Recent progress in the understanding of vacuum expectation values and of
infrared divergences in different regularization schemes is reviewed. Vacuum
expectation values are gauge and renormalization-scheme dependent quantities.
Using a method based on Slavnov-Taylor identities, the renormalization
properties could be better understood. The practical outcome is the computation
of the beta functions for vacuum expectation values in general gauge theories.
The infrared structure of gauge theory amplitudes depends on the regularization
scheme. The well-known prediction of the infrared structure in CDR can be
generalized to the FDH and DRED schemes and is in agreement with explicit
computations of the quark and gluon form factors. We discuss particularly the
correct renormalization procedure and the distinction between MSbar and DRbar
renormalization. An important practical outcome are transition rules between
CDR and FDH amplitudes.Comment: 8 pages, proceedings for Loops and Legs in Quantum Field Theory 2014,
  Weimar, German

Gnendiger, Christoph

Signer, Adrian

Sperling, Marcus

Stöckinger, Dominik

Voigt, Alexander

English

arXiv

Recent progress in the understanding of vacuum expectation values and of infrared divergences in different regularization schemes is reviewed. Vacuum expectation values are gauge and renormalization-scheme dependent quantities. Using a method based on Slavnov-Taylor identities, the renormalization properties could be better understood. The practical outcome is the computation of the ? functions for vacuum expectation values in general gauge theories. The infrared structure of gauge theory amplitudes depends on the regularization scheme. The well-known prediction of the infrared structure in CDR can be generalized to the FDH and DRED schemes and is in agreement with explicit computations of the quark and gluon form factors. We discuss particularly the correct renormalization procedure and the distinction between MS and DR renormalization. An important practical outcome are transition rules between CDR and FDH amplitudes

