Renormalization of the Abelian-Higgs Model in the R-xi and Unitary gauges and the physicality of its scalar potential

We perform an old school, one-loop renormalization of the Abelian-Higgs model in the Unitary and $R_\xi$ gauges, focused on the scalar potential and the gauge boson mass. Our goal is to demonstrate in this simple context the validity of the Unitary gauge at the quantum level, which could open the way for an until now (mostly) avoided framework for loop computations. We indeed find that the Unitary gauge is consistent and equivalent to the $R_\xi$ gauge at the level of $\beta$-functions. Then we compare the renormalized, finite, one-loop Higgs potential in the two gauges and we again find equivalence. This equivalence needs not only a complete cancellation of the gauge fixing parameter $\xi$ from the $R_\xi$ gauge potential but also requires its $\xi$-independent part to be equal to the Unitary gauge result. We follow the quantum behaviour of the system by plotting Renormalization Group trajectories and Lines of Constant Physics, with the former the well known curves and with the latter, determined by the finite parts of the counter-terms, particularly well suited for a comparison with non-perturbative studies.

Spontaneously broken gauge theories are of great physical interest, notably because of the Brout-Englert-Higgs (or simply Higgs) mechanism [1]. Even though classicaly there is a simple qualitative description of the mechanism that one can find in textbooks, at the quantum level where these theories are more precisely defined, ambiguities arise. These ambiguities are related to the question of whether one can consider the quantum scalar potential itself as a physical quantity. In this work we touch on this issue approaching it from two angles, in the context of the Abelian-Higgs model [2].
One angle of approach has to do with the basis of representation for the Higgs field. Recall that in most loop calculations the (complex) Higgs field is written in a Cartesian basis: H = φ 1 + iφ 2 [3]. The real part of the field φ 1 is the physical Higgs particle, while φ 2 represents the unphysical Nambu-Goldstone (or simply Goldstone) boson. For dimensional reasons the resulting β-functions for the Higgs mass and quartic coupling must have the form β Cart. λ are determined after the loop calculation has been performed. The former coefficient affects the behaviour of the Higgs mass under Renormalization Group (RG) flow and the latter that of the quartic coupling, that in turn affect triviality and in the presence of fermions, also instability bounds. Taking these bounds seriously at a quantitative level means that the scalar potential is considered to be a physical quantity. Here we perform our computations in a Polar basis, H = φe iχ/v , where now the physical Higgs field is φ = φ 2 1 + φ 2 2 and χ is the Goldstone boson (v is some vacuum expectation value). In this basis one expects to find β Pol. m H = c Pol. m λm 2 H + · · · and β Pol. λ agrees with c Pol. λ . We find that these coefficients do not exactly agree, introducing possibly a small but computable ambiguity in the RG flows and the above mentioned bounds. The Polar basis may have though also a deeper impact on the quantum potential, having to do with its gauge (in)dependence. Some of these issues were noticed already in [4] in the context of the effective potential.
The other angle of our approach has to do with the quantization scheme. In particular, there is an infinitum of possible gauge fixing functions that one can use during quantization, each one of them introducing at least one gauge fixing parameter, say ξ. A typical representative of these schemes is R ξ gauge fixing. It is a well known fact that any sensible quantization scheme should produce a gauge independent set of physical quantities. Such quantities are for sure the masses of physical fields and the independent dimensionless couplings like λ. Gauge independence of physical quantities in a scheme like the R ξ gauge fixing scheme is then ensured if the corresponding β-functions are ξ-independent and this is indeed the case in every consistent loop calculation. The scalar effective potential com-puted with the background field method on the other hand is notoriously known to be ξ-dependent already at one-loop, which renders its physicality (and the relevance of the precision triviality and instability bounds derived from it) a delicate matter [5,6]. It is therefore an open issue how to define an unambiguous, physical, quantum potential in spontaneously broken gauge theories. For this reason, we choose the Unitary gauge [9] as our quantization scheme. The Unitary gauge is one where only physical fields propagate and it is commonly used in textbooks in order to demonstrate the physical spectrum of spontaneously broken gauge theories at the classical level. It is rarely used though for loop calculations, in fact we are not aware of a complete renormalization work in this gauge. A possible obstruction to completing reliably such a program may be the high momentum behaviour of the Unitary gauge boson propagator resulting in integrals that often diverge worse than quadratically with a cut-off, even in a renormalizable theory. We call these integrals, "U -integrals". For this reason the Unitary gauge is sometimes called a non-renormalizable gauge. A necessary condition therefore for a Unitary gauge calculation to make sense is that the physical quantities that are ξ-independent in the R ξ gauge, to coincide with the corresponding quantities derived in the Unitary gauge. If this condition is fulfilled then through the renormalization procedure one automatically obtains a version of the scalar potential, the Unitary gauge scalar potential, that is by definition gauge independent. A question we would like to answer here is if this version of the scalar potential can be used, in principle, to derive competitive with respect to the R ξ gauge physical predictions. Of course, an important issue is to understand the connection between the Unitary and R ξ gauge potentials. The standard connection between these two gauges is to take the "Unitary gauge limit" ξ → ∞ at the level of the Feynman rules, before loop integrals have been performed. Clearly this is not what we want to do here. Instead, we would like to compare the R ξ gauge potential with the Unitary gauge potential, after loop integration. This is non-trivial, as the ξ → ∞ limit and the loop integration may not commute. For recent studies of related issues, in the context of the H → γγ decay, see [10,11,12,13] and for some earlier works on the Unitary gauge and the Abelian-Higgs model, see [4,14].
In this work, we consider the Abelian-Higgs model and perform the one-loop renormalization of the gauge boson mass and of the Higgs potential in both the R ξ and Unitary gauges. In addition to the above mentioned comparison reasons, this double computation allows us to monitor and ensure the credibility of the Unitary gauge calculation. We will be able to show that as far as the β-functions (determined by the divergent parts of the one-loop amplitudes) is concerned, the Unitary gauge is equally consistent, in fact equivalent to the R ξ gauge. Then we consider the scalar potential (determined by the finite parts of the amplitudes after renormalization) and ask whether it could also be physical. Let the finite part of the one-loop value of a quantity be defined as ( ) f with the subscript f denoting finite part. Such quantities will be the various loop corrections entering in the renormalized Higgs potential. Then schematically we have that in general ( ) f , lim ξ→∞ = lim ξ→∞ g(λ, m H , m Z , µ, ξ) , (1.1) where the behaviour of the function g at ξ = ∞ is one thing we would like to understand. The background field method in the MS scheme for example gives an effective potential where lim ξ→∞ g(ξ) = ∞ for a large class of gauge fixing functions [6], rendering the Unitary gauge limit after loop integration, singular. The singularity implies that the Unitary gauge is disconnected from the space of R ξ gauges and essentially that the Higgs potential is unphysical away from its extrema. Our calculation instead shows that lim ξ→∞ g(ξ) = 0, perhaps the most striking result of this work, as it implies that the Higgs potential can be made gauge invariant, hence physical, even away from its extrema. For formal aspects of the quantization of the Abelian-Higgs model, see [7,8] In Section 2 we review the classical Abelian-Higgs model and discuss some basic conventions in our calculation. In Section 3 we perform in detail the R ξ -gauge computation and in Section 4 the Unitary gauge computation. In Section 5 we renormalize the AH model. In Sections 6 and 7 we present a numerical analysis of our results. In Section 8 we state our conclusions. We also have a number of Appendices where auxiliary material can be found.

