Wick rotation

5.2 Wick rotation

With this section we start the prescription of the mathematical formalism of regularisation as the first step towards renormalisation.

We have shown in the third and fourth chapter that we can restrict our investigation to 1PI truncated diagrams, the so called proper vertex functions. We shall use φ⁴-theory as the most simple example of a quantum field theory which has no symmetries (especially no gauge symmetries) to be fulfilled along the procedure of regularisation. We shall also use the path integral formalism Feynman rules which corresponds to a non-normal-ordered Lagrangian.

From the point of view of renormalisation this means not more than that we have also to regularise the vacuum of the theory and this is done by an additional mass renormalisations.

The reason for that is that the path integral formalism is more simple in the case of gauge theories.

Now let us take the most simple 1-loop-diagram inφ⁴-theory, which is the tadpole contribution to the self energy shown in fig. 5.1.

−iΣ⁽¹⁾ =

l

Figure 5.1: The 1-loop tadpole contribution to the self-energy

Due to the Feynman rules we have found in the third chapter within the canonical operator formalism and in the fourth in the path integral formalism that the analytic expression of this diagram is given by

Σ⁽¹⁾= iλ 2

Z d⁴l (2π)⁴

1

l²−m²+ iη. (5.1)

The first complication we have to get rid of is the complicated pole structure for on-shell loop momenta. Now we find it again important to have used the iη-regulator³, which came into the game when we calculated the propagator in the operator formalism as well as in the path integral formalism. In the former the reason we had to plug in the iη was the time ordering operator in the definition (3.131) of the propagator, in the latter it was used to project out the vacuum expectation value as shown in section 1.10. To say it from the mathematical point of view it is the correct causal (i.e. the Feynman-St¨uckelberg-) weak limit of the propagator in momentum space.

It is clear that the iη-prescription helps us to take the correct integral over l₀. Thus we look on figure 5.2, where l₀ is depicted as a complex variable in its plane. The two poles

±ω(~p) are slightly shifted due to the iη. In (5.1) we are told to integrate over the reall₀-axis.

On the other hand the integration over the pathC vanishes due to the residuum theorem, because there are no poles inside this path thanks to the i-shifts. Now since the integrand is

∼=l0→∞1/l₀² the two quarter circles do not contribute to the integral. Thus we have Z

C

dl0f(l0) = 0 ⇒ Z ∞

−∞

dl0f(l0)− Z i∞

−i∞

f(l0) = 0, (5.2)

3Beginning with this chapter we change the regulator in the Green’s functions from ito iηbecause usually one usesd= 4−2for the space time dimension.

Iml0

Rel0

ω−i0

−ω+i0

C

Figure 5.2: The Wick rotation: The0-component of the loop momentum as a complex variable and the pole structure of the integrand in (5.1)

where the limits in the 2nd integral mean the integral along the imaginaryl₀-axis from−i∞ to i∞ (the sign comes from the fact that this path is run in the opposite direction as part of C).

Now substituting in the second integrall₀ = il₄ we find Z ∞

−∞

dl₀f(l₀) = i Z ∞

−∞

dl₄f(−il₄). (5.3)

This rule, which allows one to take the causal pole structure of the propagators into account by integrating over the complex l₀-axis instead along the real axis, is called Wick-rotation, because it can be seen as a rotation of the real path of integration to the complex axis.

Now introducing ¯l= (l₁, . . . , l₄) as a new four vector, we can write (5.1) as Σ⁽¹⁾= λ

2

Z d⁴¯l (2π)⁴

1

¯l²+m², (5.4)

where ¯l²=l²₁+· · ·l₄²is theEuclidean inner scalar product of the four-vector ¯l. Thus the Wick-rotation means to go from causal quantum field theory to Euclidean field theory. This feature of the i-description we have already seen in section 1.10 where we could either rotate the time integral over the Lagrangian to the complex axes (leading to the Euclidean description) or rotate only with a tiny angle (leading to the i-description).

Now we introduce four dimensional spherical coordinates. Since the integrand depends only on ¯l² we leave out the angular part of the integral, which will be discussed in detail in the next section. It gives only the surface of the three-dimensional sphere Ω₃ in the four-dimensional Euclidean space. Now we see the trouble with the integral explicitly, which is divergent, because the volume element readsL³dL (with L =√¯l²) while the integrand goes only with 1/L² for Large Euclidean loop momenta.

In order to calculate this most simple example to the end we make a very crude regularisation by simply cutting the integral off at a loop momentum Λ, called thecut-off. From the power

5.2 · Wick rotation

counting above we expect the integral to diverge with Λ² for Λ→ ∞. Then we can write:

Σ⁽¹⁾_reg = λ 32π⁴Ω₃

Z _Λ

0

dL L³

L²+m² = λΩ₃ 64π⁴

Λ²+m²ln m² Λ²+m²

. (5.5)

This shows that the naivepower counting was right for our simple example. The divergence is thus calledquadratic divergence.

