full gauge-art.2 Full Gauge-Parameter-Independent 29Science Diliman (January - June 2000) 12:1, 29-34 ABSTRACT INTRODUCTION Full Gauge-Parameter-Independent Higgs-Boson Self-Energy Bernd A. Kniehl1 and Caesar P. Palisoc2 1 II. Institut für Theoretische Physik, Universität Hamburg Luruper Chaussee 149, 22761 Hamburg, Germany Email: bernd.kniehl@desy.de 2 National Institute of Physics, University of the Philippines Diliman, Quezon City 1101, Philippines Email: cpalisoc@nip.upd.edu.ph We calculate the self-energy of the standard-model Higgs boson by means of the pinch technique. We work in the general Rξ gauge and show that the final result is independent of the gauge parameters. This result is useful in order to demonstrate that the threshold singularities encountered in the description of Higgs-boson production and decay in the on-shell renormalization scheme only arise at physical thresholds and are independent of the chosen gauge. Keywords: SM Higgs-boson self-energy, pinch technique, S-matrix PT framework, gauge invariance, quantum corrections, perturbation theory The standard model (SM) of Glashow, Salam and Weinberg is a non-abelian gauge field theory of the electroweak and strong interactions. It involves a Yang- Mills sector in its classical Lagrangian, which on the whole is invariant under a local SU(2) and U(1) gauge transformation. This gauge invariance property is compromised when one wishes to canonically quantize the theory. In order to achieve this, one needs to introduce gauge-fixing terms in the Lagrangian, which break the underlying gauge symmetry and introduce unphysical degrees of freedom. In order to compensate for this, one needs to add terms involving anti- commuting Faddeev-Popov ghost fields into the Lagrangian. Both these requirements for the quantization complicate matters concerning practical calculations, for example, of radiative corrections involving quantum-loop diagrams, in the vector- and scalar-boson portion of the theory (Papavassiliou & Pilaftsis, 1998). As for the Higgs-Boson, its full one- loop self-energy, when calculated from the assumption that the Higgs-Boson is an asymptotic state of the scattering (S) matrix, while gauge independent on-shell, turns out to depend on the gauge parameter ξ off-shell. This property, among others, is by no means an obstacle in conventional perturbation theory, which predicts meaningful observables independent of the gauge-fixing procedure. However, this is not the case where the conventional perturbation theory breaks down, like in the strongly coupled theory of quantum chromodynamics (QCD) and in the vicinity of resonances in a weakly coupled theory (Papavassiliou & Pilaftsis, 1996) such as the electroweak sector of the standard model. The search for a self-consistent scheme for constructing off-shell Green’s functions resulted in a formalism based on what is called the pinch technique (PT). Kniehl & Palisoc 30 As has been observed in Refs. (Kniehl, 1991; 1992; 1994), the one-loop corrected decay width of the SM Higgs-Boson as calculated in the conventional on-shell renormalization scheme exhibits singularities if the mass relations mH = 2mW or mH = 2mZ happen to be satisfied. These threshold singularities arise from the wave- function renormalization of the Higgs-Boson, which is given in terms of the derivative of the self-energy of the latter. One important application of the PT is to verify that these threshold singularities only occur at desirable properties like being resummable and process independent and complying with the unitarity of the S matrix (Papavassiliou & Pilaftsis, 1995; 1996; 1998). Conventional Higgs-Boson Self-Energy in Rξ Gauge The Feynman diagrams contributing to the conventional self-energy of the SM Higgs-Boson are depicted in Fig. 1. In Rξ gauge, we findphysical thresholds and are independent of the chosen