Electroweak Sudakov corrections to New Physics searches at the LHC and future hadron colliders

Transcription

1 UNIVERSITÀ DEGLI STUDI DI PAVIA DOTTORATO DI RICERCA IN FISICA XXVII CICLO Electroweak Sudakov corrections to New Physics searches at the LHC and future hadron colliders Mauro Chiesa Tesi per il conseguimento del titolo

2 Università degli Studi di Pavia Dipartimento di Fisica DOTTORATO DI RICERCA IN FISICA XXVII CICLO Electroweak Sudakov corrections to New Physics searches at the LHC and future hadron colliders Mauro Chiesa Submitted to the Graduate School in Physics in partial fulfilment of the requirements for the degree of DOTTORE DI RICERCA IN FISICA DOCTOR OF PHILOSPHY IN PHYSICS at the University of Pavia Adviser: Fulvio Piccinini

3 Cover: Comparison between virtual and real electroweak corrections to the process Z + 3 jets for the /H T observable at s = 100 TeV under a realistic event selection for the direct New Physics searches. ELECTROWEAK SUDAKOV CORRECTIONS TO NEW PHYSICS SEARCHES AT THE LHC AND FUTURE HADRON COLLIDERS Mauro Chiesa PhD Thesis - University of Pavia Printed in Pavia, Italy, November, 2014 ISBN:

4 ELECTROWEAK SUDAKOV CORRECTIONS TO NEW PHYSICS SEARCHES AT THE LHC AND FUTURE HADRON COLLIDERS Mauro Chiesa 2014

5 Contents 1 Introduction 1 2 Sudakov logarithms as infrared limit of electroweak corrections Infrared structure of one loop corrections for massless gauge theories Infrared limit of EW corrections and Sudakov logarithms The Denner-Pozzorini algorithm and its implementation in the ALPGEN event generator The Denner-Pozzorini algorithm: a short review DL contributions SLC contributions Implementation of the algorithm in the ALPGEN generator Virtual O(α) Sudakov corrections to New Physics searches at pp colliders Direct search for New Physics in the channel /E T + jets Technical remarks on the computation Phenomenological results Real weak corrections to Z + 2 and Z + 3 jets production Introduction: real weak corrections Real weak corrections to Z + 2 and Z + 3 jets production: a parton level analysis Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework Electroweak Sudakov corrections to the R n γ ratio Invisible Z+ jets background to New Physics searches in the channel /E T + jets Virtual O(α) Sudakov corrections to the R n γ ratio i

6 CONTENTS 7 Conclusions and future perspectives 61 List of publications 63 A Cross-checks and code validation 65 A.1 EW Sudakov corrections to Z/γ + 1 jet A.2 EW Sudakov corrections to Z + 2 jets A.3 Electroweak Sudakov corrections to di-jet production B One loop renormalization counterterms 73 B.1 On shell renormalization conditions at one loop B.2 Unrenormalized self energies B.3 Definition of the A 0, B 0 and B 1 functions C Collinear singularities in one loop radiative corrections 81 C.1 Collinear singularities in O(α) QED corrections C.1.1 Collinear limit of real O(α) corrections C.1.2 Collinear limit of virtual O(α) corrections C.2 Collinear limit of one loop EW Sudakov corrections D Sudakov corrections for external longitudinal gauge bosons in the unitary gauge 87 Bibliography 90 ii

7 Chapter 1 Introduction It is somehow a paradigm that one loop electroweak corrections are small effects, of the order a few percent, basically determined by a factor α/4π. However, at high energies and in those extreme kinematical regions in which the gauge boson masses can be neglected if compared to the kinematical invariants involved in the process, one loop electroweak corrections can be enhanced by large double and single logarithms of the invariants over the gauge boson masses. These large, energy growing, logarithms are called Sudakov logarithms and are the leading part of the one loop O(α) corrections at high energies and in the asymptotic regime described above. The typical effect of the electroweak Sudakov corrections is rather small on the integrated cross sections for standard event selections, while can be of the order of tens of percent in the extreme tails of the distributions for several interesting observables. The presence of a Sudakov logarithmic structure has first been noticed in Ref. [1 by direct inspection of the analytic expression of the virtual O(α) corrections to the production of a fermion-antifermion pair in e + e collisions. Even if Sudakov corrections are naturally encoded in any one loop electroweak computation as an asymptotic limit, a complete survey of the existing Next- To-Leading (NLO) electroweak computations is probably beyond the scope of this introduction. In Ref. [2 it has been pointed out for the first time that Sudakov logarithms are related to the infrared (IR) limit of the one loop electroweak corrections, that is when the W and the Z masses are negligible compared to all the other energy scales, so that the weak gauge bosons are basically massless and M V (V = W, Z) play the role of IR regulators like the infinitesimal mass parameters which are usually introduced in order to regularize the IR singularities appearing in massless gauge theories, like QED. The results of Ref. [2 have been the starting point of a quite rich and interesting literature. The key points are the universality of the IR limit of one loop corrections for massless theories and the well established knowledge of the IR behaviour of QED and QCD. Starting from Ref. [2 several works have been published which studied the Sudakov part of electroweak corrections in 1

8 1. Introduction analogy with the infrared limit of QCD, trying to apply the standard techniques developed in QCD and pointing out the main differences basically related to the electroweak spontaneous symmetry breaking and to the different nature of the involved charges (colour for QCD, weak isospin for the electroweak sector, where only the latter exists as physical asymptotic state). Moving from the universality of the infrared part of the virtual one loop corrections for massless gauge theories, in Refs. [3, 4, 5 a general algorithm has been developed in order to compute the electroweak Sudakov corrections (for non mass suppressed processes) in a process independent way. According to the algorithm, in the limit in which all the kinematical invariants are of the same order and much bigger than the gauge boson masses (Sudakov limit), electroweak corrections depend only on the flavours and on the kinematics of the external particles of the LO process and can be written as the sum of universal radiator functions which multiply tree level matrix elements. This algorithm has been used in Refs. [6, 7, 8, 9, 10 to compute the EW Sudakov corrections to W γ production, W W scattering and V + 1 jet (V = Z, γ, W ) production, respectively. The approach of Refs. [3, 4, 5 has been extended in Refs. [11, 12, 13, 14 to the Sudakov part of the two loop electroweak corrections. In Refs. [15, 16, 17, 18, 19, 20, 21 the universality of the infrared limit of radiative corrections has been used to write the evolution equations (IREE, infrared evolution equations) which describe the scaling of the Sudakov corrections with the energy scales considered. In particular, in Refs. [15, 16, 17, 18, 19, 20, 21 the issue of the exponentiation of the large negative Sudakov corrections has been addressed (an independent study of the exponentiation properties of the Sudakov corrections has been performed in Refs. [22, 23, 24, 25, 26, 27). More recently, in Refs. [28, 29, 30, 31 the basic ideas of Ref. [15 have been reformulated in the framework of the soft collinear effective theory (SCET) that has been used to compute the one loop corrections to V + 1 jet (V = Z, γ, W ) production in Ref. [32. In analogy with QCD, in Refs. [33, 34, 35, 36 the generalization of the Altarelli-Parisi splitting functions for the electroweak vertices has been found together with the expression of the evolution equations for the electroweak sector of the Standard Model. An algorithm to describe weak corrections in a parton shower framework has been recently proposed in Ref [37. Another very interesting topic is the role of real weak corrections. In fact, even if Sudakov logarithms represent the infrared limit of the EW corrections and the gauge boson masses act as infrared cutoffs, these cutoffs are physical parameters, so that in principle there is no need to include in the computation of the one loop weak corrections the contribution of the extra radiation of additional W and Z bosons (at variance with the case of QED or QCD, where the IR singularities of the virtual corrections are regularized by means of arbitrary regulators, which are cancelled only once the contribution of the IR divergent real corrections is included in the calculation). Moreover, real 2

9 weak corrections usually are not included into EW NLO predictions because the additional gauge bosons decay, leading to final states that in principle are not degenerate with the LO ones. However, virtual weak corrections in the Sudakov limit can be very large (of order -50% or larger), so that the effect of the partially compensating real weak corrections may not necessarily be negligible. The impact of real EW corrections have been studied in Refs. [38, 39, 27, 26, 24, 23, 40, 41 in analogy with QED (i.e. without any cut on the additional gauge bosons) and it was found that the cancellation between real and virtual Sudakov logarithms may be only partial (these Bloch Nordsieck violation effects have been pointed out in Refs. [26, 24, 23), while in Refs. [42, 40, 43 the effect of real weak corrections under realistic event selections has been considered. This thesis is mainly focused on the implementation of the algorithm of Refs. [3, 4, 5 (Denner-Pozzorini algorithm in the following) in the LO event generator ALPGEN [44 and on the analysis of the phenomenological impact of the electroweak Sudakov corrections at the LHC and at future proton-proton colliders. The basic features of the electroweak Sudakov corrections are described in chapter 2, pointing out the main similarities and differences with respect to the infrared limit of one loop corrections in massless gauge theories, such as QED and QCD. The Denner-Pozzorini algorithm and its implementation in the ALPGEN generator are described in chapter 3. In chapter 4, the algorithm of Refs. [3, 4, 5 will be used in order to study the impact of the one loop electroweak corrections in the Sudakov limit to the process Z( νν) + n jets (with n 3), which is an irreducible Standard Model background to the direct search for New Physics in the channel /E T + jets at the LHC and at future hadron colliders. The effect of real weak corrections to Z +2 and Z + 3 jets production and the partial cancellation between real and virtual weak corrections in the Sudakov limit are studied in chapter 5. In chapter 6 the electroweak Sudakov corrections to the process γ + n jets are considered in order to compute the corrections to the Rγ n = (Z + n jets)/(γ + n jets) ratio, which is the theoretical input for the partially data driven estimate of the Z( νν) + n jets background based on the experimental measurement of the differential distributions for γ + n jets events. 3

