Quantum Field Theory and Jet Phenomena · Quantum Field Theory and Jet Phenomena ... tivistic...
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Quantum Field Theory and Jet Phenomena David Elofson Mentor: Dr. Edward Deveney Department of Physics, Bridgewater State University Abstract As part of my 2015 summer NSF research experience for undergraduates (REU) at Duke University, I wrote code to analyze particle-physics experimen- tal results from the Large Hadron Collider (LHC) at CERN. I analyzed jets as part of a large scale multinational experiment called ATLAS. Jets are quite literally groups of particles with similar trajectories that have been traced by the sophisticated detectors in this experiment. Each jet and how it interacts with the known fields in each detector tell the story of the subatomic particles and their fundamental interactions in the collision. Particle physics at this ex- treme ultra-relativistic energy level can only be understood with what is called relativistic Quantum Field Theory (QFT). Unfortunately, QFT is really only encountered at the graduate level and very often skipped by Ph.D.s if they are not in theory. It is certainly not what is often encountered as an undergradu- ate. This poster will discuss some of the theory that I learned over the last two semesters and how it leads to the calculations of lifetimes and scattering cross sections of interactions related to the jets that I studied. CERN CERN is currently the largest and most powerful particle accelerator in the world. It is a collaboration of 113 countries to run particle physics experiments to test the standard model. ATLAS is one of four exper- iments and is a large, 7000 ton detector that measures the energy and momentum of particles created in collisions. Figure 1: Most of the LHC’s 27km is located in France, but the center of operation for the ATLAS detector is in Switzerland. CERN can now reach energies of 13 TeV. Creating Massive New Particles at CERN The purpose of CERN is to create new massive particles using high energy collisions of electrons or protons. The fundamental concept is illustrated by the diagram of fruit that CERN published. Here, strawberries are accelerated to extremely high velocities (faster than 0.9999c) until they have tremendous amounts of kinetic energy in opposite directions and then forced to collide. The products of the col- lision include many small acorns as well as extremely massive fruits like an apple, a pear and a banana, all of which are detected by the detector. The interaction can be written as s +¯ s → A + B + P + a + ... (1) Figure 2: Strawberries with enough kinetic energy can collide to make much larger fruits like bananas, apples and pears as well as many small acorns. Based on the law of conservation of mass, it is impossible to describe where all of the extra mass in the products came from. The answer to this question however lies in the huge amounts of kinetic energy that was created by the acceleration of the original particles (this is why CERN is so big). In the collision, the extra kinetic energy is converted to mass according to Einstein’s famous relationship E = γmc 2 (2) or this equation E 2 = p 2 c 2 + m 2 c 4 (3) which more clearly shows the relation between kinetic energy, speed (momentum) and mass. This can be written using the conservation of energy for the strawberry collison as p 2 strawberries c 2 + m 2 strawberries c 4 = m 2 fruit out c 4 . (4) Jets We can write the strawberry collision in terms of diagrams called Feyn- man diagrams. The heavier outgoing products will form ”hadronic” jets. In actual high energy physics experiments, the strawberries are e - and e + or a p and ¯ p. They produce a quark and antiquark each of which have a color charge which is carried by a gluon. These gluons then interact with the vacuum and decay into q- ¯ q pairs. As the kinetic energy of quarks start to be similar to nearby quarks, they will combine to form hadrons. Groups of three quarks form baryons while groups of one quark and one antiquark will form a meson. It is important that every final hadron is color neutral. Figure 3: The real particles that are observed/detected are represented by the exter- nal lines in each cloud. Clouds with two lines are mesons and clouds with three lines are baryons. Theory Many parameters of collisions are calculated based on cross-sections ( dσ dΩ ) . These are calculated using Fermi’s golden rule which says dσ dΩ ∝ |M| 2 × (phase space). (5) The phase space is related to the entropy and number of final states, but the real focus is on the value for |M | which stands for an amplitude or probability. It can be written as |M| = |hq 1 q 2 ...q n | ˆ S |p n ....p 2 p 1 i| (6) where p n are initial states (strawberries) and the q n ’s are the final states (apples, bananas, pears and acorns). ˆ S stands for John Wheeler’s scattering matrix which can be derived with a lot of math, but most importantly is defined as ˆ S = T h e -i R d 4 z H I (z ) i (7) where T stands for time ordered and means the process matters go- ing both forward in time as well as backwards [2]. This concept came from the Dirac equation and its solutions with