Institutionelles Repositorium der Leibniz Universität Hannover

PoS(LL2014)076Two-loop results on the renormalization of vacuumexpectation values and infrared divergences in theFDH schemeChristoph GnendigerInstitut für Kern- und Teilchenphysik, TU Dresden, GermanyAdrian SignerPaul Scherrer Institut, CH-5232 Villigen PSI, Switzerland andPhysik-Institut, Universität Zürich, SwitzerlandMarcus SperlingInstitut für Theoretische Physik, Universität Hannover, GermanyDominik Stöckinger∗Institut für Kern- und Teilchenphysik, TU Dresden, GermanyE-mail: Dominik.Stoeckinger@tu-dresden.deAlexander VoigtInstitut für Kern- und Teilchenphysik, TU Dresden, GermanyRecent progress in the understanding of vacuum expectation values and of infrared divergencesin different regularization schemes is reviewed. Vacuum expectation values are gauge andrenormalization-scheme dependent quantities. Using a method based on Slavnov-Taylor iden-tities, the renormalization properties could be better understood. The practical outcome is thecomputation of the β functions for vacuum expectation values in general gauge theories.The infrared structure of gauge theory amplitudes depends on the regularization scheme. Thewell-known prediction of the infrared structure in CDR can be generalized to the FDH and DREDschemes and is in agreement with explicit computations of the quark and gluon form factors. Wediscuss particularly the correct renormalization procedure and the distinction between MS andDR renormalization. An important practical outcome are transition rules between CDR and FDHamplitudes.Loops and Legs in Quantum Field Theory - LL 2014,27 April - 2 May 2014Weimar, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik Stöckinger1. IntroductionPrecision calculations in QCD, the electroweak Standard Model, or supersymmetric modelsrequire regularization and renormalization in intermediate steps. Here we discuss two recent workswhere progress in the understanding of fundamental issues and results of practical relevance wereobtained. The topic of [1] is the renormalization properties of vacuum expecation values (vevs) inspontaneously broken gauge theories. A method was developed which leads to a better understand-ing of the divergent renormalization of vevs and their gauge and renormalization-scale dependence.As a practical result the two-loop renormalization group β -functions of vevs in general and super-symmetric gauge theories were obtained and made available for use in spectrum generators andexplicit calculations.In [2] the structure of infrared divergences was investigated in different regularization schemes.The recent progress on computations using unitarity-inspired methods or four-dimensional ap-proaches highlights the importance of studying QCD amplitudes in regularization schemes whichdiffer from conventional dimensional regularization (CDR). Ref. [2] studies the relation betweenCDR and the four-dimensional helicity (FDH) scheme. The central results are the proof that theinfrared structure can be predicted in both schemes in a similar way, and that there are simpletranslation rules which convert an FDH-regularized amplitude into a CDR-regularized one. Theseresults are similar to results of Ref. [3], and they can be viewed as a continuation of Refs. [4], wherea systematic one-loop comparison of the CDR, HV, FDH, and DRED regularization schemes wascarried out and an earlier factorization problem of DRED was resolved.2. Vacuum expectation values and their renormalizationThe renormalization of a scalar field φ and associated vev v can generically be written asφ + v→√Zφ +√Z√Zˆv. (2.1)Here Z is the usual field renormalization constant, and Zˆ is an additional renormalization constant,which characterizes to what extent v renormalizes differently from φ . The field renormalizationleads to an anomalous dimension γ of the scalar field, and one can define a similar quantity γˆ,which is defined via Zˆ in the same way as γ is defined via Z. The renormalization group β functionfor the running vev in the MS or DR scheme is then given asβv = (γ + γˆ)v. (2.2)A surprising observation was made in the literature in Refs. [5]. In the minimal supersym-metric standard model (MSSM), there are two Higgs doublets Hu, Hd, and there is a cancellationbetween the two additional renormalization constants, i.e. δ ZˆHu − δ ZˆHd =finite at the one-looplevel. The usual field renormalization constants δZHu , δZHd don’t share this property. The impli-cation is that the β function for the parameter tanβ = vu/vd can simply be written in terms of theusual anomalous dimensions, i.e.