10 4 1. Introduction

11 Chapter 2 Sudakov logarithms as infrared limit of electroweak corrections 2.1 Infrared structure of one loop corrections for massless gauge theories In the computation of radiative corrections for massless gauge theories, such as QED and QCD, besides the ultraviolet (UV) divergences related to the behaviour of the loop integrals in the q 2 limit (q being the loop momentum) and cancelled order by order in perturbation theory by means of the renormalization procedure, also infrared (IR) singularities arise related to the presence of massless particles inside the loop diagrams. In this section the basic features of the IR limit of one loop corrections for massless gauge theories are briefly recalled (basically following Ref. [45), in order to better explain the main differences and similarities with respect to the IR limit of one loop electroweak corrections. This section is mainly focused on QED, where the IR singularities are usually regularized by means of mass regularization, that is giving an unphysical mass to the massless particles (in the case of QCD the standard regularization procedure used in the literature is the dimensional regularization for both UV and IR divergences). Strictly speaking the electron is not massless, however its mass (m in the following) is very small compared the EW scale: as a result, even if there are no IR divergences related to the electron mass, the would-be singularities show up as large logarithms of m over the typical energy scale of the process and m can be considered as an IR cutoff. A process-independent analysis of the IR singularities appearing in one loop diagrams for massless gauge theories has been performed in Ref. [46. The general expression of the N point one loop tensor integrals in D = 4 2ε dimensions reads: T N µ 1 µ N (p 0,, p N 1 ; m 0,, m N 1 ) = (2πµ)4 D iπ 2 5 d D q q µ 1 q µn N 0 N N 1, (2.1)

12 2. Sudakov logarithms as infrared limit of electroweak corrections where the parameter µ has a mass dimension and serves to keep the dimension of the integral constant as a function of D, while N k represents the denominator of the loop propagator with momentum q + p k and internal mass m k, namely: N k = (q + p k ) 2 m 2 k + iε (k = 0,, N 1). (2.2) Taking eq. (2.1) as a starting point, in Ref. [46 it has been proved that IR singularities arise from the following two situations: m 2 0 (m 0), m 2 (2.3) soft singularities are related to the diagrams in which a massless particle is exchanged between two external on shell legs and originate from the integration region where q p k (so that the momentum transfer in the propagator k is zero); 0 m 2 (m 0), 0 (2.4) collinear singularities are related to the diagrams in which an external on shell particle splits into two internal massless lines and arise from the integration region q p k + x(p k p k+1 ) (x being an arbitrary real variable), that is when the momentum flowing in the propagator k is collinear to the external momentum (p k p k+1 ). For the case of QED, soft singularities are related to the diagrams involving the exchange of a photon between two external legs and to the electron self energy (which enters the electron wave function renormalization counterterm), while collinear singularities correspond to the splitting of an external electron into an internal electron plus a photon. When regularized by means of mass regularization, giving an infinitesimal mass λ to the photon, the IR singularities appear as logarithms of the IR cutoffs λ and m. Taking eqs. (2.3)-(2.4) as a starting point, in Ref. [47 a general algorithm has been developed for the separation of the soft/collinear structure of an arbitrary N point loop integral. In particular, in Ref. [47 the loop structure in the singular regions is decomposed at the integrand level as: N 1 k=0 1 N k N 1 1 N k k=0 A soft n N n 1 N n N n+1, (2.5) k n,n+1 A coll nk N n N n+1 N k, (2.6) 6

13 2.1. Infrared structure of one loop corrections for massless gauge theories (where n is the index labelling the singular region considered) so that the IR part of the general N > 3 function reduces to a linear combination of scalar Passarino-Veltman three point functions C 0 [48, 49, 50. In Ref. [47 the universal expression of the A coefficients is also given, together with a list of the IR singular C 0 functions. Once the IR limit of the loop integrals have been isolated the virtual one loop matrix elements in the IR limit can be written in a process independent way as the sum of radiator functions which multiply tree level matrix elements. In the soft limit the factorization can be easily computed in the so called eikonal approximation (that is neglecting the photon momentum and the masses in the numerator of the propagators). First of all, the matrix element for the process i f with the radiation of an additional (also off shell) photon becomes: M i f+γ µ (p, q) q 0 M i f 0 (p) l 2eQ l p l, µ (q + η l p l ) 2 m 2 l (2.7) where the photon leg with momentum q has been truncated, M i f 0 (p) is the matrix element for the LO process i f, the factor η k is set to 1 if the photon is radiated off an outgoing fermion line k (to 1 if k labels an incoming fermion line) and the index l spans all the external fermions with momentum p l and electric charge Q l. Then the soft limit of one loop virtual corrections is obtained connecting each pair of terms appearing in eq. (2.7) with a photon propagator ig µν q 2 λ 2 +iε (in the ξ = 1 gauge): M i f Virt. soft = α 1 4π Mi f 0 Q l Q m I lm, (2.8) 2 l,m I lm = (2πµ)4 D d D 4(p l p m ) q [ iπ 2 q2 λ 2[ [ (q + η l p l ) 2 m 2 l (q + ηm p m ) 2 mm, 2 (where the factor +iε in the loop denominators is understood). The loop integrals in eq. (2.8) are the C 0 functions entering eq. (2.5), while the diagonal terms m = l lead to the single soft logarithm associated with the electron wave function renormalization counterterm. The soft limit of the virtual one loop QED corrections (2.8) is a function of the unphysical parameter λ. However, also the process i fγ when the photon is not detected because it is either soft (i.e. below the detector energy resolution ΔE) or collinear with its emitter gives rise to O(α) corrections to the LO process i f. These real O(α) corrections in the soft limit can be obtained from eq. (2.7) considering the emission of a real photon and thus connecting each pair of emitting legs in eq. (2.7) with the sum over the photon 7

14 2. Sudakov logarithms as infrared limit of electroweak corrections polarizations and integrating over the soft photon phase space: M i fγ Real soft 2 = e 2 M i f 0 2 = α 2π Mi f 0 2 l,m ΔE λ dq 0 q 0 soft 1 d 3 q (2π) 3 2q 0 1 l,m Q l η l Q m η m (p l p m ) (qp m )(qp l ) dcosθ q Q l η l Q m η m (p l p m ) E l E m (1 β l cosθ ql )(1 β m cosθ qm ). (2.9) The integrand in eq. (2.9) is again divergent in the soft limit q 0 0 and in the collinear limit θ qj 0 (in the case of photon radiation off a massless fermion, that is β j = 1 m 2 j /E2 j 1, j = l, m). While the finite electron mass prevents from the collinear singularities, soft divergences are regularized by the lower bound of integration λ, that effectively corresponds to a mass parameter for the emitted photon as the one introduced in eq. (2.8) (in eq. (2.9) the upper bound of integration is fixed by the degeneracy condition q 0 ΔE and the integral can be performed in four dimensions since no UV singularities are present in the real corrections). The integral in eq. (2.9) can be computed following Ref. [48 and turns out that it develops the same kind of logarithmic structure of eq. (2.8) as far as the dependence on the λ parameter is concerned. More precisely, the coefficients of the logarithms of the unphysical photon mass λ coming from eq. (2.8) and (2.9) are the same but with opposite sign, so that the sum of the real and virtual O(α) corrections is no longer a function of the arbitrary cutoff λ. This cancellation is an example of the Bloch-Nordsieck theorem [51, which states that in the computation of radiative corrections for abelian gauge theories all the logarithms of the IR cutoffs cancel order by order in perturbation theory in the sum of virtual and real contributions for inclusive observables. The Bloch-Nordsieck theorem also rules the cancellation of the collinear singularities in the m 0 limit: a very short overview of the basic features of the collinear limit of one loop QED corrections is given in Appendix C. As regards the IR limit of the one loop QCD corrections, the behaviour of the loop integrals in the soft/collinear regions is the same as the one described above for the case of QED, while the factorization of the one loop corrections is more involved since the radiation of additional real/virtual gluons changes the original colour structure of the considered process, due to the fact that the underlying gauge symmetry group SU(3) C is non abelian. As a result, at variance with the QED case where the IR part of both the real and the virtual corrections factorizes on the same LO matrix element, the O(α S ) QCD corrections in the IR limit factorize into radiator functions which multiply tree level matrix elements that are the LO amplitude and its colour correlated amplitudes. However, the IR singularities again cancel in the sum of real and virtual corrections for inclusive observables if, besides the sum over the final state colours, also the average over the colours of the initial state particles is considered, as stated by the KLN theorem [46, [52. 8

15 2.2. Infrared limit of EW corrections and Sudakov logarithms 2.2 Infrared limit of EW corrections and Sudakov logarithms As pointed out in Refs. [1, 2, Sudakov logarithms correspond to the IR limit of the electroweak radiative corrections. In fact, when the considered energy scales are much larger than the weak gauge boson masses, M W and M Z can be regarded as IR regulators for the loops involving the exchange of weak bosons and play the same role as the mass parameter λ, that was introduced in the previous section in order to regularize the QED IR singularities arising from loop diagrams in which a soft/collinear photon is connected to external on shell particles (with the following difference: in mass regularized QED it is assumed that λ < m in the intermediate steps of the calculations, while for EW corrections M V > m f, m f being the mass of a light fermion and V = W, Z). While the logarithmic structure of the EW Sudakov corrections shares the same features as the one arising from the IR limit of QED or QCD in mass regularization, since it is determined by the singular behaviour of the loop integrals, the factorization properties of the Sudakov corrections are similar to the ones of the IR part of the QCD corrections, due to the non abelian structure of the underlying gauge symmetry group SU(2) L U(1) Y. For these reasons, in the literature, Sudakov logarithms have been mainly studied in analogy with the IR limit of QCD. There are, however, several differences between the IR limit of EW and the one of QCD corrections, basically related to the spontaneous symmetry breaking of the EW theory and to the nature of the weak charges. The spontaneous symmetry breaking of the EW theory implies that the gauge boson masses are physical parameters. In particular, this means that, even if from a formal point of view in the high energy limit M W and M Z act as IR regulators for the loop diagrams, they cannot be set to zero and the one loop virtual weak corrections are always finite (but potentially very large since they are enhanced by large logarithms of the gauge boson masses over the energy scale of the considered process). As stressed in Refs. [3, 4, 5, another less trivial consequence of the spontaneous symmetry breaking of the EW theory is that also the scalar sector enters the corrections both because of the correspondence between longitudinal gauge bosons and would-be Goldstone bosons and because the collinear factorization is based on the BRS invariance of the full Standard Model. Since the one loop virtual weak corrections are finite by themselves, they can be computed separately from the O(α) contributions associated with the diagrams obtained from the ones of the LO process with the additional radiation of a real weak boson. Moreover, the additional gauge bosons decay so that in principle their decay products lead to final states which are not degenerate with the considered signature. For this reason the contribution of real weak corrections usually is not included in the computation of one loop EW corrections. 9