negative energy - Feyn- man claimed that antiparticles are the same as particles moving back- wards in time and therefore have what appear to be negative energies. The Dirac and Schr¨ odinger equations fail to fully explain this process however because both are only single particle theories. Combining equations 5, 6 and 7, it can be said that dσ dΩ ∝ hq 1 q 2 ...q n |T h e -i R d 4 z H I (z ) i |p n ....p 2 p 1 i . (8) Following Griffiths [1] and using the Dyson expansion, The first stage of a jet is the interaction e - + e + → γ → q +¯ q (9) which is ordinary quantum electrodynamics. This would be distin- guished in equation 8 by changing H I to H I QED and only using two states in the bra and ket (p 1 ,p 2 and q 2 ,q 1 ). The Lagrangian for QED is as follows: L = 1 4 F μν F μν + ¯ ˆ ψ ( iγ μ ∂ μ - m ) ˆ ψ - q ¯ ˆ ψγ μ A μ ˆ ψ (10) where only the last term, -q ¯ ˆ ψγ μ A μ ˆ ψ is the QED interaction term. Us- ing Wick’s theorem, the first order (tree level) term for |M | is derived as |M| = Qg 2 e p 1 + p 2 ¯ v (p 2 )γ μ U (p 1 ) ¯ U (p 3 )γ μ v (p 4 ) (11) which represents p 1 p 3 p 2 q p 4 e - e + ¯ q q Figure 4: An electron and positron collide creating a photon which decays into a quark and antiquark. This value can now be plugged into equation 5 to calculate a value for dσ dΩ . The cross sections are then integrated to yield a σ total and the ratio of cross sections can be compared to experimental data as shown by Griffiths and given here. Figure 5: R is plotted against electron energy (in GeV). Griffiths used this experi- mental data to confirm the theoretically calculated ratios.[1] Conclusion In order to do the theoretical calculations relating to jet experiments, I had to, in addition to taking quantum mechanics I and II, also take a year of quantum field theory. QFT introduces the second quantiza- tion which turns the wave-like quantum fields of first quantization into operator-valued particle fields that are inherently many particle. Rela- tivistic quantum field theory was followed though second quantization, Wick’s theorem, Feynman diagrams, renormalization, gauge fields and spontaneously broken symmetry. Using these concepts I was able to follow a complete jet calculation from Griffiths. References [1] David J. Griffiths. Introduction to Elementary Particles. John Wiley and Sons, Inc., New York City, New York, 1987. [2] Tom Lancaster and Stephen J. Blundell. Quantum Field Theory for the Gifted Amateur. Oxford University Press, Oxford, United Kingdom, 2014.
Transcript of Quantum Field Theory and Jet Phenomena · Quantum Field Theory and Jet Phenomena ... tivistic...
Quantum Field Theory and Jet PhenomenaDavid Elofson
Mentor: Dr. Edward DeveneyDepartment of Physics, Bridgewater State University
Abstract
As part of my 2015 summer NSF research experience for undergraduates(REU) at Duke University, I wrote code to analyze particle-physics experimen-tal results from the Large Hadron Collider (LHC) at CERN. I analyzed jets aspart of a large scale multinational experiment called ATLAS. Jets are quiteliterally groups of particles with similar trajectories that have been traced bythe sophisticated detectors in this experiment. Each jet and how it interactswith the known fields in each detector tell the story of the subatomic particlesand their fundamental interactions in the collision. Particle physics at this ex-treme ultra-relativistic energy level can only be understood with what is calledrelativistic Quantum Field Theory (QFT). Unfortunately, QFT is really onlyencountered at the graduate level and very often skipped by Ph.D.s if they arenot in theory. It is certainly not what is often encountered as an undergradu-ate. This poster will discuss some of the theory that I learned over the last twosemesters and how it leads to the calculations of lifetimes and scattering crosssections of interactions related to the jets that I studied.
CERN
CERN is currently the largest and most powerful particle accelerator inthe world. It is a collaboration of 113 countries to run particle physicsexperiments to test the standard model. ATLAS is one of four exper-iments and is a large, 7000 ton detector that measures the energy andmomentum of particles created in collisions.
Figure 1: Most of the LHC’s 27km is located in France, but the center of operationfor the ATLAS detector is in Switzerland. CERN can now reach energies of 13 TeV.
Creating Massive New Particles at CERN
The purpose of CERN is to create new massive particles using highenergy collisions of electrons or protons. The fundamental concept isillustrated by the diagram of fruit that CERN published.
Here, strawberries are accelerated to extremely high velocities (fasterthan 0.9999c) until they have tremendous amounts of kinetic energy inopposite directions and then forced to collide. The products of the col-lision include many small acorns as well as extremely massive fruitslike an apple, a pear and a banana, all of which are detected by thedetector. The interaction can be written as
s + s→ A + B + P + a + ... (1)
Figure 2: Strawberries with enough kinetic energy can collide to make much largerfruits like bananas, apples and pears as well as many small acorns.
