Coleman-Weinberg potential

Quantum Field Theory-II Problem Set n. 9 - Solutions

UZH and ETH, FS-2020 Prof. G. Isidori

Assistants: C. Cornella, D. Faroughy, F. Kirk, J. Pag`es, A. Rolandi www.physik.uzh.ch/en/teaching/PHY552

Coleman-Weinberg potential

This exercise is based on the ground-breaking paper published by S. Coleman and E. Weinberg in 1973 “Radiative Corrections as Origin of Spontaneous Symmetry Breaking”. We will study how to compute the effective potential in a massless λφ⁴ theory and also discuss how quantum corrections can give rise to spontaneous symmetry breaking.

As it was seen in class the effective action is the generator of the 1-PI correlation functions and is defined as

Γ[φ_c] =Z[J]− Z

d⁴x J(x)φ_c(x) . (1)

Where φ_c(x) = ^δZ[J]_δJ(x) is the classical field. By assuming that the ground states of the theory we are studying are translational invariant (so that the vacuum does not break conservation of momentum) we can compute the effective potential with the following equation

V(φ_c) = −X

n

1 n!

Γ˜⁽ⁿ⁾(0)φⁿ_c , (2)

where ˜Γ⁽ⁿ⁾(0) is the momentum space 1-PI n-point correlation function evaluated at 0 external momenta (n = 2 corresponds to the inverse propagator). More precisely we have

Γ⁽ⁿ⁾(x1, ..., x_n) = ih0|T{φ(x1)...φ(x_n)}|0i_{1P I} . (3) Eq. (2) can be found by pluggingφ_c(x) =φ_c = const in the field and momentum expansions of Γ.

Consider the massless λφ⁴ theory:

L= 1

2(∂_µφ)²− λ

4!φ⁴ . (4)

where φ is a real scalar field. Notice that this Lagrangian has a discrete Z₂ (internal) symmetry corresponding to the field transformation φ(x)→ −φ(x).

I. Compute the one-loop effective potentialV(φ_c) for the theory regularizing the UV divergences with a momentum cutoff Λ.

We set V(φ_c) = P

n≥0V_n(φ_c). WhereV_n contains only the terms corresponding n-th loop compu- tation.

At tree level only two diagrams can give possible contributions. The first being the propagator

p ⁼

−i

p² , (5)

therefore ˜Γ⁽²⁾(p) =−p² and the contribution of this diagram is exactly 0.

For the quartic interaction instead we get

=iλ × (external propagators) (6)

which leads to ˜Γ⁽⁴⁾(0) =−λ by eq. (3). Therefore V₀(φ_c) = λ

4!φ⁴_c , (7)

which is, as expected, the classical potential in the Lagrangian.

At one loop the diagrams giving contributions at 0 external momentum are infinite. There is one for each polygon, in which we have two external lines attached to each vertex of the polygon:

+ + + + . . . (8)

Therefore only the even-numbered n-point 1-PI correlation functions will be non-zero. We will label them byn= 2k with k the number of vertices in the polygon andn the number of external lines. Each graph comes with a symmetry factor which is notn! because rotations and reflections of these polygons do not give genuinely new graphs. Instead we have (k−1)![(2k−1)(2k−3)...1]:

1

n!(k−1)![(2k−1)(2k−3)...1] = (k−1)!

(2k)(2k−2)(2k−4)...2

= (k−1)!

(2k)(2k−2)(2k−4)...2

= 1 2^k

(k−1)!

(k)(k−1)(k−2)...1

= 1 2^kk.

(9)

Furthermore, Bose statistics impose that exchanging two line at the same vertex does not lead to

a new graph, therefore we have to multiply the result by 1/2.

p1

p₂

pk ···

=−

Z d⁴p (2π)⁴

λ p²₁

λ p²₂...λ

p²_k

× (ext.props.) , (10)

where each p_i depends linearly on p and external momenta. They can be easily computed using momentum conservation. In the case where all external momenta are 0 we get

p₁ p2

p_k ···

ext.momenta = 0

=−

Z d⁴p (2π)⁴

λ^k p^2k

× (ext.props.) . (11)

Therefore

1 (2k)!

Γ˜^(2k)(0) =−i 2

1 2^kk

Z d⁴p (2π)⁴

λ^k

p^2k . (12)

This allows us to compute the 1-loop terms of the effective potential V₁(φ_c) =−X

n

φⁿ_c n!