gauge. This will be exhibited in a forthcoming paper (Kniehl et al., in preparation), which will also explain how the threshold singularities are removed by the consequent application of the pole definitions of mass and width. In this paper, we derive the full Higgs-boson self-energy in the PT framework. We work in the general renormalizable Rξ gauge (Fujikawa et al., 1972) and explicitly show that the result obtained is independent of the gauge parameters. Our result generalizes a previous analysis (Papavassiliou & Pilaftsis, 1998). where G is related to the Fermi constant by G = GF/ (2π√2), s is the Higgs-boson virtuality, mH and mW are the masses of the Higgs and W bosons, respectively, ξW is the gauge parameter associated with the W boson, the functions A0 and B0 are the so-called scalar one- and two-point functions (Kniehl, 1994), respectively, and the term (W → Z) signifies the contribution involving the Z boson, which is obtained from the one involving the W boson by replacing mW 2 and ξW with mZ 2 and ξZ, respectively. In the ’t Hooft-Feynman gauge, with ξW = ξZ = 1, Eq. (1) agrees with the corresponding result given in Eqs. (B.2) and (B.3) of Ref. (Kniehl, 1991). Pinch Technique The PT is an algorithm that renders one-loop gauge- boson self-energies gauge independent. It has rich applications in QCD (Cornwall & Papavassiliou, 1989) and in the electroweak sector of the SM (Degrassi & Sirlin, 1992; Papavassiliou, 1990; Papavassiliou & Sirlin, 1994). At the one-loop level, the technique unravels self-energy contributions from vertex and box diagrams that are otherwise excluded in the conventional manner of computing the self-energy. By themselves, these PT self-energy contributions are gauge dependent. When combined with the conventional self-energy, these contributions exactly cancel the gauge-dependent terms of the former rendering the combined self-energy gauge independent. The resulting PT self-energy satisfies ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,,,2 2 2 ,, 8 9 4 3 2 1 ,, 4 1 ,,3 2 2 1 3 2 22 0 22 0 2 22 0 42 0 2 22 0 42 22 0 42 2 2 0 22 0 2 ⎭ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−− ++ →+−− ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +−+ ⎩ ⎨ ⎧ −−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +=Π ∑ ffff f ff HHHHH WWWWH WWWW WWHWWHH mmsBm s mAmN mmsBmmAm ZWmmsBms mmsBmsm s mAmsmAm sG s ξξ ξ π (1) Full Gauge-Parameter-Independent 31 Figure 1. Feynman diagrams pertinent to the conventional self-energy of the SM Higgs boson in Rξ gauge. Kniehl & Palisoc 32 Full PT Contribution in Rξ Gauge In calculating the full PT contribution to the Higgs-boson self-energy in Rξ gauge, we use the S-matrix PT framework elaborated in Ref. (Degrassi & Sirlin, 1992). The scattering process considered is a four-fermion process with the Higgs-Boson as intermediate state. In the formulation of Ref. (Degrassi & Sirlin, 1992), the relevant amplitudes reflecting the gauge-boson and external-fermion interactions are described in terms of matrix elements of Fourier transforms of time-ordered products of current operators. Through successive current contractions with the longitudinal four-momenta found in the propagators of the massive vector bosons, Ward identities are triggered, after which the relevant pinch contribution is identified upon application of appropriate equal-time commutators of currents. The relevant Feynman diagrams are depicted in Fig. 2. Setting aside the details, the full PT contribution is found to have the following form, DISCUSSION OF THE RESULTS Some comments on Eqs. (1) and (2) are in order. As expected (Papavassiliou & Pilaftsis, 1998), their sum, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,,,2 2 2 ,, 8 9 4 3 2 1 ,,32 4 3 2 22 0 22 0 2 22 0 42 0 2 22 0 422 2 2 2 0 2 2 ⎭ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−− ++ →+⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +−⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += ∆Π+Π=Π ∑ ffff f ff HHHHH WWWWW H H WW H HHHH PT HH mmsBm s mAmN mmsBmmAm ZWmmsBmsmm m m mAm mG sss π ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) . 