Some Remarks on the Spectral Functions of the Abelian Higgs Model

PHYSICAL REVIEW D 100, 065009 (2019)

Some remarks on the spectral functions of the Abelian Higgs model

† ‡ ∥ D. Dudal,1,2,* D. M. van Egmond ,3, M. S. Guimarães,3, O. Holanda,3,§ B. W. Mintz,3, L. F. Palhares,3,¶ †† G. Peruzzo,3,** and S. P. Sorella3, 1KU Leuven Campus Kortrijk-Kulak, Department of Physics, Etienne Sabbelaan 53 bus 7657, 8500 Kortrijk, Belgium 2Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium 3Universidade do Estado do Rio de Janeiro, Instituto de Física-Departamento de Física Teórica-Rua Sáo Francisco Xavier 524, 20550-013 Maracaná, Rio de Janeiro, Brasil

(Received 31 May 2019; published 16 September 2019)

We consider the unitary Abelian Higgs model and investigate its spectral functions at one-loop order. This analysis allows us to disentangle what is physical and what is not at the level of the elementary particle propagators, in conjunction with the Nielsen identities. We highlight the role of the tadpole graphs and the gauge choices to get sensible results. We also introduce an Abelian Curci-Ferrari action coupled to a scalar field to model a massive photon which, like the non-Abelian Curci-Ferarri model, is left invariant by a modified non-nilpotent BRST symmetry. We clearly illustrate its nonunitary nature directly from the spectral functionviewpoint. This provides a functional analogue of the Ojima observation in the canonical formalism: there are ghost states with nonzero norm in the BRST-invariant states of the Curci-Ferrari model.

DOI: 10.1103/PhysRevD.100.065009

I. INTRODUCTION Many properties of the nonperturbative infrared region In recent years there has been an increasing interest in the of QCD are coherently described today by lattice simu- properties of the spectral function (Källen-Lehmann den- lations [10]. Analytically, however, despite the progress sity) of two-point correlation functions, especially in non- made in the last decades, the achievement of a satisfactory Abelian gauge theories such as quantum chromodynamics understanding of the infrared (IR) region is still a big (QCD). It was found [1–5], in lattice simulations for the challenge. The gluon propagator is not gauge invariant, and minimal Landau gauge, that the spectral function of the therefore one needs to fix a gauge using the Faddeev-Popov gluon propagator is not non-negative everywhere, which (FP) procedure, introducing ghost fields, while trading the means that there is no physical interpretation for this local gauge invariance with the Becchi-Rouet-Stora-Tyutin propagator like there is for the photon propagator in (BRST) symmetry. In the perturbative ultraviolet region of quantum electrodynamics (QED). This behavior of the QCD, the FP gauge fixing procedure works, giving results gluon spectral function is commonly associated with in excellent agreement with experiments. However, an the concept of confinement [6–9]. The nonpositivity of extrapolation of the perturbative results to low energies the spectral function then becomes a reflection of the is plagued by infrared divergencies caused by the existence inability of the gluon to exist as a free physical particle, of the well-known Landau pole. In the same region, it is i.e., as an observable asymptotic state of the S-matrix. known that the standard FP procedure does not fix the gauge uniquely: several field configurations satisfy the same gauge condition, e.g., the transverse Landau gauge, *[email protected] leading to the so-called Gribov copies [11,12]. Over the † [email protected] years, various attempts have been made to deal with this ‡ [email protected] problem in the continuum functional approach; see for §[email protected] ∥ example [13–18]. [email protected] The problem of the Landau pole in asymptotically free ¶[email protected] **[email protected] theories can be provisionally circumvented, with an †† [email protected] adequate renormalization scheme, by the introduction of an effective infrared gluon mass [19]. This is in accordance Published by the American Physical Society under the terms of with the lattice data, which show that the gluon propagator the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to reaches a finite positive value in the deep IR for space-time the author(s) and the published article’s title, journal citation, Euclidean dimensions d>2; see e.g., [20–28]. Of course, and DOI. Funded by SCOAP3. analytically one would like to recover the massless

2470-0010=2019=100(6)=065009(22) 065009-1 Published by the American Physical Society D. DUDAL et al. PHYS. REV. D 100, 065009 (2019) character of the FP gauge fixing theory in the perturbative states in the sm-invariant subspace (“the would-be physical UV region. The use of such theory which implements an subspace”) was confirmed. However, it is worth mention- effective mass only in the IR region was first proposed ing here that the goal of [37] and follow-up works was not in [29] based on the idea of a momentum-dependent or to introduce a theory for massive gauge bosons but to dynamical gluon mass [30,31]. For this, the Schwinger- discuss a relatively simple and useful effective description Dyson equations are employed in order to get a suitable gap of some nonperturbative aspects of QCD. Unitarity of the equation that governs the evolution of the dynamical gluon gauge bosons sector is not so much an issue, as one expects mass mðp2Þ, which vanishes for p2 → ∞. This setup them to be undetectable anyhow, due to confinement. preserves both renormalizability and gauge invariance. Within this perspective the existence of an exact nilpotent However, the Schwinger-Dyson equations are an infinite BRST symmetry becomes a quite relevant property when set of coupled equations which require a truncation trying to generalize the action (1) to other gauges than procedure; see for example [32–36] for a detailed presen- Landau gauge, as next to unitary one should also expect tation of the subject. that the correlation functions of gauge-invariant observ- Recently, a more pragmatic approach was taken in ables are gauge-parameter independent. We will come back [16,37,38], or in other works like [39–41]. Instead of a to this question in a separate work. model justified from first principles, the observations In this context, it is worthwhile to investigate the spectral obtained from lattice simulations were used as a guiding properties of massive gauge models to try to shed some principle. The massive gluon propagators observed in light on the infrared behavior of their fundamental fields. lattice simulations for Yang-Mills theories led to consid- A direct comparison with a massive model that preserves ering the following action: the original nilpotent BRST symmetry, such as the Higgs- Z Yang-Mills model, can be particularly enlightening. In any 1 2 d a a a a a a m a a case, the explicit determination of the spectral properties of S¼ d x FμνFμν þc¯ ∂μðDμcÞ þib ∂μAμ þ AμAμ ; 4 2 Higgs theories and the study of the role played by gauge symmetry there are interesting to pursue on their own. ð1Þ For the Uð1Þ Higgs model, one can fix the gauge by ’ ξ → ∞ which is a Landau gauge FP Euclidean Lagrangian for pure means of t Hooft Rξ-gauge. For the formal limit we gluodynamics, supplemented with a gluon mass term. This end up in the unitary gauge, which is considered the term modifies the theory in the IR but preserves the FP physical gauge as it decouples the nonphysical particles. perturbation theory for momenta p2 ≫ m2. The action (1) However, this gauge is known to be nonrenormalizable [51]. is a particular case of the Curci-Ferrari (CF) model [42]. Here, we refer the reader to [52] for a recent analysis of the The mass term breaks the BRST symmetry of the model, unitary gauge. The Landau gauge, on the other hand, ξ → 0 which turns out to still be invariant under a modified BRST corresponds to . Most articles on massive Yang-Mills symmetry [43], models employ the renormalizable Landau gauge, although it was noticed that this gauge might not be the preferred a ab b gauge in nonperturbative calculations [53]. In fact, as was s Aμ ¼ −Dμ c ; ð2Þ m recently established in [54], the use of the Landau gauge in

a g abc b c the massive Yang-Mills model (1) leads to complex pole smc ¼ 2 f c c ; ð3Þ masses, which will obstruct a calculation of the Käll´en- Lehmann spectral function. Indeed, if at some order in a a smc¯ ¼ ib ; ð4Þ perturbation theory (one loop as in [54] for example) a pair of Euclidean complex pole masses appear, at higher order a 2 a 2 smb ¼ im c ; ð5Þ these poles will generate branch points in the complex p - plane at unwanted locations, i.e., away from the negative 2 which is, however, not nilpotent since smc¯ ≠ 0. real axis, deep into the complex plane, thereby invalidating The legitimacy of the model (1) depends, of course, on a Käll´en-Lehmann spectral representation. This can be how well it accounts for the lattice results. In [19,37,38],it appreciated by rewriting the Feynman integrals in terms of has been shown that, both at one- and two-loop order, the Schwinger or Feynman parameters, whose analytic proper- model reproduces quite well the lattice predictions for the ties can be studied through the Landau equations [55]. Let gluon and ghost propagator. For other applications of (1), us also refer to [56,57] for concrete examples. we refer to [44–47]. However, from the Kugo-Ojima Understanding the different gauges and their influence criterion [48], it is known that nilpotency of the BRST on the spectral properties is a delicate subject. This gave us symmetry is indispensable to formulate suitable conditions further reason to undertake a systematic study of the for the construction of the states of the BRST-invariant spectral properties of Higgs models. In this paper, we physical (Fock) subspace, providing unitarity of the present the results for the simplest case: that of the Uð1Þ S-matrix. Indeed, in [49,50] the existence of negative norm Abelian Higgs model. In fact, it turned out that this model is

