baryon number violation detectable at future high-energy colliders, with potential signals in six-quark interactions and Lambda-antiLambda transitions.

Contribution

It proposes a novel connection between exotic instantons and baryon-violating signatures, analyzing their detectability at colliders beyond the LHC, especially at energies around 20-100 TeV.

Findings

Lambda-antiLambda transitions require extremely high luminosity to detect.

Exotic instanton-induced six-quark interactions could be tested at future colliders like SppC.

Model is compatible with existing neutron-antineutron and meson decay bounds.

Abstract

We discuss possible implications of Exotic Stringy instantons for Baryon-violating signatures in future colliders. In particular, we discuss high-energy quarks collisions and $\Lambda-\bar{\Lambda}$ transitions. In principle, $\Lambda-\bar{\Lambda}$ 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 $NN\rightarrow \pi\pi,KK$. On the other hand, (exotic) instanton-induced six quark interactions can be tested in near future high-energy colliders beyond LHC, at energies around $20-100\, \rm TeV$. Super proton-proton collider (SppC) is 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 $NN\rightarrow \pi\pi,KK$ bounds.

Figures

Equations

$$Y=q_a{Q}_a+q_b{Q}_b+q_c{Q}_c,$$

$${\mathcal{L}}_{eff}\sim k_f^{(1)}{U}_f^i{\tau }_i\alpha +k_f^{(2)}{D}_f^i{\tau }_i\beta ,$$

$${\mathcal{W}}_{E2-D6-\widehat{D6}}$$

$${\mathcal{O}}_{n\bar{n}}+{\mathcal{O}}_{\Lambda \bar{\Lambda }}$$

$$e^{-S_{E2}}=e^{-{\mathcal{V}}_{E2}/g_s+i{\sum }_r{q}_r{a}_r},$$

$$|e^{-S_{E2}}|\le e^{-\frac{2}{g_s}}$$

$$\frac{1}{g_{YM}^2}=\frac{M_S^3{\mathcal{V}}_3}{(2\pi {)}^4{g}_s},$$

$$M_{Pl}^2=\frac{M_S^8{\mathcal{V}}_6}{(2\pi {)}^7{g}_s^2},$$

$$\mathcal{A}(q_{f_1}^c{q}_{f_2}^c\to {\bar{q}}_{f_3}^c{\bar{q}}_{f_4}^c{\bar{q}}_{f_5}^c{\bar{q}}_{f_6}^c)\simeq {\mathcal{Y}}_{f_1{f}_2{f}_3{f}_4{f}_5{f}_6}^{(1)}\frac{{\Lambda }^{-3}}{M_{SUSY}^2}$$

$$\mathcal{A}(q_{f_1}^c{q}_{f_2}^c\to {\overline{\stackrel{~}{q}}}_{f_3}^c{\overline{\stackrel{~}{q}}}_{f_4}^c{\overline{\stackrel{~}{q}}}_{f_5}^c{\overline{\stackrel{~}{q}}}_{f_6}^c)\simeq {\mathcal{Y}}_{f_1{f}_2{f}_3{f}_4{f}_5{f}_6}^{(1)}{\Lambda }^{-3}$$

$$d{\sigma }_{q_{f_1}^c{q}_{f_2}^c\to {\overline{\stackrel{~}{q}}}_{f_3}^c{\overline{\stackrel{~}{q}}}_{f_4}^c{\overline{\stackrel{~}{q}}}_{f_5}^c{\overline{\stackrel{~}{q}}}_{f_6}^c}=d\hat{\Omega }\frac{\mathcal{C}}{4{\Lambda }^6}{\mathcal{P}}_2(s,s_{12},s_{34}),$$

$$\mathcal{C}=\frac{C_4}{(8{\pi }^2{)}^3}Tr\left\{{\mathcal{Y}}^{(1{)}^{†}}{\mathcal{Y}}^{(1)}\right\},$$

