On the relationship between gauge dependence and IR divergences in the $\hbar$-expansion of the effective potential

On the relationship between gauge dependence and i r divergences in the ħ h -expansion of the effective potential

Andreas Ekstedt

^*

and Johan Löfgren

^† Department of Physics and Astronomy, Uppsala University Box 516, SE-751 20 Uppsala, Sweden

2 0 1 8 - 1 0 - 0 4

Perturbative calculations of the effective potential evaluated at a broken minimum, Vmin, are plagued by difficul- ties. It is not straightforward to get a finite and gauge invariant result for Vmin. In fact, the methods proposed to deal with gauge dependence and i r divergences are orthogonal in their approaches. Gauge dependence is dealt with through the ħh-expansion, which establishes and maintains a strict loop-order separation of terms. On the other hand, i r divergences seem to require a resummation that mixes the different loop orders. In this paper we test these methods on Fermi gauge Abelian Higgs at two loops. We find that the resummation procedure is not capable of removing all divergences. Surprisingly, the ħh-expansion seems to be able to deal with both the divergences and the gauge dependence. In order to isolate the physical part of Vmin, we are guided by the separation of scales that motivated the resummation procedure; the key result is that only hard momentum modes contribute to Vmin.

1 Introduction

One of the most important tools for studying spontaneous symmetry breaking within q f t is the effective potential V , which can be considered as the quantum-corrected version of the classical potential V

0

. The effective potential is given by V = V

0

+ħhV

1

+ħh

²

V

2

+. . . with factors of ħh inserted in order to em- phasize that this quantity is usually calculated perturbatively. If the theory allows for spontaneous symmetry breaking through the scalar field φ, its vacuum expectation value, or vev, would be found by extremizing the effective potential

^∂

V |

φ=φ^m

= 0. In this way the effective potential allows us to find the quantum corrected minimum, φ

^m

, and the corresponding background energy density V

min

≡ V (φ

^m

).

It can be hard to extract physical information from the ef- fective potential. In particular, V

min

is in principle a measurable quantity, yet there are difficulties in obtaining a physical value of V

min

in perturbation theory.

One difficulty is gauge dependence. The effective potential is in general gauge dependent, but is guaranteed to be gauge independent when evaluated at its extremum φ

^m

. However, as recently pointed out by Andreassen, Frost, and Schwartz [1], and by Patel and Ramsey-Musolf [2], gauge invariance of V

min

relies on a strict ħ h power counting. To establish a strict counting the aptly named ħ h-expansion is used.

Another difficulty is that there are families of gauges for which the effective potential diverges near the broken mini- mum. These divergences come from Goldstone bosons becom- ing massless and signal a breakdown of the perturbative ex- pansion. It has been proposed that a resummation might be required to obtain a finite result. Resummation of i r diver- gences has been discussed by Martin in [3], and by Elias-Miró, Espinosa, and Konstandin in [4].

Both the gauge and i r problems relate to the perturbative expansion, but the solution of the gauge invariance and i r divergence issues stand in contrast to each other. Gauge invari- ance requires a strict separation of loop orders, and i r diver- gences suggest that contributions from all loop orders should

* Andreas.Ekstedt@physics.uu.se

† johan.lofgren@physics.uu.se

be included. In this paper we propose a solution to these dif- ficulties. We argue that in a general model with spontaneous symmetry breaking, the ħ h-expansion is capable of treating both the gauge invariance and the i r divergence issues. We argue that the known resummation procedure is problematic in the presence of certain gauge dependent singularities. However, we find that certain techniques used for resummation are also useful when performing an ħ h-expansion.

Similar considerations as those in this paper have arisen pre- viously in the context of the Coleman-Weinberg (c w) model [5], see references [1, 6]. We discuss this model from the point of view given by our results for the Abelian Higgs model.

In section 2 relevant notation is introduced and the theoret- ical background is reviewed. Section 3 summarizes our main results. To illustrate our procedure, the full 2-loop effective po- tential is calculated in the Abelian Higgs model. Details and proofs can be found in the appendix. Finally, conclusions are given in section 4.

2 Background

The section starts with a summary of conventions, presented for the Abelian Higgs model. We review the origins of gauge de- pendence and i r divergences. Methods for dealing with these issues are discussed: a consistent ħ h-expansion and daisy resum- mation respectively. The general conventions are then collected in appendix A.

2 .1 a b e l i a n h i g g s

The Abelian Higgs model is a useful toy-model because it ex- hibits the issues that we want to discuss: gauge dependence and i r divergences. The model consists of a U(1) gauge field A

^µ

together with a complex scalar Φ =

^p1₂

(φ

1

+ iφ

2

) charged under this symmetry. The Lagrangian, using the conventions of [1], is

L =L

AH

+ L

g.f.