Γ˜⁽ⁿ⁾_1−loop(0)

=−X

k

φ^2k_c (2k)!

Γ˜^(2k)_1−loop(0)

= i 2

Z d⁴p (2π)⁴

X

k

1 k

λφ²_c 2p²

k

= −i 2

Z d⁴p (2π)⁴ ln

1− λφ²_c 2p²

,

(13)

where in the last line we used ln(1−x) = −P

kx^k/k. This last integral is quite clearly UV divergent. Introducing a cut-off atk² = Λ², performing a Wick rotation and defining a= ^λφ₂^2c we

have

V₁(φ_c) = 1 2

Z

k²=Λ²

d⁴k_E (2π)⁴ ln

1 + a k²

= 1

16π² Z Λ

0

dk k³ln 1 + a

k²

= 1

16π² Z Λ

0

dk k³ln k²+a

− Z Λ

0

dk k³ln k²

= 1

64π² −a²ln Λ²+a

+aΛ² + Λ⁴ln Λ² +a

+a²log(a)−Λ⁴log Λ²

= 1

64π²

2aΛ²+a²

ln a Λ² − 1

2

= λΛ²

64π²φ²_c+ λ²φ⁴_c 256π²

lnλφ²_c

2Λ² − 1 2

,

(14)

where we used the expansion ln(a+ Λ²) = ln(Λ²)

a

Λ² − _2Λ^a2₄ +O(a³/Λ⁶)

and dropped all terms that go to 0 as Λ goes to infinity.

Combining eqs. (7) and (14) we get the effective potential up to 1-loop with the cut-off V(φ_c) = λ

4!φ⁴_c+ λΛ²

64π²φ²_c+ λ²φ⁴_c 256π²

lnλφ²_c

2Λ² − 1 2

. (15)

II. After introducing the necessary counterterms, renormalize the theory imposing the following renormalization conditions:

d²V(φ_c) dφ²_c

φc=0

= 0 , d⁴V(φ_c) dφ⁴_c

φc=µ

=λ , Z(φ_c=µ) = 1 , (16) where Z(φ) is the wave function renormalization constant. Discuss the minima of the potential and their meaning.

Introducing the renormalized field and couplings we have L = 1

2(∂_µφ₀)²−λ0

4!φ⁴₀ = 1

2(∂_µφ_R)²− λ_R

4!φ⁴_R+1

2A(∂_µφ_R)²− 1

2Bφ²_R− 1

4!Cφ⁴_R . (17) Withφ₀ =√

Zφ_R,Z = 1 +A, λ₀ =λ_R+C and m²_R= 0 +B. Note that the mass renormalization term is present, even though we are studying a massless theory. That is because the theory has no symmetry that guaranteesa priori a vanishing bare mass in the limit of a vanishing renormalized mass.

In the following we will takeφ_R(x) =φ_c= const (instead ofφ0(x) = φ_cas in partI) to simplify the notation. In terms of the renormalized quantities the tree level terms do not change because, by

definition,A,B andC are 0 at tree level. For the 1-loop level the counter-terms can be computed in a similar fashion as their tree-level counterparts and we find

V(φ_c) = λ

4!φ⁴_c+ 1

2Bφ²_c+ 1

4!Cφ⁴_c+ λΛ²

64π²φ²_c+ λ²φ⁴_c 256π²

lnλφ²_c

2Λ² − 1 2

. (18)

Firstly, by imposing ^d2V_dφ^(φ₂^c)

c

φc=0

= 0 we get

B =−λΛ²

32π² . (19)

Secondly, with the arbitrary renormalization scale µ, ^d4V_dφ^(φ₄^c)

c

φc=µ=λ requires C =− 3λ²

32π²

lnλµ² 2Λ² +11

3

. (20)

Taking eqs (19) and (20) into eq. (18) we find the renormalized effective potential:

V(φ_c) = λ

4!φ⁴_c+ λ²φ⁴_c 256π²

lnφ²_c

µ² − 25 6

. (21)

Note that the 1-loop contribution is proportional to λ², in fact it turns out that the nth-loop contribution (after renormalization) will be proportional to λⁿ⁺¹. By factoring out λφ⁴_c/4! we get that the 1-loop multiplicative correction is _256π^4!λ₂

ln^φ_µ^2c₂ − ²⁵₆

. Which reveals the assumptions underlying the loop expansion: the expression in eq. (21) is valid as long as|λln^φ_µ^2c2| 1, which implies that eq. (21) is not valid at high and low φ²_c. Furthermore perturbation theory requires

|λ| 1.

