# Improved bounds on the dipole moments of the tau-neutrino at high-energy and collisions: ILC and CLIC

###### Abstract

In this work we study the potential of the processes and at a future high-energy and high-luminosity linear electron positron collider, such as the ILC and CLIC to study the sensibility on the anomalous magnetic and electric dipole moments of the tau-neutrino. For integrated luminosity of 590 and center-of-mass energy of 3 , we derive limits on the dipole moments: and in the collision mode and of and with the collision mode, improving the existing limits.

###### pacs:

14.60.St, 13.40.EmKeywords: Non-standard-model neutrinos, Electric and Magnetic Moments.

## I Introduction

In the Standard Model (SM) S.L.Glashow (); Weinberg (); Salam () extended to contain right-handed neutrinos, the neutrino magnetic moment induced by radiative corrections is unobservably small, , where is the Bohr magneton Fujikawa (); Shrock (). Current limits on these magnetic moments are several orders of magnitude larger, so that a magnetic moment close to these limits would indicate a window for probing effects induced by new physics beyond the SM Fukugita (). Similarly, a neutrino electric dipole moment will also point to new physics and will be of relevance in astrophysics and cosmology, as well as terrestrial neutrino experiments Cisneros (). In the case of the magnetic moment of the the best bound is derived from globular cluster red giants energy loss Raffelt (),

(1) |

is far from the SM value. The best current laboratory constraint

(2) |

is obtained in elastic scattering experiment GEMMA Bed (), which is an order of magnitude larger than the constraint obtained in astrophysics Raffelt ().

For the magnetic moment of the muon-neutrino the current best limit has been obtained in the LSND experiment Auerbach ()

(3) |

In the case of the electric dipole moment Aguila () the best limits are:

(4) |

The most general expression consistent with Lorentz and electromagnetic gauge invariance, for the tau-neutrino electromagnetic vertex may be parameterized in terms of four form factors:

(5) |

where is the charge of the electron, is the mass of the tau-neutrino, is the photon momentum, and are the electromagnetic form factors of the neutrino, corresponding to charge radius, magnetic moment (MM), electric dipole moment (EDM) and anapole moment (AM), respectively, at Escribano (); Vogel (); Bernabeu1 (); Bernabeu2 (); Dvornikov (); Giunti (); Broggini (). The form factors corresponding to charge radius and the anapole moment, do not concern us here.

The current best limit on has been obtained in the Borexino experiment which explores solar neutrinos. Searches for the magnetic moment of the tau-neutrino have also been performed in accelerator experiments. The experiment E872 (DONUT) is based at elastic scattering. In the CERN experiment WA-066, a limit on is obtained on an assumed flux of tau-neutrinos in the neutrino beam. The L3 collaboration obtain a limit on the magnetic moment of the tau-neutrino from a sample of annihilation events at the resonance. Experimental limits on the magnetic moment of the tau-neutrino are shown in Table I.

Experiment | Method | Limit | C. L. | Reference |
---|---|---|---|---|

Borexino |
Solar neutrino | Borexino () | ||

E872 (DONUT) | Accelerator | DONUT () | ||

CERN-WA-066 | Accelerator | A.M.Cooper () | ||

L3 | Accelerator | L3 () |

Others limits on the magnetic moment of the are reported in the literature Gutierrez9 (); Gutierrez8 (); Data2014 (); Gutierrez7 (); Gutierrez6 (); Aydin (); Gutierrez5 (); Perez (); Gutierrez4 (); Gutierrez3 (); Larios (); Keiichi (); Aytekin (); Gutierrez2 (); Hernandez (); Maya (); Gutierrez1 (); DELPHI (); Escribano (); Gould (); Grotch ().

