An Overview of Interacting Quantum Field Theories, Renormalizability of Quantum Field Theories

Home , Coupling constant, Minimal coupling

PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 01

Monday, January 06, 2020

Topics: An Overview of Interacting Quantum Field Theories, Renormalizability of Quantum Field Theories.

An Overview of Interacting Quantum Field Theories

Free Field Theories

Among the set of all possible quantum field theories that do not contain interactions or non- linearities, two kinds of field theories immediately stand out: one involving the Klein-Gordon field and the other containing the Dirac field. Let us briefly go though them, refreshing what we have learned in the previous course, QFT 1.

The Klein-Gordon Field

The Klein-Gordon field is a scalar field φ with mass m. The Lagrangian of this field theory is

1 1 L = (∂ φ)2 − m2φ2. (1) 2 µ 2

The kinetic part of the above Lagrangian can be split into the form

2 2 2 (∂µφ) = φ˙ − (∇φ) . (2)

From the above Lagrangian we get the following equation of motion for the scalar ﬁeld

∂2 − ∇2 + m2 φ = 0 =⇒ ∂µ∂ + m2 φ = 0. (3) ∂t2 µ

The above equation is known as the Klein-Gordon equation. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

Introducing the canonical momentum density conjugate to φ(x)

π(x) = φ˙(x) (4) we can write down the Hamiltonian Z H = d3x H Z 1 1 1 = d3x π2 + (∇φ)2 + m2φ2 . (5) 2 2 2

These three terms in the above Hamiltonian represent the energy cost of moving in time, the energy cost of shearing in space and the energy cost of having the ﬁeld around at all. The Klein-Gordon equation admits plane-wave solutions

φ(x) ∼ e±ipx. (6)

It represents arbitrary number of particles, each evolving independently. We can expand the scalar ﬁeld in a series of plane waves, and ladder operators

Z 3 d p 1 −ipx † ipx φ(x) = 3 p ape + ape , (7) (2π) 2Ep ∂ π(x) = φ(x). (8) ∂t

The creation and annihilation operators obey the commutation relations

† 3 (3) [ap, aq] = (2π) δ (p − q), (9)

[ap, aq] = 0, (10) † † [ap, aq] = 0. (11)

The Feynman propagator is

Z 4 d p −ip·(x−y) DF (x − y) = DeF (p) e (2π)4 Z d4p i = e−ip·(x−y). (12) (2π)4 p2 − m2 + i

We can deﬁne this as

DF (x − y) ≡ h0|T φ(x)φ(y)|0i, (13) where T denotes the “time-ordering". Upon using the step function, we can write the time ordered product as

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We can also consider a complex valued scalar ﬁeld,

ϕ = φ1 + iφ2 (15) and with φ1 and φ2 denoting real scalar ﬁelds. We have the Lagrangian

2 2 2 L = |∂µϕ| − m |ϕ| . (16)

The equation of motion for this Lagrangian is again the Klein-Gordon equation

µ 2 ∂ ∂µ + m ϕ = 0, (17) µ 2 ∗ ∂ ∂µ + m ϕ = 0. (18)

The ﬁeld ϕ∗ can be treated as the anti-particle of ϕ and vice versa.

The Dirac Field

Let us consider the Dirac ﬁeld ψ of mass m. The Lagrangian is

µ L = ψ (iγ ∂µ − m) ψ, (19) where ψ ≡ ψ†γ0. (20)

The Euler-Lagrange equation for ψ gives the Dirac equation

µ (iγ ∂µ − m) ψ(x) = 0. (21)

The Euler-Lagrange equation for ψ gives the same equation, in Hermitian-conjugate form

µ −i∂µψ(x)γ − mψ(x) = 0. (22)

From the Lagrangian we see that the canonical momentum conjugate to ψ is iψ†. The Hamil- tonian is Z H = d3x ψ (−iγ · ∇ + m) ψ Z = d3x ψ† −iγ0γ · ∇ + mγ0 ψ. (23)

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The ﬁelds can be expanded in a plane wave basis

Z d3p 1 X ψ(x) = as us(p)e−ipx + bsvs(p)eipx , (24) (2π)3 p p p 2Ep s Z d3p 1 X ψ(x) = as†u¯s(p)eipx + bs†v¯s(p)e−ipx . (25) (2π)3 p p p 2Ep s