The classical theory and some basics
The bare Lagrangean of the AH model is Zero subscripts or superscripts denote bare quantities. As usual, the covariant derivative is D µ = ∂ µ + ig 0 A 0 µ and the gauge field strength is F 0,µν = ∂ µ A 0 ν − ∂ ν A 0 µ . The Higgs field H 0 is a complex scalar field. The Lagrangean is invariant under the U (1) gauge transformations with θ(x) a gauge transformation function and the discrete, global Z 2 symmetry We assume that both m 0 and λ 0 are positive quantities.
As it stands, the part of the Lagrangean that corresponds to a potential triggers SSB. Minimization yields the vev (2.5) The second of the above equations defines the bare vacuum expectation value (vev) parameter v 0 . The field H 0 can be expanded around its vev as where now φ 0 is the new Higgs field fluctuation and χ 0 is a massless Goldstone boson. Eq. (2.6) is the so called "Polar basis" for the Higgs field, as opposed to the "Cartesian basis" H 0 (x) = 1/ √ 2(v 0 + φ 1,0 (x) + iφ 2,0 (x)). The former is typically used for demonstrating the physical spectrum while the latter for quantization. Here we will stick to the Polar basis for both. Replacing the vev into Eq.(2.1), the Lagrangean takes the form with g µν the Minkowski space metric. The bare gauge boson (Z boson) mass is defined as m Z 0 = g 0 v 0 and the bare Higgs mass as m H 0 = √ 2m 0 = √ 2λ 0 v 0 . As implied by the above expression, we have decided to use as independent parameters the masses and the quartic coupling λ 0 . This means that we have eliminated the gauge coupling according 2λ 0 wherever they appear. Setting the Goldstone field χ 0 to zero, gives us the Unitary gauge Lagrangean which will be the topic of a separate section. For now we will keep the Goldstone in the spectrum. A consequence of having a Goldstone in the spectrum is a mixing term between the Z and χ 0 in Eq. (2.7).
We will compute all one-loop Feynman diagrams that contribute to the renormalization of the Z mass and of the scalar potential, using Dimensional Regularization (DR) [16]. We assume basic knowledge of the DR technology that we therefore do not review here, except from some necessary basic facts that can be found in the Appendices. The renormalization scale parameter of DR is denoted by µ. The small expansion parameter ε of DR is defined via Since we are using DR in our renormalization scheme, a consequence is that the trace of the metric is g µν g µν = d.
Now, each diagram comes with a symmetry factor. Consider a one-loop diagram containing n vertices with k n lines on each vertex. These k n lines are divided into k in n and k ext n for the internal and the external lines respectively. The procedure to obtain the correct symmetry factor is given for example in [15]: • For each of the n vertices, count all possible ways that the k n lines can be connected to the external legs of a given diagram. This is k ext n . Doing this for every vertex defines n O as the product of the k ext n 's. The remaining lines belong to k in n .
• At each of the n vertices there are k in n lines. Count all the possible ways the loop can be constructed using these lines. The product of the k in n 's defines n I .
• For each of the n vertices, count the number of k i lines that are equivalent. This defines i .
• Finally, for a given diagram count all the possible equivalent vertices of type j, defining v j .
The symmetry factor of the diagram is then where a, b are indicators specifying the diagram.
A large part of the one-loop diagram expressions is dominated by the set of basic integrals called Passarino-Veltman (PV) integrals [17]. We collect in Appendix B the basics of the formulation of PV integrals, following mostly [18]. In our calculation several nonstandard integrals emerge as well. The divergent ones we call collectively U -integrals and we compute them in Appendix C. Finite integrals and finite parts of divergent integrals are harder to classify systematically so we will be dealing with them as we proceed.
We also introduce some useful notation. Our convention for the name of a one-loop Feynman diagram F is F G,L E (2.10) G = R ξ , U specifies either the R ξ or the Unitary gauge where the diagram is computed. L is a list containing the field(s) running in the loop in the direction of the loop momentum flow, starting from the vertex on the left side of the diagram. In case the diagram is an irreducible Box, we start the list from the upper left vertex. E = H, Z specifies the external legs. In the case E = Z, there may be additional Lorentz indices following E. We know that one-loop Feynman diagrams can be either finite or divergent. In the first case the corresponding integrals contain only finite terms and in the second case they include both divergent and finite parts. Furthermore, in each of the above cases, in the R ξ gauge, the corresponding parts could be either gauge (i.e. ξ) dependent or gauge independent. In the Unitary gauge there is no such distinction. Therefore, every set of one-loop diagrams contributing to the same correlator, where the sum over the index L has been performed and the Lorentz indices (if present) have been appropriately contracted, can be expressed as where the square brackets denote ξ-independent part, the curly brackets denote ξ-dependent part and the subscripts ε and f denote divergent and finite part respectively. The 1/ε factor is absorbed in the definitions of [F G E ] ε and F G E ε . A word of caution here is that the above separation of diagrams into ξ-independent and ξ-dependent parts is clearly not unique. Nevertheless it is a very useful notation (once we get used to it) since it allows to perform algebra with diagrams easily and also facilitates the comparison with the Unitary gauge. In fact, in most cases the ξ-independent part of sums of diagrams is just the corresponding Unitary gauge result. The sum of the gauge independent and gauge dependent finite parts is denoted as Round, square and curly brackets appearing in any other context have their usual meaning. All entirely finite integrals are computed as described in Appendix A. We will be giving the divergent parts and the finite parts of sums of groups of diagrams explicitly in the main text, leaving some of the (increasingly cumbersome) expressions for finite (parts of) diagrams to Appendix E. There are some finite parts sitting inside the U -integrals too, which we compute together with their divergent parts and do not show explicitly, in order to avoid repetitions.
We perform renormalization away from the usual MS scheme. This means that our renormalization conditions force us to keep some non-trivial finite terms, that eventually enter in the renormalized Higgs potential. What finite terms are kept is of course not a unique choice. We have not checked if the gauge invariance of the potential could also be achieved in a pure MS scheme. If however MS (or any other scheme) would result in a gauge dependent potential, it would mean that in spontaneously broken gauge theories scheme dependence kicks in already at one loop. Recall that in QCD (a not spontaneously broken theory) scheme dependence appears at three loops. This would be strange because it would relate gauge dependence to scheme dependence. Our feeling is that any physical renormalization subtraction scheme applied to our calculation should produce a gauge invariant Higgs potential. Furthermore, we perform renormalization off-shell, at zero external momenta. External momenta will be generically denoted by p. In 2-point, 3-point and 4-point functions zero external momenta means p i = 0 with i = 1, 2, i = 1, 2, 3 and i = 1, 2, 3, 4 respectively. This choice, beyond being a huge simplifying factor, is justified since we are interested in terms of the Lagrangean without derivatives. It would be equally strange if after renormalization the Higgs potential would pick up an external momentum dependence. In other words we believe that even if we had performed renormalization on-shell for example, we would have obtained the same results, alas in a more complicated way: not only individual diagrams would become much more involved, but also non-1PI diagrams may have to be added in order to arrive at gauge invariant β-functions. A physical argument for zero external momenta could be that if one is interested in the high energy limit, is entitled to set the masses of external particles equal to zero, that is p 2 i = 0. But this choice, using momentum conservation in 2, 3 and 4-point functions is the same as setting the momenta themselves to zero. A subtle point of setting external momenta to zero is that in a diagram a term of the form p µ p ν T µν (p, m i ), with p an external momentum, m i mass parameters and T µν a tensor (DR) integral, may appear. If we set p = 0 before integration, this term is zero. If on the other hand we do the integral first, then contract and then take the limit p → 0, we may find a non-zero result. The latter procedure is the correct one. Expressions of the form of Eq. (2.11) in the main text will be thus given at zero external momenta. Nevertheless, for completeness and for potential future use, we collect some on-shell expressions in Appendix F.

