Solutions to the homogeneous RGE

Now we have found two ways to calculate perturbatively the functionsβ, γ_φ and γ_m to the homogeneous RGE (5.267). Given these functions, it is easy to obtain solutions for this equation since it is a functional extension to the flow equation

∂

∂tS(t, ~x) +~v(~x)∇~xS(t, ~x) = 0 (5.283) which describes the conservation of the quantity S along the orbit of a volume element of a fluid ~x(t;~x0) with the initial condition~x(0) =~x0. Indeed, from (5.283), we find For the RGE (5.267) Γ corresponds to S, the renormalised parameters m², φ, λ to ~x, the functions γ_m, γ_φ, β to ~v(~x). Instead of ∂_t, in (5.283) we have M ∂_M. Thus we set ¯M = Mexp(τ). Then we haveM ∂_M =∂_τ, and the defining equations (5.268) for the RGE functions β,γ_φ and γ_m read

∂τφ=−γ_φ(λ)φ, ∂τm² =−γm(λ)m², ∂τλ=β(λ). (5.286) Given the solution to the third equation,

Z ^¯λ the solutions for the first equation is given by integration:

φ¯=φexp Here ¯Z is a finite renormalisation constant, i.e., (5.288) describes the change of the field’s normalisation due to the change of the mass renormalisation scale from M to ¯M =Mexpτ. Of course, also the mass has to change according to the second equation of (5.286):

¯

Then we can apply (5.285) to the RGE (5.269):

Γ[φ, m², λ;µ²] = Γ[ ¯φ,m¯²,λ,¯ µ¯²]. (5.290) This equation is no surprise, since the generating functional for vertex functions Γ is al-ways equal to the corresponding bare functional and thus its value cannot change when the renormalisation scale M is changed. With (5.287-5.289) we have defined how the coupling and the renormalised mass have to be scaled in order to ensure this independence of the renormalisation scale.

5.13 · Asymptotic behaviour of vertex functions

From (5.290) one immediately obtains the corresponding behaviour of the vertex functions under change of the renormalisation scale:

Γ⁽ⁿ⁾_1...n(m², λ;M²) = δⁿ

δφ₁· · ·δφ_nΓ[φ, m², λ, M²]

φ=0

= ¯Z^n/2Γ⁽ⁿ⁾_1...n( ¯m²,λ; ¯¯ M²). (5.291) Here we have used the chain rule for functional derivatives together with (5.288).

A Fourier transformation shows that the same equation holds true in the momentum space representation:

Γ⁽ⁿ⁾(˜p, m², λ;M²) = ¯Z^n/2Γ⁽ⁿ⁾(˜p,m¯²,¯λ; ¯M²). (5.292) 5.12.4 Independence of the S-Matrix from the renormalisation scale From (5.292) we can immediately see that S-Matrix elements are independent of the renor-malisation scale. Due to the LSZ reduction theorem, to calculate transition matrix elements, as the first step we have to build (connected) Green’s functions from the vertices (1PI trun-cated Green’s functions) and to amputate the external legs again. This is done by connecting Γ⁽ⁿ⁾ functions with Green’s functionsG⁽²⁾= 1/Γ⁽²⁾. RescalingM to ¯M, according to (5.292), each Green’s function gets a factor 1/Z. This compensates the two factors√

Z of Γ⁽ⁿ⁾ corre-sponding to the legs which are connected. Thus the truncatedn-point Green’s function obeys the same scaling law (5.291):

W_trunc⁽ⁿ⁾ (˜p, m², λ;M²) = ¯Z^n/2W_trunc⁽ⁿ⁾ (˜p,m¯²,λ; ¯¯ M²). (5.293) To obtain the S-matrix element, for the truncated leg, labelled with k, we have to mul-tiply with the corresponding asymptotically free wave function ϕ_k(p) in momentum space, corresponding to the single-particle states in the in- and out-multi particle states. On the right-hand side we writeϕ_k(p_k) = ¯ϕ_k(p_k)Z^−1/2. Since this happens to each of thenexternal legs, finally we have

S_{f i}= Yn k=1

ϕ_k(p_k)W_trunc⁽ⁿ⁾ (˜p, m²λ;M²) = Yn k=1

¯

ϕ_k(p_k)W_trunc⁽ⁿ⁾ (˜p,m¯²,¯λ; ¯M²). (5.294) That means, theS-matrix element does not change with the renormalisation scale as it should be for physical quantities.