065009-2 SOME REMARKS ON THE SPECTRAL FUNCTIONS … PHYS. REV. D 100, 065009 (2019) already very illuminating on aspects like positivity of the massive Abelian model. We also address the residue spectral function, gauge-parameter independence of physi- computation. Section VI collects our conclusions and cal quantities, and unitarity. Of course, these properties are outlook. not unknown in the Abelian case. This article should therefore not be seen as giving any new information on II. ABELIAN HIGGS MODEL: the physical properties of the Abelian model. Rather, SOME ESSENTIALS exactly because these properties are so well known, we are in a better position to understand the problems that we We start from the Abelian Higgs classical action with a face when calculating the analytic structure behind some of manifest global Uð1Þ symmetry them within a gauge-fixed setup. This work is therefore a Z 1 λ 2 2 first attempt to understand analytically the spectral proper- 4 † † v S ¼ d x FμνFμν þðDμφÞ Dμφ þ φ φ − ; ties of a Higgs-gauge model in contrast to those of a 4 2 2 nonunitary massive model. As such, it is laying the ð6Þ groundwork for future work on these properties in the non-Abelian SUð2Þ Higgs case as well as in the massive where model of [37], Eq. (1), investigating the origin of the complex poles structure reported in [54,58,59]. Fμν ¼ ∂μAν − ∂νAμ; The Uð1Þ Higgs model is known to be unitary [60,61] φ ∂ φ φ and renormalizable [62]. In this work, we consider two Dμ ¼ μ þ ieAμ ð7Þ propagators: that of the photon and that of the Higgs scalar field. They are obtained through the calculation of the one- and the parameter v gives the vacuum expectation value ℏ φ loop corrections to the corresponding 1PI two-point (vev) of the scalar field to first order in , h i0 ¼ v. The spontaneous symmetry breaking is implemented by functions. After adopting the Rξ-gauge, we are left with an exact BRST nilpotent symmetry. Of course, the corre- expressing the scalar field as an expansion around its lation function of BRST-invariant quantities should be vev, namely independent of the gauge parameter. Since the transverse 1 φ ¼ pffiffiffi ððv þ hÞþiρÞ; ð8Þ component of the photon propagator is gauge invariant, we 2 should find that the one-loop corrected transverse propagator does not depend on the gauge parameter. As a where the real part h is identified as the Higgs field and ρ is consequence, neither does the photon pole mass. This the (unphysical) Goldstone boson, with hρi¼0. Here we property has been proven before by the use of the Nielsen choose to expand around the classical value of the vev, so – identities, [63] (see also [64 66]) but never in a direct that hhi is zero at the classical level but receives loop calculation. The same goes for the Higgs particle propa- corrections.1 The action (6) now becomes gator: the gauge independence of its pole mass was proven Z 1 1 1 in [63] but never in a direct loop calculation, to our 4 ∂ ∂ ∂ ρ∂ ρ − ρ∂ knowledge. We underline here the importance of properly S ¼ d x 4 FμνFμν þ 2 μh μh þ 2 μ μ e μhAμ taking into account the tadpole contributions [67,68] or, 1 ∂ ρ 2 2 ρ2 equivalently, the effect on the propagators of quantum þ eðh þ vÞAμ μ þ 2 e Aμ½ðh þ vÞ þ Aμ corrections of the Higgs vacuum expectation value. Armed 1 with the one-loop results, we are able to calculate the þ λðh2 þ 2hv þ ρ2Þ2 ; ð9Þ spectral properties of the respective propagators for differ- 8 ent values of the gauge parameter. An additional aim of this and we notice that both the gauge field and the Higgs field work is to compare our results with those of a nonunitary have acquired the following masses: massive Abelian model, to clearly pinpoint, at the level of spectral functions, the differences (and issues) of both m2 ¼ e2v2;m2 ¼ λv2: ð10Þ unitary and nonunitary massive vector boson models. h This article is organized as follows. In Sec. II, we review With this parametrization, the Higgs coupling λ and the 1 the Uð Þ Higgs model and its gauge fixing, as well as the parameter v can be fixed in terms of m, mh and e, whose tree-level field propagators and vertices. In Sec. III,we values will be suitably chosen later in the text. calculate the one-loop propagator of both the photon field and the Higgs field, showing the gauge-parameter inde- 1There is of course an equivalent procedure of fixing hhi to pendence of the transverse photon propagator and of the zero at all orders, by expanding φ around the full vev: 1 Higgs pole mass up to one-loop order. In Sec. IV,we φ pffiffi φ ρ ¼ 2 ððh iþhÞþi Þ. In Appendix B we explicitly show calculate the spectral function of both propagators, and in that—as expected—both procedures give the same final results Sec. V we compare our results with those of a nonunitary up to a given order.

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Even in the broken phase, the action (9) is left invariant sAμ ¼ −∂μc; sc ¼ 0; by the following gauge transformations: sφ ¼ iecφ;sφ† ¼ −iecφ†;

† † sh ¼ −ecρ;sρ ¼ ecðv þ hÞ; δAμ ¼ −∂μω; δφ ¼ ieωφ; δφ ¼ −ieωφ ; sc¯ ¼ ib; sb ¼ 0: ð13Þ δh ¼ −eωρ; δρ ¼ eωðv þ hÞ; ð11Þ Importantly, the operator s is nilpotent, i.e., s2 ¼ 0, where ω is the gauge parameter. allowing us to work with the so-called BRST cohomology, a useful concept to prove unitarity and renormalizability of A. Gauge fixing the Abelian Higgs model [48,62,69]. Quantization of the theory (9) requires a proper gauge We can now introduce the gauge fixing in a BRST- fixing. We shall employ the gauge fixing term invariant way via Z ξ Z d 1 S ¼ s d x −i cb¯ þ c¯ð∂μAμ þ ξmρÞ ; ð14Þ 4 2 gf 2 Sgf ¼ d x ð∂μAμ þ ξmρÞ ; ð12Þ 2ξ Z ξ d 2 2 ¼ d x b þ ib∂μAμ þ ibξmρ þ c¯∂ c ’ 2 known as the t Hooft or Rξ-gauge, whichR has the pleasant 4 property of canceling the mixed term d xðevAμ∂μρÞ in − ξm2cc¯ − ξmechc¯ : 15 the expression (9). Of course, (12) breaks the gauge ð Þ invariance of the action. As is well known, the latter is replaced by the BRST invariance. In fact, introducing the Notice that theghosts ðc;¯ cÞ get a gauge parameter-dependent FP ghost fields c;¯ c as well as the auxiliary field b, for the mass while interacting directly with the Higgs field. BRST transformations, we have The total gauge fixed BRST-invariant action then becomes

Z 1 1 1 1 1 4 ∂ ∂ ∂ ρ∂ ρ − ρ∂ ∂ ρ 2 2 2 2 ρ2 S ¼ d x 4 FμνFμν þ 2 μh μh þ 2 μ μ e μhAμ þ ehAμ μ þ 2 m AμAμ þ 2 e Aμ½h þ vh þ Aμ 1 2 2 2 2 1 2 2 ξ 2 2 2 λ h ρ ρ 4 m μ∂μρ b ∂μ μ ξ ρ ¯ ∂ − ξ ¯ − ξ ¯ þ 8 ð þ Þðh þ þ hvÞþ2 hh þ mA þ 2 þ ib A þ ib m þ cð Þc m cc m ecch ; ð16Þ with contributions which are nonzero and the resulting tadpole diagrams have to be included in the quantum corrections sS ¼ 0: ð17Þ for the propagators.3 Of course the final result for the propagators would be In Appendix A we collect the propagators and vertices the same had we chosen to expand the φ field around its full corresponding to the action (16) of the Abelian Higgs vev and required hhi¼0. In fact, including the tadpole model in the Rξ gauge. diagrams in our formulation has the same effect as shifting the masses of the fields to include the one-loop corrections φ 0 III. PHOTON AND HIGGS PROPAGATORS to the Higgs vev h i, calculated by imposing hhi¼ (see AT ONE LOOP Appendix B for the technical details). These diagrams can actually be seen as a correction to the tree-level mass term: In this section we obtain the one-loop corrections to the in the spontaneously broken phase the gauge boson mass is photon propagator, as well as to the propagator of the Higgs given by m ¼ ehφi, thus depending on hφi, which receives 2 boson. This requires the calculation, in Secs. III A and III B, quantum corrections order by order. Therefore, the full of the Feynman diagrams as shown in Figs. 1 and 2. inverse photon propagator can be written as Notice that the last four diagrams in Figs. 1 and 2 vanish for hhi¼0. Since we have chosen to expand the φ 3 field around its classical vev v [cf. (8)], hhi has loop The diagrams with tadpole balloons are not part of the standard definition of one-particle irreducible diagrams that contribute to the self-energies. However, since the momentum 2We have used the techniques of modifying integrals into flowing in the vertical h-field (dashed) line is zero, they can be “master integrals” with momentum-independent numerators effectively included as a momentum-independent term in the self- from [70]. energies.

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FIG. 1. Contributions to one-loop photon self-energy in the Abelian Higgs model, including tadpole contributions in the second line. Wavy lines represent the photon field, dashed lines the Higgs field, solid lines the Goldstone boson, and double lines the ghost field.

FIG. 2. Contributions to the one-loop Higgs self-energy. Line representations are as in Fig. 1.