In this work we study the sensibility of the anomalous magnetic and electric dipole moments of the tau-neutrino through the processes and at a future high-energy and high-luminosity linear electron positron collider, with a center-of-mass energy in the range of 500 to 1600 , such as the International Linear Collider (ILC) Abe (), and of 3 to the Compact Linear Collider (CLIC) Accomando (). Not only can the future linear collider be designed to operate in collision mode, but it can also be operated as a and collider. This is achieved by using Compton backscattered photons in the scattering of intense laser photons on the initial beams. The other well-known applications of linear colliders are to study new physics beyond the SM through and collisions. A quasi-real photon emitted from one of the incoming or beams can interact with the other lepton shortly after, and the subprocess can generate. Hence, first, we calculate the main reaction by integrating the cross section for the subprocess . Also, photons emitted from both and beams collide with each other, and the subprocess can be produced. Second, we find the main reaction by integrating the cross section for the subprocess . In both cases, the quasi-real photons in and collisions can be examined by Equivalent Photon Approximation (EPA) Budnev (); Baur (); Piotrzkowski (), that is to say, using the Weizsacker-Williams approximation. In EPA, photons emitted from incoming leptons which have very low virtuality are scattered at very small angles from the beam pipe and because the emitted quasi-real photons have a low virtuality, these are almost real. These processes have been observed experimentally at the LEP, Tevatron and LHC Abulencia (); Aaltonen1 (); Aaltonen2 (); Chatrchyan1 (); Chatrchyan2 (); Abazov (); Chatrchyan3 (). In particular, the most stringent experimental limit on the anomalous magnetic dipole moment of the tau lepton is obtained through the process by using multiperipheral collision at the LEP DELPHI1 ().

In Refs. Sahin (); Sahin1 (), the electromagnetic properties of the neutrinos were examined via the Weizsacker-Williams approximation at the LHC. In Ref. Sahin () nonstandard couplings and were investigated via production in the process . In addition, the potential of collisions at the LHC was studied via the reaction to probe neutrino-photon coupling by Ref. Sahin1 ().

With these motivations, we study the potential of the processes and and derive limits on the dipole moments and at and level ( and C.L.) via Weisacker-Williams approximation, and at a future high-energy and high-luminosity linear electron positron collider, such as the ILC and CLIC to study the sensibility on the anomalous magnetic and electric dipole moments of the tau-neutrino.

For this we calculate the main reaction by integrating the cross section for the subprocess . The acceptance cuts will be imposed as for pseudorapidity and for transverse momentum cut of the final state lepton, respectively. For the second process we calculate the main reaction . Neutrinos in this process are not detected directly in the central detector. Therefore we do not apply any cuts for the final state particles. The corresponding Feynman diagrams for the main reactions as well as for the sub-processes which give the most important contribution to the total cross-section are shown in Figs. 1-4.

To illustrate our results for both processes we show the dependence of the total cross-section as a function of anomalous couplings and for three different values of the center-of-mass energies and , respectively. The variation of the cross-section as a function of and for different values of (Weizsacker-Williams photon virtuality) and center-of-mass energy of and is evaluated. We also include a contours plot for the upper bounds of the anomalous couplings and with C.L. at the with corresponding maximum luminosities for both processes. The sensitivity limits on the magnetic moment and the electric dipole moment of the tau-neutrino for different values of photon virtuality, center-of-mass energy and luminosity are also calculated.

This paper is organized as follows. In Section II, we study the dipole moments of the tau-neutrino through the processes in the collision mode and through the collision mode. Finally, we present our results and conclusions in Section III.

## Ii Cross-section of and

In this section we present numerical results of the cross-section for both processes and as a function of the electromagnetic form factors of the neutrino and . In addition, to see the sensitivity of the magnetic moment and the electric dipole moment to new physics, we plot versus . We carry out the calculations using the framework of the minimally extended standard model at next generation linear and collisions: ILC and CLIC.

We use the CompHEP Pukhov () packages for calculations of the matrix elements and cross-sections. These packages provide automatic computation of the cross-sections and distributions in the SM as well as their extensions at tree-level. We consider the high energy stage of possible future linear and collisions with and 3 and design luminosity 230, 320 and 590 according to the data reported by the ILC and CLIC Abe (); Accomando (). In addition, we consider the acceptance cuts of for pseudorapidity and for transverse momentum cut of the final state lepton, respectively.

### ii.1 Magnetic moment and electric dipole moment via

The corresponding Feynman diagrams for the main reaction , as well as for the subprocess which give the most important contribution to the total cross-section are shown in Figs. 1-2. From Fig. 2, the Feynman diagrams (1)-(3) correspond to the contribution of the standard model, while diagram (4) corresponds to the anomalous contribution, that is to say, for the collisions there are SM background at the tree-level so the total cross-section is proportional to , respectively.