By takingφ_c→0 we can see thatV(0) = 0 and that forφ²_c <1 the logarithmic term is negative.

This seems to imply that eq (21) may have a local maximum at φ_c = 0, meaning that quantum loop corrections have turned the classical minima at the origin to a maxima, meaning that the minima of the potential have to be somewhere else. Indeed, from the vanishing of the derivative of dV /dφ_c= 0, it turns out that there are two symmetric minima atφ_c=±hφi satisfying

λlnhφi² µ² = 11

3 λ− 32

3 π² , (22)

where hφi denotes the value at which φ_c is at the minimum of the potential, i.e. the vacuum expectation value (VEV) of φ_c. The presence of these two minima would indicate spontaneous breaking of the Z₂ symmetry caused by the 1-loop radiative corrections! To see this one just needs to shift the field φ → φ⁰ = φ− hφi such that the minima of φ⁰ is at the origin (hφ⁰i = 0).

Plugging this field redefinition back into the Lagrangian give rise to cubic terms inφ⁰ that violate the originalφ → −φ symmetry.

Unfortunately, there is a big problem when inspecting eq. (22) more carefully. This expression raises two giant red flags: we find that λln^hφi_µ2² =O(1) which violates the |λln^hφi_µ2²| 1 require- ment for the loop expansion, and even worse, at higher loop orders we get higher powers of the large logarithm term ln^hφi_µ2², which implies that athφi we can no longer neglect these higher order contributions. The main problem is that we just concluded that the three points defining the curve of the effective potential are out of the range of our approximation and we can see that with higher orders (which will give O[λⁿ⁺¹(ln^φ_µ²₂)ⁿ]) we have the same problem. Therefore we clearly cannot trust the result of eq. (21). To address this problem we will need to find a way to consistently “resum” all these large logarithms using RGE’s.

Let’s now look instead at another theory where the 1-loop computation can be trusted and spon- taneous symmetry breaking does indeed occur: massless scalar QED. This theory consists in a massless complex scalar field φ =φ₁ +iφ₂ minimally coupled to a gauge boson (a photon). The Lagrangian of this theory is:

L=−1

4F_µνF^µν+1

2(∂_µφ₁−eA_µφ₂)²+1

2(∂_µφ₂+eA_µφ₁)²− λ

4!(φ²₁+φ²₂)² + counterterms . (23) This Lagrangian has now a continuousU(1) gauge symmetry. One can notice that the non-kinetic terms all depend on |φ|², therefore by taking the field to be translation-invariant the effective potential is a function of |φ| = const. Then we get V₀(φ) = _4!^λ|φ|⁴+e²|φ|²A_µA^µ. The diagrams contributing to the 1-loop terms of the effective potential are the same as those in λφ⁴ but with φ and A_µ in the internal lines. After renormalization, the 1-loop approximation of the effective potential is

V(φ_c) = λ 4!φ⁴_c+

5λ²

1152π² + 3e⁴ 64π²

φ⁴_c

lnφ²_c

µ² − 25 6

. (24)

In this situation we can notice that the minima are well within the range of validity of eq. (24). In pureλφ⁴ the minimum of the effective potential was found by balancing aλ term with aλ²log^φ_µ^2c₂ term. For λ 1 this cannot be realised at the same time as λlog ^φ_µ^2c2 1. Though for massless scalar QED we have an additional e⁴log^φ_µ^2c₂ term to balance the term in λ. The important thing to point out is that despite the fact that the term in e⁴log_µ^φ2c₂ arises formally at a higher order than the term in λ there is absolutely no reason why λ and e⁴ cannot be of the same order of magnitude. Actually this is what we should expect if we think of the quartic interaction in scalar QED as being forced on us to renormalize the divergence in Coulomb scattering which is itself of ordere⁴ (in a similar procedure as the one adopted in exercise sheet 4). Here the main difference is that we can trust eq. (24) in its range of validity because it contains at least 2 local minima of the curve which are some of the points that constitute its shape. The higher orders do not affect these minima because the log^φ_µ^2c₂ can be kept small. Therefore in massless scalar QED there are 2 local minima in the effective potential away from the origin. If they turn out to be global minima then wewould have spontaneous symmetry breaking of U(1) gauge symmetry. In this theory, the scalar field acquires via radiative corrections a mass given by m²_φ = _8π^3e42hφi² and the photon also gets a mass given bym²_A=e²hφi². This means that massless scalar QED, at the 1-loop level is no

longer “massless” and it is no longer QED because the photon acquires a mass! This phenomenon is also known as dimensional transmutation because the initial dimensionless parameters of the Lagrangian e and λ have now been traded for e and a mass parameter m_φ.

