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.