To illustrate our results we show the dependence of the cross-section on the anomalous couplings and for in Fig. 5 for three different center-of-mass energies and Dvornikov (), respectively. The cross-section is sensitive to the value of the center-of-mass energies, as well as to . The sensitivity to increases with the collider energy, as well as with reaching a maximum at the end of the range considered: . In Fig. 6, we show again the total cross-section, but now for different values ââof Dvornikov () and center-of-mass energies of . We observed that the variation of the cross-section for as a function of the anomalous couplings and it is clear for all case.

In Figures 7 and 8 we present the dependence of the sensitivity limits of the magnetic moment and the electric dipole moment with respect to the collider luminosity for three different values of the Weizsacker-Williams photon virtuality and center-of-mass energies of . In these figures, we observe one variation of in all the interval of , and it is almost independent of the value of .

As an indicator of the order of magnitude, in Tables II-III we present the bounds obtained on the magnetic moment and electric dipole moment for , and at and , respectively. We observed that the results obtained in Tables II and III are competitive with those reported in the literature DONUT (); A.M.Cooper (); L3 (). For the electric dipole moment our limits compare favorably with those reported by K. Akama, et al. Keiichi () and R. Escribano, et al. Escribano () ,

In Fig. 9 we used three center-of-mass energies planned for the ILC and CLIC accelerators in order to get contours limits in the plane for and the planned luminosities of and Weizsacker-Williams photon virtuality . For the collision, we perform analysis at since the number of SM events is greater than 10.

, | ||
---|---|---|

C. L. | ||

(8.73, 3.35, 1.60) | (16.8, 6.46, 3.08) | |

(9.30, 3.30, 1.53) | (17.9, 6.36, 2.95) |

, | ||
---|---|---|

C. L. | ||

(8.22, 2.88, 1.32) | (15.8, 5.56, 2.54) | |

(8.97, 3.14, 1.44) | (17.3, 6.06, 2.78) |

### ii.2 Magnetic moment and electric dipole moment via

In this subsection we study the dipole moments of the tau-neutrino via the process for energies expected at the ILC and CLIC Abe (); Accomando (). The corresponding Feynman diagrams for the subprocess which give the most important contribution to the total cross-section are shown in Figs. 3 and 4. In this case, the total cross-section of the subprocess depends only on the diagrams (1) and (2) with anomalous couplings, and there is no contribution at tree level of the standard model, that is to say .

For the study of the subprocess in Fig. 10, we show the total cross-section as a function of the electromagnetic form factors of the neutrino and for three different center-of-mass energies and Dvornikov (), respectively. We can see from this figure that the total cross-section changes strongly with the variation of the and values.

As in subsection A, we show the and dependence of the total cross-section for in Fig. 11. From this figure we observed a significant dependence of the cross-section with respect to and , and different values of center-of-mass energy and . In Figures 12 and 13 we present the dependence of the sensitivity limits of the magnetic moment and the electric dipole moment with respect to the collider luminosity for three different values of Dvornikov () and center-of-mass energies of .

In Tables IV and V we present the bounds obtained on the magnetic moment and electric dipole moment for , and at and . We observed that the results obtained in Tables IV-V improve the bounds reported in the literature DONUT (); A.M.Cooper (); L3 ().

In the case of the electric dipole moment the C. L. sensitivity limits at 0.5, 1.5 and 3 center of mass energies and integrated luminosities of 230, 320 and , respectively can provide proof of these bounds of order , that is to say, they are improved by one order of magnitude than those reported in the literature: Keiichi () and , C. L. Escribano ().

Finally, in Fig. 14 we summarize the respective limit contours for the dipole moments in the plane for . Starting from the top, the curves are for and ; and ; and , respectively. We have used . In this case for the collision, we perform Poisson analysis at since the number of SM events is smaller than 10.

, | ||
---|---|---|

C. L. | ||

(10.90, 5.70, 3.50) | (2.10, 1.09), 6.75 | |

(11.60, 6.10, 3.70) | (2.24, 1.18), 7.14 |

, | ||
---|---|---|

C. L. | ||

(9.90, 5.20, 3.10) | (1.91, 1.00), 5.98 | |

(10.60, 5.54, 3.40) | (2.04, 1.07), 6.56 |

## Iii Conclusions

Even though and processes require new equipment, and are realized spontaneously at linear colliders without any equipment. These processes will allow the next generation linear collider to operate in three different modes, , and , opening up the opportunity for a wider search for new physics. Therefore, the and linear collisions represents an excellent opportunity to study the sensibility on the anomalous magnetic moment and electric dipole moment of the tau-neutrino.