III. Apply the Callan-Symanzik equation to the effective potential to resum all the 1-loop 1PI diagrams in order to obtain theβ-function for λ and the renormalization group improved expres- sion for the Coleman-Weinberg potential. Discuss the minima of the potential in this case.

In eq. (21) we found the renormalized 1-loop approximation of the effective potential for massless λφ⁴. The main thing to note that the expression depends on the renormalization scale µ. This scale is completely arbitrary and we only used it to define the scale of the renormalized field and coupling constant. A small change in the renormalization scaleµcan be compensated by a small change in λ and a small rescaling of the field. It is quite easy to see that, at 1-loop, by taking µ→µ⁰ then we only have to shift

λ→λ⁰ = d⁴V(φ_c) dφ⁴_c

φc=µ⁰

=λ+ 3λ²

32π² log µ⁰²

µ² (25)

to obtain the same effective potential, but in terms ofλ⁰ andµ⁰. This is a consequence of the Callan- Symanzik equation for the effective potential. In order to find the Callan-Symanzik equation for the effective potential we must look for the Callan-Symanzik equation of the effective action because Γ[φ_c(x) = φ_c= const] = −δ(0)(2π)⁴V(φ_c). As it was seen in class it is quite clear that we have the following relation between the bare Green functionG⁽ⁿ⁾₀ and renormalized Green function G⁽ⁿ⁾:

G⁽ⁿ⁾₀ ({x_i}, λ) =h0|T{φ₀(x₁)...φ₀(x_n)}|0i

=Z(µ)^n/2h0|T{φ(x₁)...φ(x_n)}|0i

=Z(µ)^n/2G⁽ⁿ⁾({x_i}, λ, µ) .

(26)

To get a similar expression for the 1-PI correlation functions we need to consider how they are defined: we start with a 1-PI connected diagram and truncate all the propagators. A 1-PI con- nected diagram has nothing different from a regular connected diagram in regards to number of external fields, therefore if G⁽ⁿ⁾_0,c =Z^αG⁽ⁿ⁾_c then Γ⁽ⁿ⁾₀ = ^Z_Z^α_nΓ⁽ⁿ⁾ because we have to divide by Z for each propagator we divided fromG⁽ⁿ⁾c . A connected Green function is computed from the generic Green function by subtracting all the disconnected parts as a product of Green functions of lower order, but in each term of the addition we have the same number of external fields. Therefore eq.

(26) also holds for connected green functions which implies thatα=n/2. Thus

Γ⁽ⁿ⁾₀ ({x_i}, λ) = Z(µ)^−n/2Γ⁽ⁿ⁾({x_i}, λ, µ) . (27) From here the derivation is identical to the one for the generic Green function (see lecture notes) except for the sign of the exponent of Z, which changes a sign in the final Callan-Simanzik equation:

µ ∂

∂µ +β(λ) ∂

∂λ −nγ(λ)

Γ⁽ⁿ⁾({x_i}, λ, µ) = 0 , (28)

with β(λ) = µ^dλ_dµ and γ(λ) = √^µ Z

d√ Z

dµ . Using the expansion of the effective action in terms of the 1-PI correlations functions

Γ([φ_c], λ, µ) =X

n

1 n!

Z

d⁴x₁...d⁴x_nφ_c(x₁)...φ_c(x_n)Γ⁽ⁿ⁾({x_i}, λ, µ) , (29) we can find the Callan-Simanzik for Γ([φ_c], λ, µ) (here [φ_c] denotes that Γ is a functional in φ_cbut a regular function for µand λ). Taking the result of eq. (28) and plugging it into the RHS of eq.

(29) we get 0 = X

n

1 n!