$$d\hat{\Omega }=d\cos \theta \frac{d\varphi }{2\pi }d\cos {\hat{\theta }}_{12}\frac{d{\hat{\varphi }}_{12}}{2\pi }d\cos {\hat{\theta }}_{34}\frac{d{\hat{\varphi }}_{34}}{2\pi }.$$

$${\mathcal{P}}_2(x,y,z)=-{\zeta }_1(y){\zeta }_1(z)[-4y\log {\zeta }_2^{+}(x,y,z)$$

$$+x\log {\zeta }_2^{-}(x,y,z)+y^2+{\zeta }_3(x,y,z)]$$

$${\zeta }_1(x)=\sqrt{1-\frac{2(m_1+m_2)}{x}+\frac{(m_1-m_2{)}^2}{x^2}}$$

$${\zeta }_2^{\pm }(x,y,z)=-\frac{x^2-2x(x+y)+(y-z{)}^2-x\pm y\mp z}{{\Lambda }^2}$$

$${\zeta }_3(x,y,z)=\frac{1}{3}\left[x^2+5x(y+z)-\frac{2(y-z{)}^2}{x}\sqrt{{\zeta }_2^{+}+x-y+z}\right]$$

$${\mathcal{A}}_{s>M_S^2}^{tree-level}={\mathcal{A}}^{QFT}{e}^{-{\alpha }^{\prime }s\log {\alpha }^{\prime }s+...}$$

$${\mathcal{A}}_{s>M_S^2}^{N-loops}={\mathcal{A}}^{QFT}{e}^{-\frac{1}{N}{\alpha }^{\prime }s\log {\alpha }^{\prime }s+...}.$$

$$|{\Lambda }_H⟩=p|\Lambda ⟩+q|\overline{\Lambda }⟩,$$

$$|{\Lambda }_L⟩=p|\Lambda ⟩-q|\overline{\Lambda }⟩,$$

$$m\equiv \frac{m_H+m_L}{2},\Delta m\equiv m_H-m_L$$

$$\Gamma \equiv \frac{{\Gamma }_H+{\Gamma }_L}{2},\Delta \Gamma \equiv {\Gamma }_H-{\Gamma }_L,$$

$$x_{\Lambda }\equiv \frac{\Delta m}{\Gamma },y_{\Lambda }\equiv \frac{\Delta \Gamma }{2\Gamma }.$$

$$\mathcal{R}=\frac{\mathcal{N}\Big{(}J/\psi\rightarrow\Lambda\overline{\Lambda}\rightarrow(p\pi^{-})(p\pi^{-})\Big{)}}{\mathcal{N}\Big{(}J/\psi\rightarrow\Lambda\overline{\Lambda}\rightarrow(p\pi^{-})(\overline{p}\pi^{+})\Big{)}}=\frac{x_{\Lambda}^{2}+y_{\Lambda}^{2}}{2}.$$

$$\delta m_{\Lambda \overline{\Lambda }}=\frac{1}{\sqrt{2}}\sqrt{\mathcal{R}}\Gamma .$$

$$\delta m_{\Lambda \bar{\Lambda }}\simeq y_2\frac{{\Lambda }_{QCD}^6}{{\mathcal{M}}_{\Lambda \bar{\Lambda }}^5},$$

$$\frac{{\mathcal{M}}_{\Lambda \bar{\Lambda }}}{{\mathcal{M}}_{n\bar{n}}}=\frac{y_2}{y_1}=10^{-18}÷10^{18}.$$

$$⟨{\Phi }_{a_1{b}_1}...{\Phi }_{a_N{b}_N}{⟩}_{\mathcal{B}}=\frac{e^{\mathcal{K}/2}{Y}_{{\Phi }_{a_1{b}_1},...,{\Phi }_{a_M{b}_M}}}{\sqrt{K_{a_1{b}_1,...,a_M{b}_M}}},$$

$$⟨{\Phi }_{a_1{b}_1},...,{\Phi }_{a_M{b}_M}{⟩}_{\mathcal{B}}\simeq \int d^{4}x{d}^2\theta \sum _{conf}\prod _a(\prod _{i=1}^{I^{+}}d{\lambda }_a^i)(\prod _{i=1}^{I^{-}}d{\bar{\lambda }}_a^i)$$