16 2. Sudakov logarithms as infrared limit of electroweak corrections In the high energy limit, where the size of the negative one loop virtual weak corrections can be of the order of several tens of percent, the positive contribution of real weak corrections can lead to significant compensations between real and virtual corrections. Of course, for realistic event selections the size of the cancellation is strongly dependent on the specific observables and cuts considered, as will be discussed in chapter 5. However, also in the QED or QCD-like case of observables which are completely inclusive over the additional radiation of real gauge bosons, the cancellation of the Sudakov logarithms in the sum of virtual and real contributions may be only partial. This feature of the O(α) EW corrections is known as Bloch Nordsieck violation [26, 24, 23 and it is related to another important difference between QCD and the EW theory. The cancellation between real and virtual IR singularities for inclusive observables in QCD takes place according to the KLN theorem only after that the sum over the colours of the final state particles and the average over the colours of the initial state ones is performed. This sum/average over the colours is needed in QCD because the colour charges do not exist as free asymptotic states, while this is not the case for the weak isospin that basically corresponds to the EW charge. As a result, Bloch Nordsieck violations arise from the incomplete sum/average over the weak isospins of the external particles of the considered process. 10

17 Chapter 3 The Denner-Pozzorini algorithm and its implementation in the ALPGEN event generator 3.1 The Denner-Pozzorini algorithm: a short review As already stated in the introduction, the main topic of this thesis is the phenomenological study of the impact of the one loop electroweak corrections in the Sudakov limit to the Z and γ plus jets production processes which are of great interest for the direct search for New Physics at the LHC and at future pp colliders. These corrections have been computed by means of the Denner-Pozzorini algorithm which has been implemented in the ALPGEN generator [44. In this section the basic formulas proved in Refs. [3, 4, 5 are very briefly summarized in order to better explain the technical aspects of the implementation of the algorithm. The starting point is the fact that electroweak Sudakov logarithms are the infrared limit of virtual one loop EW corrections and the IR structure of one loop corrections is universal, depending only on the flavour and on the kinematics of the particles of the considered process at the LO. Formally, the asymptotic IR limit is defined by the requirement that all the kinematical invariants involved in the process under consideration are of the same order and much larger than the W and Z boson masses (Sudakov limit). Actually, there are also single Sudakov logarithms related to the one loop renormalization: they also are universal, since they are ruled by the renormalization group equations, and are naturally encoded in the usual expression of the EW renormalization counterterms (collected in Appendix B) that can be found for example in Ref. [53. Concerning the IR structure of virtual one loop corrections for massless gauge theories, as already discussed in the previous chapter, in Ref. [46 it has 11

18 3. The Denner-Pozzorini algorithm and its implementation in the ALPGEN event generator been proved that infrared singularities arise from two classes of diagrams: from those diagrams in which a massless propagator is connected to two external on shell legs and from the ones in which an on shell external leg splits into two internal massless lines. Once the singularities are regularized giving an infinitesimal mass to the massless propagators, they appear as logarithms of these masses. Of course, this is a gauge dependent statement which rules for the ξ = 1 gauge: all the intermediate steps of the calculation are performed assuming the ξ = 1 gauge, while the final result is gauge independent. In the Sudakov limit all the particles of the Standard Model are basically massless compared to the other energy scales involved, so that the W and the Z bosons can be considered as almost massless and their masses play the role of (physical) IR cutoffs. Only internal bosonic propagators lead to the Sudakov logarithms, as can be easily understood by direct inspection of the SM vertices and neglecting all the mass dependent ones (since they would give contributions like M 2 log(s/m 2 ), in the limit M 2 0). This kind of argument, however, requires some care when external longitudinally polarized gauge bosons are considered: for this reason, in Refs. [3, 4, 5 longitudinal gauge bosons are replaced with the would-be Goldstone bosons by means of the Goldstone boson equivalence theorem (GBET in the following). In Refs. [3, 4, 5 EW Sudakov corrections are grouped into three classes: the double logarithmic (DL) part, the single logarithmic collinear (SLC) part and the remaining single logarithms associated with the renormalization counterterms. The DL part has the form: δ DL M NLO i 1 i n = N δkl DL M LO i 1 j k j l i n, (3.1) k=1 l>k where the indexes k and l span all the electroweak charged external legs of the LO order process, N is the number of these legs, δkl DL is a radiator function which depends on the flavours of the particles i k and i l and contains the double logarithm of their invariant mass over the gauge boson masses, while M LO i 1 j k j l i n is the tree level matrix element involving the external legs i 1 j k j l i n (j k = i k and j l = i l in the case of the exchange of a photon or a Z boson between the two legs k and l, otherwise j k and j l are the SU(2) transformed of i k and i l, respectively). The single logarithmic part of the corrections coming from the unrenormalized one loop matrix element is: δ SLC M NLO i 1 i n = N k=1 δk SLC M LO i 1 j k i n, (3.2) where the notation is the same as in eq. (3.1), the radiator functions containing the single logarithms factorize on each external leg separately and M LO i 1 j k i n differs from the matrix element of the LO process only for the possible mixing between neutral gauge bosons or scalars. 12

19 3.1. The Denner-Pozzorini algorithm: a short review The single logarithms associated with the fields renormalization are obtained from the fields wave function renormalization counterterms by setting the renormalization scale to the one of the kinematical invariants of the process in order to extract the mass singular part of these counterterms. In the case of external longitudinally polarized gauge bosons replaced by the corresponding would-be Goldstone bosons, the one loop corrections to the GBET are also included, as described in Refs. [3, 4, 5. Once added the SLC part, these contributions can be written as δ SL M NLO i 1 i n = N k=1 δk SL M LO i 1 j k i n. (3.3) The remaining single logarithms come form the running of the dimensionless parameters (the renormalization of the mass parameters leads only to mass suppressed contributions): i1 i n δ P R M NLO i 1 i n = δe δmlo δe + δc W δm LO i 1 i n δc W + δh t δm LO i1 i n δh t + δh eff δm LO i 1 i n H, (3.4) δh H where h t = m t /M W, h H = M 2 H /M 2 W and c W = M W /M Z. As for the fields wave function renormalization, the logarithmic structure of eq. (3.4) is obtained from the expression of the counterterms by setting the renormalization scale to the one of the kinematical invariants of the process (in δh eff also the Higgs tadpole contribution has to be included [54). The next two subsections summarize the most important steps of the derivation of eqs. (3.1) and (3.2) DL contributions The DL contributions arise from the one loop diagrams in which a soft-collinear gauge boson is exchanged between two external legs. Using the graphical notation of Refs. [3, 4, 5, eq. (3.1) reads: δ DL M(i 1 i n ) = V a =γ,z,w ± k l > k j l, j k i k i l V a j k j l, (3.5) where the blob represents the sum of all the tree level diagrams for the external flavour string (i 1 j k j l i n ) with amputated off shell legs j l and j k. j l and j k are equal to i l and i k if V a = Z or γ, otherwise they are the SU(2) transformed of i l and i k, respectively. As for the case of QED, the soft-collinear limit of each of the diagrams in eq. (3.5) can be computed in the so called eikonal approximation (i.e. neglecting the masses and the loop momentum in the numerators), while the expressions for the vertices V a i k j k and V a i l j l can be worked out using the Dirac 13

20 3. The Denner-Pozzorini algorithm and its implementation in the ALPGEN event generator equation for external fermions, or the transversity condition together with the GBET for external transverse gauge bosons. Once all the diagrams in eq. (3.5) have been computed in the eikonal approximation, eq. (3.5) becomes: δ DL M(i 1 i n ) = N k=1 l<k V a =γ,z,w ± I V a j k,i k I V a j l,i l M 0 (i 1 j k j l i n )δ V a k,l, (3.6) where I V a j m,i m is the gauge group generator corresponding to the V a i m j m vertex (m = k, l), M 0 (i 1 j k j l i n ) is the matrix element for the process involving the external particles (i 1 j k j l i n ) and δ V a k,l is the radiator function associated to the exchange of a V a boson between the two external particles k and l. Of course, the external legs for the NLO process are still (i 1 i k i l i n ), while in the r.h.s. of eq. (3.6) the tree level matrix elements are evaluated for the flavour string (i 1 j k j l i n ), but the replacement of the external wave functions for the fields i k and i l with the ones for j k and j l leads only to mass suppressed effects (once the GBET is used for the longitudinal gauge bosons). In eq. (3.6) the loop contributions have been factorized as: δ V a k,l = ie 2 d 4 q (2π) 4 4(p k p l ) (q 2 M 2 V a)[(p k + q) 2 m 2 j k [(p l q) 2 m 2 j l, (3.7) which is a standard Passarino-Veltman scalar three point function [48, 49, 50 that can be computed in the Sudakov limit as described in Ref. [55 (the result can also be found in Ref. [47) SLC contributions The SLC contributions are associated to the collinear splitting i k j k V a, where i k is an external on shell leg and the gauge boson V a becomes collinear to the internal line j k. With the notation of Refs. [3, 4, 5, the SLC terms arise from the contributions of the form: j k i k, V a (3.8) where the blob is again the sum of all the tree level-like diagrams with external legs (i 1 j k i n, V a ) and j k, V a are off shell. Each of the diagrams in eq. (3.8) can be written as: I = i(4π) 2 µ 4 D d D q N(i k, j k, V a ; q) (2π) D (q 2 MV 2 + iε)[(p a q)2 Mj 2 k + iε, (3.9) where N(i k, j k, V a ; q) represents a contribution to the blob in eq. (3.8). 14

21 3.2. Implementation of the algorithm in the ALPGEN generator Since part of the collinear singularities appearing in eq. (3.9) have already been computed in eqs. (3.6)-(3.7), in order to isolate the SLC part of the corrections the results of eqs. (3.6)-(3.7) must be subtracted from eq. (3.9), obtaining: = δ SLC M(i 1 i k i n ) = V a =A,Z,W ± j k i k j k V a l k j l trunc. i k i l V a j k j l eik. coll. = j k j k δ SLC (j k, i k )M 0 (i 1 j k i n ), (3.10) where the last line is obtained by means of the collinear Ward identities proved in Ref. [4 after that the collinear logarithmic part of the integral in eq. (3.9) has been factorized out (as described in Appendix C), M 0 (i 1 j k i n ) is the matrix element for the flavour string (i 1 j k i n ) (with j k i k in the case of mixing between neutral gauge bosons or scalars), while δ SLC (i k, j k ) is the single logarithmic radiator function corresponding to the external leg i k. The proof of the collinear Ward identities of Ref. [4 relies on the BRS symmetry of the spontaneously broken electroweak theory. Since the details of the proofs are different for external scalars, fermions and gauge bosons, they are not included in this very short overview of the algorithm. 3.2 Implementation of the algorithm in the ALPGEN generator According to the Denner-Pozzorini algorithm, electroweak corrections in the Sudakov limit can be written in a factorized form as the sum of contributions that consist of radiator functions multiplied by tree level matrix elements. The radiator functions are universal due to the universality of the infrared structure of the virtual one loop corrections, while the tree level matrix elements needed are the ones of the considered LO process and its SU(2) correlated amplitudes. The algorithm is thus well suited for the implementation in a LO event generator such as ALPGEN which provides automatically all the matrix elements needed. ALPGEN is basically a collection of process-specific packages for several hard processes interfaced to a process-independent section of code. In the following chapters only the vbjet package will be considered: it deals with the production of nw + mz + jγ + lh + k jets with n + m + j + l + k 8 and k 3. The process-dependent packages generate the phase space points together with the flavour and the helicity configurations, while the common part 15

