### 1, i, j = 1

```Exercises for QUANTUM FIELD THEORY II - A.A. 2014/15
Problem 1. Let T a ≡ (T a )ij , a = 1, · · · , N 2 − 1, i, j = 1, · · · , N , be a basis for the
fundamental representation of the Lie algebra of SU (N ), with
[T a , T b ] = f abc T c ,
tr(T a T b ) = −
δ ab
,
2
tr(f a f b ) = −N δ ab ,
tr(f a f b f c ) =
N abc
f .
2
a) Show that the completely symmetric tensor
dabc ≡ i tr(T (a T b T c) )
(1)
- which can be defined for an arbitrary Lie group G and for any representation - is an
invariant real tensor.
b) Noting that the matrices
δ ab
(a b)
1
i T T +
2N
are antihermitian and traceless and that the T a form a basis for such matrices, prove the
identity
1
δ ab
T a T b = f abc T c −
1 − 2idabc T c .
(2)
2
2N
Verify this relation explicitly for the case of SU (2).
c) Using (2) conclude that the Casimir invariant of the fundamental representation, defined by T a T a = −CN 1, is CN = (N 2 − 1)/2N .
Problem 2. Le ϕ ≡ ϕi be complex scalar fields transforming in the fundamental representation of SU (N ).
analyzing the multiple product N × N × N × N , show that there are only two independent
quartic (particle number preserving) invariants, i.e. invariants of the kind ϕ† ϕ† ϕϕ.
b) Show that the real polynomials
(ϕ† ϕ)(ϕ† ϕ),
(ϕ† T a ϕ)(ϕ† T a ϕ),
(ϕ† T a T b ϕ)(ϕ† T a T b ϕ)
are SU (N )-invariant. NB: these polynomials correspond to the two invariant tensors
δ im δ jn and (T a )im (T a )jn , contracting ϕ†i ϕm ϕ†j ϕn .
Problem 3. Let ϕ ≡ ϕa be complex scalar fields in the adjoint representation 8 of SU (3).
a) Using that 8×8 = (1+8+27)S +(8+10+10)A , see e.g. [Slansky, Phys. Rep.], determine
the number of independent real invariant quartic interactions of the kind ϕ† ϕϕ† ϕ, as in
Problem 2. Hint: the product R1 × R2 contains a singlet only if R1 = R2 .
b) Write three independent invariant quartic interaction terms.
Problem 4. Consider the lagrangian of scalar QCD with gauge group SU (N )
1
1
L = − F αµν F a µν + (Dµ ϕ)† Dµ ϕ − m2 ϕ† ϕ − P (ϕ, ϕ† ) ≡ − F αµν F a µν + Lϕ ,
4
4
1
(3)
where P is an invariant quartic polynomial and the scalars transform in a generic irreducible representation Θa of SU (N ):
Dµ ϕ = (∂µ − gAµ )ϕ,
Aµ = Aaµ Θa .
a) Determine the covariantly conserved color currents J aµ and the conserved (N¨other)
color currents j aµ .
b) Writing the renormalized scalar lagrangian as
Lren
= Zϕ (∂µ ϕ† ∂ µ ϕ − (m2 + δm2 )ϕ† ϕ) + gZ1ϕ (ϕ† Aµ ∂µ ϕ − ∂µ ϕ† Aµ ϕ)
ϕ
−g 2 Z2ϕ ϕ† (Aµ Aµ )ϕ − Pe(ϕ, ϕ† ),
(4)
derive the relations, alias Slavnov-Taylor identities, involving Zϕ , Z1ϕ and Z2ϕ , analogous
to Z3 /Z1 = ZΨ /Z1Ψ etc.
c) Draw all one-loop Feynman diagrams that contribute to the renormalization of the
four-gluon amplitude.
d) Assuming that the scalars transform in the fundamental representation N , discuss the
relation between Pe(ϕ, ϕ† ) and P (ϕ, ϕ† ). In particular, how many independent coupling
constants appear in these polynomials, if you want the theory to be strictly renormalizable?
e) Draw all one-loop Feynman diagrams contributing to the renormalization of the quartic
scalar interaction. Analyze the group-theoretical structure of their divergent parts and
check explicitly if they are consistent with the answer to point d).
Problem 5. Consider the lagrangian of a gauge theory with generic gauge group G, with
Nf fermions and Ns complex scalars transforming respectively in the representations T a
and Θa ,
1
Lf s = − F αµν F a µν + Ψ(iγ µ Dµ − M )Ψ + (Dµ ϕ)† Dµ ϕ − m2 ϕ† ϕ − P (ϕ, ϕ† ),
4
(5)
where
Dµ ϕ = (∂µ − gAaµ Θa )ϕ,
Dµ Ψ = (∂µ − gAaµ T a )Ψ,
P (ϕ, ϕ† ) = CIJM N ϕ†I ϕJ ϕ†M ϕN .
a) Write down the Feynman rules of the theory.
b) Show that, in Lorenz-Feynman gauge with λ = 1, using dimensional regularization
and relying on the minimal subtraction scheme, at one loop the gluon wave-function
renormalizes according to
10
8
2
g2
Tadj − Nf Tf − Ns Ts ,
Z3 = 1 +
(4π)2 ε 3
3
3
where Tf and Ts are the Dynkin indices of the representations T a and Θa . Hint: use the
known results of a theory without scalars.
c) Using the known results of a theory without scalars, derive the one-loop β-function
11
4
1
g3
β(g) = −
Tadj − Nf Tf − Ns Ts .