We have done an analysis of the total cross-section of the processes and as a function of the anomalous coupling and . The analysis is shown in Figs. 5, 6, 10 and 11 for different center-of-mass energies and several values of the Weizsacker-Williams photon virtuality. In all cases, the cross-section shows a strong dependence on the anomalous couplings and .

The correlation between the luminosity of the collider and the anomalous magnetic moment and the electric dipole moment is presented in Figs. 7 and 8. In both cases, we see that there is a strong correlation between and the dipole moments, the same is also observed in Figs. 12 and 13 as well as in Tables II-V.

We also include contours plots for the dipole moments at the in the plane for and in Figures 9 and 14. The contours are obtained from Tables II-V.

It is worth mentioning that our bounds obtained in Tables II-V on the anomalous magnetic moment for the processes and for , and at and C. L. compare favorably with the bounds obtained in Table I by DONUT DONUT (), WA66 A.M.Cooper () and L3 Collaboration L3 (), as well as those reported by K. Akama, et al. Keiichi () and R. Escribano et al. Escribano () While in the case of the electric dipole moment our results obtained in Table II-V are improved by one order of magnitude than those reported in the literature Keiichi () and , Escribano ().

In conclusion, we have found that the processes and in the and collision modes at the high energies and luminosities expected at the ILC and CLIC colliders can be used to probe for bounds on the magnetic moment and electric dipole moment of the tau-neutrino. In particular, we can appreciate that for integrated luminosities of 590 and center-of-mass energies of 3 , we derive limits on the dipole moments: and for the process and of and for , better than those reported in the literature.