22 3. The Denner-Pozzorini algorithm and its implementation in the ALPGEN event generator of the code uses these variables as input to compute the corresponding tree level matrix element through a call to the matrix routine (which implements the ALPHA algorithm [56, 57, 58) and takes care of the integration procedure. ALPHA is based on the Dyson-Schwinger method to recursively define one-particle off-shell Green s functions, which are computed numerically. In order to implement the Denner-Pozzorini algorithm in the ALPGEN generator, the program work-flow is left unchanged, simply the LO matrix element in the integral is multiplied by a factor Re δ EW. The computation of the correction δ EW is briefly described in the following paragraphs. The algorithm of Refs. [3, 4, 5 has been implemented in the vbjet package of the ALPGEN generator basically following eqs. (3.1), (3.3) and (3.4). First of all the double logarithmic part of the corrections is computed. For each pair of external EW charged legs the radiator contribution associated with the exchange of either a Z boson or a photon is computed as a function of the flavours and the kinematics of the two particles. For those pairs of legs k, l for which the exchange of a W boson is also allowed, the code works as follows: the original LO flavour string (i 1 i k i l i n ) is mapped in the SU(2) transformed one (i 1 j k j l i n ); the kinematics of the event is rescaled in order to put the two transformed legs k, l on their mass shell preserving the transverse and the longitudinal momenta of all the final state particles. More precisely, the kinematics is modified only if one of the two legs k, l is a final state vector boson V (V = Z, γ, W ), since the vbjet package only deals with light (massless) quarks: when m(j a ) m(i a ) (a = k, l), the energy of the leg a is rescaled according to the mass-shell condition m 2 (j a ) = E 2 (a) p(a) 2, this of course changes the total energy in the final state, so that the energy and the longitudinal momentum of the initial state particles are fixed by the four-momentum conservation. The tree level matrix element for the SU(2) transformed flavour string (i 1 j k j l i n ) is evaluated numerically by means of the ALPHA algorithm through a call to the matrix routine. While needed in order to provide on shell particles as input for the matrix routine, the transformation of the kinematics has basically negligible impact in the applications discussed in chapters 4, 5 and 6 for the event selections considered. The SL part of the correction contains both the SLC terms defined in subsection and the logarithmic part of the wave function renormalization counterterms. The SL contribution is computed in analogy with the DL one: for each EW charged external leg k, the corresponding radiator function is 16

23 3.2. Implementation of the algorithm in the ALPGEN generator computed together with the transformed tree level matrix element needed (in the case of the mixing between neutral gauge bosons or scalars) after the usual transformation of the kinematics of the event. It is worth noting that eq. (3.3) was obtained in the Sudakov limit in which all the kinematical invariants are basically the same (2p k p l r for each pair of external legs k and l), and the scale r in the argument of the single logarithms of eq. (3.3) is naturally fixed to the one of the kinematical invariants. For the realistic applications of the algorithm, however, even if the invariants should be of the same order, they are not identical, so that in the implementation of the algorithm the scale of the SL part is chosen event by event as the mean of the largest and the smallest of the invariants: clearly it is an arbitrary choice, however different scale choices lead to the same predictions up to single logarithms of the ratio of the invariants, which are not enhanced for event selections close to the Sudakov limit, as the ones considered in chapters 4, 5 and 6. While the SLC part of the correction and the logarithmic structure of the wave function renormalization counterterms share the same factorization properties and can be cast in the form (3.3), in general this is not the case for the single logarithms coming from parameter renormalization. However, in Appendix E of Ref. [5 has been shown that for processes involving the production of an arbitrary number of transverse vector bosons in fermion anti-fermion annihilation also the logarithms related to the parameter renormalization can be included in eq. (3.3): the argument of Ref. [5 can be easily extended to the processes V +N jets considered in chapters 4, 5 and 6 for the most relevant partonic subprocesses of order α N jets S α. Concerning the present implementation of the Denner-Pozzorini algorithm in the ALPGEN generator another remark is in order: the matrix elements provided by ALPHA for the processes Z + N jets and γ + N jets (N = 1, 2, 3) considered in the phenomenological studies of the following chapters contain both contributions of order α N jets S α and of order α 3 (for N > 1) which cannot be disentangled. Of course the parameter α N jets 2 S renormalization for the two classes of processes is different: in the present version of the code the contribution of the parameter renormalization is correctly implemented only for the dominant order α N jets S α subprocesses (the impact of this approximation will be discussed in subsection 4.2). As discussed in chapter 4, for the phenomenological results shown in this thesis the Sudakov corrections for the longitudinally polarized gauge bosons have been neglected. However, also these contributions can be included as described in Ref. [7. Since the tree level matrix elements provided by ALPHA are computed in the unitary gauge, the amplitudes involving external would-be Goldstone bosons are not directly available. However the Sudakov corrections for the case of longitudinally polarized gauge bosons can be obtained by using the tree level matrix elements for external longitudinal gauge bosons with the formulas of Refs. [3, 4, 5 for the radiator functions associated with external would-be Goldstone bosons, once these formulas have been modified in order to account for the phase transformation which relates the tree level amplitudes 17

24 3. The Denner-Pozzorini algorithm and its implementation in the ALPGEN event generator for the longitudinal gauge bosons and the ones for the corresponding would-be Goldstone bosons (see for example Appendix D). 18

25 Chapter 4 Virtual O(α) Sudakov corrections to New Physics searches at pp colliders 4.1 Direct search for New Physics in the channel /E T + jets One of the main tasks of the LHC proton-proton collider is the search for Physics beyond the Standard Model. Among the possible extensions of the SM, Supersymmetry (SUSY) is probably one of the most theoretically explored and appealing, since it addresses the main issues of the SM like the mass hierarchy and the naturalness problem. SUSY relates fermionic and bosonic degrees of freedom postulating for each SM particle a supersymmetric partner with the same quantum numbers but opposite spin-statistics. According to R-parity conserving SUSY models, SUSY particles should be produced in pairs and decay into SM particles plus light stable supersymmetric particles (LSP). If the LSP are weakly interacting and neutral, like the neutralino, they also provide a possible candidate for Dark Matter. According to these theories, at the LHC coloured SUSY particles (squarks and gluinos) should be produced in pairs, interact strongly producing jets, and decay into LSP which escape detection, leading to signatures with missing transverse energy /E T and several jets. The amount of /E T and the p T scale of the jets should be of hundreds of GeV, in order to fulfil the lower limits on the SUSY particles masses set by the previous results of the direct searches for New Physics (NP) performed at LEP and TEVATRON. Both the ATLAS [59, 60 and CMS [61, 62 Collaborations published their results on the direct search for NP in the channel /E T + multi-jets. These analyses require at least two or three jets at high p T and a large amount of missing momentum. Additional cuts on the total transverse momentum of the jets H T = j pj T and on the angular separation between the jet p T s and the 19

26 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders missing transverse momentum are also imposed. The SM background to the NP searches in the channel /E T + multi-jets is made up of pure QCD multi-jets events (with large missing momentum coming from leptonic decays of heavy-flavour hadrons inside the jets or jet energy mismeasurement), W + jets events (when the W boson decays leptonically and the charged lepton is not detected), Z + jets events (when the Z boson decays into neutrinos) and t t events. Among these processes, Z + multi-jets with Z νν turns out to be the most important irreducible SM background for the event selections with up to three jets. The background processes V +jets (V = W, Z) have been widely studied in the literature. Exact NLO QCD corrections to Z + 4 jets and W + 4 jets, computed by means of the package BlackHat and interfaced to the parton shower generator SHERPA can be found in Refs. [63, 43 and [64, respectively. Fixed-order (NLO) QCD predictions for the production of a vector boson in association with 5 jets at hadron colliders are presented in Ref. [65. Leading and next-to-leading logarithmic EW corrections to the processes V = γ, Z, W + 1 jet, with on-shell W and Z bosons, can be found in Refs. [9, 8, 10, where two-loop Sudakov corrections are also investigated. Very recently EW and QCD corrections to the same processes have been computed using the soft and collinear effective theory in [32. The exact NLO EW calculation for V = W, Z + 1 jet, with on-shell W, Z bosons can be found in Refs. [66, 67, 68, and the same with W, Z decays has been published in Refs. [69, 70, 71. The NLO EW calculation for Z(ν ν)+ 2 jets, for the partonic subprocesses with one fermion current only (i.e. including only gluon-gluon contributions to Z + 2 jets), has been completed and can be found in Ref. [72. In Ref. [73 one loop EW corrections for Z + 2/3 jets production in the Sudakov limit including all partonic subprocesses have been computed using the Denner- Pozzorini algorithm [3, 4, 5 implemented in the ALPGEN multiparticle event generator [44. Recently exact O(α) corrections for the process Z + 2 including also four quark subprocesses have been presented in Ref. [74. In this chapter the results of Refs. [73 and [75, 76, 77 are collected, showing the effect of one loop virtual weak corrections in the Sudakov limit for the processes Z + n jets (with n = 1, 2, 3) as SM background to the direct search for NP at the LHC and future proton-proton colliders. In the following the results for Z + 2 jets are given for a set of cuts which mimics the ATLAS event selection of Ref. [59, namely: m eff > 1 TeV /E T /m eff > 0.3 p j1 T > 130 GeV pj2 T > 40 GeV η j < 2.8 Δφ( p j T, / p T ) > 0.4 ΔR (j1,j2) > 0.4, (4.1) where the effective mass is defined as m eff = i p T i + /E T and j 1, j 2 are the hardest and next-to-hardest jet, respectively. For Z + 3 jets the CMS baseline 20