+ L

ghost

, L

_AH

= − 1

4 F

_µν

F

^µν

− (D

^µ

Φ)

^†

D

_µ

Φ − V

0

Φ, Φ

^†



arXiv:1810.01416v1 [hep-ph] 2 Oct 2018

where F

µν

=

^∂µ

A

_ν

−

∂_ν

A

_µ

is the U(1) field strength, D

µ

=

∂_µ

+ ieA

µ

is the covariant derivative, L

g.f.

+ L

ghost

contains the details of gauge fixing and the ghost sector, and V

0

[Φ, Φ

^†

] is the classical scalar potential. Expressed in terms of the real degrees of freedom ~ φ = (φ

1

, φ

2

), the classical potential, V

0

, is

V

₀

[φ

1

, φ

2

] = − 1

2 m

²

φ

₁²

+ φ

2²

+ 1

4! λ φ

²₁

+ φ

²2

2

.

The Lagrangian, L

AH

, is invariant under global and lo- cal U(1) transformations. However, the remaining terms L

_g.f.

, L

_ghost

, explicitly break the gauge invariance. Common gauge fixing choices are discussed in [1]; we note that the commonly used R

ξ

gauges,

L

_g.f.

= − 1

2ξ

^∂µ

A

^µ

+ ξφφ

2

2

,

L

_ghost

= −¯c



∂²

− ξe

²

φ

²

 1 + h

φ

‹‹

c,

explicitly breaks the remnant global U(1) symmetry. The break- ing of this symmetry complicates calculations involving Gold- stone bosons, which is what we are interested in. To simplify calculations we restrict ourselves to the Fermi gauges, which preserve the global U(1) symmetry. In these gauges, the gauge fixing and ghost terms are

L

_g.f.

= − 1

2ξ

^∂µ

A

^µ

2

, L

_ghost

= −c

^∂µ∂µ

c.

In Fermi gauges ghosts are free. A slight complication with these gauges is kinetic mixing between the longitudinal gauge boson mode and the Goldstone boson.

Because the potential is invariant under the global U(1) sym- metry, we have the freedom to place the vev of our vector ~ φ in any direction. Choosing the vev φ to lie in the φ

1

direction, that is, shifting φ

1

→ φ + φ

1

, the field-dependent masses squared in the presence of the background field are

H = −m

²

+ 1 2 λφ

²

, G = −m

²

+ 1

6 λφ

²

, A = e

²

φ

²

.

The masses denote the tree-level mass of the fields: the Higgs mass H, the Goldstone mass G, and the “photon” mass A — with the notation that the mass-squared of field X is also denoted as X . Due to the kinetic mixing between the longitudinal mode of A

^µ

and G, it is useful to introduce the masses G

₊

and G

₋

,

G

_±

= 1

2 €G ± ÆG (G − 4ξA)Š.

The masses G

₊

and G

₋

depend explicitly on the gauge fixing parameter ξ.

There are different possible scenarios depending on the value of m

²

; it is assumed that λ > 0.

• m

²

< 0: This model is called scalar qed. Scalar qed consist of two scalars, with mass

  m

²



 , and a massless gauge boson.

This model is not considered in this work because there is no spontaneous symmetry breaking.

• m

²

= 0: This is the cw model mentioned in the introduc- tion. Classically, this model does not exhibit spontaneous

symmetry breaking, but masses can be generated through quantum corrections. The generation of mass needs a careful treatment of perturbation theory — the coupling λ neccesar- ily scales as ħ h, as is discussed in [1]. The model is discussed in subsubsection 3.3.2.

• m

²

> 0: This model is called Abelian Higgs. There is sponta- neous symmetry breaking because the classical potential has a minimum located at φ

0

= p

6m

²

/λ; the masses evaluated at this field point are

H|

φ₀

= 2m

²

, G

_±

|

φ₀

= 0,

A|

φ₀

= 6 e

²

λ m

²

.

The Abelian Higgs model is the main focus in this paper. The Feynman rules relevant for deriving the effective potential are given in appendix A.

2 . 2 t h e e f f e ct i v e p ot e n t i a l

In this subsection conventions for the 1-loop potential are given.

The ħ h-expansion and daisy resummation methods are reviewed.

2.2.1 The 1-loop potential

The 1-loop contribution of a scalar field with mass X is in m s given by 1

V

1

(φ) ∼ 1 4 X

²



L

X

− 3 2

‹

, (2.1)

where we introduced the same shorthand as in [6], L

X

≡ log X /µ

²



with µ the m s renormalization scale. To simplify the formula, ħ h is rescaled with a factor of 16π

²

.