Full text

CEPC, SppC; Exotic Instanton; Baryon number violation

I Introduction

In recent companion papers, implications of exotic stringy instantons in $B-L$ violating rare processes and baryogenesis were explored Addazi:2014ila ; Addazi:2015ata ; Addazi:2015rwa ; Addazi:2015hka ; Addazi:2015eca ; Addazi:2015fua ; Addazi:2015oba ; Addazi:2015goa ; Addazi:2015ewa ; Addazi:2015yna ; Addazi:2016xuh ; Addazi:2016mtn

As known, in open string theories, instantons are (Euclidean) D-branes wrapping n-cycles on the Calabi-Yau (CY) manifold. These solutions were calculated and classified in the literature (see Bianchi:2009ij for a complete review on these subjects).

In this paper, we will suggest that the exotic instantons can be tested in collider physics. As shown in Addazi:2015goa six quarks $\Delta B=2$ violating transitions can be generated by one exotic instanton in the context of a low scale string theory scenario with $M_{S}\simeq 10\div 100\,\rm TeV$. The model suggested in Addazi:2015goa 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 Kuzmin:1985mm ; Kuzmin:1987wn . 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 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 $\Lambda-\bar{\Lambda}$ transitions, and $qq\rightarrow\bar{q}\bar{q}\bar{q}\bar{q}$ $qq\rightarrow\bar{\tilde{q}}\bar{\tilde{q}}\bar{\tilde{q}}\bar{\tilde{q}}$ 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 neutron-antineutron 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 are Euclidean D-branes (or E-branes), intersecting physical D-brane stacks. In our letter, we will consider IIA string-theories 111For classic papers on open string theories see Sagnotti0 ; Sagnotti1 ; Pradisi ; Sagnotti4 ; Sagnotti5 . In open string theories, calculations of scattering amplitudes are much simpler than F-theories or heterotic string theories. . 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 $U(3)_{a}\times U(2)_{b}\times U(1)_{c}$ or $U(3)_{a}\times Sp(2)_{b}\times U(1)_{c}\times U^{\prime}(1)_{s}$ or $U(3)_{a}\times U(2)_{b}\times U(1)_{c}\times U^{\prime}(1)_{d}$ or eventually higher node extensions 222See E1 ; E4 ; E7 ; DB2 ; DB10 ; DB19 ; DB25 for (incomplete) litterature on intersecting D-branes models. . Let us remark that the basic fundamental elements are: i) $D6$-branes wrapping 3-cycles in the Calabi-Yau $CY_{3}$; ii) $\Omega$-planes; iii) $E2$-instantons; iv) open strings 333Before 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 Bianchi:2013gka for a discussion of Unoriented quivers with Flavour branes. .

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 $\alpha_{s}^{\prime}\rightarrow 0$. 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 $U(3)_{a}\times U(2)_{b}\times U(1)_{c}$,

[TABLE]

where $Q_{a,b,c}$ corresponds to $U(1)_{a}\subset U(3)_{a}$ and $U(1)_{b}\subset U(2)_{b}$ and $U(1)_{c}$ respectively. On the other hand, two linear independent combinations of $U(1)$’s orthogonal to (1) will be anomalous 444In string theory, anomalous $U(1)$s can be cured through the generalized Green-Schwarz mechanism. The two anomalous vector bosons $Z^{\prime},Z^{\prime\prime}$ get a mass of the order of the String scale, through Stückelberg mechanism. Peculiarly, they have to interact with $Z,\gamma$ through generalized Chern- Simon (GCS) terms in order to have anomalies’ cancellations. See Stuck1 ; Stuck11 for an extensive discussion on these aspects. . 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 the $\hat{U}(1)$-stack (image of $U(1)$ with respect to $\Omega$). For $\alpha_{s}^{\prime}\rightarrow 0$, this construction generate , effective Lagrangian terms among ordinary superfields $U^{c},D^{c}$ and fermionic moduli (also called modulini):