2 1 ,, 4 1 ,, 4 1 2 1 22 0 42 22 0 222 2 0 2 0 2 ⎭ ⎬ ⎫ →+−+ ⎥⎦ ⎤ ⎢⎣ ⎡ ++−− ⎩ ⎨ ⎧ −−=∆Π ZWmmsBms mmsBmmsms mAmAms G s WWWWH WWWHH WWWHHH ξξ ξ π(2) (3) which represents the full PT self-energy of the Higgs- Boson, is manifestly independent of ξW and ξZ for all values of s. In the special case of s = mH 2, ΠHH(s) is separately gauge independent, while ∆ΠHH(s) vanishes. In fact, this ensures that the total decay width of the Higgs-Boson, which is proportional to the absorptive part of its self-energy, is gauge independent and does not receive additional contribution due to the application of the PT. In the ’t Hooft-Feynman gauge, our full PT result agrees with that reported in Ref. (Papavassiliou & Pilaftsis, 1998). For gauges other than the ’t Hooft-Feynman gauge, our result only differs from the one in Ref. (Papavassiliou & Pilaftsis, 1998) with respect to the first term in Eq. (2). However, consistency between the two results remains and has not been compromised, since the authors of Ref. (Papavassiliou & Pilaftsis, 1998) have explicitly omitted contributions from tadpole and seagull graphs, which are included in our consideration and are the ones responsible for the presence of the first term in Eq. (2). On the other hand, their significance must not be underestimated, since they render the dispersive part of the Higgs-Boson self- energy gauge independent as well. Full Gauge-Parameter-Independent 33 Figure 2. Feynman diagrams pertinent to the pinch parts of the self-energy of the SM Higgs boson Kniehl & Palisoc 34 ACKNOWLEDGMENTS The work of B.A.K. was supported in part by the Bundesministerium für Bildung und Forschung under Contract No. 05 HT9GUA 3, and by the European Commission through the Research Training Network Quantum Chromodynamics and the Deep Structure of Elementary Particles under Contract No. ERBFMRXCT980194. The work of C.P.P. was supported by the German Academic Exchange Service (DAAD) through Grant No. A/97/00746. REFERENCES Cornwall, J. M. & J. Papavassiliou, 1989. Gauge Invariant Three Gluon Vertex in QCD. Phys. Rev. D40: 3474-3485. Degrassi, G. & A. Sirlin, 1992. Gauge-Invariant Self- Energies and Vertex parts of the Standard Model in the Pinch Technique Framework. Phys. Rev. D46: 3104- 3116. Fujikawa, K., B. W. Lee, & A. Sanda, 1972. Generalized Renormalizable Gauge Formulation of Spontaneously Broken Gauge Theories. Phys. Rev. D6: 2923-2943. Kniehl, B. A., 1991. Radiative Corrections for Η → ZZ in the Standard Model. Nucl. Phys. B352: 1-26. _______, 1991. Radiative Corrections for H→W+W - (δ) in the Standard Model. Nucl. Phys. B357: 439-466. _______, 1992. Radiative Corrections for H→f f (δ) in the Standard Model. Nucl. Phys. B356: 3-28. _______, 1994. Higgs Phenomenology at One Loop in the Standard Model. Phys. Rep. 240: 211-300. Kniehl, B. A., C. P. Palisoc, & A. Sirlin. July 2000. hep-ph/0007002. Papavassiliou, J., 1990. Gauge Invariant Proper Self- Energies and Vertices in Gauge Theories with Broken Symmetry. Phys. Rev. D41: 3179-3191. Papavassiliou, J. & A. Pilaftsis, 1995. Gauge Invariance and Unstable Particles. Phys. Rev. Lett. 75: 3060-3063. _______, 1996. A Gauge-Independent Approach to Resonant Transition Amplitudes. Phys. Rev. D53: 2128- 2149. _______, 1996. Gauge-Invariant Resummation Formalism for Two-Point Correlation Function. Phys. Rev. D54: 5315-5335. _______, 1998. Gauge-and Renormalization-Group- Invariant Formulation of the Higgs-Boson Resonance. Phys. Rev. D58: 053002-1-053002-26. Papavassiliou, J. & A. Sirlin, 1994. Renormalizable W Self-Energy in the Unitary Gauge via the Pinch Technique. Phys. Rev. D50: 5951-5957. −