27 4.2. Technical remarks on the computation selection of Ref. [61 is considered: H T > 500 GeV / H T > 200 GeV p j T > 50 GeV η j < 2.5 ΔR (ji,j k ) > 0.5 Δφ( p j1,j2 T, / H T ) > 0.5 Δφ( p j3 T, / H T ) > 0.3, (4.2) where H T = i p T i, /HT = i p T i and j 1, j 2, j 3, are the hardest, the next-tohardest and the remaining jet, respectively. Finally, for the numerical results at s =33, 100 TeV also a rescaled version of the cuts in eqs. (4.1), (4.2) is considered: while the angular separation and the centrality requirement for the jets are kept fixed (as they are essentially determined by the geometry of the detector), the p j T and m eff (or /H T and H T ) cuts are rescaled as the centre of mass energy grows form 14 to 33 or 100 TeV. The rescaled event selection adopted for Z + 2 jets is: m eff > 2(7) TeV, ( s = 33(100) TeV), p j1 T > 260(910) GeV, ( s = 33(100) TeV), p j2 T > 80(280) GeV, ( s = 33(100) TeV), η j < 2.8, ΔR (j1,j 2) > 0.4, Δφ( p j T, / p T ) > 0.4, (4.3) while the rescaled cuts for Z + 3 jets are: H T > 1(3.5) TeV, ( s = 33(100) TeV) / H T > 0.4(1.4) TeV, ( s = 33(100) TeV) p j T > 100(350) GeV, ( s = 33(100) TeV), η j < 2.5, Δφ( p j1,j2 T, H/ T ) > 0.5, Δφ( p j3 T, H / T ) > 0.3. (4.4) 4.2 Technical remarks on the computation In this chapter the electroweak Sudakov corrections to the processes Z + 2 and Z + 3 jets are computed using the Denner-Pozzorini algorithm [3, 4, 5. The underlying hypothesis of the algorithm is that all the kinematical invariants of the electroweak charged legs are of the same order and larger than the W mass. Figure 4.1 (left plot) shows the maximum invariant mass distributions for the processes Z + 2, 3 jets at s = 7, 14 TeV, respectively, obtained by considering, on an event-by-event basis, all possible combinations of invariant masses between electroweak charged particles at the parton level: most of the events are characterized by at least one invariant mass above few hundreds GeV. For the event selections considered, the approximation of Refs. [3, 4, 5 is still expected to hold, since the radiator contributions depending on large kinematical invariants are reliable, whereas those depending on small kinematical invariants (which are at least of order M W, as shown in the plot on the right of fig. 4.1) may lead to unreliable contributions, which, however, are 21

28 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders ATLAS 7 TeV ATLAS 14 TeV CMS 7 TeV CMS 14 TeV [ pb GeV dmmax 10 5 ATLAS 7 TeV ATLAS 14 TeV CMS 7 TeV CMS 14 TeV [ pb GeV dmmin m MAX [GeV m min [GeV Figure 4.1: Maximum (left plot) and minimum (right plot) invariant mass distributions for Z + 2 and Z + 3 jets under the ATLAS and CMS cuts of eqs. (4.1)-(4.2) numerically below the stated accuracy, since the arguments of the involved logarithms are of order one. Figure 4.1 can be considered as a first assessment of the applicability of the method: a detailed description of the validation of the calculation can be found in Appendix A. Figures show the relative importance of the partonic subprocesses at the leading order. While the subprocesses with one fermion current (curve A) are the main part of the cross section, the relative importance of the subprocesses with four identical quarks (curve B) and with two pairs of identical quarks in the same isodoublet (e.g. ud udz(g), curve C) in the tails of the considered distributions is of order 45, 50%. The contribution of the other four quark subprocesses is small even if their sum is not negligible. The Denner-Pozzorini algorithm has been implemented in the vbjet.f routine: this means that the LO subprocesses considered are both of order αs 2 α and of order α 3 for Z + 2 jets (αs 3 α and α Sα 3 for Z + 3 jets). While the double and the single collinear logarithmic part of the Sudakov corrections are the same for the two classes of subprocesses, the single logarithms coming from the running of the electroweak parameters are different: in the present implementation of the algorithm it is assumed that the dominant subprocesses are of order α N jets S α. At 14 TeV the size of the purely weak LO subprocesses in the tails of the distributions can be conservatively estimated to be of order 8, 10% by comparing the left and the right plots of figs (in the plots on the right only the order α N jets S α contributions have been considered): in principle 22

29 4.2. Technical remarks on the computation s = 7 TeV A B C s = 7 TeV QCD only A B C i / LO dmeff dmeff i / LO dmeff dmeff m eff [GeV m eff [GeV Figure 4.2: Relative importance of the most relevant partonic subprocesses for Z +2 jets under the cuts of eq. (4.1) at s = 7 TeV. Curves A, B, C are the subprocesses with one fermion current (e.g. ug ugz), with four identical quarks (e.g. uu uuz) and with two pairs of identical quarks in the same isodoublet (e.g. ud udz), respectively. The plots on the right are obtained considering only the QCD LO contributions. the corrections to this part of the cross sections are overestimated, however in the tails of the distributions the double logarithmic part of the correction (which is properly taken into account also for the weak LO subprocesses) is by far the most relevant one, so that the overall result is still within the stated accuracy. For the leading order α N jets S α partonic subprocesses the Z boson can only be emitted by a quark line. As a result, the contribution of longitudinally polarized Z bosons is suppressed at high energy (fig. 4.6), where the longitudinal gauge bosons behave like the would-be Goldstone bosons according to the Goldstone boson equivalence theorem. For this reason, the Sudakov corrections to the longitudinally polarized Z bosons are not included in the computation. As a conclusive remark, on one hand QED corrections are a gauge invariant subset of the one loop electroweak corrections to Z + 2 and Z + 3 jets for the O (α N jets S α) partonic subprocesses, while on the other hand they turn out to be rather small for fully inclusive setup (see for instance the results of Refs. [10, 67). Since this chapter is mainly focused on the effect of weak corrections, following Ref. [8 the QED contribution is not included in the numerical results. 23

30 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders s = 14 TeV A B C s = 14 TeV QCD only A B C i / LO dmeff dmeff i / LO dmeff dmeff m eff [GeV m eff [GeV Figure 4.3: Relative importance of the most relevant partonic subprocesses for Z + 2 jets under the cuts of eq. (4.1) at s = 14 TeV. Same notation and conventions as in fig Phenomenological results In this section the results of Refs. [73, 75, 76 are collected for the virtual electroweak Sudakov corrections to the processes Z+2 and Z+3 jets considered as Standard Model background to the direct search for New Physics in the channel /E T + jets at hadron colliders. For the numerical results the default ALPGEN input parameters and PDFs sets have been used, namely: CTEQ6l (PDFs), G F = GeV 2, M W = GeV, M Z = GeV (electroweak input parameters), µ 2 = MZ 2 + jet p2 T jet (factorization and renormalization scale). Since no cuts are applied to the decay products of the Z, the Z boson is produced on shell. As in Refs. [73, 75, 76, in this section only a purely partonlevel analysis has been considered: this means, in particular, that the jets are represented by tree-level partons, while the missing transverse momentum corresponds to the transverse momentum of the Z boson. Figure 4.7 shows the effect of weak Sudakov corrections on the effective mass distribution (m eff = Njets j=1 p T j + /E T ) for Z + 2 jets under the ATLAS cuts of eq. (4.1). As can be seen, the size of virtual weak corrections on the total cross section is very small, while in the tails of the m eff distribution at the LHC at s = 7 TeV is of order 20, 25% and it grows up to 40, 45% at s = 14 TeV. 24

31 4.3. Phenomenological results s = 7 TeV A B C s = 7 TeV QCD only A B C / LO d /HT 0.6 / LO d /HT 0.6 i d /HT 0.4 i d /HT /H T [GeV /H T [GeV Figure 4.4: Relative importance of the most relevant partonic subprocesses for Z + 3 jets under the cuts of eq. (4.2) at s = 7 TeV. Same notation and conventions as in fig Figure 4.8 shows the results for Z + 3 jets under the CMS cuts of eq. (4.2). As in the case of Z + 2 jets, the size of virtual weak corrections in the tails of the /H T distribution (which is the most interesting region for the direct NP searches) at s = 7 and 14 TeV is of order 25% and 45%, respectively. Recently, several proposals have been presented for future proton-proton colliders that will operate after the second run and the high luminosity run of the LHC (HL-LHC). These colliders will work at a s that goes from 33 TeV for the high energy LHC (HE-LHC) [78 to 100 TeV for the pp mode of TLEP [79. The main tasks of these future hadron colliders (hh-fccs) will be the study of the Higgs self-interaction and, if no NP signals will appear in the next run of the LHC, the direct search for New Physics. In this latter scenario it seems natural that the event selections used for the NP searches will be modified in order to look at higher NP mass scales. For this reason, in figs. 4.9 and 4.10 the selection cuts of eqs. (4.3)-(4.4) have been considered: while the acceptance and the angular separation cuts are the same of eqs. (4.1)- (4.2) the p T, m eff, H T and /H T cuts have been rescaled by a factor two at s = 33 TeV and by a factor seven at 100 TeV. Figure 4.9 shows the effect of the virtual weak Sudakov corrections to Z + 2 jets for the effective mass distribution under the selection cuts of eq. (4.3). While the size of the Sudakov corrections in the tail of the distribution was of order 45% at the LHC at s = 14 TeV, at future colliders it grows up to 70 and 85% at 33 and 100 TeV, respectively. The impact of the Sudakov corrections to Z + 3 jets for the /H T distributions under the cuts of eq. (4.4) is shown in fig. 4.10: again the corrections in the tails of the distributions grow as the centre of mass energy increases and reach the value of 60% at 33 TeV 25

32 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders s = 14 TeV A B C s = 14 TeV QCD only A B C / LO d /HT 0.6 / LO d /HT 0.6 i d /HT 0.4 i d /HT /H T [GeV /H T [GeV Figure 4.5: Relative importance of the most relevant partonic subprocesses for Z + 3 jets under the cuts of eq. (4.2) at s = 14 TeV. Same notation and conventions as in fig and 80% at 100 TeV. In principle, at 100 TeV even larger values of m eff and /H T can be reached, however the PDFs extrapolation in these regions becomes rather uncertain, so that in the present analysis 20 TeV and 8 TeV have been chosen as maximum values of m eff and /H T, respectively. Even if the Sudakov corrections in the tails of the distributions increase with s, it should be noticed that the correction itself is almost independent of the collider energy, i.e. for a given bin of the m eff or /H T distribution the correction remains essentially the same at 14, 33 or 100 TeV, as shown in figs where the same selection cuts have been applied for the three energy setup. The correction is also quite independent of the details of the event selection, as can be seen by comparing figs with figs and with figs In the tails of the distributions the size of the Sudakov corrections grows with the collider energy moving from 14 to 33 and 100 TeV simply because as the centre of mass energy increases, more and more extreme kinematical configurations are involved and for these configurations the corrections are expected to be very large. This behaviour has been found for other processes studied in Refs. [75, 76, such as di-jet, di-boson and inclusive single vector boson production. In conclusion, in Refs. [73, 75, 76 the one loop weak Sudakov corrections have been computed to the process Z + n jets (with n 3), which is an irreducible Standard Model background to the direct search for New Physics at the LHC in the signatures with multi-jets and missing transverse momentum. At the LHC the effect of virtual weak corrections in the event selections considered is large and it becomes even larger at future proton-proton colliders (where the higher energy allows to look at more and more extreme kinematical regions). 26

