課程名稱︰粒子物理一
課程性質︰選修
課程教師︰蔣正偉
開課學院：理學院
開課系所︰物理學研究所
考試日期（年月日）︰2020年11月09日
考試時限（分鐘）：130分鐘
（如未明確表示，則不予發放）
試題 :
PHYS8033 Particle Physics I
Midterm Exam
INSTRUCTIONS
This is an open-notes (your own notes only), open-book (Thomson only), 120-
minute exam. To avoid any misunderstanding, ask if you have any question about
the problems or notations. The convention of summation over repeated indices
is used throughout, unless otherwise noted. The score of each sub-problem is
given by the number in square brackets at the beginning.
PROBLEMS
1.[20 points] The Pauli matrices has the following fundamental product rule
σ_iσ_j=δ_{ij}+iε_{ijk}σ_k,
where i,j,k,∈{1,2,3}.
(a)[10] Prove the completeness relation for the Pauli matrices:
σ^i_{αβ}σ^i_{γδ}=2δ_{αδ}δ_{βγ}-δ_{αβ}δ_{γδ}
(b)[10] The fundamental representation of the SU(2) generators is given in
terms of the Pauli matrices S_i=\vec{σ}/2. Show that the fundamental
representation is a real representation.
2.[30 points total] Quarks and anti-quark form respectively triplet and anti-
triplet representations inder the gauged color SU(3) group. Gluons as the
gauge bosons form a color octet. As we know, normal matter existing in Nature
has to be color neutral.
(a)[10] Explain whether the bound state of two gluons can or cannot exist in
Nature.
(b)[10] The pentaquark discovered in July 2015 is P^+_c with the content
c\bar{c}uud through the decay process of Λ^0_b→P^+_cK^-. (The contents
of Λ^0_b and K^- are bud and s\bar{u}, respectively.) Explain why such a
state is allowed by color SU(3). Also, explain whether the baryon number
and isospin are conserved in the reaction.
(c)[10] Suppose twe ignore the mass gap between the charm quark and the three
light quarks and enlarge the flavor SU(3) symmetry to flavor SU(4) symme-
try. What are three smallest possible flavor SU(4) representations to
which P^+_c could belong?
3.[20 points total] Consider the Lorentz group in the following problems.
(a)[10] Derive that for the vector representation, the Lorentz generators
(J^{μν})^ρ_σ = i[g^{μρ}δ^ν_σ-g^{νρ}δ^μ_σ] .
(b)[10] Show explicitly that the charge conjugation of a left chiral Weyl
field transforms like a right chiral Weyl field.
4.[30 points total] The free theory of a real scalar field φ is given by
L=1/2(∂_μφ)(∂^μφ)-1/2m^2φ^2.
Suppose the theory has a global symmetry under spacetime translations
x^μ→x^μ+ε^νδ_ν^μ.
(a)[10] What is the form variation, δ_Pφ≡φ'(x)-φ(x), of hte scalar field
φ?
(b)[10] What is the variation in L due to the above form variation of φ?
(c)[10] If m^2=0, the theory has the symmetry φ→φ+a, where a is some real
constant. What is the conserved current associated with this symmetry?