2.2.2 The ħ h-expansion

Even though the Nielsen identity [7] guarantees that the phys- ical quantity V

min

is gauge invariant, care must be taken when V

_min

is calculated in perturbation theory. The issue is that the Nielsen identity is a non-perturbative statement, but in pertur- bation theory things are more subtle.

A consistent ħ h-expansion is necessary in order to establish this gauge invariance. This ħ h-expansion has recently been dis- cussed by Patel and Ramsey-Musolf in [2], but see also [8]

and [9] for earlier applications. The key point is in how the minimum φ

^m

is treated. The minimum is found by solving the equation

∂

V |

_φ=φ^m

= 0. (2.2)

Because the potential V is calculated perturbatively, equa- tion (2.2) should also be solved perturbatively, order by order in ħ h. This gives φ

^m

= φ

0^m

+ ħhφ

1^m

+ ħh

²

φ

₂^m

+ . . ., where the con- tributions φ

1^m

, φ

^m₂

, . . ., are found by inserting this expansion in equation (2.2),

∂

V

₀

|

_φ^m₀

= 0,

∂

V

1

|

φ₀^m

+ φ

1^m∂²

V

0





φ^m₀

= 0 =⇒ φ

1^m

= −

^∂

V

₁

∂²

V

₀

   

φ^m₀

.

. . .

1 For a more complete discussion of the 1-loop potential we recommend [1].

When φ

^m

is evaluated perturbatively, V

min

can be consistently calculated order by order in perturbation theory. The first few terms of V

_min

are

V (φ

^m

) = V

0

|

_φ^m₀

+ ħh V

1

|

_φ^m₀

+ ħh

²

 V

2

− 1

2 φ

₁^m

2

∂²

V

0

‹   

φ₀^m

+ . . .

It has been shown that V (φ

^m

) evaluated in this way is gauge invariant order by order in ħ h [2].

In the Fermi gauges it can be shown that φ

1^m

and higher order contributions have i r divergences of the form ξ log[G]

and worse [6]. However, these divergences are not a problem because the vev φ

^m

is not a physical quantity. Furthermore, the divergence of φ

^m

ensures that the effective potential is finite. All gauge dependent divergences are guaranteed to cancel order by order in ħ h.

2.2.3 i r divergences in the effective potential

The effective potential evaluated at φ

0

has i r divergences when calculated perturbatively (because the Goldstone bosons are massless). These i r divergences point to a problem with the perturbative expansion, suggesting that it might be necessary to perform a resummation. To illustrate the idea, consider Landau gauge — where the mixed G-A

^µ

propagator vanishes, and the Goldstone propagator is D

G

(k) = i/(k

²

− G). The most severe divergences are given by daisy diagrams [3, 4],

1

2

^ΠΠ

, 1

4

^{Π ΠΠ}

, 1

6

^Π

Π

Π

Π

, . . .

The daisies are built from Goldstone propagators and insertions of the Goldstone self-energy Π(k

²

). These diagrams, for a given loop order L, contain the worst divergences in the limit G → 0 because they have the maximum number of Goldstone propa- gators within the same loop. The value of a daisy diagram with n “petals” can be written

1 2n

Z (dk)

 1

k

²

− G

‹

n

(−Π(k

²

))

ⁿ

,

where we introduced a shorthand notation for the integration, R (dq) = Q

^2ε

R

d

^d

q/ (2π)

^d

, with Q the renormalization scale, and d = 4 − 2ε. ir divergences come from the limit G → 0.

The daisy integrand, for soft momenta k

²

∼ G, scales as ∼ G

²

(Π(0)/G)

ⁿ

. The daisy diagrams are i r divergent for n ≥ 2, which corresponds to 3 loops and higher. In this argument we used Π(0) instead of Π(k

²

), because for soft momenta the momentum dependence of Π(k

²

) simply corresponds to sub- divergences,

Π(k

²

) ∼ Π(0) + GΠ

⁰

(0) + . . .

It is possible to sum up the leading divergences of the daisies.

The result of the daisy resummation is X

∞

n=1

1 2n

Z

(dk)(−1)

ⁿ⁺¹

 1

k

²

− G

‹

n

(Π(0))

ⁿ

= 1

2 Z

(dk) log 

k

²

− G − Π(0) − 1 2

Z

(dk) log  k

²

− G 

.