It is clear that, for perturbatively calculated matrix elements, again this holds true only up to the order of the expansion parameter taken explicitly into account.

5.13 Asymptotic behaviour of vertex functions

Now we use the fact that in momentum space representation the mass dimension of Γ⁽ⁿ⁾(p) is 4−d. So these functions must fulfil the following scaling law

Γ⁽ⁿ⁾(e^τp,˜ e^2τm², λ,M¯²) = exp[(4−n)τ)]Γ⁽ⁿ⁾[˜p, m², λ;M²]. (5.295) On the left-hand side, we have used the definition ¯M = e^τM. Making use of (5.291) we find Γ⁽ⁿ⁾(e^τp,˜ e^2τm², λ; ¯M²) = exp[τ(4−n)] ¯Z^n/2Γ⁽ⁿ⁾(˜p,m¯²,λ; ¯¯ M²). (5.296)

If we substitutem² with e^−2τm², according to (5.289) also ¯m²= e^−2τm¯². The same time we This means, a scaling of the external momenta ˜p can be compensated by a redefinition of m² with e^−2τZ¯_mm² and of λ by ¯λ. Here we have to take ¯m and ¯λ as functions of m and λ. The scaling behaviour of the vertex functions is not the naively expected, i.e., it does not scale with the canonical dimension 4−n but with one corrected by the factor in (5.297).

Also m² does not rescale in the canonical way, namely just by a factor exp(−2τ) but with an additional factor ¯Zm. Since λ is dimensionless, naively we expect it to be unchanged by a rescaling of the renormalisation mass scale, but as we learnt now instead it is multiplied by a factor ¯Z_λ/Z¯². For this reason, the RGE coefficients β,γ_φ and γ_m often are also called anomalous dimensions.

We like to check (5.297) for the one-loop result. As an example, we take as Γ⁽⁴⁾. Since γ_φ=O(~²) the overall scaling factor is simply 1. Thus we need to calculate ¯λ and ¯Z_m only.

The perturbative expression for the four-point function is (in the MIR scheme) Γ⁽⁴⁾(s, t, u, m², λ;M²) =−λ+~λ² After some algebra, one can verify the relation

Γ⁽⁴⁾(e^2τs,e^2τt,e^2τu, m², λ;M²) = Γ⁽⁴⁾(s, t, u,Z¯_m²m²,λ;¯ M²) (5.301) to order~ which is identical with 5.297.

5.13.1 The Gell-Mann-Low equation

For sake of completeness, we like to derive also the RGEs for other renormalisation schemes.

We start with a MOM scheme (momentum-subtraction scheme). Here the renormalisation conditions usually are given by

Σ(p² =m², m², λ;M²) = 0,

∂_p2Σ(p², m², λ;M²)|p²=−M² = 0, Γ⁽⁴⁾(˜p, m², λ;M²)|s=t=u=−4/3M² =−λ

(5.302)

5.13 · Asymptotic behaviour of vertex functions

The subtraction point p² = −M² for the second condition, determining the wave-function renormalisation, should be space-like and independent of mfor the case we like to study the massless theory, i.e., m = 0. For m >0, one can choose a value for M² ∈]−4m²,∞[. The first condition chooses the physical mass of the particle to be given by m. Using a BPHZ renormalisation, modified to fulfil (5.302), in (5.263) again δm² = 0. Here also γ_m vanishes since the unrenormalised integrand for Σ does not depend on M and the subtractions for Σ are taken atp² =m², according to the first line in (5.302), the overall counterterms δZ_mm² to the mass do not depend onM² either. So here we have only two RGE coefficients, namely

β(λ, m/M) =M ∂_Mλ, γ_φ(λ, m/M) =−M

φ ∂_Mφ. (5.303)

The corresponding renormalisation group equation is derived in the same way as (5.267), i.e., from

Γ[φ²₀, m²₀, λ₀] = Γ[φ, m², λ;M²] ⇒

M ∂_M +β(λ, m/M)∂_λ −γ_φ(λ, m/M) Z

d⁴xφ(x) δ δφ(x)

Γ[φ, m², λ;M²] = 0. (5.304) This is the Gell-Mann-Low RGE. Expanding with respect to the field φ one obtains the Gell-Mann-Low RGE for the proper vertex functions in the same way as we derived (5.271):

[M ∂_M +β(λ, m/M)∂_λ−nγ_φ(λ, m/M)] Γ⁽ⁿ⁾(˜p, m², λ;M²) = 0. (5.305) It shows that also in this scheme the S-matrix elements are independent of the choice of the momentum renormalisation scale M². The proof is exactly the same as given in Sect. 5.12.4 for the homogeneous RGE.