[TABLE]

where $\tau^{i}$ modulini live at $U(3)$-$E2$ intersections, $\alpha$ modulini at $U(1)-E2$ and $\beta$ at $\hat{U}(1)-E2$. Here, we consider 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

[TABLE]

where $\mathcal{Y}^{(1)}_{f_{1}f_{2}f_{3}f_{4}f_{5}f_{6}}=k^{(1)}_{f_{1}}k^{(1)}_{f_{2}}k^{(2)}_{f_{3}}k^{(2)}_{f_{4}}k^{(2)}_{f_{5}}k^{(2)}_{f_{6}}$ 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 (II) can generate diagrams like the one in Fig.2. In particular, $n-\bar{n}$ and $\Lambda-\bar{\Lambda}$ transitions are generated by the two effective operators

[TABLE]

where $\mathcal{M}_{E2}^{3}=e^{+S_{E2}}M_{S}^{3}$, $y_{1}=\mathcal{Y}^{(1)}_{111111}$ and $y_{2}=\mathcal{Y}^{(1)}_{112112}$. More details and analysis of explicit quivers were extensively discussed in Addazi:2015goa . These operators correspond to the generation of an effective Majorana mass for the neutron and for the $\Lambda$ baryon.

$e^{-S_{E2}}$ is the effective action of $E2$-brane wrapping 3-cycles on the $CY_{3}$. It is related to the string coupling as

[TABLE]

where $\mathcal{V}_{E2}$ 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 non-perturbative effects:

[TABLE]

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 $g_{s}=0.25\div 1$ suggest a coupling strong as $e^{-10}\div e^{-2}\simeq 5\times 10^{-5}\div 1.1\times 10^{-1}$, with a small volume $\mathcal{V}_{E2}$ This result is peculiarly different with respect to non-perturbative classical configurations in field theories, like sphalerons, usually suppressed as $e^{-1/g_{YM}^{2}}$. 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 $M_{Pl}/M_{S}$ ratio. Let us remind that, by dimensional reduction to 4d,

[TABLE]

[TABLE]

where $\mathcal{V}_{3}$ is the volume wrapped by $D6$-branes (stacks) corresponding to YM bosons; $\mathcal{V}_{6}$ 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 $\mathcal{V}_{3}\sim\sqrt{\mathcal{V}_{6}}$. 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 $\mathcal{Y}^{(1)}$ flavor components 555However, 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 collapse1 ; collapse2 . 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..

II.1 Exotic instantons in high energy collisions

In this section, we will consider two quarks high energy collisions induced by exotic instantons. For $s<<\Lambda^{2}$, the scattering amplitude is just reduced to a contact six quark interaction

[TABLE]

with $q^{c}=u^{c},d^{c},....t^{c}$ RH quarks and $\Lambda^{-3}\simeq M_{S}^{-3}e^{-S_{E2}}$. A less suppressed channel is $q^{c}q^{c}\rightarrow\bar{\tilde{q}}^{c}\bar{\tilde{q}}^{c}\bar{\tilde{q}}^{c}\bar{\tilde{q}}^{c}$. In the low energy limit, its amplitude is

[TABLE]

The corresponding quarks-squarks cross section can be evaluated by integrating the squared of the amplitude all over the 4-dimensional phase space $d\Phi_{4}$. 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 $s=E_{CM}^{2}$. One finds that

[TABLE]

where

[TABLE]

with $C_{4}$ a combinatorial factor depending on the number of equal particles in the final state, and $d\hat{\Omega}$ the integral all over scattering angles

[TABLE]

The notation $\hat{\phi}_{ij}$ indicates variables evaluated with respect to the rest frame of $q_{ij}$. $\mathcal{P}_{2}$ is a complicated polynomia of $s,s_{12},s_{34}$ and subleading logaritimic functions of expression later:

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

with $m_{1,2,3,4}$ squark masses in the final state.