(4π)2 3
3
3
2
d) Derive the one-loop β-function β(α) for the strong coupling constant α = g 2 /4π.
e) How many color-scalars Ns in the adjoint (or fundamental) representation of SU (3)
could we add at most to the Standard Model, to keep QCD asymptotically free?
Problem 6. Consider the lagrangian (5) of Problem 5, where P (ϕ, ϕ† ) is the most general
G-invariant quartic polynomial in the scalars.
a) Is Lf s strictly renormalizable? Consider, as example, real scalars ϕ ≡ ϕa in the adjoint
representation of G, with (Θa )bc = −f abc , and consider the interaction terms
X = ΨT a Ψϕa ,
Y = dabc ϕa ϕb ϕc ,
where dabc is given in (1). Can X (a Yukawa coupling) and Y appear as divergent one-loop
counterterms? Can they appear at higher loops? Answer using a symmetry argument.
b) Consider the lagrangians
L1 = Lf s + λX,
L2 = Lf s + µY.
where λ and µ are coupling constants. Are the lagrangians L1 and L2 strictly renormalizable? (Hint: remember the concept of renormalizable and super-renormalizable
interactions). If the answer is negative draw some divergent Feynman diagram whose
renormalization requires the introduction of interaction terms not present in L1 and L2 .
c) Specify the discussion of the previous points to the particular case G = SU (2).
Problem 7. Consider the lagrangian L1 of Problem 6.
a) Draw all one-loop Feynman diagrams that contribute to the renormalization of the
three-point function ΨΨA.
b) Introducing the renormalized minimal-interaction lagrangian Lren = −igZ1Ψ Ψγ µ Aµ Ψ,
compute the renormalization constant Z1Ψ at one-loop order. Hint: using the known
result in the absence of scalars, one concludes that
Z1Ψ = 1 −
g2
2
(2C
+
2C
)
+
o
λ
,
f
(4π)2 ε
where Cadj and Cf are the Casimir invariants of the adjoint and fermion representations
respectively.
c) Draw all one-loop Feynman diagrams that contribute to the renormalization of the
three-point function ϕ† ϕA.
d) Draw all one-loop Feynman diagrams that contribute to the renormalization of the
four-point function ϕ† ϕAA.
e) Introduce the renormalization constants Z1ϕ and Z2ϕ for the scalar interactions, as in
(4). Show that they have the general one-loop structure
Z1ϕ = 1 +
1
a1 g 2 + b 1 λ 2 ,
ε
Z2ϕ = 1 +
1
a2 g 2 + b 2 λ 2 ,
ε
where ai and bi are numerical constants.
f) Prove that b1 = b2 . Hint: use the Salvnov-Taylor identities derived in Problem 4, together with the fact that at one loop scalar fields do not contribute to the renormalization
a
of the ghost lagrangian ∂µ C Dµ C a .
3
Problem 8. Consider the gauge group G = SO(N ) and denote the YM potentials by
IJ
JI
AIJ
µ , where Aµ = −Aµ and I, J = 1, · · · , N . Choose the fermions in the fundamental
“vector” representation of SO(N ), Ψ ≡ ΨI , and denote the local SO(N ) transformation
parameters by ΛIJ (x) = −ΛJI (x). For an infinitesimal transformation one has thus
δΨI = ΛIJ ΨJ .
(6)
a) Derive the form of the infinitesimal transformations of the potentials, δAIJ
µ , imposing
that the covariant derivative is given by
J
Dµ ΨI = ∂µ ΨI − AIJ
µ Ψ .
IJ
.
b) Derive the form of the YM field-strength Fµν
IJ
I
IJ
related to the general construction of
c) How are the expressions of Dµ Ψ , δAµ and Fµν
non-abelian gauge theories.
d) Add real scalar fields ϕI transforming in the fundamental representation, too, and
I
I
construct the most general renormalizable lagrangian using the fields AIJ
µ , Ψ and ϕ . Is
a Yukawa coupling allowed? Hint: SO(N ) is a euclidean version of the Lorentz-group,
and as the latter it has only two invariant tensors for “vector”-indices, i.e. δ IJ and εI1 ···IN .
e) Consider the gauge group G = SO(10) with fermions in the “spinor”-representation 16,
Ψ ≡ Ψi , i = 1, · · · , 16. If you want to couple these fields to scalars via a Yukawa coupling,
which irreducible representations could you choose for the scalars? In particular, would
a (real) scalar multiplet of the form ϕIJKM , completely antisymmetric in all four vector
indices, do the job? How many independent Yukawa couplings could you construct for
each chosen representation of the scalars? Hint: use the products of SO(10) irreducible
representation listed in [Slansky].
f) Find the BRST-transformation δC IJ of the ghost-field C IJ (replacing ΛIJ ) and verify
that it is nihilpotent, i.e. δ 2 C IJ = 0.
4
```