The creation and annihilation operators obey the anti-commutation relations

r s† r s† 3 (3) rs {ap, aq } = {bp, bq } = (2π) δ (p − q)δ ., (26) with all other anti-commutators equal to zero. The Feynman propagator is

Z 4 d p −ip·(x−y) SF (x − y) = SeF (p) e (2π)4 Z d4p i(p/ + m) = e−ip·(x−y). (27) (2π)4 p2 − m2 + i

We can deﬁne the propagator as a time ordered product of ﬁelds

SF (x − y) ≡ h0|T ψ(x)ψ(x)|0i. (28)

We have  ψ(x)ψ(y) if x0 > y0, T ψ(x)ψ(y) = (29) −ψ(y)ψ(x) if y0 > x0.

Turning on Interactions

The ﬁeld equations given above, the Klein-Gordon equation and the Dirac equation, are linear equations. There are no interactions. What this is telling us that the individual particles of the theory exist in their isolated modes. That is, they are not aware of the existence of each other, and particles of other species. However, the formalism of free theory is extremely important. The free theory forms the basis for doing perturbative calculations in the interacting theory. We need to turn on interactions to model the real world. We can add some interaction terms to the Lagrangian. However, these terms must respect causality and Lorentz invariance. The terms describing the interactions will be of the form Z 3 Hint = d x Hint [φ(x)] Z 3 = − d x Lint [φ(x)] . (30)

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Imposing the condition that the interaction Lagrangian is local, we have

[Lint(x), Lint(y)] = 0, (31) for space-like separation. We will also restrict ourselves to theories in which the interaction terms contain only functions of ﬁelds, not their derivatives. We can treat them using a perturbative expansion when interaction strength (or coupling pa- rameter) is weak. Let us look at the three commonly occurring interacting theories.

The Phi-Fourth Theory

The Lagrangian is 1 1 λ L = (∂ φ)2 − m2φ2 − φ4, (32) 2 µ 2 4! where λ, a dimensionless coupling constant, is a measure of the strength of the interaction. Note that if we consider a φ3 interaction, then the energy would not be positive definite. The phi-fourth theory is the simplest of all interacting quantum field theories. The self- interaction of the Higgs field in the standard electroweak theory contains this term. The phi-fourth theory also arises in statistical mechanics. The equation of motion for phi-fourth theory is

λ (∂2 + m2)φ = − φ3, (33) 3! which is a non-linear equation.

Quantum Electrodynamics

The Lagrangian is

LQED = LDirac + LMaxwell + Lint 1 = ψ i∂/ − m ψ − (F )2 − eψγµψA . (34) 4 µν µ

In the above Aµ is the electromagnetic vector potential, Fµν = ∂µAν − ∂νAµ is the electromagnetic ﬁeld tensor, and e = −|e| is the electron charge. If we want to describe a fermion of charge Q, then we need to replace e with Q. If we want to describe several species of charged particles at once, then we simply need to duplicate LDirac and Lint for each additional particle species. We can write QED Lagrangian more simply as

1 L = ψ(iD/ − m)ψ − (F )2, (35) QED 4 µν

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where Dµ is the gauge covariant derivative

Dµ ≡ ∂µ + ieAµ(x). (36)

The QED Lagrangian is invariant under the local (or spacetime dependent) gauge transformation

ψ(x) → eiα(x)ψ(x), (37) ψ(x) → e−iα(x)ψ(x), (38) 1 A → A − ∂ α(x). (39) µ µ e µ

That is, the theory has a symmetry under phase rotations. It is the U(1) gauge symmetry. The equation of motion for fermion is

(iD/ − m)ψ(x) = 0. (40)

This is the Dirac equation coupled to the electromagnetic ﬁeld by the minimal coupling pre- scription. That is, ∂ → D. The Euler-Lagrange equation for the gauge ﬁeld is

µν ν ∂µF = eψγ ψ = ejν. (41)

These are the inhomogeneous Maxwell’s equations, with the current density

jν = ψγνψ. (42)

The Yukawa Theory

The Lagrangian is

LYukawa = LDirac + LKG − gψψφ. (43)

The above Lagrangian is similar to QED but with the photon replaced by a scalar particle φ. Yukawa originally invented this theory to describe nucleons (ψ) and pions (φ). The Standard Model contains Yukawa interaction terms as the couplings of the scalar Higgs ﬁeld to quarks and leptons. Most of the free parameters in the Standard Model are Yukawa coupling constants.