33 4.3. Phenomenological results 10 4 total LO long. Z 10 3 total LO long. Z [ pb GeV dmeff Z + 2j ATLAS 7 TeV [ pb GeV dmeff Z + 2j ATLAS 14 TeV m eff [GeV total LO long. Z m eff [GeV total LO long. Z [ pb GeV d /HT Z + 3j CMS 7 TeV /H T [GeV [ pb GeV d /HT Z + 3j CMS 14 TeV /H T [GeV Figure 4.6: LO m eff ( / H T ) distributions for Z + 2 jets (Z + 3 jets) at s = 7, 14 TeV compared to the purely longitudinal Z contributions. With such large negative effects, also the possible compensation of real heavy gauge boson radiation and the higher-order electroweak contributions (beyond one-loop) require further investigation. 27

34 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders [ pb GeV LO NLO Virt. dmeff 10 6 s = 7 TeV δ Virt m eff [GeV [ pb GeV dmeff s = 14 TeV LO NLO Virt δ Virt m eff [GeV Figure 4.7: Weak corrections to Z +2 jets in the ATLAS setup of eq. (4.1) at the LHC centre of mass energies of 7 and 14 TeV. The upper panels show the effective mass distribution at LO (solid blue line) and at NLO including only virtual one loop weak corrections (dotted red line). The lower panels show the relative effect (δ EW = NLO LO ) of virtual weak corrections (dotted red LO line). 28

35 4.3. Phenomenological results [ pb GeV LO NLO Virt. d /HT 10 6 s = 7 TeV δ Virt /H T [GeV [ pb GeV d /HT s = 14 TeV LO NLO Virt δ Virt /H T [GeV Figure 4.8: Weak corrections to Z + 3 jets in the CMS setup of eq. (4.2) at the LHC centre of mass energies of 7 and 14 TeV, with the same notation and conventions as in fig The change in the slope of the distributions at /H T around 500 GeV is an effect of the H T cut of eq. (4.2) 29

36 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders [ nb TeV dmeff Z + 2j s = 33 TeV LO NLO Virt. δ Virt [ nb TeV dmeff Z + 2j s = 100 TeV m eff [TeV LO NLO Virt δ Virt m eff [TeV Figure 4.9: Weak corrections to Z + 2 jets in the rescaled ATLAS setup of eq. (4.3) at the hh-fcc energies of 33 and 100 TeV, with the same notation and conventions as in fig

37 4.3. Phenomenological results LO NLO Virt. [ nb TeV d /HT Z + 3j s = 33 TeV δ Virt /H T [TeV [ nb TeV d /HT Z + 3j s = 100 TeV LO NLO Virt. δ Virt /H T [TeV Figure 4.10: Weak corrections to Z + 3 jets in the rescaled CMS setup of eq. (4.4) at the hh-fcc energies of 33 and 100 TeV, with the same notation and conventions as in fig

38 4. Virtual O(α) Sudakov corrections to New Physics searches at pp colliders [ nb TeV TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt. dmeff Z + 2j TeV δ Virt. 33 TeV δ Virt. 100 TeV δ Virt m eff [TeV Figure 4.11: Comparison between the Sudakov corrections to Z + 2 jets at the LHC s = 14 TeV (blue curve) and at FCC energies of 33 (green curve) and 100 TeV (red curve) for the m eff distribution under the cuts of eq. (4.1). In the lower panel are shown the relative electroweak Sudakov corrections defined in fig. 4.7: the corrections for the three energy setup overlap. [ nb TeV d /HT TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt Z + 3j TeV δ Virt. 33 TeV δ Virt. 100 TeV δ Virt /H T [TeV Figure 4.12: Same plot as in fig but for Z + 3 jets under the cuts of eq. (4.2). As in fig. 4.11, the corrections for the three energy setup are basically the same. 32

39 Chapter 5 Real weak corrections to Z + 2 and Z + 3 jets production 5.1 Introduction: real weak corrections In the computation of one loop virtual electroweak corrections, the diagrams involving virtual photons can lead to infrared and/or collinear singularities that have to be regularized either by dimensional regularization or by introducing an unphysical mass parameter for the photon. This way virtual O(α) corrections become functions of the arbitrary infrared regulators, and the dependence on the infrared regulators only cancels when also real photonic corrections are included (i.e. all the diagrams obtained from the LO ones with the emission of an additional real photon). The extra emission of additional W and Z bosons usually is not included in the computation of real O(α) corrections for two reasons: first of all the gauge bosons decay, so that in principle they lead to final states which are not degenerate with the LO ones; the second reason is that real purely weak corrections are always finite, in the sense that they do not depend on unphysical parameters (in fact, even in the Sudakov limit, when the corrections are dominated by the Sudakov logarithms which are the IR limit of the weak corrections, the gauge boson masses act as physical IR regulators). However, in those regions in which the Sudakov corrections become very large ( 50% or larger), the effect of the partially compensating radiation of real gauge bosons may lead to significant positive contributions. The issue of real weak boson emission has been addressed by several authors. For example, Refs. [38, 39, 27, 26, 24, 23, 40, 41 studied the effect of real weak corrections in analogy with QED or QCD: all the diagrams obtained from the LO ones with the emission of an additional W or Z boson are considered as real corrections, the additional gauge boson is produced on-shell and integrated over the full phase space. The result is a significant cancellation between real and virtual corrections, which however may be incomplete due to the incomplete average on the isospin of the initial state particles (this effect 33

40 5. Real weak corrections to Z + 2 and Z + 3 jets production ZW ( ν l ν l jj) + jj ZZ( ν l ν l jj) + jj W W ( ν l ljj) + jj ZW ( ν l ν l ν l l) + jj ZW ( ν l lll) + jj ZZ( ν l ν l ll) + jj ZZ( ν l ν l ν l ν l ) + jj W W ( ν l ν l ll) + jj ZW ( ν l ljj) + jj ZW ( ν l ν l jj) ZW ( ν l ljj) ZZ( ν l ν l jj) W W ( ν l ljj) ZW ( ν l ljj) + j ZW ( ν l ν l jj) + j ZZ( ν l ν l jj) + j W W ( ν l ljj) + j Table 5.1: Vector boson radiation processes contributing to the considered signatures. In brackets vector boson decay channels are specified, while outside the brackets j stands for a matrix element QCD parton. The above processes are for the Z + 2 jet final state, whereas for three jet final states the processes are the same ones plus an additional QCD parton. is known as Bloch Nordsieck violation [26, 24, 23). For example, in e + e collisions the cancellation would require also the processes with e + ν e, e ν e and ν e ν e as initial states, while in pp collisions the cancellation would take place if u and d type quarks were weighted by the same PDFs. The approach of Refs. [42, 40, 43 is more phenomenological: the additional gauge bosons decay and are included in the real corrections only when the final states are degenerate with the LO ones. Again the cancellation between real and virtual weak corrections is only partial and moreover it is strongly dependent on the event selection considered. Following the approach of Refs. [42, 40, 43, in Ref. [73 a first estimate of the impact of real weak corrections to the processes Z + 2 and Z + 3 jets with the event selection of eqs. (4.1) and (4.2) has been computed. In the following sections the results of Ref. [73 are collected. 5.2 Real weak corrections to Z+2 and Z+3 jets production: a parton level analysis In Ref. [73 any contribution to the experimental event selection of O(α 2 α n S ) with n 2 for Z +2 jets (n 3 for Z +3 jets) has been considered as real weak radiation. From a purely perturbative point of view, only the processes with n = 2 (for Z + 2 jets, n = 3 for Z + 3 jets) should be considered as real O(α) corrections (upper panel of Table 5.1), however also the processes in the lower panel of Table 5.1 contribute to the same experimental signature and moreover they are the most relevant ones among the real EW radiation contributions. In Ref. [73 jets coming from vector bosons decay are distinguished from the other jets (called matrix element jets) and the latter are always required within the acceptance cuts in order to avoid infrared QCD singularities: this can be considered as a LO prediction of the real contributions, which provides at least a first estimate of the effect of real weak corrections to the considered processes. For the real weak corrections to Z + 2 jets the event selection of eq. (4.1) is 34

41 5.2. Real weak corrections to Z + 2 and Z + 3 jets production: a parton level analysis considered. While for the computation of virtual corrections of chapter 4 there were exactly two matrix element jets generated with p T > 40 GeV, η < 2.8 and ΔR separation larger than 0.4, this is not the case for the real corrections when one of the two gauge bosons decays into quarks. As a first step, all the final state partons closer than 0.4 in ΔR are combined into a single jet. Then the jets and the final state charged leptons (if present) are divided into two categories: the tagged jets (or charged leptons) and the untagged ones, where the tagged jets are defined by the requirement p j T > 40 GeV and η < 2.8, while the charged leptons are tagged if p l T > 20 GeV and ηl < 2.4. Finally, in order to mimic in a simplified way the event reconstruction procedure of Ref. [59, in the case of overlap between tagged jets and tagged charged leptons, if a tagged jets and a tagged lepton are closer than ΔR = 0.2 the jet is removed from the tagged ones and the event is discarded if it does not contain exactly two tagged jets. At the very end, if a tagged charged lepton is closer to a tagged jet than ΔR = 0.4, the lepton is removed from the tagged ones and the event is discarded if any tagged charged lepton survives. The effective mass is then computed as the sum of the p T s of the tagged jets together with the missing momentum (neutrinos plus untagged leptons and untagged jets with η < 4.5). For the real weak corrections to Z +3 jets, the event selection of eq. (4.2) is considered. In Ref. [73 a much more simplified event reconstruction procedure has been considered: as a first step, charged leptons and final state partons closer than ΔR = 0.2 are merged into a single jet and if any charged lepton with p l T > 10 GeV and ηl < 2.5 survives the event is discarded, then the jets are merged together if ΔR jj 0.5 and the event is considered only if contains exactly three jets within the cuts of eq. (4.2). After this selection, /H T is defined as minus the sum of the p T s of these three jets. Figures and show the effect of the real weak corrections to Z + 2 jets (under the cuts of eq. (4.1)) and Z + 3 jets (under the cuts of eq. (4.2)), respectively, when the processes of Tab. 5.1 are considered as real corrections and the additional experimental-like cuts described in the previous two paragraphs are imposed. As can be seen, the requirement that the final states of the real correction processes are degenerate with the signal leads to rather small real contributions (ranging from 5% in the tail of the m eff distribution at the LHC s of 7 TeV up to 15% at the FCC centre of mass energy of 100 TeV for Z + 2 jets, while the real corrections to Z + 3 jets in the tails of the /H T distributions range from 6% to 20% moving from 7 to 100 TeV), so that the cancellation between real and virtual weak corrections is quite small for all the considered energy setup ( s = 7, 14, 33 and 100 TeV). Following the approach of Refs. [42, 40, 43, the contribution of real weak corrections turns out to be small once realistic selection cuts are imposed on the decay products of the vector bosons. This picture changes completely if a more theoretical or QED-like point of view as the one of Refs. [38, 39, 27, 26, 24, 23, 40, 41 is considered: that is, if real corrections to Z +n jets are defined 35