The Goldstone 1-loop contribution is

¹₂

R (dk) log k

²

− G  : the resummation implies that the Goldstone mass in the 1-loop potential should be shifted according to G → G = G + Π(0).

Because G is the inverse of the Goldstone two-point function

evaluated at zero momentum, it is possible to calculate it from the effective potential,

G =

^∂2_φ2

V (φ

1

, φ

2

) = 1

φ

^∂

V (φ), (2.3) where the last equality follows from the Goldstone theorem.

However, we have above argued that V has i r divergences.

Therefore, by shifting G → G =

_φ^1∂

V (φ), we will not remove the i r divergences, but instead just shuffle them around. Is there a consistent way to identify the i r divergent part of G?

This question was investigated in [4], and answered by Espinosa and Konstandin in [10].

The idea is to introduce a separation of scales, in which fields with momentum that scale as k

²

∼ G are soft; fields with momentum that scale as k

²

 G are hard. It is then possible to separate the effective potential into a hard and a soft part, V (φ) = V

^h

(φ) + V

^s

(φ), where V

^h

only contains contributions from hard fields, and V

^s

from soft and hard. We refer to this separation as the hard/soft split. From G =

_φ^1∂

V (φ) the Goldstone self-energy can be separated into a hard part ∆ and a soft part Σ, Π(0) = ∆ + Σ, with ∆ =

φ¹∂

V

^h

, Σ =

φ¹∂

V

^s

.

At one loop the hard/soft splitting is simple. For Abelian Higgs in Landau gauge the potential splits as

V

₁^h

(φ) = 1 4

• H

²



L

_H

− 3 2

‹ + 3A

²

 L

_A

− 5

6

‹˜

,

V

₁^s

(φ) = 1 4 G

²



L

_G

− 3 2

‹ .

The Goldstone self-energy splits as

∆

₁

= 1 2

•

λH (L

H

− 1) + 3e

²

A  L

_A

− 1

3

‹˜

, (2.4)

Σ

₁

= 1

6 λG (L

G

− 1) . (2.5)

Because i r divergences come from soft momenta, all i r diver- gences of the effective potential (G) are contained in V

^s

(Σ). In order to consistently resum the leading divergences, only the hard part of the self-energy should be used, that is

G = G + ∆. (2.6)

It was shown in [10] that this resummation is consistent to all orders in perturbation theory in Landau gauge. Our discus- sion has so far been about how to resum the 1-loop potential, but the authors of [10] demonstrate how it is possible to re- sum the i r divergences of any loop order. In the process of resummation, diagrams with soft Goldstone lines and hard self- energy insertions are stripped off at each loop order to be resummed. The result is to shift the Goldstone mass G → G in the remaining soft diagrams, which have all lines soft. The authors of [10] show through power counting that these all- lines-soft diagrams are finite in the limits G → 0, G → 0, ensuring that all i r divergences cancel consistently. In prac- tice, the i r resummation is implemented by subtracting the k-loop soft diagrams s

k

from V

^s

, and resumming them into V

^s

, V

^s

(φ) = V

^s

(φ) + V

^s

(φ) − P

_∞

k=1

ħ h

^k

s

_k

(φ)

, curing the ir divergences of V

^s

.

In section 3 we demonstrate that the method of [10] is insuf-

ficient at two loops in Fermi gauge Abelian Higgs — there are

additional i r divergences that need to be cancelled through

the ħ h-expansion. The cw model is a unique exception that is

reviewed.

3 Results

We show that there are two different classes of i r divergences in the Fermi gauges, and that the resummation procedure de- fined in equation (2.6) is incapable of dealing with all i r diver- gences. We argue that the ħ h-expansion is free of ir divergences to all orders.

3 .1 i r d i v e r g e n c e s i n t h e f e r m i g au g e s

Let us discuss the origin of i r divergences in Fermi gauges, to parallel how daisy diagrams are resummed in Landau gauge (see subsubsection 2.2.3). Before turning to the Fermi gauges, it is useful to review power counting of the soft scale G in Landau gauge.

3.1.1 Power counting in Landau gauge

Our main concern are diagrams where the momenta on all lines are soft. These diagrams remain after the leading diver- gent diagrams with hard self-energies have been subtracted; it’s important that these all-line-soft diagrams are finite in the limit G → 0. The power of G that a generic soft vacuum diagram D scales with is denoted by P

G

(D)|

ξ=0

. Denoting the number of A-G-H vertices V

^AGH

, and similarly for other vertices, it can be shown that

P

G

(D)|

_ξ=0

= 2+V

^AGH

+V

^GGAA

+2V

^HHAA

+V

^HHGG

+V

^HHH

+2V

^HHHH

. P

_G

(D)|

ξ=0

shows that all-line-soft diagrams, at all loop levels, scale with a positive power of G. It’s then safe to shift G → G in these diagrams.

