Confronting the Faster-Than-Light Neutrinos? Utpal Sarkar Physical Research Laboratory, Ahmedabad 380 009, INDIA [email protected] http://www.prl.res.in/ utpal Fundamental Theories: Newtonian Mechanics Classical Mechanics is not applicable to fast moving particles: Special Theory of Relativity Uncertainty Principle at short distances: Quantum Mechanics Interactions of Relativistic particles at short distances: Quantum Field Theory Newton’s Law of gravitation fails for relativistic particles: General Theory of Relativity How well do we know Special Theory of Relativity, Quantum Mechanics, General Theory of Relativity (and hence, the Equivalence Principle), CPT theorem, Spin-Statistics theorem ??? Structure of the talk ² Recent experiment on faster-than-light neutrino ² It’s implication on Special Theory of Relativity and Lorentz invariance ² Confronting the experiment using phenomenological constraints ² Testing the violation of Lorentz invariance from Cosmic Ray data ² Constraining equivalence principle violation ² The Standard Model and the Higgs scalar ² Cosmology and beyond the standard model Recent Experiment on faster-than-light neutrino: A muon neutrino º¹beam was produced at CERN, Geneva: A 400 GeV/c proton beam ! charged pions and kaons ! ¼ ! ¹ +º ¹¹ K ! ¹ +º ¹¹ OPERA Collaboration at the Gran Sasso Underground Laboratory (LNGS), detected the º¹from CNGS at a distance of 730 kms away. The arrival time reported by the OPERA collaboration is earlier than expected from luminal speed by a time interval ±t = (60:7 § 6:9stat § 7:4syst) ns: This implies a superluminal propagation velocity for neutrinos: (vº ¡ c) ¡5 ±cº = = (2:48 § 0:28stat § 0:30syst) £ 10 c for an average energy of 17 GeV. Special Theory of Relativity: 1 The principle of relativity: The laws of physics are not affected by the motion of any inertial reference frames, i.e. the frames that are moving with constant velocity with respect to each other. 2 The principle of invariant light speed: Speed of light in empty space (c) is constant and same in all inertial frames. O P B P A B O A (a) (b) Both the observers at O and P will find that light travelled by a distance c in all directions to the circle in 1 sec. (a) Observer O will find himself at the center of the circle, (b) while observer at P will also find himself at the center of the circle. Particles with v > c are the Tachyons. Tachyon: Consider energy-momentum relationship: E2 = ~p 2c2 + m2c4 m2c4 For m ¿ ~p, we can express: E = ~pc + 2~pc dE m2c4 From the Hamilton-Jacobi equation: v = = c ¡ 2 d~p 2~p c Ordinary massive particles : m2 > 0 ) v < c Light and other massless particles m2 = 0 ) v = c T achyons m2 < 0 ) v > c Tachyons will find light travelling with velocity ¡c ) t !¡t ¡5 If º¹ s are tachyons, for the OPERA value of cº¹ = 2:5 £ 10 SN1987A neutrinos would have reached us four years earlier. Apparently observed faster-than-light neutrinos cannot be Tachyons. ¡9 Observed supernovae SN1987A neutrinos implies: j±cºj < 2 £ 10 Lorentz Invariance and OPERA: Assume energy-momentum is conserved in a particular reference frame A Lorentz transformation for an observer moving at velocity ~v yields ~p = γ m ~v ; and E = γ m c2 ; p where γ = 1= 1 ¡ v2=c2 is the Lorentz factor. ¡ ¢ 2 ¹ E 2 2 2 2 The invariant momentum is defined as p = p¹p = c ¡ j~pj = m c ; so that E2 = j~pj2c2 + m2c4 For any massive particle we thus require v < c, which contradicts OPERA result. For a muon neutrino with mass around m » 1 eV and energy E » 17 GeV (c ¡ v)=c » 10¡20. Violation of Lorentz Invariance: Gauge invariant Lorentz non-invariant interaction (Coleman & Glashow) h i y i u D0 ¡ i(1 ¡ ²=2)~σ ¢ D~ u : The parameter ² is a measure of violation of Lorentz invariance. This implies a modified energy-momentum dispersion relation: p2c2 = E2 ¡ j~pj2c2 = m2c4 + ²j~pj2c2 : In other words, writing the dispersion relation as E2 = j~pj2c2 + m2 c4 : º¹ º¹ º¹ the maximum attainable velocity, and the mass are now modified: c2 = (1 + ²)c2 = (1 + ²); m = m=(1 + ²) º¹ º¹ q With the new Lorentz factor γ = 1= 1 ¡ v2=c2 , we have º¹ º¹ 2 2 (cº¹ ¡ v) m E = γº mº c = γº m with ¼ ; for very large v ¹ ¹ º¹ ¹ 2 cº¹ 2E . The present result from OPERA gives (v ¡ c) = v ¡ 1 = (2:48 § 0:28 § 0:3) £ 10¡5 : c When Lorentz invariance is violated, the Lorentz factor contains (c ¡ v) m2 º¹ ¼ ¼ 10¡20 : 2 cº¹ 2E Since c < v < cº¹ , the present measurement of the º¹ velocity v becomes consistent with the modified Lorentz factor and the º¹ remains time-like. Different particles can have different limiting velocities (ci), if Lorentz