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Gravity,Cosmic Rays and the LHC Richard Shurtleff∗February 2,2008Abstract The high energy proton beams expected when the Large Hadron Collider (LHC)comes online should provide a pass/fail test for a gravity-related explanation of ultra-high energy cosmic rays.The model predicts that particles have two kinds energies,equal for null gravitational potentials and,in the potential at the Earth,differing sig-nificantly above one TeV.If correct,a 7TeV trajectory energy proton at the LHC would deliver a 23.5TeV particle state energy in a collision.PACS -11.10.-z,29.20.D-,96.50.S Keywords -field theory,proton accelerators,Large Hadron Collider,cosmic rays As proton beams in accelerators achieve energies previously the exclusive domain of cosmic rays (CRs),experiments in the lab can begin to sort out the various explanations of what many consider to be the mysterious behavior of ultrahigh energy CRs.The 7TeV proton beam expected shortly at the Large Hadron Collider cross a gravitational threshold that should confirm or discard at least one CR explanation.That explanation is prented in Ref.[1],hereafter referred to as ‘I’.To give some idea of what it is about,obrve that the effects of electromagnetic fields on protons in a particle
accelerator and in interstellar space produce nearly classical trajectories.And a classical trajectory in quantum mechanics maximizes or minimizes the pha of the quantum field.The term ‘trajectory (four-)momentum’refers to the spatial momentum and energy of the proton moving along its trajectory.
Now consider a cond kind of momentum.Also from quantum mechanics we know that particle states can be expanded over a basis of momentum eigenstates.Each eigenstate
2 is proportional to a plane wave who pha is the scalar product of momentum p and Minkowski event coordinates x,exp(ip·x).Then momentum is the rate of change of pha with respect to distance and time,which we call the‘particle state(four-)momentum’.
In the quantumfield theory of free massive particles the phas of the quantumfield that determine the trajectory and the phas of the particle states match so that the trajectory momentum and the particle state momenta correspond.In I and a previous work[2],a well-known construction of quantumfields inflat spacetime[3]is modified by including more general translation reprentations.This should be allowed since translations along with rotations and boosts form the group of spacetime symmetries connected to the identity.The paration of particle state and trajecto
ry momenta can be traced back to the fact that any translation prerves all coordinate differences.
郁金香小说The modification introduces additional free parameters.Then assumptions are made in I so that the trajectory momentum and particle state momentum can explain the cosmic ray spectrum by having one energy greater than the other in a gravitationalfield.江南水乡国画
The time component of momentum is the total energy.For the high energy particles considered here,the total energy is much larger than the rest energy and the kinetic energy is just a little less than the total energy.So we can refer to either total or kinetic energy as simply the‘energy’.
In I,the particle state energy¯E depends on the trajectory energy E and the gravitational potentialφ.To the accuracy needed here,one has from I the relation
¯E=E(1−4φγ2),(1) where the gravitational potential includes a factor of the square of the speed of light c to make the quantity unitless andγis the relativistic gamma,γ=1+E/mc2≈E/mc2,since the rest energy mc2is small compared to E.The value of the gravitational potential at the Earth’s surface is−1.06×10−8,referenced to a null potential in interstellar space.Becau |φ|is so small,terms of higher order inφhave been dropped.
To apply Eq.(1)to a CR proton primary,it is argued that the trajectory energy E is the proton’s energy upon acceleration most likely in interstellar space at a supernova remnant’s shock wave.[4,5]The proton then travels in the disk of the Galaxy with this same trajectory energy E,trapped by the galactic magneticfield.As it approaches the Earth,the trajectory energy E increas by a negligible amount due to the conventional effects of gravity.Thus the CR proton strikes the Earth’s atmosphere with a trajectory energy E,esntially unchanged since its acceleration at the supernova remnant.