Renormalizability of Quantum Field Theories

We can ask the question: Is the list of interactions inﬁnite?

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Maybe we could write the interactions for the scalar ﬁeld theory as

λ L = n φn. (44) int n!

However, one simple and reasonable axiom comes to our help. It eliminates all but a few of the possible interactions. That axiom is that the theory be renormalizable. In quantum field theories we need to introduce an energy cutoff Λ in the calculations - in the Feynman diagrams involving internal loops. That is, we need to cutoff the integral at some finite momentum Λ. At the end of the calculation we take the limit Λ → ∞, hoping that physical quantities turn out to be independent of Λ. If this is indeed the case, then the theory is said to be renormalizable. Suppose we have interactions where the coupling constants have the dimensions of mass to some negative power. Then, to obtain a dimensionless scattering amplitude, this coupling constant must be multiplied by some quantity of positive mass dimension. It turns out that this quantity is none other than Λ. Such a term diverges as Λ → ∞ rendering the theory non-renormalizable. Any theory containing a coupling constant with negative mass dimension is not renormalizable. Such theories require an infinite number of input parameters to make sense.

Dimensional Analysis to Rule Out Terms

We can use a bit of dimensional analysis to throw out nearly all candidate interactions. Let us use the units where ~ = c = 1. In these units, the action Z S = dx4 L (45) is dimensionless. That is [S] = M 0. (46)

Thus L must have dimension (mass)4. That is the Lagrangian has mass dimension 4. We write it as [L] = M 4. (47)

From the kinetic terms of the various free Lagrangians we look at in the previous section, we note that the scalar and vector ﬁelds, φ and Aµ, have mass dimension 1. That is,

[φ] = M, [Aµ] = M. (48)

We can also see that [ψ] = [ψ] = M 3/2. (49)

We can now tabulate all of the allowed renormalizable interactions. For theories involving only

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µφ3 and λφ4. (50)

The coupling constant µ has mass dimension 1 and λ is dimensionless. We note that terms of the form φn, (51) for n > 4, are not allowed. The reason is that their coupling constants would have mass dimension 4 − n. Let us look at interaction terms including spinor ﬁelds. We see that spinor self interactions are not allowed. The term ψ3 violates Lorentz invariance. It also has mass dimension 9/2. The only allowable new interaction is the Yukawa term

gψψφ. (52)

Let us consider interaction terms involving vector ﬁelds. Possible interactions are

µ eψγ ψAµ. (53)

The scalar QED Lagrangian 2 2 2 L = |Dµϕ| − m |ϕ| , (54) contains µ ∗ 2 2 2 eA ϕ∂µφ and e |ϕ| A . (55)

We can add mass terms to vector ﬁelds

1 m2A Aµ. (56) 2 µ

But care must be taken not to break gauge symmetry. Question: Consider a scalar ﬁeld theory in d spacetime dimensions with Lagrangian density

N 1 1 X 1 L = (∂ φ)2 − m2φ2 − λ φn. (57) 2 µ 2 n! n n=3

1 Show that the mass dimension of λn is d − 2 n(d − 2). It is possible to show that however complicated a fundamental theory appears at very high energies, the low energy approximation to this theory, that we see in experiments, should be a renormalizable quantum ﬁeld theory.

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In practice it is extremely diﬃcult to solve interacting quantum ﬁeld theories exactly. Exceptions are some lower dimensional theories. That also using symmetry properties.

A much simpler and generally applicable approach is to treat the interaction term Hint as a perturbation, and then compute its eﬀects as far as in perturbation theory as is practicable. Some examples of renormalizable theories are: QED, non-abelian gauge theories, phi-fourth theory. Examples of non-renormalizable theories include non-linear sigma models, the V − A current theory of weak interactions, and Einstein’s theory of gravity.

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

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