在這項研究中，我們提出由宇宙背景微中子引起的新訊號。我們研究了具有
韌致輻射的微中子-微中子散射，這是一個沒有運動學臨界值且不依賴於共振的
過程。原則上，這一過程將允許人們測量宇宙背景微中子的速度分佈。作為一
個特定的例子，我們計算了這個過程的橫截面，並提供了其發出的韌致輻射光
子的能量和角度分佈，來自於殘留宇宙微中子與太陽微中子散射。

In this work, we present a new signature induced by the Cosmic Neutrino
Background CNB. We study the neutrino-neutrino scattering with bremsstrahlung
emission, which is a process without a kinematic threshold and does not depend
on resonance. In principle, this process would allow one to measure the velocity
distribution of the CNB neutrinos. As a particular example, we calculate the
cross section for this process and provide the energy and angular distributions of
the emitted bremsstrahlung photon as well as the total expected rate for solar
neutrinos scattering from a relic neutrino emitting a photon.

Contents
Abstract (Chinese)I
Acknowledgements (Chinese)II
AbstractIII
AcknowledgementsIV
ContentsV
List of FiguresVII
List of TablesVIII
1 Introduction1
2 Background3
2.1The Cosmic Neutrino Background . . . . . . . . . . . . . . . . . . .
3
2.2The Solar Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3Electromagnetic Properties of Neutrinos . . . . . . . . . . . . . . .
8
2.4Bremsstrahlung Radiation . . . . . . . . . . . . . . . . . . . . . . . 11
2.5Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Bremsstrahlung From Neutrino-Neutrino Scattering
3.1
14
The 2 to 2 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 14
V3.2Matrix Elements of Neutrino-Neutrino Scattering . . . . . . . . . . 16
3.3Matrix Elements of Neutrino-Antineutrino Scattering . . . . . . . . 25
3.4No Divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5From Matrix Elements to the Cross Section . . . . . . . . . . . . . 34
3.6From Cross Section to Event Rate . . . . . . . . . . . . . . . . . . . 37
4 Results
40
4.1Constant Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2Energy Distributions of the Bremsstrahlung . . . . . . . . . . . . . 41
4.3Angular Distribution of the Bremsstrahlung . . . . . . . . . . . . . 42
4.4The Total Expected Rate . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Summary and Conclusion 46
Bibliography 48