δ ZˆHu −δ ZˆHd = finite ⇒ βtanβ = (γHu − γHd ) tanβ (2.3)2PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik Stöckingerat the one-loop level, with no γˆ contributions.Furthermore, one may ask why the additional renormalization constant Zˆ is necessary in thefirst place. Clearly, in non-gauge theories, i.e. theories with only a global symmetry, it is notrequired. The point is that local gauge theories can only be quantized using some kind of gaugefixing. Depending on which gauge fixing is chosen, the additional renormalization constant Zˆcan be necessary. The method employed in Refs. [1] aims to make this explicit. In the processit explains the observation mentioned above, and it allows the explicit computation of βv at thetwo-loop level in generic gauge theories.The essence of the method, introduced in Refs. [6] for a slightly different purpose, is to replacethe vev by a background (classical) field φˆ , carry out the renormalization process (in particular thestudy of Slavnov-Taylor identities and their implications on the required independent renormaliza-tion constants) in presence of these background fields, and specialize to φˆ = v =const only at theend. Using the background fields, the usual Rξ gauge fixing term for the abelian Higgs model canbe written asF = ∂ µAµ + ieξ (φˆ†φ −φ†φˆ ). (2.4)As long as φˆ is treated as a background field which transforms covariantly under global gaugetransformations, this gauge fixing term breaks only local, but not global gauge invariance. Asa consequence only renormalization constants in agreement with global gauge invariance can benecessary. Zˆ can appear as the field renormalization of φˆ . The main trick is to define a BRStransformation of the background field, assφˆ = qˆ, sqˆ = 0. (2.5)Then the Slavnov-Taylor identity provides a useful relation for Zˆ: this renormalization constant canbe directly determined from the Green function ΓqˆKφ , because the counterterm contribution to thisGreen function is ΓctqˆKφ = − 12δ Zˆ. Here Kφ is the source for the BRS transformation of φ , so thisGreen function corresponds to the coupling of the composite operator (sφ) to the background fieldqˆ. It is an unphysical Green function but a very useful technical tool.Fig. 1 (left) shows the single one-loop diagram in a generic gauge theory. The only vertexof Kφ follows from the structure of the BRS transformation of scalar fields: Kφ couples to onescalar field and one Faddeev-Popov ghost; the coupling is the gauge coupling. The only vertex of qˆoriginates in the gauge fixing: qˆ couples to the prefactor of φˆ in the gauge fixing, i.e. the Feynmanrule is proportional to ξ and to the gauge coupling. Hence the one-loop contribution to this Greenfunction (and thus to δ Zˆ) is proportional only to the gauge parameter ξ and to the squared gaugecoupling of the scalar field. In a generic gauge theory the MS result can be written asδabδ Zˆ(1)(a) =1(4pi)22g2ξC2ab(S) ·1ε, (2.6)where C2ab(S) = TAacTAcb with the generators TA acting on the scalar fields φa. Hence the result isproportional to ξ and to the squared gauge couplings of the scalar field. Since all two-loop diagramsmust involve the same two vertices as the ones in Fig. 1 (left), this proportionality extends to thetwo-loop result; Fig. 1 (right) shows a sample two-loop diagram.3PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik StöckingerKφ qˆ Kφ qˆFigure 1: Left: The single one-loop diagram to ΓqˆKφ . The dotted line is a Faddeev-Popov ghost propagator;the dashed line a scalar field propagator. Right: Sample two-loop diagram with a fermion loop (solid line),which leads to corrections involving Yukawa couplings.This result already explains two statements made earlier. First we see that in Landau gauge,where ξ = 0, we obtain δ Zˆ = 0. This reflects the fact that in a gauge which does not break globalgauge invariance, the additional renormalization constant Zˆ is not needed. Second, for ξ 6= 0, Zˆ isneeded but is proportional to squared gauge couplings. This explains the observation (2.3) sincethe squared SU(2) and U(1) gauge couplings of the two Higgs doublets in the MSSM are equal.The full two-loop result for δ Zˆ, γˆ and βv has been obtained in Refs. [1]. Here we provide theresult for γ and γˆ for general supersymmetric gauge theories:γ(1)ab (S)∣∣∣DRSUSY=1(4pi)2[g2 (1−ξ )C2ab(S)−12Y ∗apqYbpq], (2.7)γˆ(1)ab (S)∣∣∣DRSUSY=1(4pi)22g2ξ ξ ′C2ab(S) , (2.8)γ(2)ab (S)∣∣∣DRSUSY=1(4pi)4{g4[(94− 53ξ − 14ξ 2)C2(G)−S2(S)]C2ab(S) (2.9)−2g4C2ac(S)C2cb(S)+12Y ∗arcYrpqY∗pqdYbcd+g2[C2ac(S)Y∗cpqYbpq−2Y ∗apqC2pr(S)Ybrq]},γˆ(2)ab (S)∣∣∣DR/MSSUSY=ξ ξ ′(4pi)4{g4[7−ξ2C2(G)C2ab(S)−2(1−ξ )C2ac(S)C2cb(S)](2.10)−g2C2ac(S)Y ∗cpqYbpq}.Here Yabc are conventionally normalized