For $s\rightarrow\Lambda^{2}$, cross sections rapidly grows up. $\bar{s}=\Lambda^{2}$ corresponds to the bound of the unitarity break-down. In other word, our effective cross-section would badly breaks the Froissart bound.

For $s\simeq\Lambda^{2}$, 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 $s\simeq\Lambda^{2}$ corresponds to an infinite series of stringy amplitudes with six open-strings’ insertions $\sum_{g=0}^{\infty}\mathcal{A}^{g}(z_{1},z_{2},z_{3},z_{4},z_{5},z_{6})$, 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 $\Lambda=e^{+S_{E2}}M_{S}>M_{S}$. As a consequence, an infinite tower of stringy higher spins states are excited at $\Lambda$ 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

[TABLE]

while N-loops amplitudes behaves as 666The result found by Gross-Mende in Gross:1987ar is valid for four open string amplitude. Their result is expected to be qualitatively the same as for six points amplitudes in $2\rightarrow 4$ fixed angle scatterings. Gross:1987ar

[TABLE]

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 $\Lambda$-scale. For $E_{CM}\simeq\Lambda$ the process lies into the fully non-perturbative 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 $s>>\Lambda^{2}$.

We will return to phenomenological implications in the next sections.

III Phenomenology of $\Lambda-\bar{\Lambda}$ transitions

Now let us discuss the implications of $\Lambda-\overline{\Lambda}$ oscillations on the phenomenological side 777Of course, in the standard model analysis, these baryon-violating effects need not to be considered, e.g., see the $CP$ violation of $\Lambda_{c}\rightarrow\Lambda\pi$ decay Lambdac and also the conventional $N\bar{N}$ scattering KangHM1 ; KangHM2 ; KangHM3 ; KangHM4 .. 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 hadron-hadron colliders. As discussed in Ref. Kang , BES detector with huge $J/\psi$ data sample has been proposed to measure the possible $\Lambda-\overline{\Lambda}$ oscillation by studying the quantum coherent $\Lambda\overline{\Lambda}$ states. Following the treatment of $D^{0}-\overline{D}^{0}$ mixing PDGrev . To avoid the complications due to mixing, the charged $B$ was used as an example in Ref. Bl4 . The eigenstates emerging from the oscillations can be written as

[TABLE]

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 $\sqrt{1+z}/\sqrt{2}$ and $\sqrt{1-z}/\sqrt{2}$, respectively, where $z$ involves the oscillation mass $\delta m_{\Lambda\overline{\Lambda}}$ appearing in the off-diagonal elements of the effective Hamiltonian used to describe the time evolution of the $\Lambda-\overline{\Lambda}$ system. For details, see Ref. Kang or the general consideration in PDGrev . One can define the following mass and width:

[TABLE]

from which the mixing parameters are defined as

[TABLE]

For the free $\Lambda$ case without external magnetic fields, $\Delta m=2\delta m_{\Lambda\overline{\Lambda}}$. For simplicity, we will confine ourselves to this case. The influence of external magnetic field is discussed in Ref. Kang . The dominant decay mode of $\Lambda$ is $\Lambda\rightarrow p\pi^{-}$ with branching ratio $64\%$ PDG , and correspondingly $\overline{\Lambda}\rightarrow\overline{p}\pi^{+}$ assuming $CP$ invariance. In the case that $\Lambda-\overline{\Lambda}$ oscillation happens, the process $\overline{\Lambda}\rightarrow\Lambda\rightarrow p\pi^{-}$ will be possible. This phenomenon can be probed by counting the event number $\mathcal{N}$ for $J/\psi\rightarrow\Lambda\overline{\Lambda}\rightarrow(p\pi^{-})(p\pi^{-})$ –wrong-sign decay– while the right-sign decay would be $J/\psi\rightarrow\Lambda\overline{\Lambda}\rightarrow(p\pi^{-})(\overline{p}\pi^{+})$. Time-integrated decay rate for the wrong-sign decay relative to the right sign is found to be