Based on the law of conservation of mass, it is impossible to describewhere all of the extra mass in the products came from. The answer tothis question however lies in the huge amounts of kinetic energy thatwas created by the acceleration of the original particles (this is whyCERN is so big). In the collision, the extra kinetic energy is convertedto mass according to Einstein’s famous relationship
E = γmc2 (2)
or this equation
E2 = p2c2 + m2c4 (3)
which more clearly shows the relation between kinetic energy, speed(momentum) and mass. This can be written using the conservation ofenergy for the strawberry collison as
p2strawberriesc
2 + m2strawberriesc
4 = m2fruitout
c4. (4)
Jets
We can write the strawberry collision in terms of diagrams called Feyn-man diagrams. The heavier outgoing products will form ”hadronic”jets.
In actual high energy physics experiments, the strawberries are e−
and e+ or a p and p. They produce a quark and antiquark each ofwhich have a color charge which is carried by a gluon. These gluonsthen interact with the vacuum and decay into q-q pairs. As the kineticenergy of quarks start to be similar to nearby quarks, they will combineto form hadrons. Groups of three quarks form baryons while groups ofone quark and one antiquark will form a meson. It is important thatevery final hadron is color neutral.
Figure 3: The real particles that are observed/detected are represented by the exter-nal lines in each cloud. Clouds with two lines are mesons and clouds with three linesare baryons.
TheoryMany parameters of collisions are calculated based on cross-sections(dσ
dΩ
). These are calculated using Fermi’s golden rule which says
dσdΩ∝ |M|2 × (phase space). (5)
The phase space is related to the entropy and number of final states,but the real focus is on the value for |M | which stands for an amplitudeor probability. It can be written as
|M| = |〈q1q2...qn|S|pn....p2p1〉| (6)where pn are initial states (strawberries) and the qn’s are the final states(apples, bananas, pears and acorns). S stands for John Wheeler’sscattering matrix which can be derived with a lot of math, but mostimportantly is defined as
S = T[e−i
∫d4zHI(z)
](7)
where T stands for time ordered and means the process matters go-ing both forward in time as well as backwards [2]. This concept camefrom the Dirac equation and its solutions with negative energy - Feyn-man claimed that antiparticles are the same as particles moving back-wards in time and therefore have what appear to be negative energies.The Dirac and Schrodinger equations fail to fully explain this processhowever because both are only single particle theories.
Combining equations 5, 6 and 7, it can be said that
dσdΩ∝∣∣∣〈q1q2...qn|T
[e−i
∫d4zHI(z)
]|pn....p2p1〉
∣∣∣. (8)
Following Griffiths [1] and using the Dyson expansion, The firststage of a jet is the interaction
e− + e+→ γ → q + q (9)which is ordinary quantum electrodynamics. This would be distin-guished in equation 8 by changing HI to HIQED and only using twostates in the bra and ket (p1, p2 and q2, q1).
The Lagrangian for QED is as follows:
L =1
4FµνF
µν +¯ψ(iγµ∂µ −m
)ψ − q ¯
ψγµAµψ (10)
where only the last term,−q ¯ψγµAµψ is the QED interaction term. Us-
ing Wick’s theorem, the first order (tree level) term for |M | is derivedas
|M| = Qg2e
p1 + p2
[v(p2)γµU(p1)
][U(p3)γµv(p4)
](11)
which represents
p1 p3
p2
q
p4
e−
e+
q
q
Figure 4: An electron and positron collide creating a photon which decays into aquark and antiquark.
This value can now be plugged into equation 5 to calculate a valuefor dσ
dΩ. The cross sections are then integrated to yield a σtotal and theratio of cross sections can be compared to experimental data as shownby Griffiths and given here.
Figure 5: R is plotted against electron energy (in GeV). Griffiths used this experi-mental data to confirm the theoretically calculated ratios.[1]
ConclusionIn order to do the theoretical calculations relating to jet experiments,I had to, in addition to taking quantum mechanics I and II, also takea year of quantum field theory. QFT introduces the second quantiza-tion which turns the wave-like quantum fields of first quantization intooperator-valued particle fields that are inherently many particle. Rela-tivistic quantum field theory was followed though second quantization,Wick’s theorem, Feynman diagrams, renormalization, gauge fields andspontaneously broken symmetry. Using these concepts I was able tofollow a complete jet calculation from Griffiths.
References[1] David J. Griffiths. Introduction to Elementary Particles. John Wiley
and Sons, Inc., New York City, New York, 1987.[2] Tom Lancaster and Stephen J. Blundell. Quantum Field Theory
for the Gifted Amateur. Oxford University Press, Oxford, UnitedKingdom, 2014.