3.1.2 Power counting in the Fermi gauges

The power counting is a bit more involved in the Fermi gauges.

Consider a daisy diagram at L loops, 1

2 (−1)

^L

L − 1 Z

(dk)

 k

²

− ξm

²_A

(k

²

− G

₊

)(k

²

− G

₋

)



L−1

× ∆

^L−1

, (3.1)

where ∆ is the hard part of the Goldstone self-energy, as de- scribed in subsubsection 2.2.3. Here i r divergences come from the momentum region k

²

∼ G

_±

∼ 0. For soft momenta k

²

∼ G

_±

this diagram scales as ∼ (G

±

)

²

(G

±

− ξA)

^L−1

∆

^L−1

/(G

_±²

)

^L−1

. Log- arithmic divergences now appear already at two loops, and are gauge dependent. At three loops there are divergences of the forms ξ

²

/G

_±⁴

, ξ/G

_±²

and log G

±

. The 3-loop i r divergence proportional to log G

_±

is not gauge dependent and is the same divergence we found in Landau gauge.

We now try to extend the resummation to general Fermi gauges. Resumming daisy diagrams in Fermi gauges leads to the shift G

_±

→ G

_±

in the soft potential, where G

_±

is defined by the shift G → G in G

±

. Does this resummation method resolve all the divergences? Consider how a generic all-lines-soft diagram scales with G

_±

. Note that some propagators (photon, Goldstone, mixed) have two terms with different scaling, and that the most severely i r divergent terms are those that scale with the lowest power of G

±

. We will for the moment focus on diagrams with the most severe i r divergences. These diagrams scale with G

_±

as

P

G_±

(D) = 2 − 2V

^{g g g g}

− 2V

^{g gh}

− V

^{g gγγ}

+ V

^hhh

+ 2V

^hhhh

+ V

^hhγγ

.

Because some of the vertices contribute negatively to P

G_±

(D), it’s possible to have diagrams that scale with non-positive pow-

ers of G

±

. The powers of G

±

can be reshuffled to show the interplay between i r divergences and gauge dependence,

P

G_±

(D) = P

G

(D)|

ξ=0

− N

ξ

(D), (3.2)

where N

ξ

(D) denotes the power of ξ in the most severely ir divergent term in diagram D. The class of diagrams discussed are possible at two loops or higher. Because they are not guar- anteed to be finite in the G

_±

→ 0 limit, it’s of no use to shift G

_±

with G

_±

. The resummation procedure is hence not able to remove all i r divergences. In subsubsection 3.3.1 we show a specific diagram where this happens.

The i r divergent diagrams with all soft Goldstone lines can be thought of as coming from soft Goldstones with Π

^s

insertions which we argued should not be resummed in Landau gauge.

The gauge dependent i r divergences can also be thought of as artifacts from neglecting a proper perturbative expansion.

From equation (3.2) we note that divergences appearing in Fermi gauge are necessarily gauge dependent. Hence, these divergences are guaranteed to cancel when V

min

is evaluated order by order in ħ h.

To summarize, there are two classes of i r divergences in the effective potential. The i r divergences in the first class are gauge dependent and cancel when V

min

is evaluated order by order in ħ h. This class of ir divergences is not resummable. The second class of i r divergences consists of gauge independent i r divergences which are present in Landau gauge, and can be resummed. A priori, it is not clear how this second class fares under the ħ h-expansion.

These two classes of i r divergences seem to require different methods to deal with them. Is it possible to deal with both classes at once? We explore this in the next subsection.

3 . 2 a st r at e gy fo r c a lc u l at i n g V

_{m i n}

Because there are gauge dependent i r divergences in Fermi gauges that can not be removed through resummation, the need for resumming the effective potential is not evident.

However, such gauge dependent divergences are guaranteed to cancel when evaluating V

min

perturbatively through the ħ h- expansion. At three loops and higher there are additional i r divergences that are present even in Landau gauge. Naively these divergences should be resummed. How are these issues reconciled? In this subsection we will discuss the i r fitness of the ħ h-expansion.

Using the hard/soft split introduced in [4, 6, 10], we prove that the three loop evaluation of V

min

through the ħ h-expansion is finite. We also show that all the leading singularities cancel to all orders. We argue that the ħ h-expansion is ir fit to all orders.