[TABLE]

A discussion of time-dependent observables is also presented in Kang , and the BES-III detector is capable to access this time information. Assuming $y_{\Lambda}=0$ – it is indeed a quite small quantity – one can get an estimate of the mixing mass parameter $\delta m_{\Lambda\overline{\Lambda}}$ from Eq. (19) as

[TABLE]

According to the designed luminosity of BEPC-II in Beijing BESdesign , $10\times 10^{9}$ $J/\psi$ and $3\times 10^{9}$ $\psi^{\prime}$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 $\mathcal{R}\leq 4\times 10^{-7}$ and correspondingly $\delta m_{\Lambda\overline{\Lambda}}\leq 10^{-15}\,\text{MeV}$ at the 90% of confidence level (C.L.) Kang , inferring from the knowledge of interval estimate for very rare signal. Currently, a sample of $1.31\times 10^{9}$ $J/\psi$ events has been collected Jpsinumber , then the aforementioned upper limit will be increased by a factor of $\sqrt{10/1.31}\approx 2.8$, i.e. $\delta m_{\Lambda\overline{\Lambda}}\leq 3\times 10^{-15}\,\text{MeV}$. Consequently, the oscillation time will be bounded by the upper limit as $2.5\times 10^{-7}$ s at 90% confidence level. This would be the first search in the experiment. See also Kang for more details on this analysis.

Note that $\Upsilon(4S)$ also has the same quantum numbers as $J/\psi$, i.e., $I^{G}(J^{PC})=0^{-}(1^{--})$, and can also decay to coherent $\Lambda\overline{\Lambda}$ states. Belle and BaBar detectors could be used to probe this process as well. Currently, there are $772\times 10^{6}$ $\Upsilon(4S)$ data available in Belle Belle and $471\times 10^{6}$ for BaBarBaBar . Taking into account the fact that the non-$B\overline{B}$ decay mode only constitutes less than 4% (at 95% confidence level) of the total decay rate, much less $\Lambda\overline{\Lambda}$ events are expected. Otherwise assuming that most of $\Upsilon(4S)$ decays into $\Lambda\overline{\Lambda}$, the event number can increase by one or two orders lager more than BES. Although the above upper limit for $\Lambda-\bar{\Lambda}$ oscillation could be accessed from the beginning in the experiment, we require an extremely higher luminosity to get a comparable bound to the $n-\bar{n}$ case. One can find this point from Eq. (20) – the oscillation mass is proportional to the square root of the luminosity. $n-\bar{n}$ oscillation time is constrained to be $10^{7}\div 10^{8}$ smaller than the $\Lambda-\bar{\Lambda}$ 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 CEPC . According to the design agenda, the electron-positron collider will be converted into a proton-proton collider, with an unprecedented center-of-mass energy of 50-90 TeV at the second phase. This is the project of super proton-proton collider (SppC) Wangyf . 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, $n-\bar{n}$ and proton-proton colliders

An operator $\mathcal{O}_{\Lambda\bar{\Lambda}}$ generates not only $\Lambda-\bar{\Lambda}$ transition but also $NN\rightarrow KK$ transition. As discussed in the previous section $\delta m_{\Lambda\bar{\Lambda}}$ can be constrained up to $10^{-6}\,\rm eV$ by next generation of experiments. However, $NN\rightarrow KK$ is just constraining $\mathcal{O}_{\Lambda\bar{\Lambda}}$ up to $\delta m_{\Lambda\bar{\Lambda}}<10^{-21}\,\rm eV$. $\delta m_{\Lambda\bar{\Lambda}}$ is related to a New Physics scale by