R ξ gauge
Quantization of Eq. (2.7) requires gauge fixing. The gauge fixing term that removes the Goldstone-gauge boson mixing defines the R ξ gauge and under the standard Faddeev-Popov procedure introduces the extra ghost contribution The sum L R ξ = L 0 + L gf + L ghost yields the final expression from which the R ξ gauge Feynman rules can be derived. Notice that the Goldstone has acquired an unphysical, gauge dependent mass m χ 0 = √ ξm Z 0 . With this gauge fixing choice the ghost fields have apart from a kinetic term, a mass term equal to that of the Goldstone boson's but they are not coupled to the Higgs field or to the gauge boson.
The Feynman rules arising from Eq.(3.3) are the following: In the above rules, k a,b denotes the momenta of the Goldstone bosons in a vertex. In the case that we have two Goldstones in a diagram, we choose the convention where one of them gets in and the other gets out of the vertex (since we will encounter the Goldstone only in loops).
The momentum dependence of the vertices including at least one Goldstone boson is a direct consequence of our choice to use the Polar instead of the Cartesian basis for the Higgs field, as in the latter case there are no such vertices. As a result, the corresponding loop integrals will have extra loop momentum factors in their numerator and this triggers the appearance of U -integrals (also) in the R ξ gauge.
In what follows, we present only the final expressions for the divergent and finite parts of the various sectors given at zero external momenta. The explicit calculation steps that we followed are given in the Appendix D.

Tadpoles
One-point functions are also called tadpoles. The first such diagram is and analytically evaluates to The next tadpole comes with a gauge boson loop: and it is equal to The last tadpole has a Goldstone loop: is calculated we obtain that the above diagram is equal to The total tadpole value is the sum of the above three contributions: T procedure we obtain that the quantity that enters in the renormalization of the mass of the Z gauge boson is therefore We now start computing the one-loop Feynman diagrams contributing to M R ξ Z,µν . The first contributing diagram has a Higgs running in the loop: and, in DR, it is equal to Next we meet a couple of "sunset" diagrams. The first is In DR and using Eq. (B.35), it can be expressed as Notice that the C-type integral above is a special PV case, computed in Appendix B as well.
The next sunset diagram is the last that contributes to the one-loop correction of the gauge boson propagator: and its explicit form reads Adding up all contributions we obtain The contraction that we need is Eq. (3.10), which for general p has the form (3.14) Specializing now to p = 0, we can express the result as The anomalous dimension of Z can be also determined now, through the relation Evidently, all divergent parts in this sector are ξ-independent.
Regarding the finite parts, we have It is easy to check that all of the above three diagrams are reducible Tadpoles corresponding to the three Tadpoles of Section 3.1.
A few vacum polarization diagrams are in order. The first is The Goldstone loop contribution where the a = 1, 2, 3 superscripts on the C 0 -integrals correspond to the different combinations of the denominators according to Eq. (B.12) of Appendix B. The D µν integrals are defined in Eq. (B.39).
The last contribution to the one-loop correction of the Higgs mass comes from the sunset where using again the standard steps, we are allowed to write this as This sector turns out to be ξ-independent.

Corrections to the Higgs cubic vertex
Triangle diagrams with Higgs external legs yield corrections to the Higgs cubic vertex. Such corrections will play a crucial role in the definition of the one-loop scalar potential. The external momenta are taken to be all inflowing, thus satisfying p 1 + p 2 + p 3 = 0. Triangle diagrams can be split in "reducible" and "irreducible" kinds. Reducible are those that are expressible in terms of 2-point function diagrams and irreducible are those that are not.

Reducible Triangles
The first reducible Triangle diagram is: As anticipated, it is not an independent diagram. It is the same loop-diagram as in Eq. (D.35) (with the same symmetry factor), divided by v 0 . We can therefore write directly the result: The factor of 3 is due to two additional diagrams, obtained from the above by cyclically permuting the external momenta. These contributions however, evaluated at either p 1 = p 2 = p 3 = 0 or p 2 1 = p 2 2 = p 2 3 = m 2 H give an identical result. The next diagram is one with a Goldstone in the loop: and it is equal to Eq. (D.37) divided by v 0 : The factor of 3 has a similar origin as before.
The diagram with a gauge boson loop (3.36) The last reducible Triangle is: It is the same as Eq. (D.42) divided by v 0 :

Irreducible Triangles
We now turn to the irreducible Triangles. All irreducible Triangle diagrams can be labelled by the momenta P 1 = p 1 and P 2 = p 1 + p 3 . A simplifying consequence of renormalizing at p i = 0 is that we can set P i = 0, hence we can extract both divergent and finite parts, using The limit should be carefully taken, as explained in Sect. 2. Regarding denominators, from now on, we start following the notation of Eq. (A.3). Here, apart from finite integrals of the E-type, we will also see the appearance of several divergent U -integrals. All finite integrals and U -integrals here and in the following are computed (sometimes without further notice) in Appendices E and C respectively. Finite diagrams do not play a role in the renormalization program but they contribute to the scalar potential.
The first contribution to the irreducible Triangle class involves a Higgs loop and it is finite. It is the diagram with a symmetry factor S 5 K H = 1. In DR it can be expressed as As explained, There is another finite diagram, the one with a gauge boson loop: with a symmetry factor S 6 K H = 1. Its expression in DR is where we have used the notation for finite (parts of) diagrams, explained in Eq. (2.12). Following again previous arguments, at zero momentum we obtain This is the first of several irreducible diagrams whose explicit form is not particularly illuminating, so we directly transfer it to Appendix E.
Now, let us move on to diagrams that have both an infinite and a finite part. The first such diagram is with a symmetry factor S 7 K H = 1. In DR it becomes where the mass arguments of the D 1,2,3 denominators are easily recovered from the ZχZ superscript structure of the diagram: D 1 (m Z ), D 2 (m χ ) and D 3 (m Z ). We are not done yet since there are three different ways to insert the Goldstone propagator in the loop. Therefore, there are two more contributing diagrams of the same kind as K Performing the calculations at zero external momenta, the above diagrams have identical divergent and finite parts with K Next, we have the diagram with two Goldstones and one gauge boson in the loop: It is given by the relation with symmetry factor S 8 K H = 1. In DR it reads (3.52) Now, similarly to the previous case there are three ways to insert the gauge boson in the loop, which means that there are two more diagrams of the same kind as K Again here, at zero external momenta all three diagrams have the same divergent and finite parts: The last one-loop correction to the three-point vertex comes from the irreducible Triangle given by the expression with symmetry factor S 9 K H = 1. There is only one diagram of this kind and its explicit form in DR reads Summing up all the irreducible Triangles, we find that at zero external momenta We see that the irreducible Triangles do not contribute to the divergent part of the 3-point function.
It is worth looking a bit closer at the cancellation of the gauge fixing parameter ξ from the finite part of this sector. The finite parts, collected according to the loop propagators, are It is easy to see the cancellation of ξ in the sum.
We now add reducible and irreducible contributions into K

Corrections to the quartic coupling
Diagrams with four external Higgs fields contribute through their divergent parts to the running of the Higgs quartic self coupling λ and through their finite parts they contribute to the one-loop scalar potential. They are collectively called "Box diagrams", denoted as B H and come in three classes. The first two of these classes contain reducible diagrams and the third class contains irreducible Box diagrams, called "Square (S)-Boxes". The reducible class is further split in two subclasses, called "Candy (C)-Boxes" and "Triangular (T )-Boxes". They have the following structure: Regarding the momentum flow, we take all four external momenta p i = 1, 2, 3, 4 to be inflowing and satisfying p 1 +p 2 +p 3 +p 4 = 0. Candies and S-Boxes come in three versions, corresponding to the usual s, t and u channels, where T -Boxes come in six versions instead because they are not invariant under a reflection with respect to the axis passing through the centre of the loop in the diagram. There are two inequivalent topologies and each topology comes with s, t and u channels. Any U -integral that may appear is dealt with in Appendix C and finite integrals of the E, F , G and H-type in Appendix E.