−1 2 2 2 2 1 GAAðp Þ¼p þ e v þð PI diagramsÞ A. Corrections to the photon self-energy þðdiagrams with tadpolesÞ The first diagram contributing to the photon self-energy is the Higgs boson snail (first diagram in the first line of 2 2 φ 2 1 ¼ p þ e h i þð PI diagramsÞ; ð18Þ Fig. 1), and it gives a contribution where the equalities are to be understood up to a given −4 2 Γ 2 − 2 d−2 2 e ð d= Þ mh order in perturbation theory and a similar reasoning can be Γ 1ðp Þ¼ δμν: ð19Þ AμAν; ð4πÞd=2 2 − d 2 drawn for the Higgs propagator. In what follows, we proceed with the expansion adopted in (8) and include the tadpole diagrams explicitly in our The second diagram is the Goldstone boson snail (second self-energy results. diagram in the first line of Fig. 1) The calculations are done for arbitrary dimension d.In d 4 − ϵ Sec. III C we will analyze the results for ¼ , making 2 2 d=2−1 2 −4e Γð2 − d=2Þ ðξm Þ use of the techniques of dimensional regularization in the Γ 2ðp Þ¼ δμν: ð20Þ AμAν; d=2 2 − 2 MS scheme. ð4πÞ d

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Being momentum independent, the only effect of these first two diagrams is to renormalize the mass parameters 2 2 ðmh;m Þ. The third term contributing to the photon propagator is the Higgs-Goldstone sunset (third diagram in first line of Fig. 1) Z 4 2 Γ 2 − 2 1 Γ 2 e ð d= Þ 2 ξ 2 P AμAν;3ðp Þ¼ 2 dx Kd=2−1ðmh; m Þ μν ð4πÞd= 2 − d 0 2 2 ð2 − dÞ 2 2 2 2−1 ξ 1 − 4x 1 − x p 2−2 ξ Lμν ; þ Kd= ðmh; m Þþ 4 ð ð ÞÞ Kd= ðmh; m Þ ð21Þ where we used the definitions

2 2 2 2 2 α Kαðm1;m2Þ ≡ ðp xð1 − xÞþxm1 þð1 − xÞm2Þ ; ð22Þ and

pμpν Pμν ¼ δμν − ; ð23Þ p2

pμpν Lμν ¼ ; ð24Þ p2 which are the transversal and longitudinal projectors, respectively. The fourth term contributing to the photon propagator is the Higgs-photon sunset (fourth diagram in first line of Fig. 1) Z 4 2 Γ 2 − 2 1 Γ 2 e ð d= Þ 2 − 2 2 2 2 2 − 2 ξ 2 P AμAν;4ðp Þ¼ 2 dx½ðð dÞm Kd=2−2ðmh;m ÞþKd=2−1ðmh;m Þ Kd=2−1ðmh; m ÞÞ μν ð4πÞd= 2 − d 0 2 − 2 2 2 2 2 − 2 ξ 2 þðð dÞm Kd=2−2ðmh;m ÞþKd=2−1ðmh;m Þ Kd=2−1ðmh; m Þ 2 − 2 2 2 2 − 2 ξ 2 L þð dÞp x ðKd=2−2ðmh;m Þ Kd=2−1ðmh; m ÞÞÞ μν: ð25Þ

Γ 2 Finally, we have four tadpole (balloon) diagrams. The AμAν;7ðp Þ Higgs boson balloon (first diagram of the last line in Fig. 1) Z 2 d 1 ξ 2 2 m d k − 1 δ ¼ e 2 2 2 ðd Þþ 2 2 μν; 2 m ð2πÞd k þ m k þ ξm 2 4e Γð2 − d=2Þ 3 d=2−1 h Γ 5ðp Þ¼ m δμν; ð26Þ AμAν; ð4πÞd=2 ð2 − dÞ 2 h ð28Þ the Goldstone boson balloon (second diagram of the last and finally, the ghost balloon (fourth diagram of the last line line in Fig. 1) in Fig. 1) Z 4 2 Γ 2 − 2 1 2 d ξ 2 e ð d= Þ d=2−1 2 2 m d k Γ 6ðp Þ¼ ðξmÞ δμν; ð27Þ Γ 8ðp Þ¼−2e δμν: ð29Þ AμAν; 4π d=2 2 − 2 AμAν; 2 2π d 2 ξ 2 ð Þ ð dÞ mh ð Þ k þ m the photon balloon (third diagram of the last line in Fig. 1) Combining all these contributions (19)–(29), we find

Z 4e2 Γ 2 − d=2 1 md Γ 2 ð Þ 2 − 2 2 2 2 2 d−2 − 1 P AμAνðp Þ¼ 2 dx ð dÞm Kd=2−2ðm ;mhÞþKd=2−1ðm ;mhÞþmh þ 2 ðd Þ μν ð4πÞd= 2 − d 0 m Z h 4 2 Γ 2 − 2 1 2 − e ð d= Þ d 1 − 4 2 2 ξ 2 2 − 2 2 2 2 2 þ 2 dx ð xÞp Kd=2−2ðmh; m Þþð dÞðm þ p x ÞKd=2−2ðmh;m Þ ð4πÞd= 2 − d 0 4 md 2 2 d−2 − 1 L þ Kd=2−1ðmh;m Þþmh þ 2 ðd Þ μν: ð30Þ mh

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λ Γ 2 − 2 Defining 2 ð d= Þ d−2 Γ 1ðp Þ¼−3 m ; ð35Þ hh; ð4πÞd=2 ð2 − dÞ h Γ Π⊥ 2 P Πk 2 L AμAν ¼ AAðp Þ μν þ AAðp Þ μν; ð31Þ the Goldstone boson snail (second diagram in the first line it follows that of Fig. 2)

⊥ ∂ξΠ ¼ 0: ð32Þ λ Γ 2 − 2 AA 2 ð d= Þ 2 d=2−1 Γ 2ðp Þ¼− ðξm Þ ð36Þ hh; ð4πÞd=2 ð2 − dÞ As expected, Eq. (32) expresses the gauge parameter independence of the gauge-invariant transverse component and the photon snail (third diagram in the first line of Fig. 2) of the photon propagator [63]. Then, the connected transverse form factor, 2 Γhh;3ðp Þ ⊥ 2 1 1 ⊥ 2 4 2 Π O e Γð2 − d=2Þ −2 2 2−1 GAAðp Þ¼ 2 2 þ 2 2 2 AAðp Þþ ðe Þ; ð33Þ ¼ −2 ððd − 1Þmd þ ξðξm Þd= Þ: p þ m ðp þ m Þ ð4πÞd=2 ð2 − dÞ can be rewritten in terms of the resummed form factor as ð37Þ 1 G⊥ p2 : Next, we observe a couple of sunset diagrams: the Higgs AAð Þ¼ 2 2 − Π⊥ 2 O 4 ð34Þ p þ m AAðp Þþ ðe Þ boson sunset (fourth diagram in the first line of Fig. 2)

2 9 λ Γð2 − d=2Þ 2 B. Corrections to the Higgs self-energy Γ 4ðp Þ¼ ð2 − dÞm hh; 2 4π d=2 ð2 − dÞ h The first diagrams contributing to the Higgs self-energy ðZ Þ 1 are of the snail type, renormalizing the masses of the 2 2 × dxKd=2−2ðmh;mhÞ; ð38Þ internal fields. 0 The Higgs boson snail (first diagram in the first line of Fig. 2) the photon sunset (first diagram in the second line of Fig. 2)

Z Γ 2 − 2 1 1 4 Γ 2 2 ð d= Þ 2 − 2 2 − 1 2 2 p 2 2 hh;5ðp Þ¼e 2 dx ð dÞ m ðd Þþ p þ 2 Kd=2−2ðm ;m Þ 2 − d ð4πÞd= 0 2m 4 2 p 2 2 2 2 2 2 2 − ð2 − dÞ 2p þ þ ξ m þ 2p ξ − 2ξm þ m K 2−2ðm ; ξm Þ m2 d= 4 2 2 2 p 2 2 2 d=2−1 2 d=2−1 þð2 − dÞ 2ξp þ 2ξ m þ K 2−2ðξm ; ξm Þþ2ðξ − 1Þðm Þ þ 2ð1 − ξÞðξm Þ ; ð39Þ 2m2 d= the ghost sunset (second diagram in the second line of Fig. 2) Z 2 Γ 2 − 2 1 Γ 2 − e ð d= Þ 2 − 2ξ2 ξ 2 ξ 2 hh;6ðp Þ¼ 2 ð dÞm dxKd=2−2ð m ; m Þ; ð40Þ ð4πÞd= ð2 − dÞ 0 the Goldstone boson sunset (third diagram in the second line of Fig. 2) Z 1 λ Γ 2 − d=2 1 Γ 2 ð Þ 2 − 2 ξ 2 ξ 2 d=2−2 hh;7ðp Þ¼ 2 ð dÞmh dxKd=2−2ð m ; m Þ ð41Þ 2 ð4πÞd= ð2 − dÞ 0 and a mixed Goldstone-photon sunset (fourth diagram in the second line of Fig. 2) Z Γ 2− 2 1 1 4 Γ 2 2 ð d= Þ 2− 2 2 p ξ2 2 2 2ξ−2ξ 2 2 2 ξ 2 hh;8ðp Þ¼e 2 dx ð dÞ p þ 2 þ m þ p m þm Kd=2−2ðm ; m Þ 2−d ð4πÞd= 0 m 4 2 2 2 2 p 2 2 2 p 2 d=2−1 p 2 d=2−1 −ð2−dÞ ξ m þ þ2p ξ K 2−2ðξm ;ξm Þþ2 1−ξ− ðm Þ þ2 2ξ−1þ ðξm Þ : ð42Þ m2 d= m2 m2

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Finally, we have the tadpole diagrams: the Higgs balloon (first diagram on the third line of Fig. 2) λ Γ 2 − 2 2 ð d= Þ d−2 Γ 9ðp Þ¼9 m ; ð43Þ hh; ð4πÞd=2 ð2 − dÞ h the photon balloon (second diagram on the third line of Fig. 2)

2 Γ 2 − 2 2 e ð d= Þ d−2 2 d=2−1 Γ 10ðp Þ¼6 ððd − 1Þm þ ξðξm Þ Þ; ð44Þ hh; ð4πÞd=2 ð2 − dÞ the Goldstone boson balloon (third diagram on the third line of Fig. 2) λ Γ 2 − 2 2 ð d= Þ 2 d=2−1 Γ 11ðp Þ¼3 ðξm Þ ð45Þ hh; ð4πÞd=2 ð2 − dÞ and the ghost balloon (fourth diagram on the third line of Fig. 2)

2ξ Γ 2 − 2 2 e ð d= Þ 2 d=2−1 Γ 12ðp Þ¼−6 ðξm Þ : ð46Þ hh; ð4πÞd=2 ð2 − dÞ

Putting together Eqs. (35) to (46) we find the total one-loop correction to the Higgs boson self-energy, Z Γ 2 − 2 1 1 4 Π 2 ≡ Γ 2 ð d= Þ 2 − 2 2 2 − 1 2 2 p 2 2 hhðp Þ hhðp Þ¼ 2 dx ð dÞe m ðd Þþ p þ 2 Kd=2−2ðm ;m Þ 2 − d ð4πÞd= 0 2m 9 2 2 2 2 2 p 2 d=2−1 2 d=2−1 þ λð2 − dÞm K 2−2ðm ;m Þþe −2 þ 4ðd − 1Þ ðm Þ þ 6λðm Þ 2 h d= h h m2 h 4 λ 2 p 2 2 2 2 p 2 2 d=2−1 þð2 − dÞ − e þ m K 2−2ðξm ; ξm Þþ2 e þ λ ðξm Þ : ð47Þ 2m2 2 h d= m2