[TABLE]

so that $\mathcal{M}_{\Lambda\bar{\Lambda}}\simeq 100\,\rm TeV$ scale. On the other hand, $\mathcal{O}_{n\bar{n}}$ is actually constrained by $\delta m_{n\bar{n}}<10^{-23}\,\rm eV$, corresponding to $\mathcal{M}_{n\bar{n}}>300\,\rm TeV$ NNbar0a ; NNbar0b . The next generations of experiments will test 888See Buchoff:2015qwa for a discussion on perturbative renormalization corrections of $n-\bar{n}$ operators. $\mathcal{M}_{n\bar{n}}\simeq 1000\,\rm TeV$ NNbar3 . In our model,

[TABLE]

Such a large hierarchy range is naturally understood by couplings arising from mixed disk amplitudes. For example the number $10^{-18}$ is understood as $\frac{k^{(1)}_{1}k^{(1)}_{1}k^{(2)}_{1}k^{(2)}_{1}k^{(2)}_{2}k^{(2)}_{2}}{k^{(1)}_{1}k^{(1)}_{1}k^{(2)}_{1}k^{(2)}_{1}k^{(2)}_{1}k^{(2)}_{1}}\simeq 10^{-3}\times 10^{-3}\times 10^{-3}\times 10^{-3}\times 10^{-3}\times 10^{-3}$. As a consequence, a $100\,\rm TeV$-scale test of $\Lambda-\bar{\Lambda}$ is motivated from the theoretical side, independently from $n\leftrightarrow\bar{n}$ limits. A $100\,\rm TeV$ scale scenario for $\Lambda-\bar{\Lambda}$ can be naturally obtained with $e^{+S_{E2}}\simeq 1$ (small 3-cycles wrapped by the $E2$-brane), $M_{S}\simeq M_{SUSY}\simeq 100\,\rm TeV$. In this case, a final test-bed for this model can be provided by future proton-proton $100\,\rm TeV$ colliders beyond LHC. In fact a $uds\rightarrow\bar{u}\bar{d}\bar{s}$ transition will be directly tested, mediated by an exotic instanton in collision. However, a more intriguing scenario can be opened in the case of $10\,\rm TeV<M_{SUSY}\simeq M_{S}<100\,\rm TeV$. In this case, planned proton-proton colliders can test other flavor amplitudes like $ud\rightarrow\bar{c}\bar{s}\bar{s}\bar{s},\bar{c}\bar{s}\bar{b}\bar{b}$ that are less constrained by $n-\bar{n}$ or $NN\rightarrow\pi\pi,KK$ processes. In particular, the predicted experimental processes for a high energy proton-proton collider are $pp\rightarrow 4\tilde{q}$. This leads to several different channels. This is the case of $pp\rightarrow 4q+4\chi^{0}$ leading to four jets and missing transverse energy $pp\rightarrow 4j+E_{M.T}$ or $pp\rightarrow 3j+t+E_{M.T}$ (with standard top decays) and so on. Other interesting channels are coming from stops productions and successive decays like $\tilde{t}\rightarrow W+\tilde{b}\rightarrow W+b+\chi^{0}$, $\tilde{t}\rightarrow b+\chi^{+}\rightarrow b+W+\chi^{0}$.

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 $\Lambda$ for $s>>\Lambda^{2}$.