In interstellar space,the particle state energy¯E is the same as the trajectory energy E becauφvanishes in interstellar ¯E=E whenφ=0.As the CR proton approaches the Sun and Earth,the particle state energy begins to increa becau the
A PROBLEMS3 potentialφdrops below zero.When the proton interacts with the atmosphere it delivers the energy¯E creating a CR shower.For a trajectory energy of E=2PeV=2×1015 eV,the proton deposits¯E=300EeV=3×1020eV of particle state energy as can be verified with Eq.(1).The trajectory energy2PeV is near the limit of expected energies from supernova remnant accelerations,and the300EeV energy is the energy of the most energetic CR particle so far detected.[6]Thus an ultrahigh energy CR proton primary is in an ultrahigh energy particle state only when near the Sun and Earth.
One indication that the model in I is wrong is the obrved anisotropy of incident CRs. The Pierre Auger Collaboration recently determined that ultrahigh energy CRs em to arrive from Active Galactic Nuclei or other extragalactic sources.[7]
However,a stronger test of the model is expected when the Large Hadron Collider(LHC) [8]becomes operational.See Fig.1for a graph of the energy ratio¯E/E from Eq.(1)in the upper energy region of currently operating or propod proton accelerators.
For an E=7TeV proton trajectory energy at the LHC,Eq.(1)predicts a proton particle state energy of23.5TeV.The proton bunches move,as designed,through the magnets and other electromagneticfields with nominal7TeV trajectories.But when they interact with other protons or a target,the particle state energy applies and they deliver23.5TeV per proton to the target.By comparison,an E=1TeV trajectory proton at the Tevatron[9] would gain only5%and deliver1.05TeV per proton,not much more than expected.The predicted factor of more than3in beam energy from7to23.5TeV at the LHC should make it obvious whether or not the assumptions made in I could be valid.
A Problems
1.According to the LHC website[8],a proton beam can have2808bunches with1.15×1011 protons each.Find the beam energy in megajoules and liters of gasoline if each proton has
(a)7TeV of energy and(b)23.5TeV of energy.
2.To the level of approximation in Eq.(1),the spatial momentum has the same ratio of particle state to trajectory values.Thus¯pµ=pµ(1−4φγ2),where the index runs overµ∈{1,2,3,4}={x,y,z,t}in a Minkowski coordinate system.Find the particle state mass¯m for a proton with a trajectory energy of7TeV.[The particle states in I are‘effective particle states’derived from the proton particle states.]
3.By arching experimental results,find theflux of cosmic rays incident on the Earth’s atmosphere that have an energy of23.5TeV.Give the answer in units of the number of particles per square meter per steradian per TeV per day.
A PROBLEMS 4
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3.35log E E in eV E E Particle State Energy to Trajectory Energy Ratio
Figure 1:The Two Energies for a Proton.A proton or other massive particle,in Ref.[1],‘I’,has two energies who ratio is ¯E/E =1−4φγ2,where φis the gravitational potential divided by the square of the speed of light and γis the relativistic gamma of the proton’s trajectory.The trajectory energy E and the particle state energy ¯E
are nearly equal up to about 1TeV.Doubling is often ud as a threshold;here doubling occurs at E =4.6TeV giving ¯E
=9.2TeV.Given 7TeV as a possible proton energy at the LHC,collisions should deliver a particle state energy per proton of 23.5TeV (=3.35×7).For protons at the highest energy obrved for cosmic rays,the ratio is on the order of 105.
太平轮彼岸
REFERENCES5 References
[1]R.Shurtleff,on-line article,arXiv:0801.0071[astro-ph],(2008).
[2]R.Shurtleff,on-line article,arXiv:hep-th/0702023,(2007).
[3]S.Weinberg,The Quantum Theory of Fields,Vol.I(Cambridge University Press,Cam-
bridge,1995),Chapter5and references therein.
[4]See,for example,Y.Uchiyama et al.,Nature449,p.576(2007)and references therein.
[5]See,for example,A.M.Hillas arXiv:astro-ph/0607109v2,(2006)and references therein.
[6]D.J.Bird,et al,Phys.Rev.Lett.71,3401(1993).
[7]The Pierre Auger Collaboration(9November2007)Science318(5852),938;ArXiv
preprint:arXiv:0711.2256v1[astro-ph]
[8]The Large Hadron Collider Project[h/lhc].
[9]Fermi National Accelerator Laboratory,Tevatron Department,
女生素描v/tevatron/.