Z

d⁴x₁...d⁴x_nφ_c(x₁)...φ_c(x_n)

µ ∂

∂µ+β(λ) ∂

∂λ −nγ(λ)

Γ⁽ⁿ⁾({x_i}, λ, µ) . (30) Notice how we can bring out of the integral and sum µ_∂µ^∂ , β(λ)_∂λ^∂ and γ(λ) but not n. To solve this we can use that

Z

d⁴yφ_c(y) δ

δφ_c(y)Γ([φ_c], λ, µ) = Z

d⁴yφ_c(y)X

n

n n!

Z

d⁴x1...d⁴x_n−1φ_c(x1)...φ_c(x_n−1)Γ⁽ⁿ⁾({x1, ..., x_n−1, y}, λ, µ) . (31) Therefore we get the Callan-Symanzik equation for the effective action:

µ ∂

∂µ +β(λ) ∂

∂λ −γ(λ) Z

d⁴xφ_c(x) δ δφ_c(x)

Γ([φ_c], λ, µ) = 0 . (32) By applying eq. (32) to the case φ_c=φ_c = const we getR

d⁴xφ_c(x)_δφ^δ

c(x) =φ_c∂φ^∂

c and

µ ∂

∂µ +β(λ) ∂

∂λ −γ(λ)φ_c ∂

∂φ_c

V(φ_c, λ, µ) = 0 , (33) which is the Callan-Symanzik equation for the effective potential. Using dimensional arguments it is straightforward to see that V(φ_c, λ, µ) =φ⁴_cf(φ_c/µ, λ) wheref is a adimensional homogeneous function (for more details see Peskin & Schroeder). Therefore eq. (33) becomes

µ ∂

∂µ +β(λ) ∂

∂λ −4γ(λ)−γ(λ)φ_c ∂

∂φ_c

f(φ_c/µ, λ) = 0 . (34) Using that for an arbitrary functionh(a/b) we have _∂b^∂h(a/b) = −_b^a2 ∂

∂xh(x)

x=a/b and _∂a^∂ h(a/b) =

1 b

∂

∂xh(x)

x=a/b we find that b_∂b^∂h(a/b) =−a_∂a^∂h(a/b). Therefore

β(λ) ∂

∂λ −4γ(λ)−[γ(λ) + 1]φ_c ∂

∂φ_c

f(φ_c/µ, λ) = 0 , (35)

or

φ_c ∂

∂φ_c + 4¯γ(λ)−β(λ)¯ ∂

∂λ

f(φ_c/µ, λ) = 0 , (36)

with ¯β(λ) = _1+γ(λ)^β(λ) and ¯γ(λ) = _1+γ(λ)^γ(λ) . We now replace the variable φ_c/µ with t = ln^φ_µ^c. Using that _∂t^∂ =φ_c∂φ^∂

c we find

∂

∂t+ 4¯γ(λ)−β(λ)¯ ∂

∂λ

f(t, λ) = 0 . (37)

As was seen in class, we introduce the function ¯λ(t, λ) such that (∂

∂tλ(t, λ) = ¯¯ β(¯λ(t, λ)),

¯λ(0, λ) =λ . (38)

Then this function verifies (see lecture notes for proof) ∂

∂t−β(λ)¯ ∂

∂λ

λ(t, λ) = 0¯ . (39)

Then the general solution to eq. (37) is

f(t, λ) =g(¯λ(t, λ))e^{−4R0tdt0γ(¯¯ λ(t,λ))} , (40) whereg can be any function of ¯λ. Though it can be determined by imposing the renormalization conditions. Furthermore we do not fully know ¯β and ¯γ, but only their 1-loop approximation. This approximation makes it that we can find ¯λ by integrating ¯β but the result we find remains valid only if ¯λ remains small! We did not compute Z in II but it is easy to see that at Z = 1 at one loop, therefore γ(λ) = 0.

The way to findβ is by using eq. (33) and eq. (21) to see how a small change inµis compensated by a small change in λ and φ_c, which is what we did with eq. (25). We saw that there is no change inφ_c (thusγ = 0) and that λ →λ+_32π^3λ2₂ log ^µ_µ⁰²₂. Therefore by takingµ⁰ =µ+δµ we find the 1-loop approximation ofβ

β(λ) =¯ β(λ) = µdλ

dµ = 3λ²

16π² . (41)

Therefore we have _∂t^∂λ(t, λ) =¯ _16π^3¯λ2₂ and the solution to eq. (38) is

¯λ(t, λ) = λ

1−3λt/16π² . (42)

We can see that ¯λ remains small for all values of t <0 and has a Landau pole att= ^16π_3λ² 1.