invariance is violated. If the limiting velocities of the three neutrinos are different, that would lead to neutrino ¡22 oscillations. Present experimental constraints imply jcºi ¡ cºj j < 10 for i 6= j; i; j = e; ¹; ¿. Simplest solution is to assume, cºe = cº¹ = cº¿ = cº. OPERA then requires ¡5 cº = 2:5 £ 10 . Phenomenological Constraints: Consider a decay process: A ! B + C This is allowed if MA > MB + MC. 2 In case of VLI, one may write effective mass of the particles as mX = mX + ®~p . Thus if A is superluminal, the decay condition becomes 2 2 2 MA + ®A~p A > MB + ®B ~p B + MC + ®C ~p C Consider pion decay: ¼ ! ¹ + º¹ If º¹ is superluminal (®¼ = ®¹ = 0 and ®º¹ = ®), at some high energies 2 2 (m¼ ¡ m¹)=® < mº¹ pions cannot decay. 2 On the other hand, at some even higher energies, when mº¹ + ®~p , º¹ can decay º¹ ! ¼ + ¹ These processes imply stringent constraints. But they are approximate and not correct. The decay constraint is obtained using energy-momentum conservation in the rest frame of pions, not in the laboratory frame. In VLI models they are not equivalent. Pion Decay Kinematics D E @E Defining, in general (for massless particles and LIV), @p = 1 + ® as the effective average from pmin up to the typical momenta of » 17 GeV of neutrinos, we write the energy-momentum relation Eº = pº(1 + ®) ; where ® corresponds to VLI. To avoid other constraints, assume all particles (except the neutrinos) have terminal 2 2 1=2 velocity c = 1, and satisfy, Ei = (pi + mi ) ; The four vector of the decaying pion can then be written as (E¼; p¼; 0; 0) and those of the final neutrino and muon can be given by ¡ ¢ (Eº; pº`; pºt; 0) and E¹; p¹`; p¹t; 0 : The longitudinal components of momenta are pº` = ´p¼ and p¹` = (1 ¡ ´)p¼ and the transverse components are pºt = ¡p¹t = pt. 2 The energy conservation E¼ = Eº + E¹ ; then gives (for large p¼ and positive pt : · ¸ 1 m2 ®´ · m2 ¡ ¹ 2 ¼ 2p¼ (1 ¡ ´) In OPERA average neutrino energy is » 17 GeV that arise from the decay of pions with a mean energy of » 60 GeV, so that the typical value of h´i ¼ 0:3. We thus obtain a bound that rules out all models of VLI: · ¸ 1 m2 ® · m2 ¡ ¹ » 4 £ 10¡6 : OP ERA 2 ¼ 0:6p¼ 0:7 Cosmic ray muon spectra at higher energies give stronger constraint on VLI The observed differential energy spectrum is well represented by · ½ ¾ » ¯¡1 h1 ¡ ´i¼E¼ f¹(E) = A¼h1 ¡ ´i ¼ E + h1 ¡ ´i E ½ ¼ ¼ ¾¸ ¯¡1 h1 ¡ ´iKEK ¡¯ AKh1 ¡ ´iK E ; E + h1 ¡ ´iKEK where ¯ = spectral index of the cosmic ray spectrum » 2:65; E¼ = h0(θ)m¼=c¿¼; EK = h0(θ)mK=c; ¿K; A¼=K = Constants. 5 h0(θ) = 7 £ 10 sec θ cm, the scale height of the Earth’s atmosphere at a zenith angle θ ¿¼=K = decay lifetimes of pions/kaons at rest h1 ¡ ´i¼=K = the fractional momenta carried away by the muons in pion/kaon decay averaged over the spectrum of cosmic rays, around the energy band of interest. Considering the pion and kaon contributions and ´ distributions, we can arrive at the following bounds: Muon spectra at Baksan fits very well with this theoretical estimate up to » 4 £ 104 GeV. Thus the muon data imply ® < 10¡11 : Similarly the atmospheric neutrino spectrum up to » 400 TeV by the Ice-Cube Collaboration imply a constraint ® < 10¡13 : To explain OPERA we need ® » 10¡5, which is not allowed by cosmic ray data. Violation of Equivalence Principle In weak field limit with Newtonian potential Á = GM=r ¿ 1 consider effective metric: g44 = 1 ¡ 2®Á gij = ¡±ij(1 + 2γÁ) : In GTR, ® = γ = 1. Assuming γ = 1 and different ®i (VEP), the dispersion relation becomes: m2 E0 = p + ¡ ®Áp ; 2p µ ¶ ®1Áp If two VEP states jºGi = differ from the flavor eigenstates ®2Áp jºF i = U(θG)jºGi, there can be neutrino oscillations. In presence of both mass oscillation and VEP oscillation (with same mixing angles θM = θG), the oscillation length becomes £ ¤¡1 L = 2¼ ¢m2=2p + pÁ¢® Cosmic rays data constraints VEP, because the dispersion relation is modified. The Standard Model and the Higgs Scalar: Strong Interaction: Force between the quarks – constituents of nucleons and mesons (hadrons) Electromagnetic Interaction: Force between charged particles Weak Interaction: Neutron decay n ! p + e¡ +º ¹. Only left-handed particles take part in this interaction (charged current). At energies above 250 GeV, they combine into Strong and Electroweak Interaction: The Standard Model Symmetry group SU(3)c £ SU(2)L £ U(1)Y breaks into SU(3)c £ U(1)Q At higher energies before the symmetry breaking SU(3)c: 8 massless Gluons mediating strong interaction among quarks §;3 SU(2)L: 3 massless gauge bosons W , that interact with only the left-handed particles.