Reducible Boxes
Candies have the generic momentum dependence B t and √ u for the three channels. Their total contribution is then a sum over P 1 .
The first Candy is the famous diagram which solely determines the β-function in pure scalar theories, in the case where the Higgs is expressed in the Cartesian basis, all other diagrams being finite. Here in the Polar basis we will see that this is still the case alas in a non-trivial way. The result for this diagram can be obtained from Eq. (3.34) (without the factor of 3) divided by v 0 and evaluated at P 1 : The Goldstone Candy is analogously equal to Eq. (3.35) divided by v 0 (without the factor of 3), evaluated at P 1 : The gauge Candy Finally, there is a mixed Candy The full contribution of the Candies is obtained by adding Eq. (3.63), Eq. (3.64), Eq. (3.65) and Eq. (3.66) (summed over the three channels) and can be expressed as for p i = 0. We now turn to the T -Boxes that have three propagators in the loop. Each of the six channels of a given T -Box is determined by two linear combinations of the external momenta that we call P 1 and P 2 . A consistent choice for (P 1 , P 2 ) for the channels T 1,··· ,6 is for example The generic form of a T -Box is therefore B R ξ ,T H (P 1 , P 2 ) and the total contribution is obtained by a sum over the six different pairs (P 1 , P 2 ). Note that the Mandelstam variables enter also via the inner products All T -Boxes can be obtained from their corresponding irreducible Triangles. They share a common symmetry factor and analytical structure with all the difference encoded in the values that the pair (P 1 , P 2 ) can take. Hence, in this sector we give directly the formal expression in order to avoid unnecessary repetitions. To begin, we have two finite diagrams: There are diagrams that have both divergent and finite parts. There are three with one Goldstone and two gauge bosons in the loop: and iB respectively. And there are three with one gauge boson and two Goldstones in the loop: Adding up all the T -Boxes we find that at zero external momenta

Irreducible Boxes
Irreducible, or S-Boxes, have a channel structure determined by three linear combinations of momenta, called P 1 , P 2 and P 3 . The resulting three independent channels are the usual s, t and u channels with the momenta P 1,2,3 determined easily from the above figure: s −channel : Irreducible Boxes have therefore the generic form B R ξ ,S H (P 1 , P 2 , P 3 ) and receive a contribution from the s, t and u channels. In order to take into account all channels, after considering the different diagram topologies, we sum over the (P 1 , P 2 , P 3 ) according to the above rule. All S-Boxes have symmetry factor 1. Several annoying for the eye expressions for finite terms (B We start the computation of the S-Boxes with the finite Higgs loop diagram with the D i in the denominator defined in Appendix A. This is just a finite D 0 PV integral: The next diagram is also finite. It is and is equal to In DR it takes the form Another finite diagram is and in DR to , giving different, but finite result. The total contribution of these diagrams too is obtained by summing over the three possible channels (P 1 , P 2 , P 3 ), i.e. the s, t and u channels, according to Eq. (3.74).
We turn to the S-Boxes which have both infinite and finite parts. The first one of this type is and is given by the expression

(3.82)
A portion of the finite part of the above diagram is actually built in the D B4 -integrals (defined at the end of Sect. B.2) while the rest of it is given in Appendix E. There are in total six different topologies for this diagram corresponding to the different ways the two Goldstones can be distributed in the loop. All these diagrams have exactly the same divergent part as B R ξ ,ZZχχ H but not the same finite part.
Next, we have and in DR evaluates to ) f,2 are entirely finite, given in Appendix E. The rest of the finite part is encoded in the PV and U -integrals.
The last divergent S-Box is the one with only Goldstone bosons inside the loop: (3.85) Summing up all topologies and channels, at zero external momenta, we find for the S-Boxes the relations We note the interesting facts that B has only a {· · · } ε part and that The cancellation of ξ from the finite parts is more illuminating when results from individual loop structures are shown: The cancellation of ξ is now evident.
The final step here is to collect all Boxes and sum them up. Adding Eq.(3.68), Eq.(3.72) and Eq.(3.87), we obtain at p i = 0: A couple of final comments are in order. First, each block of box diagrams, Candies, T and S-Boxes is by itself ξ-independent. Moreover, looking at the results from the three sectors, one notices that they have an explicit s, t and u-dependence. So, one would expect that [B R ξ H ] ε could also be channel dependent. Nevertheless, Eq.(3.90) shows that the s, t and u-dependence cancels when the full contribution of the box diagrams is taken into account.

Unitary Gauge
In the previous section, we computed one-loop processes in the Abelian Higgs model when the gauge symmetry is broken using an R ξ gauge fixing term. However one of our main goals here is to investigate this model in the Unitary gauge. This is interesting since, in the Unitary gauge only physical degrees of freedom are present, in contrast to the R ξ gauge. Moreover, there are statements in the literature that argue that the Unitary gauge may be problematic at the quantum level, so by comparing it to the R ξ gauge, we will try to clarify the correctness of these arguments.
In the Unitary gauge, no gauge fixing is needed, therefore there is no need for ghosts. The Unitary gauge Lagrangean can be simply obtained from Eq. (3.3) by removing gauge fixing and ghost terms and setting χ 0 = 0 in the remaining. Doing so, we obtain the Unitary gauge Lagrangean from which the Feynman rules can be derived: In the Unitary gauge, integrals of the U -type are ubiquitous. But such integrals we have already seen in the R ξ gauge due to the momentum dependent vertices of the Polar basis for the Higgs field. In the Unitary gauge we do not have momentum dependent vertices, the U -integrals arise only because of the form of the propagators.
In the following sections we compute one-loop diagrams, setting the external momenta to zero at the end, as in the R ξ gauge. Again for completeness, we present some on-shell results in Appendix F. We will directly insert symmetry factors here since they are the same as in the corresponding R ξ calculation. In the Unitary gauge, since there is no gauge fixing parameter, we have trivially { } ε = { } f ≡ 0. As before, we will consistently move nasty expressions for finite parts to Appendix E.

Tadpoles
The Higgs tadpole is and in DR reads The gauge tadpole is and following Appendix D.2, this becomes Adding up the two results, we have The above expressions are identical to the R ξ expressions Eq. (3.8) and Eq. (3.9).

Corrections to the gauge boson mass
The Z boson mass receives its first correction from Next is the Higgs sunset The sum of these two corrections is (4.10) As before, the proper contraction we are after is which can be easily shown to be equal to (4.14) The wave function renormalization factor is now straightforward to compute: Glancing back on Sect.3.2 we see the first sign of the consistency of the calculation by noticing that M U Z and M R ξ Z have the same gauge-independent divergent parts and divergent parts of their respective anomalous dimensions.
Regarding the finite parts we have and It is interesting to notice that the last two expressions for the finite parts can be obtained from the corresponding R ξ gauge expressions in Eq. (3.19) and Eq. (3.20) for ξ = 1 and not for ξ → ∞.