Since ¯γ = 0 we have that the solution to the Callan-Symanzik equation is, by eq. (40), f(t, λ) = g(¯λ(t, λ)). Any function g would satisfy eq. (37) but when we expand ¯λ(t, λ) in orders of λ our solution has to match with the loop computation of the effective potential given in eq. (21). It would seem that this would bring us back to the same problems we had before, but actually it does not. Because the loop computation did not take into account the running of the coupling

constants and contained the arbitrary scale, while here we find a solution which depends only on the running coupling constant. We can rewrite eq. (21) as

V(φ_c) = 1 4!φ⁴

λ+ 3λ²t

16π² − 25λ² 64π²

. (43)

Notice that if we expand ¯λ up toO(λ³) we find the first two terms in the parenthesis of eq. (43).

Then it is clear that the solution to the Callan-Symanzik equation is g(¯λ) = 1

4!

λ¯− 25¯λ² 64π²

, (44)

therefore we find that the effective potential is V(φ_c) =

¯λ 4!φ⁴_c

1− 25¯λ 64π²

. (45)

The range of validity of this version of the effective potential is much larger than the one given from the loop computation. Before the range of validity was |λt| 1 therefore excluding small and large momenta, but now we added allt < 0 to the range therefore eq. (45) is valid from the momenta of the renormalization scale to 0. We can see this considering that the error in eq. (45) is of order O(¯λ³), as φ_c goes to 0 also ¯λ goes to 0 so the result becomes more and more accurate.

We can clearly see that eq. (45) predicts a local minimum at φ_c = 0 and it is clear that the parenthesis always stays positive for small ¯λ. Therefore, assuming that the potential continues to grow when we get out of the range of eq. (45), we do not have spontaneous symmetry breaking.

The procedure we just used is called “the renormalization group improvement”, it successfully re-summed the large logarithms and can also be used for correlation functions or other quantities.

In this exercise the predictions of loop computation did not “survive” the renormalization group improvement: even though the minima were out of the range it seemed that (in the range) the effective potential decreased asφ_cincreased, though we found that this is not the case: it increases.

Though, in massless scalar QED the predictions survive. This is sort of expected because, as we said above, the points defining the curve of the effective potential are within the range of the prediction. In this case the procedure is very similar as to the λφ⁴, but it is done with two coupling constants (the details can be found in the paper cited at the very beginning). We start with the 1-loop computation of the effective potential in eq (24). The Callan-Symanzik equation for the effective action is

µ ∂

∂µ +β ∂

∂λ +β_e ∂

∂e −γ_e Z

d⁴xA_µc(x) δ δA_µc(x)

− γ Z

d⁴xφ_1c(x) δ δφ_1c(x) +

Z

d⁴xφ_2c(x) δ δφ_2c(x)

Γ = 0 . (46)

Here we have Z = 1 + _8π^3e22t, with t= ln^φ_µ, and we find

¯

γ = 3e²

16π² , (47)

β¯= (5

6λ²−3e²λ+ 9e⁴)/4π² , (48)

β¯_e=−e¯γ_e = e³

48π² . (49)

Therefore, by approximating the differential equations, we get

¯

e² = e²

1−e²t/24π² , (50)

¯λ= 1 10¯e²h√

719 tan√

719 ln ¯e+θ + 19i

, (51)

whereθ is an integration constant to be chosen such that ¯λ=λwhen ¯e=e. From the running of the coupling constants we can see that at large and positive values oft¯ebecomes large and that at large and negative values oft ¯λ becomes large, since the argument of the tangent passes through a multiple of π. Therefore there is no interest in expressing V in terms of ¯λ and ¯e because the range of validity does not become bigger. But we can still gain some information: we can change the argument of the tangent by 2π by varying t so that ¯e changes. In the course of this variation

¯λgoes through all the real axis therefore we cannot trust the parts where ¯λis not small. But this allows us to move ¯λ from any small value to any other small value, in particular we can bring it back to the same value, but ¯e will have changed. Therefore there is no obligation for ¯e⁴ to be of the same order as ¯λ, but the result of eq. (24) is valid for any smalleand λ. Because ife⁴ is not of same order asλ, we can appropriately change the renormalization scale (thus changet) to make it so that it is the case. Therefore the effective potential of massless scalar electrodynamics develops a minimum away from the origin and spontaneous symmetry breaking occurs, for arbitrary small λ and e.