Acknowledgements

A. G. R. acknowledges support from CONACyT, SNI, PROMEP and PIFI (México).

## References

- (1) S. L. Glashow, Nucl. Phys. 22, 579 (1961).
- (2) S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
- (3) A. Salam, in Elementary Particle Theory, Ed. N. Svartholm (Almquist and Wiskell, Stockholm, 1968) 367.
- (4) K. Fujikawa and R. Shrock, Phys. Rev. Lett. 45, 963 (1980).
- (5) Robert E. Shrock, Nucl. Phys. B206, 359 (1982).
- (6) M. Fukugita and T. Yanagida, Physics of Neutrinos and Applications to Astrophysics, (Springer, Berlin, 2003).
- (7) A. Cisneros, Astrophys. Space Sci. 10, 87 (1971).
- (8) G. G. Raffelt, Phys Rep. 320, (1999) 319.
- (9) A. G. Bed, et al., [GEMMA Collaboration] Adv. High Energy Phys. 2012, (2012) 350150.
- (10) L. B. Auerbach, et al., [LSND Collaboration] Phys Rev. D63, (2001) 112001, hep-ex/0101039.
- (11) F. del Aguila and M. Sher, Phys Lett. B252, (1990) 116.
- (12) R. Escribano and E. Massó, Phys. Lett. B395, 369 (1997).
- (13) P. Vogel and J. Engel, Phys. Rev. D39, 3378 (1989).
- (14) J. Bernabeu, et al., Phys. Rev. D62, 113012 (2000).
- (15) J. Bernabeu, et al., Phys. Rev. Lett. 89, 101802 (2000); Phys. Rev. Lett. 89, 229902 (2002).
- (16) M. S. Dvornikov and A. I. Studenikin, Jour. of Exp. and Theor. Phys. 99, 254 (2004).
- (17) C. Giunti and A. Studenikin, Phys. Atom. Nucl. 72, 2089 (2009).
- (18) C. Broggini, C. Giunti, A. Studenikin, Adv. High Energy Phys. 2012, (2012) 459526; arXiv:1207.3980 [hep-ph] and references therein.
- (19) C. Arpesella, et al., [Borexino Collaboration], Phys. Rev. Lett. 101, 091302 (2008).
- (20) R. Schwinhorst, et al., [DONUT Collaboration], Phys. Lett. B513, 23 (2001).
- (21) A. M. Cooper-Sarkar, et al., [WA66 Collaboration], Phys. Lett. B280, 153 (1992).
- (22) M. Acciarri et al., [ L3 Collaboration], Phys. Lett. B412, 201 (1997).
- (23) A. Gutiérrez-Rodríguez, Int. J. Theor. Phys. 54, 236 (2015).
- (24) A. Gutiérrez-Rodríguez, Advances in High Energy Physics 2014, 491252 (2014).
- (25) K. A. Olive, et al., [Particle Data Group], Chin. Phys. C38, 090001 (2014).
- (26) A. Gutiérrez-Rodríguez, Pramana Journal of Physics 79, 903 (2012).
- (27) A. Gutiérrez-Rodríguez, Eur. Phys. J. C71, 1819 (2011).
- (28) C. Aydin, M. Bayar and N. Kilic, Chin. Phys. C32, 608 (2008).
- (29) A. Gutiérrez-Rodríguez, et al., Phys. Rev. D74, 053002 (2006).
- (30) M. A. Pérez, G. Tavares-Velasco and J. J. Toscano, Int. J. Mod. Phys. A19, 159 (2004).
- (31) A. Gutiérrez-Rodríguez, et al., Phys. Rev. D69, 073008 (2004).
- (32) A. Gutiérrez-Rodríguez, et al., Acta Physica Slovaca 53, 293 (2003).
- (33) F. Larios, M. A. Pérez, G. Tavares-Velasco, Phys. Lett. B531, 231 (2002).
- (34) K. Akama, T. Hattori and K. Katsuura, Phys. Rev. Lett. 88, 201601 (2002).
- (35) A. Aydemir and R. Sever, Mod. Phys. Lett. A16 7, 457 (2001).
- (36) A. Gutiérrez-Rodríguez, et al., Rev. Mex. de Fís. 45, 249 (1999).
- (37) J. M. Hernández, et al., Phys. Rev. D60, 013004 (1999).
- (38) M. Maya, M. A. Pérez, G. Tavares-Velasco, B. Vega, Phys. Lett. B434, 354 (1998).
- (39) A. Gutiérrez-Rodríguez, et al., Phys. Rev. D58, 117302 (1998).
- (40) P. Abreu, et al., [DELPHI Collaboration], Z. Phys. C74, 577 (1997).
- (41) T. M. Gould and I. Z. Rothstein, Phys. Lett. B333, 545 (1994).
- (42) H. Grotch and R. Robinet, Z. Phys. C39, 553 (1988).
- (43) T. Abe, et al. (Am. LC Group), arXiv:hep-ex/0106057; G. Aarons et al., (ILC Collaboration), arXiv: 0709.1893 [hep-ph]; J. Brau et al., (ILC Collaboration), arXiv: 0712.1950 [physics.acc-ph]; H. Baer, T. Barklow, K. Fujii et al., arXiv:1306.6352 [hep-ph].
- (44) E. Accomando, et al. (CLIC Phys. Working Group Collaboration), arXiv: hep-ph/0412251, CERN-2004-005; D. Dannheim, P. Lebrun, L. Linssen et al., arXiv: 1208.1402 [hep-ex]; H. Abramowicz et al., (CLIC Detector and Physics Study Collaboration), arXiv:1307.5288 [hep-ph].
- (45) V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, Phys. Rep. 15, 181 (1975).
- (46) G. Baur, et al., Phys. Rep. 364, 359 (2002).
- (47) K. Piotrzkowski, Phys. Rev. D63, 071502 (2001).
- (48) A. Abulencia, et al., [CDF Collaboration], Phys. Rev. Lett. 98, 112001 (2007).
- (49) T. Aaltonen, et al., [CDF Collaboration], Phys. Rev. Lett. 102, 222002 (2009).
- (50) T. Aaltonen, et al., [CDF Collaboration], Phys. Rev. Lett. 102, 242001 (2009).
- (51) S. Chatrchyan, et al., [CMS Collaboration], JHEP 1201, 052 (2012).
- (52) S. Chatrchyan, et al., [CMS Collaboration], JHEP 1211, 080 (2012).
- (53) V. M. Abazov, et al., [D0 Collaboration], Phys. Rev. D88, 012005 (2013).
- (54) S. Chatrchyan, et al., [CMS Collaboration], JHEP 07, 116 (2013).
- (55) J. Abdallah, et al., [DELPHI Collaboration], Eur. Phys. J. C35, (2004) 159.
- (56) I. Sahin, Phys. Rev. D85, 033002 (2012).
- (57) I. Sahin and M. Koksal, JHEP 03, 100 (2011).
- (58) Pukhov A, et al, CompHEPâa package for evaluation of Feynman diagrams and integration over multiparticle phase space, Report No. INP MSU 98-41/542, arXiv:hep-ph/9908288.