Corrections to the Higgs mass
The first correction to the Higgs mass comes from Next is The Higgs vacuum polarization diagram The corresponding gauge loop is where we have defined (4.24) Using the reduction formulae in Appendix B and summing up all contributions we can extract The anomalous dimension is then

Corrections to the Higgs cubic vertex
As before we split the diagrams in two classes. The first class consists of the reducible Triangle diagrams, while the second one consists of the irreducible Triangles. The contribution of the various topologies and channels is contained in their P 1 and (P 1 , P 2 ) dependence respectively, exactly as in the R ξ -gauge calculation.

Reducible Triangles
The first reducible Triangle is and it is the same loop as in Eq. (4.21) with p → P 1 , with the same symmetry factor, divided by v 0 : where the three different channels are obtained by summing over P 1 , with P 1 = p 1 , p 2 , p 3 (resulting to an overall factor of 3, as in the R ξ -gauge).
Next is the gauge loop (4.32) There are again two more channels with P 1 = p 2 and P 1 = p 3 , yielding identical contributions.
Adding up all the reducible Triangles, we have at p i = 0.

Irreducible Triangles
The first irreducible Triangle is (4.36) and in DR In the Unitary gauge there is only one more divergent irreducible Triangle, which is and in DR where in this case P 1 = p 1 and P 2 = p 1 + p 3 . The contracted PV integral C µ K3 is defined in Appendix B.2.
In total, we find for the irreducible Triangles Finally, adding reducible and irreducible contributions, we have These are the same results as the ones found in Eq. (3.60) and Eq. (3.61) in the R ξ -gauge.

Corrections to the quartic coupling
The separation of the Box diagrams into C, T and S-Boxes in the Unitary gauge holds exactly like in the R ξ gauge. The same goes for the labelling of the various channels by the momenta P 1 (for Candies), (P 1 , P 2 ) for T -Boxes and (P 1 , P 2 , P 3 ) for S-Boxes.

Reducible Boxes
The Higgs Candy The gauge Candy is slightly different due to the different gauge boson propagator. The diagram is These are the only two contributions in this sector and adding them up, we have We turn to the T -Boxes. In the Unitary gauge we have only two contributing diagrams, one finite and one divergent. The finite diagram is The s channel contribution is equal to The t and u channels can be obtained as explained in the R ξ -gauge calculation. The divergent diagram in this sector is and can be easily be obtained from Eq.(4.39) by dividing by v 0 : (4.52) After summing over all channels by summing over all (P 1 , P 2 ), we obtain

Renormalization
In this section we will renormalize the Abelian-Higgs model in both the R ξ and Unitary gauges. In fact, since we will be concerned here mainly with the Z-mass and scalar potential we will be able to perform the renormalization program simultaneously for both.
One of the results of the calculation of the previous two sections is that the one-loop corrections to the Z-mass and the scalar potential have identical divergent parts in the R ξ and Unitary gauges. This means that the two gauges have the same β-functions for the masses and quartic coupling. Now the Lagrangean in the Unitary gauge can be obtained from the R ξ -gauge Lagrangean by dropping the gauge fixing and ghost terms and simply setting the Goldstone field to zero. Since we are interested here only in the common subsector that consists of the Z-mass term and the scalar potential, we can carry out the renormalization program on the Unitary gauge Lagrangean and only when we arrive at the stage where we analyse the finite, renormalized scalar potential where finite corrections become relevant, we may have to distinguish between the two gauges if necessary. Thus for now and until further notice, we drop the "U" superscript that denotes Unitary gauge.

Counter-terms
We introduce the counter-terms In the Abelian-Higgs model there is a non-zero anomalous dimension for the scalar (as opposed to φ 4 theory and the linear sigma model) and likewise a non-zero anomalous dimension for the gauge boson. Therefore, we also introduce Substituting the above in Eq. (2.1) and then expressing the Higgs in the Polar basis, we obtain with L tree the renormalized tree-level Lagrangean and L count. the counterterm Lagrangian with the computed one-loop corrections also added with appropriate factors. Corrections to interaction terms between the Higgs and the Z (M Z3 and M Z4 ) we have not computed but we are not concerned with those here.
The Feynman rules for the counterterms deriving from the above expression are The renormalization conditions are in order. All conditions are imposed at zero external momenta. Regarding the gauge boson sector, our renormalization condition is that the mass of the gauge boson be m Z = gv 0 . Diagrammatically this condition is and as an equation This is an independent condition from the rest that can be directly solved for δg: At zero external momentum, this becomes where we have absorbed in δg all the finite parts. Just one comment on the Higgs-Z interaction terms that we have not computed here: they become finite provided that Now we turn to the scalar sector. In order to avoid the tadpoles contaminating the oneloop vev for the Higgs field, we impose a vanishing tadpole condition. Diagrammatically it is and as an equation The second condition is the requirement that the only term that remains in the quadratic part of the potential, is m H . This means that in the quadratic term of the potential we absorb, together with the divergent part, the entire finite term as well: These two conditions fix completely our freedom. The solution to the system of Eq. (5.10) and Eq. (5.11) is For later reference note that δm H = 2δm . (5.14) Divergences must cancel automatically from the rest. Absence of divergences in the cubic coupling means Substituting Eq. (5.12) and Eq. (5.13) in the above, we find that the divergent part cancels as expected and we are left with the finite terms Substituting again Eq. (5.12) and Eq. (5.13) in the above, we again observe the cancellation of the divergent part and we collect the finite piece The subscript f in the various one-loop quantities denotes finite part, in either the R ξ or the Unitary gauge. For example, The finite one-loop Higgs potential we are left with after renormalization is then The Higgs field anomalous dimension has apparently cancelled from the renormalized potential. We are now ready to minimize this potential. There are three extrema. Our preferred "physical" solution for the global minimum is for which the potential satisfies V 1 (v) = m 2 H . The quantities C φ 3 and C φ 4 are examples of the quantities in Eq. (1.1) of the Introduction. Note finally that V 1 (v) = 0.

Physical quantities and the β-functions
Let us denote by α 0 a generic bare coupling. Its counter-term δα(µ) is introduced via the relation where α(µ) is the renormalized running coupling. At one-loop the counter-term has the form The indices i and k are counting the fields running in the loop. The entire divergent part has been collected in the term proportional to C a . In the notation of the previous sections, we can identify (µ ε /16π 2 )C α = ε(δα) ε = ε[δα] ε + ε{δα} ε and (δα) f = [δα] f + {δα} f containing the finite logarithms and the non-logarithmic finite term. As already noted, in the Unitary gauge we have {δα} ε = {δα} f = 0 by definition.
It is useful to review the calculations of β-functions in the presence of multiple couplings. We introduce the boundary condition α(µ = m phys. ) ≡ α . (5.23) We also use the following standard definitions For a general coupling we recall the successive relations where, since δ α ∼ O( ), we have performed an expansion in in order to get rid of terms of O( 2 ) like δ α · ∂δα ∂µ . In the case of the AH model where we have three couplings, we will have a system of equations: (5.26) This system can be rewritten as Inverting the matrix we obtain Thus, all that we need to do in order to obtain the various β-functions, is to identify from the explicit form of the counterterms the quantities C α defined in Eq.(5.22) and build its β-function, according to Eq.(5.29).
Moreover, solving the differential equation for the running coupling yields the Renormalization Group flow of the coupling α: The Landau pole associated with the coupling α is In the AH model we found from our one-loop calculation that (5.32) The above expressions hold for both R ξ and Unitary gauges. It is interesting to notice that these β-functions are not identically the same as those that one would compute in a Cartesian basis for the Higgs. For a comparison see the next section.