The difference to the homogeneous renormalisation group equation is that, in general, its solution is not so easy since the coefficients β and γ_φ depend on both, λ and explicitly on m²/M². Only in the limitm→0 the equation becomes solvable as easy as the homogeneous one.

For the perturbative calculation of β andγ_φ we can use the same technique as shown in Sect.

5.12.1. One only has to use (5.305) for the renormalisation parts Γ⁽²⁾ and Γ⁽⁴⁾ and solve for the linear system of equations for β and γ_φ. Forφ⁴-theory, we obtain in lowest~-order

γ_φ^(MOM) =O(~²), β^(MOM)= 3λ²~

16π²

1− 3m²

√3m²M²+M⁴artanh

M²

√3m²M²+M⁴

+O(~²). (5.306) 5.13.2 The Callan-Symanzik equation

In the original BPHZ renormalisation scheme, the renormalised mass has to be fixed tom >0.

The renormalisation conditions read

Σ_BPHZ(s= 0, m², λ) = 0,

∂_sΣ_BPHZ(s, m², λ)|s=0 = 0, Γ⁽⁴⁾_BPHZ(s, t, u, m², λ)|s=t=u=0 =−λ.

(5.307)

Here we have neither a momentum- nor a mass-renormalisation scale as in the MOM or MIR schemes. Here the physical mass of the particle has to be defined by the zero of Γ⁽²⁾_BPHZ(m²_phys, m², λ) = 0.

Thus, within the BPHZ-scheme we interpret m² as a free mass scale, which can be varied without changing the theory, provided the wave function and coupling is renormalised such that

Γ_BPHZ[φ, m², λ] = Γ_BPHZ[φ,m¯²,λ].¯ (5.308) Instead of deriving the corresponding RGE from scratch, it is more convenient to use the homogeneous RGE (5.267). For this, we modify our MIR scheme conditions (5.264) slightly and denote the new scheme with MIR:

Σ_MIR(p² = 0, m², λ;M²)|m²=M² = 0 ⇒ δm² =M²δZ_m⁰ ,

∂_m2Σ_MIR(p²= 0, m², λ;M²)|m²=M² = 0 ⇒ δZ_m,

∂_p2Σ_MIR(p²= 0, m² =M², λ;M²) = 0 ⇒ δZ(λ), Γ⁽⁴⁾

MIR(s=t=u= 0, m² =M², λ;M²) =−λ ⇒ δλ(λ).

(5.309)

The only difference is that δm² 6= 0, but it is independent of m. The reasoning is the same as with the original MIR scheme: The only dimensionful quantity appearing in the non-subtracted Feynman integrands at the renormalisation point for the self-energy m² = M², p² = 0 is M² and thus δm² ∝ M². The factor can only depend implicitly on M over the dependence on λ.

In the following we denote the mass parameter in the BPHZ scheme with ˜m, that of the MIR scheme withm. Then we have

ΓBPHZ[φ,m˜², λ] = Γ_MIR[φ, m², λ;M²]|m=M= ˜m. (5.310) The homogeneous RGE for the MIR scheme reads the same as before, but the RGE coefficients β,γ_φ and γ_m are different from those of the MIR scheme. Thus we write ¯β, ¯γ_m, ¯γ_φ:

˜ m ∂

∂m˜ + ¯β(λ) ∂

∂λ −γ¯_m(λ)m² ∂

∂m² −γ¯_φ(λ) Z

d⁴xφ(x) δ δφ(x)

Γ_MIR[φ, m², λ; ˜m²] = 0.