[1] Planck Collaboration, N. Aghanim et. al., Planck 2018 results. VI.
Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [1807.06209].
[Erratum: Astron.Astrophys. 652, C4 (2021)].
[2] WMAP Collaboration, E. Komatsu et. al., Five-Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,
Astrophys. J. Suppl. 180 (2009) 330–376 [0803.0547].
[3] R. J. Michney and R. R. Caldwell, Anisotropy of the Cosmic Neutrino
Background, JCAP 01 (2007) 014 [astro-ph/0608303].
[4] F. de Bernardis, A. Melchiorri, L. Verde and R. Jimenez, The Cosmic
Neutrino Background and the Age of the Universe, JCAP 03 (2008) 020
[0707.4170].
[5] A. C. Melissinos, Possible and impossible experiments, in Conference on
Probing Luminous and Dark Matter: Adrian Fest, pp. 262–285, 9, 1999.
[6] M. Bauer and J. D. Shergold, Relic neutrinos at accelerator experiments,
Phys. Rev. D 104 (Oct, 2021) 083039.
[7] V. Domcke and M. Spinrath, Detection prospects for the Cosmic Neutrino
Background using laser interferometers, JCAP 06 (2017) 055 [1703.08629].
48[8] J. D. Shergold, Updated detection prospects for relic neutrinos using coherent
scattering, JCAP 11 (2021), no. 11 052 [2109.07482].
[9] V. Brdar, P. S. B. Dev, R. Plestid and A. Soni, A new probe of relic
neutrino clustering using cosmogenic neutrinos, Phys. Lett. B 833 (2022)
137358 [2207.02860].
[10] R. Mohammadi, J. Khodagholizadeh, M. Sadegh, A. Vahedi and S.-s. Xue,
Cross-correlation Power Spectra and Cosmic Birefringence of the CMB via
Photon-neutrino Interaction, 2109.00152.
[11] S. Weinberg, Universal Neutrino Degeneracy, Phys. Rev. 128 (1962)
1457–1473.
[12] PTOLEMY Collaboration, E. Baracchini et. al., PTOLEMY: A Proposal
for Thermal Relic Detection of Massive Neutrinos and Directional Detection
of MeV Dark Matter, 1808.01892.
[13] A. G. Cocco, G. Mangano and M. Messina, Probing low energy neutrino
backgrounds with neutrino capture on beta decaying nuclei, JCAP 06 (2007)
015 [hep-ph/0703075].
[14] M. Blennow, Prospects for cosmic neutrino detection in tritium experiments
in the case of hierarchical neutrino masses, Phys. Rev. D 77 (2008) 113014
[0803.3762].
[15] S. Betts et. al., Development of a Relic Neutrino Detection Experiment at
PTOLEMY: Princeton Tritium Observatory for Light, Early-Universe,
Massive-Neutrino Yield, in Community Summer Study 2013: Snowmass on
the Mississippi, 7, 2013. 1307.4738.
49[16] A. J. Long, C. Lunardini and E. Sabancilar, Detecting non-relativistic
cosmic neutrinos by capture on tritium: phenomenology and physics
potential, JCAP 08 (2014) 038 [1405.7654].
[17] E. Vitagliano, I. Tamborra and G. Raffelt, Grand Unified Neutrino Spectrum
at Earth: Sources and Spectral Components, Rev. Mod. Phys. 92 (2020)
45006 [1910.11878].
[18] S. Hannestad and J. Brandbyge, The Cosmic Neutrino Background
Anisotropy - Linear Theory, JCAP 03 (2010) 020 [0910.4578].
[19] E. Bertschinger, COSMICS: cosmological initial conditions and microwave
anisotropy codes, astro-ph/9506070.
[20] J. N. Bahcall and R. K. Ulrich, Solar models, neutrino experiments, and
helioseismology, Rev. Mod. Phys. 60 (Apr, 1988) 297–372.
[21] TEXONO Collaboration, H. T. Wong et. al., A Search of Neutrino
Magnetic Moments with a High-Purity Germanium Detector at the
Kuo-Sheng Nuclear Power Station, Phys. Rev. D 75 (2007) 012001
[hep-ex/0605006].
[22] The Nobel Prize in Physics 2015 , www.nobelprize.org, .
[23] W. Pauli, On the Earlier and more recent history of the neutrino, Camb.
Monogr. Part. Phys. Nucl. Phys. Cosmol. 1 (1991) 1–25.
[24] C. Giunti and A. Studenikin, Neutrino electromagnetic interactions: a
window to new physics, Rev. Mod. Phys. 87 (2015) 531 [1403.6344].
[25] S. Weinzierl, Introduction to Monte Carlo methods, hep-ph/0006269.
[26] J.-C. Walter and G. Barkema, An introduction to monte carlo methods,
Physica A: Statistical Mechanics and its Applications 418 (2015) 78–87.
50[27] K. Yao and J. Gao, Law of large numbers for uncertain random variables,
IEEE Transactions on Fuzzy Systems 24 (2016), no. 3 615–621.
[28] J. A. M. Vermaseren, New features of FORM, math-ph/0010025.
[29] J. Kuipers, T. Ueda, J. A. M. Vermaseren and J. Vollinga, FORM version
4.0, Comput. Phys. Commun. 184 (2013) 1453–1467 [1203.6543].
[30] K. Asteriadis, A. Q. Triviño and M. Spinrath, Bremsstrahlung from the
Cosmic Neutrino Background, 2208.01207.
[31] Particle Data Group Collaboration, P. A. Zyla et. al., Review of Particle
Physics, PTEP 2020 (2020), no. 8 083C01.
[32] A. Ringwald and Y. Y. Y. Wong, Gravitational clustering of relic neutrinos
and implications for their detection, JCAP 12 (2004) 005 [hep-ph/0408241].