2 So, for the Higgs boson resummed connected propagator, 2 4πμ˜ μ ¼ γ ; ð50Þ we find e E we find for the divergent part of the transverse photon 2 1 Ghhðp Þ¼ 2 2 − Π 2 : ð48Þ 2-point function p þ mh hhðp Þ 2 e2 p2 g2 1 Π⊥ ðp2Þ¼ þ 6 − m2 þ 3m2 ; ð51Þ AA;div ϵð4πÞ2 3 λ 2 h C. Results for d =4− ϵ and these infinities are, following the MS scheme, canceled For d ¼ 4, the 2-point functions are divergent. We by the corresponding counterterms. therefore follow the standard procedure of dimensional The renormalized correlation function is then finite in the regularization, as we have no chiral fermions present. Thus, limit d → 4, and we find for the inverse propagator we choose d ¼ 4 − ϵ with ϵ an infinitesimal parameter and analyze the solution in the limit ϵ → 0. 1 p2 m2 Let us start with the photon 2-point function, given for ⊥ 2 ¼ þ GAAðp Þ arbitrary dimension d by (30). The mass dimension of the Z e2 1 K m2;m2 coupling constant e is e 2 − d=2 ϵ=2, and redefining − 2 2 2 1 − ð hÞ ½ ¼ ¼ 2 dx Kðm ;mhÞ ln 2 ϵ=2 2−d=2 ð4πÞ 0 μ e → eμ˜ ¼ eμ˜ we put the dimension on μ˜, while e is dimensionless. Using m2 m4 m2 þ m2 1 − ln h þ 1 −3ln h μ2 m2 μ2 2 2 h 4e Γð2−d=2Þ d→4−ϵ e 2 2 2 2 ¼ −2 þ1þlnðμ Þ ; ð49Þ 2 Kðm ;m Þ 4π d=2 2− 4π 2 ϵ 2m h ; ð Þ d ð Þ þ ln μ2 ð52Þ

2 2 2 2 where we defined where we set Kðm1;m2Þ ≡ K1ðm1;m2Þ.

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In the same way, we find the divergent part of the Higgs boson 2-point function: 1 1 Π ðp2Þ¼− ðe2ð12p2 − 4ξp2Þþλð8m2 − 4ξm2ÞÞ; ð53Þ hh;div 2ϵ ð4πÞ2 h which is canceled by the corresponding counterterm. Therefore, the inverse Higgs boson propagator reads Z 1 1 1 2 2 2 2 2 2 2 1 − m − 2 Kðm ;m Þ 2 ¼ p þ mh þ 2 dx e p ln 2 ln 2 G ðp Þ ð4πÞ 0 μ μ hh p4 Kðm2;m2Þ m2 Kðm2;m2Þ − ln − 6m2 1 − ln þ ln 2m2 μ2 μ2 μ2 1 m2 Kðm2;m2Þ λ m2 −6 6 h − 9 h h þ 2 h þ ln μ2 ln μ2 ξm2 p4 m2 Kðξm2; ξm2Þ − ξðe2p2 þ λm2Þ 1 − ln − e2 − λ h ln : ð54Þ μ2 2m2 2 μ2

1 Notice that the dependence on the Feynman parameter x in parameter values m ¼ 2 GeV, mh ¼ 2 GeV, μ ¼ 10 GeV, the integrals (52) and (54) is restricted to functions of the e 1 μ e R 2 2 ¼ 10. Notice that by choosing and , we are implicitly 1 Kðm1;m2Þ Λ μ ≪ Λ type 0 dx ln μ2 . These functions have an analytical fixing the Landau pole , with ; see [52] for more solution, depicted in Appendix C. details. We have checked that results are as good as independent from the choice of μ over a very wide range of μ-values. IV. SPECTRAL PROPERTIES OF We start by calculating the pole mass in Sec. IVA. The THE PROPAGATORS pole mass is the actual physical mass of a particle that In this section we will investigate the spectral properties enters the energy-momentum dispersion relation. It is an corresponding to the connected propagators of the last observable for both the photon and the Higgs boson and section. Strictly speaking, the calculation of the spectral should therefore not depend on the gauge parameter ξ. properties should only be done to first order4 in ℏ, since the We will also discuss the residue to first order and compare one-loop corrections to the propagators have been evalu- these with the output from the Nielsen identities [63].In ated up to this order. In practice, however, for small values Sec. IV B we show how to obtain the spectral function to of the coupling constants the higher-order contributions first order from the propagator. In Sec. IV C we will discuss become negligible, and one could treat the one-loop some more details about the Higgs spectral function. solution as the all-order solution without a significant numerical difference. Even so, when looking for analytical A. Pole mass, residue and Nielsen identities rather than numerical results—for example a gauge parameter dependence—we should restrict ourselves to the first- The pole mass for any massless or massive field order results. We shall see the crucial difference between excitation is obtained by calculating the pole of the both approaches. resummed connected propagator To plot the spectral properties of our model we choose 1 m; m ; μ;e 2 some specific values of the parameters f h g.We Gðp Þ¼ ; ð55Þ want to restrict ourselves to the case where the Higgs p2 þ m2 − Πðp2Þ 2 4 2 particle is a stable particle, so we need mh < m . Furthermore, given the Abelian nature of the model, and where Πðp2Þ is the self-energy correction. The pole of the thus a weak coupling regime in the infrared, we can choose propagator is thus equivalently defined by the equation an energy scale μ that is sufficiently small w.r.t. the elusive Landau pole (that is exponentially large) and a correspond- 2 2 2 p þ m − Πðp Þ¼0; ð56Þ ing small value for the coupling constant e. For the rest of this section and the next, we will therefore choose the 2 − 2 and its solution defines the pole mass p ¼ mpole.As 4This would correspond to first order in the gauge coupling e2 consistency requires us to work up to a fixed order in and in the Higgs coupling λ neglecting the implicit coupling perturbation theory, we should solve Eq. (56) for the pole dependence in the masses. mass in an iterative fashion. To first order in ℏ, we find

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2 2 − Π1−loop − 2 O ℏ2 mpole ¼ m ð m Þþ ð Þ; ð57Þ For the Higgs boson, the Nielsen identity is a bit more complicated and is given by where Π1−loop is the first order, or one-loop, correction to ∂ −1 2 −∂ −1 2 −1 2 the propagator. ξGhh ðp Þ¼ χGY1hðp ÞGhh ðp Þ; ð64Þ Next, we also want to compute the residue Z, again up to order ℏ. In principle, the residue is given by −1 2 1 where GY1hðp Þ stands for a nonvanishing PI Green function which can be obtained from the extended 2 2 2 Z ¼ lim ðp þ mpoleÞGðp Þ: ð58Þ BRST symmetry which also acts on the gauge parameter p2→−m2 pole [65]. To be more precise, Y1 is a local source coupled to the BRST variation of the Higgs field [see (13)], We write (55) in a slightly different way Rwhile χ is coupled to the integrated composite operator 4 i d xð− 2 cb¯ þ mcρÞ. Acting with ∂χ inserts the latter 2 1 Gðp Þ¼ composite operator with zero momentum flow into the p2 þm2 −Πðp2Þ 1PI Green function hðshÞhi; see [63] for the explicit 1 expression of G−1 p2 in terms of Feynman diagrams. ¼ Y1hð Þ p2 þm2 −Π1−loopð−m2Þ−ðΠðp2Þ−Π1−loopð−m2ÞÞ 2 As a consequence, the Higgs propagator Ghhðp Þ is not 1 gauge independent, in agreement with our results (54). ¼ ; ð59Þ 2 2 −Π˜ 2 From (64) we further find p þmpole ðp Þ

−1 2 Π˜ 2 Π 2 − Π1−loop − 2 ∂ξ∂ 2 G ðp Þj 2 − 2 where we defined ðp Þ¼ ðp Þ ð m Þ. At one p hh p ¼ mpole 2 2 2 2 Π˜ − − O ℏ −1 2 −1 2 loop, expanding ðp Þ around p ¼ mpole ¼ m þ ð Þ −∂ − ∂ 2 2 2 ¼ χGY1hð mpoleÞ p Ghh ðp Þjp ¼−m ; ð65Þ gives the residue pole 1 which means that the residue is not gauge independent, as Z ¼ −1 2 2 GY1hðp Þ does not necessarily vanish at the pole. We can 1 − ∂ 2 Πðp Þj 2 − 2 p p ¼ m confirm this for the one-loop calculation; see Fig. 3. 2 2 1 ∂ 2 Π 2 2 O ℏ ¼ þ p ðp Þjp ¼−m þ ð Þ: ð60Þ Furthermore, we have

−1 2 In [63], for the Abelian Higgs model, the Nielsen identities ∂ξG ð−m Þ¼0; ð66Þ were obtained for both the photon and the Higgs boson. It hh pole was found that for the photon propagator, the transverse ξ so that the Higgs pole mass is indeed gauge independent, part is explicitly independent of to all orders of pertur- the expected result for the physical (observable) Higgs bation theory, giving the Nielsen identity mass. This can be confirmed to one-loop order by using ∂ ⊥ −1 2 0 Eq. (57); see also Fig. 6. Explicitly, in Eq. (54) for ξðGAAÞ ðp Þ¼ ð61Þ −1 − 2 Ghh ð mhÞ all the gauge parameter dependence drops out, which means that the Higgs pole mass is gauge and consequently independent to first order in ℏ. ⊥ −1 2 ∂ ∂ 2 2 2 0 ξ p ðGAAÞ ðp Þjp ¼−m ¼ ; pole Z ⊥ −1 2 ∂ξðG Þ ð−m Þ¼0; ð62Þ AA pole 1.003 confirming the gauge independence of the residue and the pole mass. Of course, this is not unexpected since the 1.002 transverse part of an Abelian gauge field propagator can be written as 1.001

T T T PμνhAμAνi ∝ hAμ Aμ i;Aμ ¼ PμνAν ð63Þ conn 1.000 T and the transverse component Aμ is gauge invariant under Abelian gauge transformations. 20 40 60 80 100 We can now compare the outcome of the Nielsen identities with our one-loop calculation (52). Indeed, to FIG. 3. Gauge dependence of the residue of the pole for the 1 the first order, Eq. (52) is an explicit demonstration of the Higgs field, for the parameter values m ¼ 2 GeV, mh ¼ 2 GeV, 1 identity (61). μ ¼ 10 GeV, e ¼ 10.