V Conclusions

In this paper, we have discussed phenomenological implications of a new class of instantons known as exotic instantons. They can generate $\Delta B=2$ violating transitions as $n\leftrightarrow\bar{n},\Lambda\leftrightarrow\bar{\Lambda}$ and $\Delta B=2$ high energy collisions like $qq\rightarrow\bar{\tilde{q}}\bar{\tilde{q}}\bar{\tilde{q}}\bar{\tilde{q}},\bar{q}\bar{q}\bar{q}\bar{q}$ –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 $n\leftrightarrow\bar{n},NN\rightarrow\pi\pi,KK$.

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 test-bed for a class of models mentioned in this paper 999This is an example of a different UV completion of six quarks operator from a non-perturbative classical configurations, rather than extra heavy fields. This has intriguing analogies with classicalization Dvali1 ; Dvali2 ; Dvali3 ; Dvali4 ; Addazi:2015ppa ; CBH1 . Reference Addazi:2015ppa is related to discussions in Addazi:2015dxa . .

iii) Our model predicts $\Lambda-\bar{\Lambda}$ transitions. The possibility to test these after in future high luminosity electron-positron colliders seems very far from our present technological possibility, if compared to the actual related limits from $NN\rightarrow KK$ transitions.

iv) $\Delta B=2$ exotic instantons can be reached in the next generation of high energy proton-proton colliders beyond LHC, well compatible with neutron-antineutron limits, $NN\rightarrow\pi\pi,KK$ and so on. We stressed how cross-section 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 Addazi:2015goa ; Addazi:2015yna 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: neutron-antineutron experiments and high-energy colliders beyond LHC. Eventually, a detection of exotic instanton-mediated processes can motivate the construction of technologically challenging high luminosity collider in order to detect a $\Lambda-\bar{\Lambda}$ transition or new rare physics experiments searching for $NN\rightarrow KK$. 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, Z. Berezhiani, G. Dvali, A. Dolgov, C Gomez and O. Kanchelli, for useful discussions on these aspects. The author XWK also thanks Hai-Bo Li for helpful advice. A. A work was supported in part by the MIUR research grant ”Theoretical Astroparticle Physics” PRIN 2012CPPYP7. XWK’s work is partly supported by the DFG and the NSFC through funds provided to the Sino-German CRC 110 Symmetries and the Emergence of Structure in QCD when he was in Jülich, and by the Ministry of Science and Technology of R.O.C. under Grant No. 104-2112-M-001-022 when he works in Taiwan from April. The work by MK was performed within the framework of the Center FRPP supported by MEPhI Academic Excellence Project (contract 02.03.21.0005, 27.08.2013), supported by the Ministry of Education and Science of Russian Federation, project 3.472.2014/K and grant RFBR 14-22-03048.

Appendix A: superpotential calculus.

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 $dx^{4}d^{2}\theta$ 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 $\langle\Phi_{a_{1}b_{1}}...\Phi_{a_{M}b_{M}}\rangle_{\mathcal{E}}$ –where $\mathcal{B}$ denoted the instantonic background and $\Phi$ are generic physical fields in the bi-fundamental representations of two generic $N^{\prime}$ and $M^{\prime}$ stacks of D6-branes

The correlator is related to the effective supergravity quantities in the action as follows Cvetic:2007ku ; Blumenhagen:2009qh :

[TABLE]

where $\mathcal{K}$ 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 Cvetic:2007ku ; Blumenhagen:2009qh :

[TABLE]

[TABLE]

[TABLE]

where $\hat{\Phi}_{a_{k}b_{k}}$ is the chain-product of all the the vertex operators at fixed value of $k$

[TABLE]

while

[TABLE]

a CFT disk correlator for $\hat{\Phi}$ and for the charged zero modes $\lambda_{a_{1}},\bar{\lambda}_{b_{1}}$ inserted in the boundary of the mixed disk amplitudes. $C_{\mathcal{E}\mathcal{D}_{b}^{*}},C_{\mathcal{E}O^{*}}$ 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:

[TABLE]

[TABLE]

where $S_{0}$ corresponds to the vacuum disk amplitude for the E2-brane:

[TABLE]

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.