As a result, all contributions coming from soft modes cancel:

only hard modes are relevant when evaluating V

min

. 3.2.1 The hard/soft split and the ħ h-expansion

The separation of the effective potential into separate contribu- tions from hard and soft momenta, V = V

^h

+ V

^s

, is valid for φ ≈ φ

₀^m

such that there is a separation of scales between G and other heavy fields.

Because the goal is to calculate V

_min

= V (φ

^m

), we separate

V

_min

into hard and soft contributions, V

min

= V

_min^h

+V

_min^s

. In order

to isolate hard/soft contributions in an efficient way, we need

to deal with the vev φ

^m

= φ

^m0

+ ħhφ

^m1

+ . . . in this separation. It

is useful to split the vev as φ

^m

= φ

^h

+ ϕ, where φ

^h

is defined

through

∂

V

^h

|

_φ=φh

= 0, and ϕ denotes a remainder which will

receive contributions both from V

^h

and V

^s

. In this way we can

isolate the contributions that are purely due to hard modes.

The remaining hard/soft modes can be dealt with separately.

Before the ħ h-expansion is performed, an expansion is per- formed around φ

^h

,

V (φ

^m

) = V (φ

^h

) + ϕ

^∂

V |

φ^h

+ 1 2 ϕ

²∂²

V 



φ^h

+ . . .

= {V

^h

(φ

^h

)}

+ V

^s

(φ

^h

) + ϕ

^∂

V

^s

|

_φh

+ 1

2 ϕ

² ∂²

V

^h

+

^∂2

V

^s

 

φ^h

+ . . . In this expansion we have isolated the purely hard part, V

^h

(φ

^h

).

This contribution is denoted as V

_min^h

, and is given by V

_min^h

= V

0^h

(φ

0^h

) + ħhV

1^h

(φ

^h0

)

+ ħh

²

 V

₂^h

− 1

2 φ

₁^h

2

∂²

V

₀^h

‹   

φ^h₀

+ . . .

The purely soft terms and the soft-hard mix terms (also referred to as soft) are denoted by V

_min^s

and are given by

V

_min^s

= ħhV

1^s

(φ

^h0

) + ħh

²

 V

₂^s

− 1

2 ϕ

²₁

+ 2φ

1^h

ϕ

1

∂²

V

₀^h

‹   

φ₀^h

+ . . .

Note that the leading order contribution to V

^h

is V

0^h

= V

0

, and that the soft potential V

^s

starts at V

₁^s

. By labeling V

0

as hard, we are ensuring that the shifted Goldstone mass can be written as G =

^∂

V

^h

/φ. That is, we identify φ

₀^h

= φ

0

, ϕ

0

= 0.

After isolating the hard and soft contributions to V

min

, we are ready to address the questions of gauge invariance and i r fitness of the ħ h-expansion. Ideally, we would like that the gauge dependence of the effective potential separates according to the hard/soft split. We perform and discuss this split in the next subsubsection. The main question is whether the soft con- tribution V

_min^s

is i r finite. Even though we are not currently able to establish the i r fitness of V

_min^s

to all orders in perturbation theory, we have shown that it holds at least up to three loops.

Additionally, leading divergences and a class of subleading di- vergences cancel to every loop order, see appendix D.

3.2.2 Gauge dependence of V

^h

In order to assess the gauge dependence of V

^h

we begin by considering the gauge dependence of the full effective potential, which can be studied through the Nielsen identity [7],

ξ

∂_ξ

+ C(φ, ξ)

^∂φ

V (φ, ξ) = 0, (3.3) where C(φ, ξ) is the Nielsen coefficient. In Fermi gauge Abelian Higgs, C(φ, ξ) is given by

C(φ, ξ) = = e

2 Z

(dk) 1 k

²

−k

µ

F

^µ

(k), (3.4)

where F

^µ

is the exact G-A

^µ

mixed propagator. The Nielsen iden- tity, taken at face value, is a non-perturbative statement that implies that the effective potential is gauge invariant when eval- uated at its extrema. However, in perturbation theory the situa- tion is more subtle. The authors of [1] point out that C(φ, ξ) can be perturbatively i r divergent when evaluated at φ

^m

, which jeopardizes the conclusion that the effective potential is gauge invariant there. On the other hand, the authors of [2] and the authors of [6] argue that equation (3.3) should be considered perturbatively. In performing an ħ h-expansion of the Nielsen

identity, any i r divergences of C(φ, ξ) will cancel against ze- ros of

^∂

V . Some of these subtleties are illustrated below, but we note that in the issue that we actually are interested in — the gauge dependence of V

^h

— these concerns turn out to play no role.