The one-loop Higgs potential
To summarize our result regarding the one-loop Higgs potential, in Eq. (5.19) we found that it is determined by the quantities C φ 3 and C φ 4 in Eq. (5.16) and in Eq. (5.18) respectively, yielding the finite expression The form of the potential is such that the ξ-independent part of its R ξ gauge expression is the same as the Unitary gauge expression. Moreover, using the standard prescription to compute the U T integral, see Eq. (C.2), the ξ-dependent part is made to vanish. In addition, the potential has no explicit µ-dependence. All the µ-dependence is implicit, through the dependence on µ of the renormalized quantities λ, m H and m Z via their RG evolution. In the following numerical analysis one can either interpret the potential in Eq. (6.1) as a ξ-independent R ξ gauge potential or simply as a Unitary gauge potential, as these are the same independently of prescriptions.
Before we proceed with the study of the gauge invariant Higgs potential Eq. (6.1), we recall the standard result used to extract physics from the Higgs potential. The usual method to compute the Higgs potential is in the context of the background field method, where a functional integration yields the so called Higgs effective potential. We will not review the details of this well known calculation; we assume that the reader has some familiarity with it. The derivation of an effective scalar potential via the background field method at one-loop is much simpler than computing Feynman diagrams. The simplicity is related among other reasons to the Cartesian basis representation of the Higgs field because in the Cartesian basis the Gaussian path integral involved, is almost trivial. The result of the calculation for the scalar potential in the Abelian-Higgs model (see for example [6]) with the same normalization of the mass and quartic coupling as in Eq. (2.1), in MS scheme with Fermi gauge fixing, is 3) The derivation of the β-function is also simple, provided that a separate diagram calculation has yielded the also well known result for the anomalous dimension γ. Then, one finds for the β-functions. These Cartesian basis results are not identically the same as the Polar basis results in Eq. (5.32). Despite however the fact that some coefficients are not the same, their physical content is similar.
In Figs. 1 and 2 we plot the RG evolution of the Higgs and the Z mass as well as that of the coupling λ, as a function of the Renormalization scale µ, as determined in Eq. (5.32). We do not produce separate figures for Eq. (6.5) because the numerical differences are quite small. The physical values at µ = 125 GeV we use are 125 GeV for the Higgs mass, 91 GeV for the Z-mass and 0.12 for λ. We stop the evolution at a certain scale µ I 3.03 · 10 46 GeV (6.6) whose physical meaning will be discussed below. We observe the usual perturbative, logarithmic evolution of the couplings.
The potential in Eq. (6.1) is gauge invariant and the one in Eq. (6.2) is manifestly gauge dependent. To quantify the effect of ξ, the different basis for the Higgs and the different subtraction schemes, we compare numerically the two results. The comparison that follows should be clearly taken with a grain of salt, as the two objects are quite different. In Fig. 3 we plot the difference between the one-loop Higgs potential Eq. (6.1) and the effective potential Eq. (6.2) at µ = 125 GeV, for ξ = 0.001, 1, 10, 100, 100000. The running values of m H , m Z and λ in the potential V 1 are determined from Eq. (5.32), while those of m, g and λ in V MS 1,eff. are determined using Eq. (6.5). A regime around φ = 0 in the plot is missing because there the effective potential is imaginary. There seem to be ways to fix this [19] but this is not our concern here. V 1 remains instead always real. What we observe is that as ξ increases, the regime where the two expressions agree shrinks. There seems to be a value of ξ for which the best agreement is achieved, which from the figure is around ξ < O(10). The limit ξ → ∞ of the effective potential is clearly singular: in Eq. (1.1) we have lim ξ→∞ g(ξ) = ∞.
From now on we concentrate on V 1 (φ) in Eq. (6.1) and we analyze it numerically. First, in Fig. 4 we plot it at the physical scale µ = 125 GeV. We observe that it has two minima, of which the one at φ = 0 is the global minimum, as claimed. The tilt in the mexican hat shape implies that the global Z 2 symmetry H 0 → −H 0 of the classical potential has been spontaneously broken by quantum effects. The breaking is small and the vacuum in the interior of the Higgs phase is stable. Going to higher scales, we observe a big desert. To illustrate this, in  one, at φ = 0, becomes shallower. Nevertheless, if we zoom in near φ = 0 we will see that the local minimum is still there. Just below µ I the picture of the potential remains the same, the local minimum becomes even deeper and there is no restoration of the global Z 2 symmetry, see right of Fig. 6. At this scale, λ 1.8, i.e. still perturbative. In Fig. 7 we plot the potential just above µ I , at 3.0 · 10 45 GeV; we observe that µ I is the approximate scale where the potential develops an instability. A related question that arises is what is the scale where perturbation theory breaks down, that is which scale produces a quartic coupling of λ 4π (according to the usual perturbative argument based on the fact that each loop introduces an additional factor of λ 2 16π 2 ). This scale turns out to be µ N P 1.5 · 10 49 GeV , (6.7) a scale slightly larger than the instability scale and remarkably close to the Landau pole of λ, µ λ L 4.3 · 10 49 GeV. The three orders of magnitude in µ between µ I and µ NP is just an order of magnitude or less on a logarithmic scale. In a gauge invariant cut-off regularization such as the lattice, this is typical in the vicinity of a phase transition and can be achieved by a moderate change of the bare couplings. We could say that an interval of scale evolution, long in perturbative time can be short in (cut-off) non-perturbative time. More on this in the next section.
In Fig. 8 we plot once more the potential in the vicinity of the Landau pole. On the left, we show the global form of the potential, where we can see the flattening of the φ = 0 regime. On the right, we zoom in the φ = 0 regime and observe that the local minimum is still there, even though the closer one gets to the Landau pole the more one must zoom in to see it, as the flattening gets closer to forming a saddle point. Note that the gauge coupling g changes very little during these scale changes thus remains perturbative. This means that beyond the Landau pole where λ turns negative lays the Coulomb phase.
The physics that we extract from the numerical analysis is that the Abelian-Higgs model remains perturbative almost all the way to the Landau pole µ λ L of λ. At a scale approximately µ I 7 · 10 −4 µ λ L an instability develops, the onset of the system trying to pass from the Higgs phase into the Coulomb phase. Above µ I there are two vacua, one is the old global minimum that has turned into a local minimum and the unstable vacuum, corresponding to the Coulomb phase. Up to the scale of 0.35 µ λ L the system is trying perturbatively to stabilize in the Coulomb phase and this is finally achieved at the Landau pole where in the last few tiny scale seconds, approximately between [0.35−1.00]·µ λ L , nonperturbative evolution takes over. The breadth of the unstable regime reflects perhaps the fact that the phase transition between the Higgs and the Coulomb phase is a strong first order phase transition [20]. It is likely that in a Monte Carlo simulation, as the phase transition is crossed, one would observe the system tunelling back and forth between the two vacua.