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6 ρ B. Obtaining the spectral function 10 . (t ) 35 We can try to determine the spectral functions them- selves to first order. To do so, we compare the Käll´en- 30

Lehmann spectral representation for the propagator 25 Z ∞ ρ 2 ðtÞ 20 Gðp Þ¼ dt 2 ; ð67Þ 0 t þ p 15 where ρðtÞ is the spectral density function, with the propagator (59) to first order, written as 10 Z 5 Gðp2Þ¼ ðp2 þ m2 − Π˜ ðp2ÞÞZ 0 t pole 0 10 20 30 40 Z ¼ 2 2 ˜ 2 2 2 ∂Π˜ ðp2Þ FIG. 4. The reduced spectral function of the photon, with t given − Π 2 2 1 p þ mpole ðp Þþðp þ mpoleÞ ∂ 2 2 2 p p ¼−m in GeV , for the parameter values m ¼ GeV, mh ¼ 2 GeV, μ 10 1 Z ¼ GeV, e ¼ 10. The first-order pole mass lies at t ¼ 2 ¼ 2 2 4.08286 GeV , and the two-particle state of one photon field p þ mpole 2 2 and one Higgs field starts at t ¼ðmh þ mÞ ¼ 6.25 GeV . ˜ 2 2 2 ∂Π˜ ðp2Þ ! Π − 2 ðp Þ ðp þ mpoleÞ ∂ 2 2 p p −m þ Z ¼ ; ð68Þ be m2 ¼ 4.08286 GeV2.InFig.4 one finds the spectral ðp2 þ m2 Þ2 AA;pole pole function for the photon propagator, which is, as expected, ξ where in the last line we used a first-order Taylor expansion positive definite, in addition to being independent. so that the propagator has an isolated pole at p2 ¼ −m2 .In The threshold (branch point) of the propagator is given by pole 2 t ¼ðmh þ mÞ , which can be read off from (52):it (67) we can isolate this pole in the same way, by defining the 2 2 2 2 corresponds to the smallest value of −p where K m ;m spectral density function as ρðtÞ¼Zδðt − m Þþρ˜ðtÞ, ð hÞ pole becomes negative. giving Z For the Higgs field, we find the pole mass up to first ∞ ℏ 2 0 251665 2 2 Z ρ˜ðtÞ order in to be mhh;pole ¼ . GeV . The Higgs Gðp Þ¼ 2 2 þ dt 2 ; ð69Þ spectral function, however, is gauge dependent, and there- p þ m 0 t þ p pole fore it cannot have any direct physical interpretation. As an and we identify the second term in each of the representations illustration, we plot the Higgs spectral function for different (68) and (69) as the reduced propagator values of the gauge parameter in Fig. 5. For small values of t, the spectral functions for different gauge parameter ˜ 2 ≡ 2 − Z values are as good as identical, with a two-particle state for Gðp Þ Gðp Þ 2 2 ; ð70Þ p m 2 2 þ pole two Higgs particles starting at t ¼ðmh þ mhÞ ¼ 1 GeV and a photon two-particle state starting at t ¼ðm þ mÞ2 ¼ so that 16 2 Z GeV . For larger t, significant differences appear ∞ ρ˜ ξ ˜ 2 ðtÞ for different values of since the threshold for the two- Gðp Þ¼ dt 2 particle state for the Goldstone boson is given by 0 t þ p pffiffiffi pffiffiffi 0 1 t ¼ð ξm þ ξmÞ2. Of course, this two-particle state ˜ 2 2 2 ∂Π˜ ðp2Þ Π − 2 ðp Þ ðp þ mpoleÞ ∂ 2 2 is highly unphysical because the Goldstone field does not B p p ¼−m C ¼ Z@ 2 2 2 A: ð71Þ represent a physical particle, something which is reflected ðp þ m Þ pole in the fact that it gives a descending contribution to the spectral function. For ξ < 3, we even find that this Finally, using Cauchy’s integral theorem in complex analy- descending behavior causes the spectral function to become ˜ 2 sis, we can find ρ˜ðtÞ as a function of Gðp Þ, giving negative. We will relate this to the asymptotic behavior of the Higgs propagator in the next section, along with 1 ρ˜ðtÞ¼ lim ðG˜ ð−t − iϵÞ − G˜ ð−t þ iϵÞÞ: ð72Þ some other salient features of the Higgs spectral properties, 2πi ϵ→0þ including its threshold. We can now extract the spectral functions for the photon and the Higgs boson. Therefore, we first calculate the pole C. Some subtleties of the Higgs spectral function masses up to first order, using Eq. (56). For our parameter In this section we discuss some subtleties that arose during values, we find the photon pole mass up to first order in ℏ to the analysis of the spectral function of the Higgs boson.

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ρ 10. (t ) p2 þ m2 − Π1−loopðp2Þ¼0: ð74Þ 6

5 The difference between the pole masses obtained by the 4 iterative method (57) and the approximation (74) is very small, of the order 10−6 GeV for our set of parameters. 3 However, it is rather interesting to notice that the pole mass 2 of the Higgs boson becomes gauge dependent in the approximation (74). This is no surprise, as the validity 1 of the Nielsen identities is understood either in an exact way or in a consistent order-by-order approximation. The 0 t 20 40 60 80 above discussed approximation is neither. In Fig. 6 one can see the gauge dependence of the FIG. 5. The reduced spectral function of the Higgs boson, with t approximated pole mass of the Higgs, in contrast with the 2 giveninGeV,forξ ¼ 2 (green, solid), ξ ¼ 3 (red, dotted), first order pole mass. Even worse, for very small values ξ 4 2 ¼ (yellow, dashed) and the parameter values m ¼ GeV, of ξ, the approximated pole mass gets complex (conjugate) 1 μ 10 1 mh ¼ 2 GeV, ¼ GeV, e ¼ 10. The first-order pole mass lies values. This is due to the fact that the threshold of the 0 251665 2 at t ¼ . GeV . The threshold for the Higgs two-particle branch cut, the branch point, for (47) is ξ dependent, as we 2 1 2 state is given at t ¼ðmh þ mhÞ ¼ GeV and for the photon will see in the next section. two-particle state at t m m 2 16 GeV2.Wefinda(neg- ¼ð þ Þ ¼ pffiffiffi pffiffiffi ative) Goldstone two-particle state at t ¼ð ξm þ ξmÞ2, corresponding to t ¼ð32 GeV2; 48 GeV2; 64 GeV2Þ for ξ ¼ð2; 3; 4Þ. 2. Something more on the branch points The existence of a diagram with two internal Goldstone 1. A slightly less correct approximation Rlines (see Fig. 2) leads to a term proportional to 1 2 1 − ξ 2 for the pole mass 0 dx lnðp xð xÞþ m Þ in the Higgs propagator (47). This means that for small values of ξ, the threshold In the previous two sections we have obtained strictly for the branch cut of the propagator will be ξ dependent too. first-order expressions. In practice, for small values of the 2 Let us look at the Landau gauge ξ ¼ 0. In this gauge, the coupling parameter e , we could think about making the above ln term is proportional to ln p2 , due to the now approximation ð Þ massless Goldstone bosons. This logarithm has a branch point at p2 ¼ 0, meaning that the pole mass will be lying on 2 1 Gðp Þ¼ 2 2 2 the branch cut. Since the first-order pole mass is real and p þ m − Πðp Þ 1−loop 2 gauge independent, this means that Π ð−m Þ is a 1 hh h ≈ singular real point on the branch cut. In the slightly less 2 2 1− 2 ; ð73Þ p þ m − Π loopðp Þ correct approximation of the last section, we will find complex conjugate poles as in Fig. 6. This is explained by 2 − 2 in which case one can fix the pole mass by locating the the fact that for every real value different from p ¼ mh, root of we are on the branch cut; see Fig. 7.

2 Re[m2 ] Im[m pole] pole 0.251667 2. 10–8

0.251666 1. 10–8

0.251665 0.005 0.010 0.015 0.020 0.025 0.030

–1. 10–8 0.251664

–2. 10–8 20 40 60 80

FIG. 6. Gauge dependence of the Higgs pole mass obtained iteratively to first order (green) and the approximated pole mass (red), for 1 1 the parameter values m ¼ 2 GeV, mh ¼ 2 GeV, μ ¼ 10 GeV, e ¼ 10. The right panel shows the real part; the left panel shows the imaginary part.