superpotential couplings. To our knowledge, the scalarfield anomalous dimension γ(2) has not been provided in the literature before. We remark that thisanomalous dimension is the one of the component scalar field in Wess-Zumino gauge, and it is notequal to the superfield anomalous dimension in a supersymmetric gauge fixing. From these resultseq. (2.2) can be used to obtain the β function for vevs and for related quantities such as tanβ .Refs. [1] provide explicit results for models of practical and phenomenological interest, such asthe MSSM, the NMSSM, and the E6SSM. The general results have been implemented in generalprograms such as Sarah [7] and FlexibleSUSY [8].3. Infrared structure of QCD amplitudes in the FDH schemeIn recent years, new methods have been developed to compute gauge theory amplitudes, andthe understanding of the infrared structure of these amplitudes has significantly improved. In par-ticular, Refs. [9] have given a prediction for the infrared 1/ε poles in conventional dimensional4PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik Stöckingerreduction for arbitrary QCD amplitudes. Specialized to form factors, this prediction readslnZ =( αs4pi)( Γ′14ε2+Γ12ε)+(αs4pi)2(−3β20Γ′116ε3+Γ′2−4β20Γ116ε2+Γ24ε)+ . . . (3.1)Here Γ′m =−2γcuspm Cq/g, Γm =+2γ im.with the cusp anomalous dimension γcusp and parton anoma-lous dimension γ i; the index i refers to either quark or gluon, and the index m refers to the coefficientof the respective quantity of (αs/4pi)m; β20 is the (αs/4pi)2-coefficient of the β function for αs.In view of the new computational methods based on unitarity, helicity and 4-dimensional alge-bra it is of high interest to study how the infrared structure depends on the regularization scheme,in particular how it is modified in schemes such as the four-dimensional helicity scheme (FDH)and dimensional reduction (DRED). An independent reason to study these regularization schemesis supersymmetry, which is broken in CDR, while FDH and DRED preserve supersymmetry to alarge extent. In Refs. [4] the differences between these schemes were clarified and it was shownthat all these schemes are consistent regularization schemes. In particular one-loop results of theearlier literature were shown to be consistent among each other and consistent with infrared factor-ization, if the schemes are used appropriately. The necessity to treat renormalization in the FDHscheme in the way advocated in [4] was also reiterated in Ref. [11].Ref. [2] can be viewed as a continuation of Refs. [4] to the two-loop level and as an extensionof Eq. (3.1) to other regularization schemes. Extending the infrared prediction (3.1) to FDH andDRED is possible by taking into account the main insight of Ref. [4], which is that FDH andDRED should be viewed as dimensional regularization with a new type of parton, a scalar fieldwith multiplicity Nε = 2ε , the ε-scalars. The ε-scalars have couplings and anomalous dimensionswhich differ from the ones of the gluons. At the two-loop level the only new coupling appearing inthe infrared prediction is αe, the coupling of ε-scalars to quarks. The FDH result corresponding toEq. (3.1) isln Z¯ =( αs4pi)( Γ¯′104ε2+Γ¯102ε)+(αe4pi)( Γ¯′014ε2+Γ¯012ε)+(αs4pi)2(−3β¯20Γ¯′1016ε3+Γ¯′20−4β¯20Γ¯1016ε2+Γ¯204ε)+(αs4pi)(αe4pi)(−3β¯e11Γ¯′0116ε3+Γ¯′11−4β¯ e11Γ¯0116ε2+Γ¯114ε)+(αe4pi)2(−3β¯e02Γ¯′0116ε3+Γ¯′02−4β¯ e02Γ¯0116ε2+Γ¯024ε)+O(α3). (3.2)Here the bars denote quantities defined in the FDH scheme, β e is the β function for αe, and Γ′i j,Γi j, βi j and βei j are the coefficients of (αs/4pi)i(αe/4pi)j of the respective quantities.The FDH formula differs in two respects from the CDR one. First, there are new structures,involving the new β function β e and involving coefficients of the order αe. Second, all quantitiesare defined in the FDH scheme and thus contain contributions from ε-scalars and differ at the orderNε from the respective CDR quantities. Ref. [3] has obtained an equivalent formula using a slightlydifferent approach.5PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik Stöckinger×ε ×εεε ×Figure 2: Sample counterterm diagrams contributing to the quark and gluon form factors in the FDHscheme. The crosses denote counterterm insertions involving δ Z¯(1)αe , δ Z¯(1)λε, and δ Z¯(2)λ , respectively.Concrete two-loop computations of form factors serve to test the prediction (3.2) and to de-termine the unknown coefficients. We have computed the two-loop quark and gluon form factors.In this way γcusp, γq and γg are determined at the two-loop level, and there are non-trivial checkssince the system is overconstrained.In the actual calculation of the two-loop form factors