Lines of Constant Physics
The following section has a more speculative character in comparison to the previous ones in an attempt to make a connection to non-perturbative properties of the Abelian-Higgs model. In a non-perturbative approach one of the first thing one does is to map the phase diagram of a model via Monte Carlo simulations. For the Abelian-Higgs model the phase diagram is defined by three axes in the space of the three bare parameters g 0 , λ 0 and m 0 . For instance in a lattice regularization, the three axes can be chosen to be with a the lattice spacing. The β above should not be confused with a beta-function, it is the conventional notation for a lattice gauge coupling. The mapping of the phase diagram has been done for the Abelian-Higgs model to some extent in the past [20] and more recently, for λ 0 = 0.15, in [21]. It seems to depend weakly on λ 0 . Here we reproduce it (semi) qualitatively:

Confined
We also note that the above phase diagram has been constructed using κ ≡ 1−2λ 0 8+a 2 m 2 0 , a slightly different normalization for κ in Eq. (7.1). This just corresponds to choosing a different normalization for the bare vev parameter v 0 . Here we will use the normalization of Eq. We now have all the ingredients to plot a first "perturbative LCP". In Fig. 9 we plot the LCP, defined by Eq. (7.3) and Eq. (7.2) with v 0 = 246 GeV, for the range µ ∈ [125, 5000] GeV. The part (lower left) of the line where the points are denser corresponds to larger values of µ. There, the parameter β = 1/g 2 0 (that decreases quite rapidly as µ increases) approaches 1, which means that most likely we are about to enter into either the Confined or the Coulomb phase. There is potentially a non-trivial message in this so it is worth redoing it this time using Eq. (7.1), i.e. to try to actually put the perturbative LCP on the non-perturbative phase diagram. The only obstacle is to relate the lattice spacing to some scale parameter in the perturbative calculation. This can be quite an involved operation, depending on our desired level of precision. For a discussion of this issue, see [22]. Here we will simplify the discussion and make the naive identification a = 1/Λ with Λ a momentum cut-off. We need to do some extra work though, as we have to estimate δm H in a cut-off regularization. Fortunately this is not too hard, it yields to leading order same relations as before. This is not a completely well defined identification but it is good enough for the information we want to extract. We call the resulting line on the β − κ plane, the "cut-off LCP". On the left of Fig. 10 we plot the cut-off LCP defined by Eq. (7.2) for v 0 = 246 GeV and projecting the λ 0 dependence on the β − κ plane. The values of the cut-off range in Λ ∈ [125, 700] GeV and they increase as we move down and left along the curve. By comparing to the Monte Carlo phase diagram we can now clearly see that around the upper limit of the Λ-range the system hits the Higgs-Coulomb phase transition. Some representative values of the parameters can be seen on Table 1. On the right of Fig. 10 we compare the cut-off LCP to the one we would have obtained   Table 2: Scale and corresponding bare parameters of the LCP defined by m Z = 91 GeV, m H = 125 GeV and λ = 0.12 with the lattice spacing identified as a = 1/µ. by the identification µ = 1/a. Some representative values of the parameters for this identification can be seen on Table 2. The numbers change but the conclusion remains: If we want to construct an LCP that keeps the values of the physical quantities close to their "experimentally measured values" (we are not in the Standard Model but the point we would like to make should be clear), we can not take the cut-off too high, since around 1 TeV or so we are forced to cross into the Coulomb phase. Notice that the analysis of the RG trajectories and of the potential in the previous section did not warn us about this. This has potentially consequences for the fine tuning of the Higgs mass. In Tables  1 and 2 we introduced the unsophisticated fine tuning parameter from which we can see the increase of fine tuning as the scale increases, with the Λ = 1/a identification naturally worse. Even so though, since the cut-off stops at a maximum value of about 1 TeV, it can not take very large values.

Conclusions
We carried out the renormalization of the Higgs potential and of the Z-mass in the Abelian-Higgs model in the R ξ and Unitary gauges. The fact that this is a sector of the action not containing derivatives, suggested us to renormalize at zero external momenta. The renormalization conditions we used imply subtractions that are necessarily different than MS. In this context, we showed that these two gauges are completely equivalent both at the level of the β-functions and at the level of the finite remnants after renormalization, that determine the one-loop Higgs potential. Moreover, we showed that after the renormalization procedure we obtain automatically the gauge independence of the potential. The cancellation of the gauge fixing parameter ξ from the scalar potential obtained via Feynman diagram calculations in an R ξ gauge fixing scheme is intriguing as the corresponding effective potential obtained via the background field method is manifestly ξ-dependent. Its equivalence to the Unitary gauge on the other hand could open a new direction in Standard Model calculations, as the number of diagrams in the Unitary gauge is significantly smaller. Apart however from simplifying already known results, computations such as done here could also render currently ambiguous results based on the dynamics of the scalar potential away from its extrema, more robust. Since this work is just a demonstration of gauge independence of the Higgs potential by construction it would be certainly worth trying to construct some sort of a more formal 'proof'.
We have analyzed numerically the Higgs potential and saw that the system generically behaves perturbatively until quite close to the Landau pole of λ. It becomes metastable already in the perturbative regime and only during its last steps towards the phase transition into the Coulomb phase turns non-perturbative. Finally, we constructed a couple of Lines of Constant Physics in the Unitary gauge and placed them on the Abelian-Higgs phase diagram in order to enhance the perturbative analysis by some non-perturbative input. This suggested us that in a Higgs-scalar system the cut-off can not be generically driven to the highest possible scales that the RG flow or the stability of the potential allows, if we want to keep the physics constant. This is because it is possible that way before those scales are reached, a phase transition may be encountered.
Regarding generalizations of this work, simple bosonic extensions should work in a similar way, without any surprises. In the presence of fermions some extra care may be needed to handle the U -integrals but this is expected to be a straightforward extra technical step. In this case though it is also expected that a new instability will appear at some intermediate scale and it would be interesting to work out in our framework how this modifies the bosonic system. Finally, if the observed consistency between the two gauges breaks down at higher loops then there is something about the quantum internal structure of spontaneously broken gauge theories that needs to be understood better.
where the range of the sum over n depends on the diagram. The n'th term of the numerator has a denominator with n factors, yields a finite result after integration over the loop momentum k and it is denoted as N . It is also diagram dependent.
In the Unitary gauge, even though this notation is redundant, we will still use it for extra clarity. The denominators are defined as and P 0 ≡ 0. Notice that the proper argument of the denominators is D i (P i−1 , m i ). Nevertheless, we do not always show both arguments systematically in the main text, apart from cases where otherwise their absence could cause ambiguities. Such a case is some finite diagrams, which generally have more than four denominators. In other cases where there can be no confusion, we show none, or only one of the arguments.
Feynman parametrization is implemented using where we defined Passing now to Dimensional Regularization, we introduce the useful shorthand notation We can then express Eq. (A.2) in DR as In a few simple cases we use the conventional notation

B Passarino -Veltman
In this Appendix we collect some standard integrals of the Passarino-Veltman type, encountered in the text.

B.1 Scalars
The simplest PV integral is the scalar tadpole integral naively quadratically divergent with a cut-off. In Dimensional Regularization, it can be expressed as (attaching the factor µ ε ) and by expanding in ε finally as The scalar, naively logarithmically diveregent PV integral is It can be computed explicitly using the formulation of Appendix A. It corresponds to n = 2 and a numerator equal to 1. In d-dimensions, this integral is Expanding in ε we get and finally We note that this integral is symmetric under m 1 ↔ m 2 . An important special case is when P 1 = 0 in which case The finite scalar integral that appears in Triangles and Boxes is of the form In DR, this integral becomes Expanding in ε it reduces to We also define some special cases of C 0 integrals that appear in the main text in various places: The last scalar is the finite Box integral The integral in d-dimensions becomes (B.14)