(5.311) Now we define

Γ^(k)_BPHZ[φ,m, λ] =˜ ∂

∂m² k

Γ_MIR[φ, m², λ; ˜m²]|m²= ˜m²

m²= ˜m²

. (5.312)

Fork= 0, according to (5.307), this is simply Γ_BPHZ. We shall come back to these functions at the end of this section and prove that it corresponds to the same diagrams of the original functional Γ^(k)_BPHZ, but with k insertions of an auxiliary two-point vertex with the the value

−iκ/2 divided by 1/κ^k. As we shall see, for the derivation of the renormalisation group equation for the BPHZ scheme, we need only the fact that eachn-point function

Γ^(n,k)_BPHZ12...n[ ˜m, λ] = δⁿ δφ1· · ·δφn

Γ^(k)_BPHZ[φ,m, λ]˜

φ=0

⇒ Γ^(n,k)_BPHZ[˜p,m˜², λ] (5.313)

5.13 · Asymptotic behaviour of vertex functions

in momentum representation has the dimension (or superficial degree of divergence) δh From (5.311) we find after some algebra

X∞

Settingm= ˜m we find theCallan-Symanzik equation or CS equation:

We see that the CS equation is of the type of a functional flow equation like the homogeneous RGE, but it isinhomogeneous due to the right-hand side. It cannot be solved as easily as the homogeneous RGE, although the approximated coefficients ¯β, ¯γ_φ and ¯γm are given explicitly from the MIR scheme.

The equation for the n-point function is derived as usual by expanding the functionals in powers of φ. With (5.313) the CS equations for the n-point vertex functions read

From (5.314) we see that the only additional renormalisation part is the two-point function with one κ insertion, i.e., Γ^(2,1)_BPHZ which is logarithmically divergent. In our approach the renormalisation condition is completely fixed by the second of the MIR conditions (5.309) which can be rewritten in terms of Γ⁽²⁾

MIR:

∂_m2Γ⁽²⁾

MIR[p² = 0, m², λ, M²]|m²=M² =−1. (5.320) Setting herein M²= ˜m² yields

Γ^(2,1)_BPHZ(p² = 0,m˜², λ) =−1. (5.321) Then the CS equation (5.319) for m= 2 gives, by setting p² = 0 the important relation

1−γ¯_φ=α ⇒ γ¯_m =−2¯γ_φ, (5.322)

where we made use of (5.318). Clearly, the latter relation can be derived directly by special-ising the homogeneous RGE (5.271) to the case n= 2 and s = 0, but now using the MIR-instead of the MIR scheme.

Deriving (5.316) j times with respect to m² and setting afterwards m² = ˜m², we find the generalised CS equation

For sake of completeness we shall prove the graphical meaning of the functional (5.312).

Since the counterterms of the MIR scheme are independent of m², we can calculate the somehow regularised unrenormalised functional Γ. Taking the derivative is not affected by the counterterms, and we have just to renormalise Γ^(n,k)[˜p] = ∂_m2Γ⁽ⁿ⁾[˜p] for k = 0 and n ∈ {2,4} (by the conditions 5.309) and for n= 2, k= 1 (by the condition 5.321) to obtain all the functions (5.312). We can write

Γ^(k)[φ, m², λ] = ∂^k_κΓ[φ, m²+κ, λ]

From this we obtain Γ[ϕ, κ] in the same way as Γ[ϕ] fromZ, namely as the functional Legendre transform of W =−i lnZ. But this doesn’t help us to make the κ-dependence explicite. To get this, we write (5.325) as

Z[J, κ] = exp

The contribution of orderκ^k in an expansion with respect to κis obviously given by Z^(k)[J, κ] = 1 Now, in the perturbative expansion of Z[J], each of the k double derivatives takes two Jj

away, joining the two propagators, connecting the external points of these external currentsJ with the (disconnected) Green’s functions, together at a new vertex pointx_j, and this vertex point stands for the expression−iκ/2. Deriving ktimes by κ simply yields

∂ That the same construction rule holds true for connected and for 1PI Green’s functions is shown in the same way as in section 4.6 for the W- and Γ functionals.

The same result can be obtained from a diagrammatic point of view from the fact that the counterterms are independent of m², as the original vertices. A derivative ∂_m2 acts only on each propagator line 1/(p²−m²+ iη), which becomes 1/(p²−m²+ iη)². Each time deriving a diagram, thus one inserts aκvertex into each propagator and sums up all these contributions.

Dans le document Introduction to Relativistic Quantum Field Theory (Page 194-0)