42 5. Real weak corrections to Z + 2 and Z + 3 jets production as the processes Z(+V ) + n jets, where the additional gauge boson (V = W, Z) is integrated over the full phase space and it is considered as not detected (in analogy with the usual treatment of real photon radiation in the standard computation of one loop electroweak corrections). With this definition of the real contributions only matrix element jets (generated within the acceptance) are involved and the missing momentum is computed as minus the sum of the p T s of the jets. Figures and show the effect of real weak corrections to Z + 2 and Z + 3 jets, respectively, according to this latter point of view: as can be seen, if no cuts are imposed on the additional gauge bosons, real weak corrections in the tails of the distributions become as large as the virtual corrections for the LHC centre of mass energies of 7 and 14 TeV, while at the FCC energies of 33 and 100 TeV real corrections turn out to be larger than the virtual ones, leading to an overcompensation of the virtual Sudakov corrections already found in Ref. [ Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework When the real weak corrections to Z + n jets (n = 2, 3) are defined as in Tab. 5.1, the contribution of those subprocesses in which the additional gauge boson decays in a quark- antiquark pair can lead to QCD infrared singularities. Considering as an example the process ZW +3 jets with Z νν and W qq, that is a real correction to Z + 3 jets, if the W boson is boosted enough its decay products will likely be merged into a single parton level jet: if this jet is tagged, then one of the three ME partons can become unresolved, thus leading to QCD infrared divergences. Of course, in section 5.2, the kind of behaviour described above is avoided by means of the generation cut imposed on the p T of the ME partons, however some criticism may arise concerning the IR safety of the separation between ME jets and jets coming from vector boson decays. One possible way to overcome this point is to generate the ZV + n jets samples (with V = Z, W, n = 0, 1, 2 and n = 1, 2, 3 for the real weak corrections to Z+2 and Z+3 jets, respectively) through the CKKW procedure [80, 81 implemented in the ALPGEN generator in the MLM framework [82. At variance with the parton level analysis of section 5.2, this requires to generate the event samples ZV + n partons that are passed to a shower Monte Carlo: after the showering and hadronization process, each of the samples can lead to an arbitrary number of jets and the proper matching condition of the samples (which avoids possible double counting) is obtained by the requirement that the hardest jets correspond to the ME partons of the original event sample considered. The sum of the showered and matched results for ZV + n partons samples gives the prediction for ZV + jets without sharp generation cuts on the p T of the jets. 36

43 5.3. Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework In this section, the process Z(νν)W (qq )+ jets is considered as a case of study for the real weak corrections to Z +3 jets in the CKKW framework: this process is the most relevant among the ones in Tab. 5.1 and allows to investigate the potential IR dependence of the parton level analysis of section 5.2, as pointed out in the previous two paragraphs. In order to study the relative effect of the real weak corrections corresponding to the Z(νν)W (qq )+ jets events, also the LO prediction for Z +3 jets have been computed in the CKKW framework from the event samples for Z + n partons (n 4). In figure 5.9 the results of the parton level analysis of section 5.2 are compared to the ones of the hadron level analysis obtained in the CKKW framework for the LHC centre of mass energy of 14 TeV. For the latter results, an event selection as close as possible to the experimental one has been considered, namely: in addition to the angular separation cuts of eq. (4.2), H T is defined as the sum of the tagged jets p T s (p j T > 50 GeV, ηj < 2.5), /H T is minus the vector sum of the transverse momenta of the jets with p T > 30 GeV and η < 5, the events containing any charged lepton with p l T > 10 GeV and η l < 2.4 are discarded (in fig. 5.9 these leptons can only come from hadron decays), jets are reconstructed by means of the anti-k t clustering algorithm [83 (implemented in the FASTJET code [84, 85) with a distance parameter of 0.5 and the number of tagged jets is required to be equal or larger than three. The results of the two analyses are not expected to be exactly the same, since they are based on quite different theoretical ingredients, however the results basically agree. Of course, this is still a preliminary result, as all the classes of processes in Tab. 5.1 should be computed in the CKKW framework, however there is no particular reason to believe that the results will change, since all of them share the same kind of potential QCD IR sensitivity. As a conclusive remark, also the virtual weak corrections to Z + 3 jets could be computed in the CKKW framework, generating the event samples for Z + n partons (n 3) according to the approximated NLO O(α) normalization. However, as can be seen from fig. 5.10, once the cuts on the p Z T and on the angular separation between the Z and the final state partons are imposed, the Sudakov corrections for the three samples are basically the same regardless of the parton multiplicities (and also of the details of the event selection considered). As a consequence, the results of chapter 4 for the virtual weak Sudakov corrections to the processes Z + 2 and Z + 3 jets will still hold for the analysis at the level of fully showered, matched and hadronized events. 37

44 5. Real weak corrections to Z + 2 and Z + 3 jets production [ pb GeV dmeff Z + 2j s = 7 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real m eff [GeV [ pb GeV LO NLO Virt. NLO Real dmeff Z + 2j s = 14 TeV δ Virt. δ Virt. + δ Real m eff [GeV Figure 5.1: Upper panels: m eff distribution for Z + 2 jets under the event selection of eq. (4.1) at LO (solid blue line), at NLO including only virtual Sudakov corrections (red dotted line) and at NLO including only the real corrections (green dash-dotted line) at the LHC centre of mass energy of 7 and 14 TeV. Lower panels: relative effect (δ EW = NLO LO ) of the virtual (red LO dotted line) and virtual+real corrections (green dash-dotted line); the size of real weak corrections is the difference between the two curves. The processes considered as real corrections are defined in Tab. 5.1 under the realistic event selection described in the text. 38

45 5.3. Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework [ nb TeV dmeff Z + 2j s = 33 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real m eff [TeV [ nb TeV dmeff Z + 2j s = 100 TeV LO NLO Virt. NLO Real δ Virt. δ Virt.+ δ Real m eff [TeV Figure 5.2: Virtual and real corrections to Z + 2 jets for the m eff observable at the FCC s of 33 and 100 TeV under the cuts of eq. (4.1) with the same notation and conventions as in fig The processes considered as real corrections are defined in Tab. 5.1 under the realistic event selection described in the text. 39

46 5. Real weak corrections to Z + 2 and Z + 3 jets production [ pb GeV Z + 3j LO NLO Virt. NLO Real d /HT 10 6 s = 7 TeV δ Virt. δ Virt. + δ Real /H T [GeV [ pb GeV LO NLO Virt. NLO Real d /HT Z + 3j s = 14 TeV δ Virt. δ Virt. + δ Real /H T [GeV Figure 5.3: Virtual and real corrections to Z + 3 jets for the /H T observable for the LHC centre of mass energies of 7 and 14 TeV under the cuts of eq. (4.2) with the same notation and conventions as in fig The processes considered as real corrections are defined in Tab. 5.1 (with an additional matrix element jet) under the realistic event selection described in the text. 40

47 5.3. Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework [ nb TeV d /HT Z + 3j s=33 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real [ nb TeV d /HT Z + 3j s = 100 TeV /H T [TeV LO NLO Virt. NLO Real δ Virt. δ Virt.+ δ Real /H T [TeV Figure 5.4: Virtual and real corrections to Z + 3 jets for the /H T observable for s = 33 and 100 TeV under the cuts of eq. (4.2) with the same notation and conventions as in fig The processes considered as real corrections are defined in Tab. 5.1 (with an additional matrix element jet) under the realistic event selection described in the text. 41

48 5. Real weak corrections to Z + 2 and Z + 3 jets production [ pb GeV Z + 2j LO NLO Virt. NLO Real dmeff 10 6 s = 7 TeV δ Virt. δ Virt. + δ Real m eff [GeV [ pb GeV dmeff Z + 2j s = 14 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real m eff [GeV Figure 5.5: Comparison between virtual and real weak corrections to Z +2 jets for the m eff observable when real corrections are defined as the contribution of the processes Z + (V ) + 2 jets (V = W, Z) and no cuts are imposed on the additional gauge boson V. For all the energy setup ( s = 7, 14 TeV) the event selection of eq. (4.1) has been considered. Same notation and conventions as in fig

49 5.3. Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework [ nb TeV dmeff Z + 2j s = 33 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real m eff [TeV [ nb TeV dmeff Z + 2j s = 100 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real m eff [TeV Figure 5.6: Comparison between virtual and real weak corrections to Z +2 jets for the m eff observable when real corrections are defined as the contribution of the processes Z + (V ) + 2 jets (V = W, Z) and no cuts are imposed on the additional gauge boson V. For all the energy setup ( s = 33, 100 TeV) the event selection of eq. (4.1) has been considered. Same notation and conventions as in fig

50 5. Real weak corrections to Z + 2 and Z + 3 jets production [ pb GeV Z + 3j LO NLO Virt. NLO Real d /HT 10 6 s = 7 TeV δ Virt. δ Virt. + δ Real /H T [GeV [ pb GeV LO NLO Virt. NLO Real d /HT Z + 3j s = 14 TeV δ Virt. δ Virt. + δ Real /H T [GeV Figure 5.7: Comparison between virtual and real weak corrections to Z +3 jets for the /H T observable when real corrections are defined as the contribution of the processes Z + (V ) + 3 jets (V = W, Z) and no cuts are imposed on the additional gauge boson V. For all the energy setup ( s = 7, 14 TeV) the event selection of eq. (4.2) has been considered. Same notation and conventions as in fig