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1–loop 2 11 Im[ hh (mh +a+iy)]* 10 spectral function, a known fact [53,71,72]. This being said, 2 at the same time we cannot trust the propagator values for p2 → ∞ without taking into account the renormalization 1 group (RG) effects and, in particular, the running of the coupling, which is problematic for nonasymptotically free gauge theories such as the Abelian Higgs model. –8 y*10 –1.0 –0.5 0.5 1.0 V. A NONUNITARY Uð1Þ MODEL –1 In this section, we discuss an Abelian model of the Curci-Ferrari (CF) type [73], in order to compare it with the –2 Higgs model (1). Both models are massive Uð1Þ models with a BRST symmetry. However, while the BRSToperator FIG. 7. Behavior of the one-loop correction of the Higgs 2 s of the Higgs model is nilpotent, this is not true for the propagator Πhhðp Þ around the pole mass, for the values a ¼ −10−6 (yellow, dashed), a ¼ 0 (red, dotted), a ¼ 10−6 CF-like model. We know that the Higgs model is unitary (green, solid). The value x is a small imaginary variation of but, by the criterion of [48], the CF model is most 2 probably not. the argument in Πhhðp Þ. Only for a ¼ 0 do we find a continuous function at x ¼ 0, meaning that for any other value, we are on the In Sec. VA we discuss some essentials for the CF-like branch cut. model: the action with the modified BRST symmetry, tree- level propagators and vertices. In Sec. VBwe discuss the one-loop propagators for the photon and scalar field and Another consequence of the fact that, for small ξ, the extract the spectral function. In Sec. VC we introduce a pole mass is a real point inside the branch cut is that local composite field operator that is left invariant by Π p2 p2 −m2 hhð Þ is nondifferentiable at ¼ h, and we cannot the modified BRST symmetry of the CF model. The extract a residue for this pole. In order to avoid such a spectral properties of this composite state’s propagator will problem, we should move away from the Landau gauge and ξ tell us something about the (non)unitarity of the model, take a larger value for , so that the threshold for the branch since for unitary models, we expect the propagator of a − 2 4ξ 2 2 cut will be smaller than mh. For this we need m >mh, BRST-invariant composite operator to be gauge parameter which, in the case of our parameters set, means requiring independent and the spectral function to be positive ξ 1 that > 64, in accordance with Fig. 6. definite.

3. Asymptotics of the spectral function A. CF-like U(1) model: Some essentials Away from the Landau gauge, we see on Fig. 5 that for We start with the action of the CF-like Uð1Þ model e.g., ξ ¼ 2 the Higgs spectral function is not nonpositive everywhere, while for e.g., ξ ¼ 4 it is positive definite, with Z ξ 3 1 2 a turning point at ¼ . How can we explain this differ- d m † S ¼ d x FμνFμν þ AμAμ þðDμφÞ Dμφ ence? The answer can be related to the UV behavior of the CF 4 2 2 propagator. For p → ∞, the Higgs boson propagator at 2 † † 2 þ mφφ φ þ λðφφ Þ one loop behaves as 2 b 2 2 − α þ b∂μAμ þ c¯∂ c − αm cc¯ ; ð77Þ 2 Z 2 G ðp Þ¼ 2 ; ð75Þ hh 2 p p ln μ2 m2 where the mass term AμAμ is put in by hand rather than with Z depending on the gauge parameter ξ. Now, one can 2 coming from a spontaneous symmetry breaking, and we show (see Appendix D) that for Z > 0, ρðtÞ becomes have fixed the gauge in the linear covariant gauge with negative for a large value of t. For our parameter set, we gauge parameter α. The mass term breaks the BRST find that for large momenta symmetry (13) in a soft way. This Abelian CF action is, 2 2 however, invariant under the modified BRST symmetry, −1 2 p lnðp Þ 2 0 5 G p → 3 − ξ ; p → ∞; smSCF ¼ , with hh ð Þ ð Þ 1600π2 for ð76Þ

5 ξ 3 Z 0 This is the Abelian version of the variation (5). For computa- so that for < , we indeed find > . This indicates that tional purposes, we have also made a rescaling ib → b. Notice the large momentum behavior of the propagator makes a that higher order α-dependent terms present in the CF model are difference around ξ ¼ 3 and determines the positivity of the absent in the Abelian limit [42,43].

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smAμ ¼ −∂μc; smc ¼ 0;smφ ¼ iecφ; † † 2 smφ ¼ −iecφ ;smc¯ ¼ b; smb ¼ −m c: ð78Þ

As noticed in our Introduction, this modified BRST 2 symmetry is not nilpotent since smc¯ ≠ 0. From the quadratic part of (77) we find the following FIG. 8. Contributions to one-loop CF photon self-energy. Wavy propagators at tree level: lines represent the photon field, and dashed lines represent the scalar field. 1 α hAμðpÞAνð−pÞi ¼ Pμν þ Lμν; p2 þ m2 p2 þ αm2 pμ hAμðpÞbð−pÞi ¼ i ; p2 þ αm2 m2 hbðpÞbð−pÞi ¼ − ; p2 þ αm2 1 FIG. 9. Contributions to the one-loop CF scalar self-energy. φ† φ − Line representation as in Fig. 8. h ðpÞ ð pÞi ¼ 2 2 ; p þ mφ 1 ¯ − − propagators in d 4 and discuss some curiosities. The hcðpÞcð pÞi ¼ 2 2 ; ð79Þ ¼ p þ αm inverse connected photon propagator, while from the interaction terms we find the vertices Z 2 1 ⊥ −1 2 2 2 e 2 2 ðG Þ ðp Þ¼p þ m þ dxKðmφ;mφÞ Γ † −p1; −p2; −p3 AA 2 Aμφ φð Þ ð4πÞ 0 2 2 2 ¼ eðp3 μ − p2 μÞδðp1 þ p2 þ p3Þ; K mφ;mφ mφ ; ; 1 − ð Þ − 2 1 − × ln 2 mφ ln 2 ; Γ † − − − − μ μ AμAνφ φð p1; p2; p3; p4Þ 2 ð81Þ ¼ −2e δμνδðp1 þ p2 þ p3 þ p4Þ;

Γφ†φφ†φð−p1; −p2; −p3; −p4Þ is independent of the gauge parameter. The threshold of the ¼ −4λδðp1 þ p2 þ p3 þ p4Þ: ð80Þ 2 branch cut is given by t ¼ −4mφ, and to avoid a pole mass 2 2 B. Propagators and spectral functions lying on the branch cut, we need to choose m < 4mφ. 1 1 The one-loop corrections to the photon and scalar self- Choosing m ¼ 2 GeV, mφ ¼ 2 GeV, μ ¼ 10 GeV, e ¼ 10, energies are given in Figs. 8 and 9. Without going through we find a positive spectral function; see Fig. 10. the calculational details, we directly give here the More interestingly, the scalar propagator

2 Z 2 2 2 2 1 K mφ; αm K mφ; 0 −1 2 2 − e 2 − α2 2 − α 2 2 1 − 2 ð Þ 4 2 0 p 1 − ð Þ GφφðpÞ¼p þ mφ 2 dx m m Kðmφ;m Þ ln 2 þ Kðmφ; Þ 2 ln 2 ð4πÞ 0 μ m μ 2 2 2 2 2 α 2 4 2 0 2 2 p Kðmφ;m Þ 2 2 p Kðmφ; m Þ p 2 Kðmφ; Þ − 2Kðmφ;m Þ 1 − ln − 2Kðmφ; αm Þ 1 − ln þ 8 x ln m2 μ2 m2 μ2 m2 μ2 4 2 2 2 2 4 2 2 2 p Kðmφ;m Þ Kðmφ;m Þ p Kðmφ; αm Þ m − 4 x2 ln − 4p2 ln − 4αp2x − αp2x2 þ 4 x2 ln − 3m2 ln m2 μ2 μ2 m2 μ2 μ2 2 2 2 αm λ 2 mφ þ αm ln þ mφ 1 − ln ð82Þ μ2 ð4πÞ2 μ2 is α dependent, and so is the iterative first-order pole mass From the fact that we find gauge-dependent (complex) 2 2 − Π1−loop − 2 mφ;pole ¼ mφ ð mφÞ. This field thus cannot re- pole masses for the scalar field, we can already draw the present a physical particle. For any value other than the conclusion that the CF model does not describe a physical Landau gauge α ¼ 0, we furthermore get complex poles; scalar field. In the next section we will explicitly verify the see Fig. 11. nonunitary of this model in yet another way.

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6 ρ 10 . (t ) A Fock space analogue of this operator was introduced 0.25 in [49], where it was established that it has a negative norm. As a consequence, it was shown that the physical subspace 0.20 relating to the symmetry generator sm was not well defined, as it contains ghost states. Several more such states were 0.15 identified later in [50]. Up to leading order, the connected propagator of the 0.10 composite operator in Eq. (83) reads 2 2 0.05 2 b 2 b 2 2 ¯ ¯ Gb 2 ¯ ðp Þ¼ þ m cc ; þ m cc 2 þm cc 2 2 t 1 50 100 150 200 2 2 4 ¯ ¯ ¼ 4 hb ;b iþm hcc; cci: ð84Þ FIG. 10. The reduced spectral function of the photon field in the Abelian CF model, with t given in GeV2, for the parameter values We thus find the propagator (84) to be 2 1 μ 10 1 m ¼ GeV, mh ¼ 2 GeV, ¼ GeV, e ¼ 10. The first-order Z 2 d pole mass lies at t ¼ 0.25151 GeV . The photon two-particle 2 3 2 d k 1 1 2 − 2 16 2 Gb 2 ¯ ðp Þ¼ m 2 2 2 2 state starts at t ¼ðmφ þ mφÞ ¼ GeV . 2 þm cc 4 ð2πÞd k þ αm ðk − pÞ þ αm 3 1 ¼ − m2 Γð2 − d=2Þ Essentially, our findings so far mean that in the CF 4 4π d=2 Z ð Þ setting, the unphysical gauge parameter α plays quite an 1 2 2 important role here, just like the coupling: different values × dxKd=2−2ðαm ; αm Þ; ð85Þ of the gauge parameter label different theories. This can 0 also be seen from another example: the one-loop vacuum and this gives, for d ¼ 4, using the MS-scheme, energy of the model will now not only depend on m but also α Z on . 2 1 2 2 2 3 m Kðαm ; αm Þ 2 Gb 2 ¯ ðp Þ¼ 2 dx ln 2 : ð86Þ 2 þm cc 4 ð4πÞ 0 μ C. Gauge-invariant operator The Abelian CF model allows us to construct a BRST- Clearly, the propagator depends on the gauge parameter α,a b2 2 invariant composite operator ð 2 þ m cc¯ Þ, with not so welcome feature for a presumably physical object. We can also find the spectral function immediately from 2 b 2 the propagator by again relying on (72). In Fig. 12, one sees s þ m cc¯ ¼ 0: ð83Þ m 2 that the spectral function is negative for different values of α. Both the α dependence and the negative definiteness 2 Although sm ≠ 0 and we can therefore no longer introduce of the spectral functions demonstrate the nonunitarity of the the BRST cohomology classes, we can still use the fact that Abelian CF model. To our knowledge, this is the first time sm is a symmetry generator, thereby defining a would- that ghost-dependent invariant operators in the physical be physical subspace as the one being annihilated by sm. subspace of a CF model have been constructed from the

2 2 Re[m ϕ,pole] Im[m ϕ,pole] α 0.24 –10 –5 5 10

0.22 –0.02

0.20 –0.04

0.18 –0.06

α –10 –5 5 10 –0.08

FIG. 11. Gauge dependence of the first order pole mass for the scalar field. The left panel shows the real part; the right panel shows the 1 1 imaginary part. The chosen parameter values are m ¼ 2 GeV, mφ ¼ 2 GeV, μ ¼ 10 GeV, e ¼ 10.