in the FDH scheme, the correct renor-malization is particularly important and non-trivial. We highlight the following two points.1. Independent couplings of ε-scalars. The couplings of ε-scalars must be treated as inde-pendent from the respective couplings of gluons. Fig. 2 shows three sample countertermdiagrams involving the renormalization constants δ Z¯(1)αe , δ Z¯(1)λε, and δ Z¯(2)λ , where the upperindex denotes the loop order. λ is the effective coupling of the Higgs boson to gluons, λε isthe corresponding coupling to ε-scalars. The renormalization of λ also appears in CDR cal-culations, and here it can be determined in the same way, although the FDH and CDR resultsdiffer. The renormalization of αe is known in the literature, but the renormalization of λε isnew. The MS value of δ Z¯λε can be obtained by an explicit one-loop off-shell calculation ofthe Higgs–ε–ε three-point function.2. Renormalization schemes MS versus DR. If CDR is used, the MS scheme is simply definedby modified minimal subtraction of UV 1/ε poles. Even in the FDH or DRED regularizationschemes, it is possible to choose MS renormalization, implying that renormalized couplingshave the same meaning as in CDR-based MS. This MS scheme is a natural and advanta-geous scheme also in the context of FDH and DRED if these regularizations are viewed asadvocated here: As long as the ε-scalars are treated as unrelated to gluons but as some newscalar fields which happen to have multiplicity Nε , the MS renormalization scheme simplyamounts to minimally subtracting all UV 1/ε poles, including the ones of the form Nε/εfrom ε-scalar loops. In this scheme, β functions and the γs involve explicit terms of theorder Nε .It is also of interest to study the case of DR renormalization. This amounts to setting Nε = 2εand only then minimally subtracting the remaining 1/ε poles. Renormalized couplings in thisscheme differ by finite shifts from the MS couplings.After consistently renormalizing the form factors either in the MS or DR schemes, we have shownthat in both cases the infrared singularities are correctly described by Eq. (3.2). The respective β6PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik Stöckingerand γ coefficients differ: in the MS case, these coefficients involve terms of the order Nε ; in the DRcase, no such terms appear, but the coefficients differ by ε-independent terms. As an illustration,we quote here the results for the quark anomalous dimension. The CDR (plus MS renormalization)results areγq10 =−3CF , (3.3)γq20 =CACF(−96154− 116pi2+26ζ (3))+C2F(−32+2pi2−24ζ (3))+CFNF(6527+pi23), (3.4)the results for FDH plus MS renormalization or DR renormalization differ by O(Nε) or by finiteterms, respectively,γ¯q10 = γq10, γ¯q,DR10 = γq10, (3.5)γ¯q01 = NεCF2, γ¯q,DR01 = 0, (3.6)γ¯q20 = γq20+Nε(167108+pi212)CACF , γ¯q,DR20 = γq20+179CACF , (3.7)γ¯q11 = Nε[112CACF −(2+pi23)C2F], γ¯q,DR11 =− β¯ e,DR11 CF , (3.8)γ¯q02 =−Nε34CFNF −N2εC2F8, γ¯q,DR02 =− β¯ e,DR02 CF , (3.9)with the non-vanishing β -coefficientsβ¯ e,DR11 = β¯e11∣∣Nε=0= 6CF , (3.10)β¯ e,DR02 = β¯e02∣∣Nε=0=−4CF +2CA−NF . (3.11)The DR results are relevant since they answer a question of the “Supersymmetry Parameter Anal-ysis” report [12]: the DR scheme is a consistent scheme not only for UV [13] but also for infrareddivergences at the multi-loop level.The MS result has the important application of transition rules between FDH and CDR reg-ularized amplitudes. For the example of the quark form factor Fq we can define the combinationQ(2) ≡ F2lq − 12(F1lq)2and obtain[Q(2)− ln Z¯(2)q]FDH=[Q(2)− lnZ(2)q]CDR+O(Nεε0), (3.12)which allows to translate e.g. an FDH-amplitude into a CDR-amplitude or vice versa.References[1] M. Sperling, D. Stöckinger and A. Voigt, JHEP 1307 (2013) 132; JHEP 1401 (2014) 068.[2] C. Gnendiger, A. Signer and D. Stöckinger, Phys. Lett. B 733 (2014) 296.7PoS(LL2014)076Two-loop results on the renormalization of vevs and infrared divergences in FDH Dominik Stöckinger[3] W. B. Kilgore, Phys. Rev. D 86 (2012) 014019.[4] A. Signer and D. Stöckinger, Phys. Lett. B 626 (2005) 127; Nucl. Phys. B 808 (2009) 88.[5] P. H. Chankowski, S. Pokorski and J. Rosiek, Nucl. Phys. B 423 (1994) 437; A. Dabelstein, Z. Phys.C 67 (1995) 495.[6] E. Kraus and K. Sibold, Z. Phys. C 68 (1995) 331; E. Kraus, Annals Phys. 262 (1998) 155; W. Hollik,E. Kraus, M. Roth, C. Rupp, K. Sibold and D. Stöckinger, Nucl. Phys. B 639 (2002) 3.[7] F. Staub, Comput. 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Two-loop results on the renormalization of vacuum expectation values and infrared divergences in the FDH scheme

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