The Nielsen coefficient C(φ, ξ) describes the gauge depen- dence of V . By the method of characteristics, it can be shown that curves φ(ξ) in the (φ, ξ) plane are curves of constant V if the curves satisfy

ξ

∂_ξ

φ = C(φ, ξ). (3.5) This is true for the vev φ

^m

(ξ). In order to find a useful ex- pression for C(φ, ξ) we use the generalized Ward identities in appendix B. We find

F

^µ

(k) = ξ p

A −k

^µ

k

²

− e G

₊

k

²

− e G

₋

, (3.6)

G e

_±

≡ 1 2



 G Π

_L

±

v u t G

Π

_L

( G Π

_L

− 4ξA)



, (3.7)

where G =

_φ^1∂

V and Π

L

(k

²

) = 1 + O(ħh) is a scalar function that parametrizes the self-energy of the longitudinal part of the (A

^µ

, G) matrix propagator. From equation (3.4) it can be seen that C(φ, ξ) has an ir divergence as G → 0. The 1-loop result is the leading contribution and is previously known (see e.g. [6]),

C

₁

(φ, ξ) = ξA 2φ

1 (G

+

− G

₋

)

 G

₊

L

_G+

− 1
− G

−

L

_G−

− 1


.

The 1-loop Nielsen coefficient C

1

(φ, ξ) is divergent in the φ → φ

^m₀

limit. Interestingly, this divergence induces a ξ de- pendent i r divergence in φ

₁^m

(ξ) through equation (3.5). This i r divergence is not problematic because it is necessary to can- cel other i r divergences in the ħ h-expansion.

Consider now the purely hard contribution. The Nielsen iden- tity for the hard potential is

ξ

∂_ξ

+ C

^h

(φ, ξ)

^∂

V

^h

(φ, ξ) = 0. (3.8) The Nielsen coefficient C

^h

can be found by using the mixed propagator given in equation (3.6) but with the shifted Gold- stone mass given by G =

^∂

V

^h

/φ. Note that the restriction to hard fields assumes k

²

 G

_±

. Hence, close to the minimum φ

^h

of the hard potential, we can expand

1 k

²

− e G

₊

k

²

− e G

₋

= 1

k

⁴

+ O G
,

because by assumption k

²

 G. 2 In this momentum region the Nielsen coefficient behaves as

C

^h

(φ, ξ)|

φ≈φ^h

= e 2

Z

(dk)k

²

ξ p A

k

⁴

+ O G
= 0 + O G
, where the last equality follows from the integral being scaleless.

In particular, at φ = φ

^h

, we find C

^h

(φ

^h

, ξ) = 0. This is in sharp contrast to the behaviour of C(φ, ξ) for the whole potential — gone are the ambiguities associated with infinities.

It is not surprising that there are no i r divergences for the hard fields because these can only come from long distance behaviour. More interesting is the fact that C

^h

(φ

^h

, ξ) = 0. First of all, this implies that the hard potential evaluated at φ

^h

is gauge invariant. Second of all, equation (3.5) implies that φ

^h 2 Because we are only considering hard fields, the “hard” self energy^∂V^H/φ is

free of any divergences, which justifies our expansion.

is gauge invariant. In other words, φ = φ

^h

is an invariant 1 d manifold in the (φ, ξ) plane under the flow defined by the Nielsen coefficient through equation (3.5). It is also understood that φ

^h

does not have any i r divergences.

To summarize, we have now shown that

• V

^h

is gauge invariant when evaluated at φ

^h

. This means that the gauge dependence of V

^h

and V

^s

separates.

• φ

^h

is gauge invariant. Note that this places heavy restrictions on the gauge dependence of G = G + ∆, because φ

^h

is defined through G 



φ^h

= 0.

3.2.3 i r fitness of the ħ h-expansion

Because V

_min^h

is finite and gauge invariant, we address the i r finiteness of V

_min^s

. Even if the soft effective potential V

^s

receives i r divergent contributions, it is not guaranteed that V

_min^s

is i r divergent. Gauge dependent divergences are guaranteed to cancel order by order in ħ h. However, it is not clear how gauge independent divergences behave. This section focuses on the gauge independent divergences found in Landau gauge (ξ = 0).

The idea is that divergences in the ħ h-expansion of V

_min^s

, V

_min^s

= ħhV

1^s

(φ

0

)

+ ħh

²

 V

₂^s

− 1

2 ϕ

₁²

+ 2φ

1^h

ϕ

₁

∂²

V

₀

‹   

φ₀

+ . . . ,

might cancel order by order. We note that in Landau gauge ϕ

1

∝ G, and will soften any ir divergence.