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3 ρ 10 . (t ) Another issue we faced was the fact that the branch t point for the Higgs propagator is ξ dependent, being 1 2 3 4 5 6 7 located at p2 ¼ 0 for the Landau gauge ξ ¼ 0. For small –0.2 values of ξ, the pole mass has a real value. However, its value is located on the branch cut, making it impossible to –0.4 define a residue at this point, and therefore a spectral function. This means that in order to formulate a spectral –0.6 function, we should move away from the Landau gauge. These issues with unphysical (gauge-variant) thresholds –0.8 are nothing new; see for example [74]. They reinforce in a natural way the need to work with gauge-invariant field –1.0 operators to correctly describe the observable excitations of a gauge theory. FIG. 12. The spectral function of the composite operator For the photon, the (transverse) propagator is gauge b2 2 2 þ m cc¯ , for α ¼ 2 (green, dotted), α ¼ 3 (red, solid), α ¼ 5 independent (even BRST invariant), and consequently so 1 (yellow, dashed). The chosen parameter values are m ¼ 2 GeV, are the pole mass, residue, and spectral function. For the μ ¼ 10 GeV. The threshold for the branch cut of the propagator is pffiffiffi pffiffiffi Higgs boson, the propagator, residue, and spectral given by t ¼ð αmþ αmÞ2, with t ¼ð2 GeV2;3 GeV2;5 GeV2Þ function are gauge dependent, while the pole mass is for ξ ¼ð2; 3; 5Þ. gauge independent, in line with the latter being an observable quantity. Notice that the residue of the two- functional viewpoint,6 complementing the (asymptotic) point function does not need to be gauge independent Fock space analyses of [49,50]. since this does not follow from the Nielsen identities, as we discussed in our main text. Rather, the Nielsen identities can be used to show that the residues of the VI. CONCLUSION AND OUTLOOK pole masses in S-matrix elements are gauge independent, In the present work we have studied the Källén-Lehmann that is, the residues of the singularities in observable spectral properties of the Uð1Þ Abelian Higgs model in the scattering amplitudes; see [75,76]. These residues can evidently be different per scattering process (and per Rξ gauge and those of a Uð1Þ CF-like model. Our main aim was to disentangle, in this analytical, different mass pole). The fact that the Higgs propagator gauge-fixed setup, what is physical and what is not at the is gauge dependent is not surprising, given that the level of the elementary particle propagators, in conjunction Higgs field is not invariant under the Abelian gauge with the Nielsen identities. Special attention was given to transformation. the role played by gauge (in)dependence of different In future work, it would be interesting to consider, even quantities and by the correct implementation of the results in perturbation theory, gauge-invariant operators and up to a given order in perturbation theory. In particular, study their spectral properties using the same techniques calculating the spectral function for the Higgs propagator in of this paper. If the elementary fields are not gauge the Uð1Þ model, it became apparent that an unphysical invariant (like the Higgs field, but also the gluon field occurrence of complex poles, as well as a gauge-dependent in QCD), these aforementioned gauge-invariant operators pole mass, is caused by the use of the resummed (approxi- will turn out to be composite in nature. Such an approach mate) propagator being exact. Indeed, for small coupling has recently been addressed in [77,78], based on constants, the one-loop correction gives a good approxi- the seminal observations of Fröhlich-Morchio-Strocchi mation of the all-order loop correction, and this is a much [79,80], in which composite operators with the same global quantum numbers (parity, spin, …) as the elemen- used method to find numerical results [19,37,54]. However, 7 for analytical purposes, one should stick to the order at tary particles are constructed. These composite states will which one has calculated the propagator. As we have enable us to access directly the physical spectrum of the illustrated, at least in the Uð1Þ Abelian Higgs model case, theory. Moreover, we notice that the spectral properties one will then find a real and gauge-independent pole mass and the behavior in the complex momentum plane of a for the Higgs boson, in accordance with what the Nielsen (gauge-invariant) composite operator will nontrivially identities dictate [63]. depend on the spectral properties of its gauge-variant constituents. This gives another motivation why it is meaningful to study spectral properties of gauge-variant 6A similar result can be checked to hold for the original non- Abelian CF model, by adding a few higher order terms to the Abelian operator introduced here. This means that a non-Abelian 7For a recent discussion of the renormalization properties of version of the operator (83), invariant under the BRST trans- higher dimensional gauge-invariant operators in Yang-Mills formation (5), can be written down. Higgs models, see the recent results by [59].

065009-16 SOME REMARKS ON THE SPECTRAL FUNCTIONS … PHYS. REV. D 100, 065009 (2019) propagators. Another nice illustrative example of this Putting this in a matrix form yields interplay is the Bethe-Salpeter study of glueballs in pure gauge theories [81], based on spectral properties of Z 1 constituent gluons and ghosts [3]. We further notice that quad 4 T S ¼ d xΨμ OμνΨν; ðA2Þ working with gauge-invariant variables will also evade bos 2 the above-mentioned problem with unphysical (gauge- variant) thresholds. Moreover, this methodology could also shed more light on how the confinement-like and where Higgs-like phases are analytically connected in the (coupling, Higgs vev) diagram in the case of a non-Abelian 0 1 Aν Higgs field in the fundamental representation, thereby B C b making contact with the lattice predictions of Fradkin- T B C Ψμ ¼ðAμ b ρ hÞ; Ψν ¼ B C; ðA3Þ Shenker [82,83]. @ ρ A In conclusion, in this work several tools have been worked out to determine spectral properties in perturbation h theory. We worked up to first order in ℏ, but everything can be consistently extended to higher orders. We paid attention and to how to avoid problems with complex poles and to the important pivotal role of the Nielsen identities, which are 0 2 2 1 intimately related to the exact nilpotent BRST invariance of ð−δμνð∂ −m Þþ∂μ∂νÞ −i∂μ m∂μ 0 the model. These tools will turn out to be quite useful for B C B i∂ν ξ imξ 0 C forthcoming work on the spectral properties of Higgs- O B C ¼ @ 2 A: Yang-Mills theories. For these theories, the Nielsen iden- −m∂ν imξ −∂ 0 tities are well established [66], with supporting lattice 000−ð∂2 −m2Þ data [84], thus providing a solid foundation with which h to compare any results. ðA4Þ

ACKNOWLEDGMENTS The tree-level field propagators can be read off from the The authors would like to thank the Brazilian agencies inverse of O, leading to the following expressions: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparoa ` Pesquisa do Estado do Rio de Janeiro (FAPERJ) for financial support. 1 ξ hAμðpÞAνð−pÞi ¼ Pμν þ Lμν; This study was financed in part by the Coordenação de p2 þ m2 p2 þ ξm2 Aperfeiçoamento de Pessoal de Nível Superior—Brasil 1 (CAPES)—Financial Code 001 (M. N. F.). This paper is hρðpÞρð−pÞi ¼ ; p2 þ ξm2 also part of the project Instituto Nacional de Ciência e 1 Tecnologia – Física Nuclear e Aplicações (INCT-FNA) − hhðpÞhð pÞi ¼ 2 2 ; Process No. 464898/2014-5. p þ mh pμ hAμðpÞbð−pÞi ¼ ; APPENDIX A: PROPAGATORS AND VERTICES p2 þ ξm2 OF THE ABELIAN HIGGS MODEL IN −im THE Rξ GAUGE hbðpÞρð−kÞi ¼ ; ðA5Þ p2 þ ξm2 1. Field propagators

The quadratic part of the action (16) in the bosonic sector pμpν pμpν where Pμν δμν − 2 and Lμν 2 are the transversal is given by ¼ p ¼ p and longitudinal projectors, respectively. The ghost propa- Z gator is quad 1 4 2 2 2 S d x Aμ −δμν ∂ − m ∂μ∂ν Aν − ρ∂ ρ bos ¼ 2 f ð ð Þþ Þ − ∂2 − 2 ¯ ∂2 − 2ξ 1 hð mhÞh þ cð m Þc ¯ − hcðpÞcð pÞi ¼ 2 2 : ðA6Þ 2 p þ ξm þ 2ib∂μAμ þ ξb þ 2imξbρ þ 2mAμ∂μρg: ðA1Þ

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2. Vertices From the action (16), we find the following vertices:

Γ − − − − δ Aμρhð p1; p2; p3Þ¼ieðpμ;3 pμ;2Þ ðp1 þ p2 þ p3Þ; Γ − − − −2 2 δ δ AμAνhð p1; p2; p3Þ¼ e v μν ðp1 þ p2 þ p3Þ; Γ − − − − −2 2δ δ AμAνhhð p1; p2; p3; p4Þ¼ e μν ðp1 þ p2 þ p3 þ p4Þ; Γ − − − − −2 2δ δ AμAνρρð p1; p2; p3; p4Þ¼ e μν ðp1 þ p2 þ p3 þ p4Þ;

Γhhhhð−p1; −p2; −p3; −p4Þ¼−3λδðp1 þ p2 þ p3 þ p4Þ;

Γhhρρð−p1; −p2; −p3; −p4Þ¼−λδðp1 þ p2 þ p3 þ p4Þ;

Γρρρρð−p1; −p2; −p3; −p4Þ¼−3λδðp1 þ p2 þ p3 þ p4Þ;

Γhhhð−p1; −p2; −p3Þ¼−3λvδðp1 þ p2 þ p3Þ;

Γhρρð−p1; −p2; −p3Þ¼−λvδðp1 þ p2 þ p3Þ;

Γchc¯ ð−p1; −p2; −p3Þ¼−mξeδðp1 þ p2 þ p3Þ: ðA7Þ

APPENDIX B: EQUIVALENCE BETWEEN Together those four contributions yield INCLUDING TADPOLE DIAGRAMS IN THE SELF-ENERGIES AND SHIFTING hφi 1 Γ 2 − d=2 1 Γ ð Þ −2 2 d−2 − 1 There is yet another way to come to (47). For this, we do hhi ¼ 2 2 ð e vm ðd Þ ð4πÞd= ð2 − dÞ m not need to include the balloon-type tadpoles in the self- h − λ ξ 2 d=2−1 − 3λ d−2 energies but rather fix the expectation value of the Higgs vð m Þ vmh Þ; ðB5Þ field hhi¼0 by shifting the vacuum expectation value of the Higgs field to its proper one-loop value. The h field which becomes, for d ¼ 4 − ϵ, one-point function has the following contributions at one- loop order: 1 1 1 2 (i) the gluon contribution − 1 μ2 ¼ 2 2 4π 2 ϵ þ þ lnð Þ mh ð Þ 2 1 2 Γ 2− 2 2 2−ϵ 2 1−ϵ 2 2−ϵ e v ð d= Þ d−2 2 d=2−1 −2 3 − ϵ − λ ξ = − 3λ − ðm ðd−1Þþξðξm Þ Þ; ×ð e vm ð Þ vð m Þ vmh ÞðB6Þ 2 4π d=2 2− mh ð Þ ð dÞ ðB1Þ 1 1 1 2 ¼ − þ 1 þ lnðμ2Þ (ii) the Goldstone boson one 2m2 ð4πÞ2 ϵ h ϵ ϵ 1 1 Γ 2 − 2 × −2e2vm2 1 − lnm2 ð3 − ϵÞ − λvξm2 1 − lnm2 ð d= Þ 2 2−1 2 2 − λv ðξm Þd= ; ðB2Þ 2 4π d=2 2 − mh ð Þ ð dÞ ϵ −3λvm2 1 − m2 : h 2ln h ðB7Þ (iii) the ghost loop

1 e2vξ Γð2 − d=2Þ We can split this into a divergent part 2 ðξm2Þd=2−1; ðB3Þ 2 4π d=2 2 − mh ð Þ ð dÞ 1 1 Γdiv ¼ ð6e2m2v þ 3m2vλ þ ξm2vÞ; ðB8Þ hhi ϵ 2 h (iv) the Higgs boson one mh

1 λv Γð2 − d=2Þ −3 md−2: ðB4Þ which we can cancel with the counterterms, and a finite part 2 4π d=2 2 − h mh ð Þ ð dÞ that reads

065009-18 SOME REMARKS ON THE SPECTRAL FUNCTIONS … PHYS. REV. D 100, 065009 (2019) 2 2 1 2 2 1 e 2 m e 2 m Γfin 1−3 v1 ¼ − v m 1 − 3ln hhi ¼ 2 2 v m ln 2 2 4π 2 μ2 m ð4πÞ μ mh ð Þ h 1 λ 2 ξ 2 1 λ v h2 ξm2 v 2 h 2 m − 3 2 1 − ξ 2 1 − þ 3m 1−ln þξm 1−ln : 2 2 mh ln 2 þ m ln 2 : 2 4π 2 2 h μ2 μ2 m ð4πÞ 2 μ μ mh ð Þ h ðB9Þ ðB16Þ Implementing this in the transverse hAAi-propagator, Now, to see how this reflects on the propagator, we can one gets rewrite our scalar field as ⊥ 2 1 1 GAAðp Þ¼ 2 2 2 ⊥ 2 ; ðB17Þ φ ¼ pffiffiffi ððhφiþhÞþiρÞ; ðB10Þ p þ e ðv þ 2ℏv1vÞ − Π ðp Þ 2 AA Π⊥ O ℏ where in the correction AA, which is already of ð Þ,we where the vacuum expectation value of the Higgs field has only include the Oðℏ0Þ part of hφi, i.e., v. tree-level and one-loop terms: We can now verify the ξ-independence of the transverse propagator hAAi. The ξ-dependent part of Π⊥ ðp2Þ is hφi¼v þ ℏv1: ðB11Þ AA −2e2 Γð2 − d=2Þ Thus the “classical” potential part of the action becomes Π⊥ ðp2Þ¼ ðξm2Þd=2−1; ðB18Þ AA;ξ d=2 2 − ð4πÞ d λ v2 2 λ φ†φ − φ 2 − 2 2 φ 2 ρ2 2 ξ 2 2 ¼ 8ðh i v þ hh iþh þ Þ ; ðB12Þ while we find the -dependent part of v1 to be [using (B5)] 1 Γ 2 − d=2 1 ð Þ λ ξ 2 d=2−1 and expanding this, we find for the shifted tree-level Higgs v1ξ ¼ 2 2 ð vð m Þ Þ: ðB19Þ ð4πÞd= ð2 − dÞ m mass h In the denominator of (B17) we now easily see that 1 2 λ 3 φ 2 − 2 λ 2 3ℏλ mh ¼ ð h i v Þ¼ v þ vv1; ðB13Þ 2 2 − Π⊥ 2 0 2 e v1ξv0 AA;ξðp Þ¼ ; ðB20Þ while the photon mass is thereby establishing the gauge independence of the trans-

2 2 2 2 2 2 verse photon propagator. m ¼ e hφi ¼ e v þ 2ℏe vv1: ðB14Þ For the Higgs propagator, we similarly find

Since now, by construction, hhi¼0, we can fix the one- 2 1 8 Ghhðp Þ¼ 2 2 2 : ðB21Þ loop correction to the Higgs vev by requiring it to absorb p þ λðv þ 3ℏv1vÞ − Πhhðp Þ the tadpole contributions: Here we observe that the ξ-dependent part of v1 has the Γfin 0 v1 þ hhi ¼ ; ðB15Þ same effect as the balloon tadpole of the Goldstone boson (45), consequently establishing the gauge parameter inde- and thus pendence of the Higgs mass pole.

APPENDIX C: FEYNMAN INTEGRALS The integrals over the Feynman parameter x that appear in the field propagators can be done in an exact way, since for general masses m1;m2, we find Z 1 2 2 2 2 2 2 Kðm1;m2Þ 1 2 m2 2 m1 2 m1m2 dxln 2 ¼ 2 m1 ln 2 þm2 ln 2 þp ln 4 0 μ 2p m1 m2 μ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2 2 − 2 −2 − 4 2 2 2 −2 2 2 − 4 −2 2 2 − 4 −1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1 þm2 p m1 þ m1m2 m1p m2 m2p p tan 4 2 2 2 2 2 2 −m1 þ2m1ðm2 −p Þ−ðm2 þp Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2 2 2 2 − 4 2 2 2 −2 2 2 − 4 −2 2 2 − 4 −1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1 þm2 þp −4 2 þ m1 þ m1m2 m1p m2 m2p p tan 4 2 2 2 2 2 2 p : −m1 þ2m1ðm2 −p Þ−ðm2 þp Þ ðC1Þ

8This procedure is also equivalent to computing hφi via an effective potential minimization up to the same order.

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þ with Kðm1;m2Þ given by we find for t → þ∞ and ϵ → 0 ,

2 2 2 2 2 Kðm1;m2Þ ≡ ðp xð1 − xÞþxm1 þð1 − xÞm2Þ: ðC2Þ

−t−iϵ −1 −tþiϵ −1 Z ðln μ2 Þ ðln μ2 Þ ρðtÞ¼ − APPENDIX D: ASYMPTOTICS 2πi −t−iϵ −tþiϵ OF THE HIGGS PROPAGATOR Z t −1 t −1 ¼ − ln −iπ þ ln þiπ At one loop, the Higgs propagator behaves like 2πit μ2 μ2 Z −1 Z t π 2 2 ¼ Im ln 2 þi G ðp Þ¼ 2 for p → ∞: ðD1Þ hh 2 p πt μ p ln μ2 Z t 2 −1=2 π ln π2 sin −arctan : D4 ¼ π μ2 þ t ð Þ For Z > 0, this can only be compatible with t lnμ2 Z ∞ 2 ρðtÞdt Ghhðp Þ¼ 2 ðD2Þ 0 t þ p ρ R From the latter expression, we can indeed infer that ðtÞ if the superconvergence relation [6,71] dtρðtÞ¼0 holds, becomes negative for large t. We find which forbids a positive spectral function. Let us support this nonpositivity of ρðtÞ by using (D1) to show that ρðtÞ is certainly negative for very large t. This argument can also −2 t→∞ Z t be found in the Appendix of [5]. ρðtÞ¼ − ln < 0 ðD5Þ t μ2 Since for a KL representation we have 1 ρðtÞ¼ lim ðGð−t − iϵÞ − Gð−t þ iϵÞÞ; ðD3Þ 2πi ϵ→0þ for Z > 0, and vice versa for Z < 0.

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