B.2 Tensors
Standard tensor PV integrals can be algebraically reduced to scalar integrals. Actually in one-loop diagrams only contractions of tensors with the metric and external momenta occur. The tadpole integral has no standard PV extension. It has an extension of the U -type, naively quartically divergent with a cut-off and it will be computed in Appendix C.
Let us introduce some shorthand notation. First, define for any P : and for any P 1 and P 2 The simplest B-tensor is the linearly divergent and it can be contracted only by a momentum The other PV tensor extension of the B-type is of the form and it is quadratically divergent. Its contraction with the metric is while its contraction with P µ P ν is The simplest C-tensor is It contracts either as The second C-tensor, logarithmically divergent, integral is Its contraction with the metric is There are three different momentum contractions that can appear. Following [18], we define the matrix and using this matrix the quantities We also define out of which we construct the four quantities This data determines the quantities and in terms of these we can express C µν itself: and from the above expression it is straightforward to compute all its momentum contractions.
We sometimes encounter C-integrals with det[G 2 ] = 0. Such integrals are computed with direct Feynman parametrization and DR (i.e. without algebraic reduction). To give an example, we may stumble on which in DR, after expanding in ε, becomes Let us now consider the linearly divergent rank 3 integral This integral could be reduced in principle algebraically, following [18]. Here, as it is only linearly divergent, we will compute it in a brute force way with Feynman parametrization. It is a case with n = 3 and N C µνα = k µ k ν k α in the language of Appendix A. It explicitly evaluates to It is now easy to compute any contraction of the above expression with the metric and/or momenta. A useful contraction is C µ K3 ≡ g να C µνα . All Box tensor integrals are computed directly as in Appendix A since they are at most logarithmically divergent. D µ , D µν , D µνα and D µναβ are computed as D µ,µν,µνα,µναβ = 6 D µνα = k µ k ν k α and N C U -integrals U -integrals are linearly, quadratically, cubically or quartically divergent diagrams that do not appear when the Higgs is expressed in a Cartesian basis and an R ξ gauge fixing is performed. If either a Polar basis for the Higgs is used and/or the computation is performed in the Unitary gauge, these integrals do appear. Clearly, the standard PV reduction formulae must be extended. At each level of n-point functions we meet at least an integral of the U -type.

C.1 U -integrals in Tadpoles
Indeed, during the Tadpole calculation in section 3.1 we find the quartically divergent contraction where V is the DR volume of space-time. The usual prescription is V = 0, in which case, the tensor-Tadpole U -integral reduces to

C.2 U -integrals in mass corrections
The basic U -integral with two denominators is the quartically divergent with the cut-off integral with D 1 = k 2 − m 2 1 and D 2 (P 1 ) = (k + P 1 ) 2 − m 2 2 as defined in Eq. (A.3). We show only the momentum argument P 1 as we would like to follow it. Adding and subtracting a term, the above integral can be rewritten as Now the second term is a standard PV integral, while the first term is still a U -integral (it is still quartically divergent) but we can compute it straightforwardly as where we have performed the shift k → k − P 1 and we have dropped the 2k · P 1 term because it is odd under k → −k. Also, we have used Eq. (C.2) for the rational part. From (C.4) and (C.5) and using Eq.(B.21), we get The reduction of U M4 was carried through using at some point a loop momentum shift. We know that in DR and in the absence of fermions, momentum shifts are ok, as long as the integral is up to naively quadratically divergent with the cut-off. U -integrals are however often cubically or quartically divergent. In order to check the validity of momentum shifts, we will now re-compute U M4 without momentum shift and compare the results.
Starting with Eq. (C.3), by adding and subtracting terms, we construct in the numerator D 2 (P 1 ) obtaining The second term in the last expression is The first term of the above relation is zero since P 1 · k term is odd under k → −k. The third term of Eq. (C.7) and the second and third terms of Eq. (C.8) are standard PV integrals. The first term of Eq. (C.7) however is still a U -integral (and still quartically divergent) but we can reduce it easily as which is the same as Eq. (C.6). Clearly, we have traded loop momentum shifts in highly divergent integrals for adding and subtracting infinities, a slightly less disturbing operation. In any case, the final results will justify or not these manipulations.

C.3 U -integrals in Triangles
In the Triangle sector we first meet It can be reduced easily as and with PV reduction further on.
The next case is the cubically divergent Following similar steps as before (C.14) Only the first term is new (and also a U -integral) but it is easy to compute it: where in the third line above we have shifted the loop momentum and then neglected the k-odd term, as before. It total, The last Triangle U -integral is quartically divergent, It is successively reduced as where we have defined U K4 = g µν U µν K4 .

C.4 U -integrals in Boxes
The lowest Box U -integrals have five momenta in their numerator. One is U µ B5 (P 1 , P 2 , P 3 , m 1 , m 2 , m 3 , m 4 ) = The pattern should start becoming obvious by now. For example, we have and its contracted version Finally we have the quartically divergent

D Explicit calculation of the diagrams
In this Appendix we present the explicit calculation of the one-loop Feynman diagrams, concentrating on the Tadpole and the two-point function categories, in both R ξ and Unitary gauges. The Triangle and Box contributions can be straightforwardly computed following similar steps, therefore there is no need to calculate them explicitly here.

D.1 R ξ -diagrams
In order to be synchronised with our main text, we start form the first set of diagrams which corresponds to the one-point functions. The first such diagram is and analytically evaluates to The symmetry factor is S 1 T = 1 2 since (n O , n l , 1 , v 1 ) = (3, 1, 3, 1). In DR this integral takes the form The "tadpole integral" A 0 is defined in Appendix B. It has mass dimension 2 and it is (external) momentum independent.
The next tadpole comes with a gauge boson loop: with symmetry factor S 2 T = 1 2 since (n O , n l , 1 , v 1 ) = (1, 1, 2, 1). Using the relation k µ k ν = gµν d k 2 under the intergral it simplifies to where the B 0 scalar integral appears at p 2 = 0 and we call it B 1 0 . Using Eq.(B.8) the gauge tadpole becomes The last tadpole has a Goldstone loop: with S 3 T = 1 2 as (n O , n l , 1 , v 1 ) = (1, 1, 2, 1). Notice that the integral in Eq. (D.6) is a U -integral (called U T ), the first in the class of highly divergent integrals that we will often encounter in the Unitary gauge. As already mentioned, the origin of such integrals emerging also in the R ξ gauge can be traced to our Polar basis choice to represent the Higgs field. Using Eq.(C.2) to calculate U T (m χ 0 ) we obtain that in DR, Finally, the total tadpole value is the sum of the above three contributions: Next we have the Z 2-point function. A gauge-boson vacuum polarization amplitude can be Lorentz-covariantly split into a transverse and a longitudinal part Contracting with p µ p ν both sides fixes Contracting with g µν gives on the other hand that can be easily solved for the transverse part in d = 4 Now, the Schwinger-Dyson equation that the dressed Z-propagator obeys is written as 14) with D µρ the tree level gauge boson propagator So, performing the contractions the Schwinger-Dyson equation becomes Contracting again with p µ p ν we have that while contracting with the metric gives The solution of the above equation is (D. 19) The quantity that enters in the renormalization of the mass of the Z gauge boson is therefore We now start computing the one-loop Feynman diagrams contributing to M R ξ Z,µν . The first contributing diagram has a Higgs running in the loop: In addition, in agreement with our notation in Appendix A, we define the following integrals H µ,µν,µνα,µναβ,µναβγδ,µναβγδ ,µναβγδ θ = 7!

F On-shell results
For completeness, in this Appendix we demonstrate the results for the divergent parts of the two-three-and four-point one-loop functions calculated on-shell, in both R ξ and Unitary gauges. We start with the case of R ξ gauge where we will demonstrate the results for the two-, three-and four-point functions calculated at on-shell, i.e. at p 2 = m 2 Z for the Z-mass and at p 2 i = m 2 H for everything else. Tadpoles are external momentum independent objects.