3-loop singularity cancellation:

To illustrate how such a cancellation can work, let us consider the logarithmic (∼ log G) divergence at three loops. In the following we will only retain possibly divergent terms; we use the symbol ' to denote ex- pressions equivalent up to finite terms. The ħ h-expansion of V

_min^s

, with all terms implicitly evaluated at φ

0

, gives

O (ħh

³

) : V

₃^s

+ φ

1^h∂

V

₂^s

+ φ

^h₁

2

2!

^∂

2

V

₁^s

!

+ ϕ

₂ ∂

V

₁^s

+ φ

1^h∂²

V

₀



.

(3.9)

The possible divergences of the expression within the paren- thesis on the first row of equation (3.9) come from the 2- and 3-loop daisy diagrams

V

₂^s

∼

^ΠΠ11(0)(0)

, V

₃^s

∼

^{Π1(0) ΠΠ11(0)(0)}

,

where the Goldstone self-energy insertions are hard. Our aim is now to translate these diagrams to the language of the ħ h- expansion. For a general l-loop daisy diagram, the contribution to the potential is

V

_l^s

= − 1 2

(Π

^h1

(0))

^l−1

l − 1

Z

(dk) (−1)

^l−1

(k

²

+ G)

^l−1

. (3.10) We can relate this expression to the soft 1-loop potential via the relation

− 1 2

Z

(dk) 1

(k

²

+ G)

ⁿ

= (−1)

^n∂nG

V

₁^s

(n − 1)! ,

which gives

V

_l^s

= 1

(l − 1)! Π

^h₁

(0)

l−1

∂^l−1_G

V

₁^s

. (3.11)

The i r divergent contributions to the 2- and 3-loop V

^s

can then be written

V

₂^s

' (Π

^h1

(0))

^∂G

V

₁^S

, V

₃^s

' (Π

^h1

(0))

^2∂

2 G

V

₁^S

2 .

By using the definition of the Goldstone mass, the derivatives with respect to G can be rewritten as

∂_G

= 3 λφ

^∂

,

∂²_G

=

 3 λφ

‹

2



− 1 φ

^∂

+

^∂2

‹ .

We give a general formula for an arbitrary number of G deriva- tives in appendix D.2.

When φ-derivatives act on the soft 1-loop potential, the result scales as

^∂n

V

₁^s

∼ G

²⁻ⁿ

. Thus the logarithmically diver- gent terms that are present at three loops can only come from

∂²

V

₁^S

. We now use Π

^h₁

(0) = −

^∂2

V

0

φ

₁^h

/φ

0

and

∂²

V

0

= λφ

²0

/3 to rewrite the divergent terms in the ħ h-expansion as

V

₃^s

+ φ

1^h∂

V

₂^s

+ φ

₁^h

2

2

^∂

2

V

₁^s

'(φ

1^h

)

^2∂2

V

₁^s

 1 2 − 1 + 1

2

‹

= 0.

The remaining terms on the second row of equation (3.9) are possibly divergent because they are proportional to ϕ

2

; we have not established whether this quantity is finite yet. It is defined by

ħ h

² ∂²

V

0

ϕ

₂

+

^∂

V

₂^s

+ φ

^h1∂²

V

₁^s

' 0.

The divergent contributions to ϕ

2

are

ϕ

₂

' − 1

∂²

V

0

∂

V

₂^s

+ φ

1^h∂²

V

^s

' −(φ

1^h

)

^2∂

2

V

₁^s

∂²

V

0

(−1 + 1) = 0, and ϕ

2

is finite. We have showed that V

_min^s

is finite (actually zero) to three loops in the ħ h-expansion.

Leading singularity cancellation:

The above example can be generalized. The idea is that we can rewrite any daisy dia- gram in terms of derivatives acting on V

1^s

. We will use this pro- cedure to show that leading singularities cancel to all orders, where leading singularities are defined as the worst possible divergence at a given loop order. In this calculation we use the symbol ' to denote equivalence up to sub-leading diver- gences (note that this definition carries over to the three loop example).

At L loops the leading singularity behaves as V

L

∼ G

^3−L

. At the ħ h

^L

order, the ħ h-expansion gives terms of the form

V

_min^s

= . . . + ħh

^L



V

_L^s

+ φ

1∂

V

_L−1^s

+ . . . + φ

₁^L−1∂^L−1

V

₁^s

(L − 1)!

 + . . .

If L ≥ 3 all the terms in the parenthesis have the same leading

singularity. This can be seen from that the leading singularity

for

∂ⁿ

V

l

is ∼ G

^3−l−n

, and l +n = L for all terms in the parenthesis.