Now we can use the recipe given in the last section. We try to absorb this divergent contri-bution of the radiation corrections to the bare parameters of the Lagrangian. This must be done by adding a counter term to the interacting Lagrangian which is of the same form as a term which is already in the bare Lagrangian (it is not important if the term in the bare Lagrangian is in the “free part” or the “interacting part”). Because this counter term should give a contribution to the self energy, it has to be∝φ².

Now we see that to make the whole contribution (i.e. the sum of (5.5) and the counter term) finite, we can set

L⁽¹⁾

CT = Σ⁽¹⁾reg+ const.

2 φ². (5.6)

This counter term has to be treated as a vertex in the interaction part of the Lagrangian, leading to the counter term Feynman-rule in figure 5.3.

= i^{Σ(1)reg+const.}₂

Figure 5.3: The 1-loop counter-term contribution to the bare Lagrangian, which compensates the infinity of the tadpole diagram.

Due to the Feynman rules the contribution of the counter term to the self energy is given by:

−iΣ⁽¹⁾_CT= i(Σ⁽¹⁾_reg+ const.). (5.7) Here we have taken into account the factor 2 from connecting the legs to the external points (the legs have of course to be amputated). We find then for the whole contribution

Σ⁽¹⁾_ren= Σ⁽¹⁾_reg−Σ⁽¹⁾_reg−const. (5.8) This is indeed an arbitrary constant contributing to the bare mass of the particles described by the quantum field φ, which is finite for Λ → ∞, because it does not depend on Λ at all.

We have expected the arbitrariness of the finite part of the counter term, because the only thing we have to fix is the divergent part, which has to be cancelled completely with help of the counter term.

Now in our case it is also simple to interpret this arbitrariness. From (4.222), Dyson’s equa-tion, we know that the approximation for the two-point Green’s function to first order in the coupling constant is given by

G⁽¹⁾(p) = 1

p²−m²−Σ⁽¹⁾ren+ i. (5.9) Now it becomes clear that the physical mass squared is given by the pole of the Green’s function of the particle, which means that we have

m²_phys =m²+ Σ⁽¹⁾_ren. (5.10)

Now choosing the constant we define a certainrenormalisation scheme. This can be done such that Σ⁽¹⁾_ren= 0, which is called thephysical renormalisation scheme or theon-shell scheme. In that case we set the bare mass equal to the physical mass order by order of the Dyson-Wick series. We may also chose another scheme. The only point is that we have to compensate the part of the regularised Feynman integrals which is infinite for Λ→ ∞ with help of a counter term in the Lagrangian. The counter term should be of the same form as a term which was in the bare Lagrangian in the beginning in order that we can absorb the infinities to the bare parameters of the theory. The numerical values of the physical parameters have to be fitted to experiments, they are not given from first principles of relativistic quantum field theory.

As we shall see later the only known principles which restrict the choice of parameters are gauge invariance and renormalisability.

From our simple example the whole idea of renormalisation can by summarised now. Our first step was to handle the pole structure of the Green’s function in order to keep the causality of the theory with help of theWick-rotation. Then we have seen that the integral is indeed divergent and to give it a definite meaning we had toregularise this integral. Here we did this by introducing a cut-off Λ for the Euclidean four-momentum. The reader should keep in mind that we have introduced amomentum scale into the theory when we keep Λ finite. The next step was to renormalise the integral making the first order Tadpole-contribution to the self energy finite for Λ→ ∞by absorbing the infinity of order Λ² for Λ→ ∞into thebare mass of the particle. After this we could take the physical limit Λ→ ∞. The physical renormalisation scheme was in this case nothing else than enforce the normal ordering description of the path integral which makes the Tadpole contribution vanish from the very beginning within the canonical operator formalism.

We can now give a further outlook of the mathematical solution of the problem of infinite Feynman integrals: For a given theory we have to show that all infinities can be cancelled with help of adding counter-terms to the bare Lagrangian which shuffle the infinities when taking the regularisation parameter (in our example the cut-off) to the physical limit (in our case this was Λ → ∞) into a finite set of bare parameters of the theory as there are masses, coupling constants and wave function normalisation constants. Thus a necessary condition for renormalisability is that only a finite set of proper amputated diagrams should be divergent. For φ⁴-theory only the 2-point and the 4-point function are allowed to be divergent. If another diagram would be divergent, and this divergence had to be compensated by a counter-term which goes withφ⁶ for example, this would violate the renormalisability of φ⁴-theory or it would at least force us to introduce a φ⁶-vertex into the bare Lagrangian from the very beginning. But we shouldn’t be forced to introduce infinite many terms into the bare Lagrangian and thus also to use an infinite set of parameters to describe the interaction of the particles involved. Although we might be forced to introduce a finite number of such bare terms, we can define a renormalisable quantum field theory such that it is possible to start with a bare Lagrangian with a finite number of parameters.

A first superficial hint which diagrams are divergent is given by power counting of the loop integrals. Beginning with the next section we shall solve the problem of renormalisation. The first step is to introduce a simple regularisation scheme which gives us a recipe to calculate systematically the Feynman integrals and to extract the infinities in order to find the correct counter terms for the bare Lagrangian. Because it is so useful for the renormalisation of gauge theories we shall use dimensional regularisation in these notes.