51 5.3. Preliminary results for the real weak corrections to Z + 3 jets in the CKKW framework [ nb TeV d /HT Z + 3j s = 33 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real /H T [TeV [ nb TeV d /HT Z + 3j s = 100 TeV LO NLO Virt. NLO Real δ Virt. δ Virt. + δ Real /H T [TeV Figure 5.8: Comparison between virtual and real weak corrections to Z +3 jets for the /H T observable when real corrections are defined as the contribution of the processes Z + (V ) + 3 jets (V = W, Z) and no cuts are imposed on the additional gauge boson V. For all the energy setup ( s = 33, 100 TeV) the event selection of eq. (4.2) has been considered. Same notation and conventions as in fig

52 5. Real weak corrections to Z + 2 and Z + 3 jets production 0.09 δ Real part. δ Real CKKW δ Real ZW + jets s = 14 TeV /H T [GeV Figure 5.9: Relative contribution Real of the real weak corrections related to LO the process Z(νν)W (qq )+ jets at the LHC at 14 TeV. δpart. Real is the correction corresponding to the sum of the results of the parton level analysis of section 5.2 for the subprocesses Z(νν)W (qq ) + n partons (n = 1, 2, 3), while δckkw Real is the correction obtained in the CKKW framework Z + 1j Z + 2j ATLAS Z + 3j CMS δ Virt. EW s = 14 TeV p Z T [TeV Figure 5.10: Relative effect of the virtual weak Sudakov corrections to Z + 1 (solid black line), Z + 2 (dotted blue line) and Z + 3 jets (dash-dotted green line). For Z + 1 jet no cuts are imposed on the final state particles, while for Z + 2 and Z + 3 jets the event selection of eq. (4.1) and (4.2) have been considered, respectively. 46

53 Chapter 6 Electroweak Sudakov corrections to the R n γ ratio 6.1 Invisible Z+ jets background to New Physics searches in the channel /E T + jets The Standard Model process Z + n jets, with Z νν, is an irreducible SM background to the direct search for New Physics (NP) in pp collisions in the channel /E T + jets. In chapter 4 the effect of the one loop virtual weak Sudakov corrections to the processes Z + 2 and Z + 3 jets has been studied considering two different sets of observables and cuts which mimic the ATLAS and CMS event selections of Refs. [59, 60 and [61, 62, respectively. For the relevant observables, the corrections have been found to be very large in the most interesting regions for the direct NP searches (of order 50% at the LHC at 14 TeV and even larger at future pp colliders). Because New Physics signals could manifest themselves as a mild deviation with respect to the large SM background, precise theoretical predictions for the processes under consideration are needed. In order to reduce the theoretical systematic uncertainties, the experimental procedure for the irreducible background determination relies on data driven methods. For instance, the most conceptually straightforward estimate of the cross section for Z( ν ν) + n jets could be done through the measured cross section for Z( l + l )+n jets times ratio, which is known with high precision from LEP1 data. With the exception of the ratio of branching ratios, the method is free of theoretical systematics. However, due to the low production rate of Z( l + l )+n jets, in particular in the signal regions, this method results to be affected by large statistical uncertainty. Other possible choices of reference processes are W +n jets and γ + n jets (Refs. [86, 87, 88, 89, 62). In both cases the statistics is not a the B(Z ν ν) B(Z l + l ) 47

54 6. Electroweak Sudakov corrections to the R n γ ratio limitation and the required theoretical input is the ratio [ [ (Z( ν ν) + n jets) (V + n jets) RV n = /, (6.1) dx dx where X is the observable under consideration and V = W, γ. For each of the reference processes, additional sources of uncertainties are related to the misidentification of either the W or the prompt γ, such as t t background (for the case of W + jets) or the photon isolation procedure (for γ+ jets). The estimate of the invisible Z+ jets background to the direct NP searches in the channel /E T + jets is obtained from the measured cross section for the process γ+ jets corrected by the Rγ n ratio. In particular, the photon is required to be detected, but it is considered as invisible: the photon p T is added to the total /E T and the selection cuts are only applied to the resulting missing momentum. The same strategy is applied to the W + jets control sample, from which an independent estimate of the Z+ jets background is obtained. Also the Z( µ + µ )+ jets sample is used in order to quantify the error on the estimate of the Z( νν)+ jets background, from the analysis of the double ratio R TH /R EXP as a function of the jet multiplicities, where R TH and R EXP are the Monte Carlo prediction and the measured value of the ratio σ(z µ + µ )/σ(γ). Recent theoretical work has been devoted to the study of the theoretical uncertainties related to the ratio in eq. (6.1). In particular, in Ref. [90 a study of the cancellation in RW n of systematic theoretical uncertainties originating from higher-order QCD corrections, including scale variations and choice of PDFs, has been presented. A first detailed analysis of the impact of higherorder QCD corrections, PDFs and scale choices to Rγ 1,2,3 has been shown in Ref. [91. More recently, the level of theoretical uncertainty induced by QCD higher-order corrections in the knowledge of Rγ 2 and Rγ, 3 relying on the comparison of full NLO QCD calculations with parton shower simulations matched to LO matrix elements, has been discussed in Refs. [92 and [43, respectively. All these studies point out that many theoretical systematics related to pqcd and PDFs largely cancel in the ratio and the corresponding theoretical uncertainty in the Rγ n ratio can be safely estimated to be within 10%. What is not expected to cancel in the ratio is the contribution of higherorder electroweak corrections, which are different for the processes Z + n jets and γ +n jets and can be enhanced by Sudakov logarithms in the typical signal regions for the NP searches. For Rγ 1 the higher-order EW corrections have been calculated in Refs. [9, 41, where it is shown that for a vector boson transverse momentum of 2 TeV they are of the order of 20%, while the EW corrections to RW 1 have been computed in Ref. [67 and turn out to be moderate (of the order of 8% for p W T = 2 TeV). In view of the forthcoming run II of the LHC, it is therefore necessary to quantitatively estimate the effects of EW corrections to Rγ, n in particular for n 2. In this chapter the results of Ref. [93 for the one loop virtual weak 48

55 6.2. Virtual O(α) Sudakov corrections to the R n γ ratio corrections to the Rγ n ratio with n 3 at the LHC at 14 TeV are collected. The effect of EW Sudakov corrections at future pp colliders is also considered. 6.2 Virtual O(α) Sudakov corrections to the R n γ ratio The one loop virtual weak corrections to the Rγ n ratio are obtained from the virtual O(α) Sudakov corrections to the processes Z + n jets and γ + n jets as [ Rγ n NLO = Rγ n LO 1 + δ EW (Z) δ EW (γ), (6.2) where δ EW (V ) = NLO (V + n jets)/dx LO (V + n jets)/dx 1, V = Z, γ, (6.3) and X is the observable under consideration. All the remarks of section 4.2 are still valid for the computation of the Sudakov corrections to γ + n jets (n 3). As in chapter 4, for the results at the LHC at 14 TeV the event selection of eqs. (4.1) and (4.2) is considered for V + 2 jets and V + 3 jets (V = Z, γ), respectively, while for the hh- FCC energies of 33 and 100 TeV the rescaled cuts of eq. (4.3) are imposed for V + 2 jets (eq. (4.4) for V + 3 jets). For n = 1 no cut on the two final state particles is imposed. As described in the previous section, for the process γ +n jets the photon p T is considered as missing momentum (and actually it is equal to the total /E T in the present parton level analysis). With respect to the calculation of chapter 4, the only difference is the choice of the renormalization and factorization scale which has been defined as µ 2 = jet p2 T jet for both the Z + n jets and the γ + n jets processes, however, the effect of this different scale choice is negligible in the m eff and /H T regions of interest. In figs. 6.1, 6.2 and 6.3 the virtual O(α) corrections to γ + 1, γ + 2 and γ + 3 jets are compared to the corrections for Z + 1, Z + 2 and Z + 3 jets for s = 14, 33 and 100 TeV. Because of the different electroweak coupling of the Z and the γ, the Sudakov corrections are larger for the processes Z + n jets (they are almost twice as big as the corrections for γ + n jets). Figures show the scaling of the corrections to γ + n jets (n = 1, 2 and 3) with the collider energy when the same event selection is considered for the three different values of s. As in the case of Z + n jets, the Sudakov corrections to γ + n jets basically do not depend on the collider energy. Since the corrections to Z + n jets and γ + n jets are different, their effect does not cancel in the Rγ n ratio, as can be seen from figs. 6.6, 6.7 and 6.8, where the Sudakov corrections to Rγ, 1 Rγ 2 and Rγ 3 are shown for the three values of s of 14, 33 and 100 TeV. The size of the O(α) virtual Sudakov corrections to the Rγ n ratio in the tails of the distributions is of order 20% at the LHC at 14 TeV, and becomes of order 40% at the FCC energy of 100 TeV. Even more important: while the ratio tends to be flat at the LO in the high H T 49

56 6. Electroweak Sudakov corrections to the R n γ ratio regions, the Sudakov corrections modify the shape of the R n γ distributions and thus should be taken into account in the extrapolation of the Z( νν)+ jets cross section from the measured cross section for γ+ jets events. 50

57 6.2. Virtual O(α) Sudakov corrections to the R n γ ratio [ nb TeV dp j T [ nb TeV dp j T [ nb TeV dp j T Z + 1j s = 14 Tev Z + 1j LO NLO Virt p j T [TeV s = 33 TeV δ Virt. LO NLO Virt Z + 1j s = 100 TeV p j T [TeV δ Virt. LO NLO Virt p j T [TeV δ Virt. [ nb TeV dp j T [ nb TeV dp j T [ nb TeV dp j T γ + 1j γ + 1j s = 14 TeV γ + 1j LO NLO Virt p j T [TeV s = 33 TeV δ Virt. LO NLO Virt s = 100 TeV p j T [TeV δ Virt. LO NLO Virt p j T [TeV δ Virt. Figure 6.1: Upper panels: jet p T distribution at LO (solid line) and at NLO in the high energy limit (dotted line), for the processes Z + 1 jet and γ + 1 jet at s = 14, 33 and 100 TeV. No cuts are imposed on the final state particles. Lower panels: relative effect of the O(α) Sudakov corrections as defined in eq. (6.3). 51

58 6. Electroweak Sudakov corrections to the R n γ ratio [ nb TeV dmeff [ nb TeV dmeff [ nb TeV dmeff Z + 2j s = 14 TeV LO NLO Virt. δ Virt m eff [TeV Z + 2j s = 33 TeV LO NLO Virt. δ Virt. [ nb TeV m eff [TeV Z + 2j s = 100 TeV LO NLO Virt. δ Virt m eff [TeV dmeff [ nb TeV dmeff [ nb TeV dmeff γ + 2j s = 14 TeV LO NLO Virt. δ Virt m eff [TeV γ + 2j s = 33 TeV LO NLO Virt. δ Virt m eff [TeV γ + 2j s = 100 TeV LO NLO Virt. δ Virt m eff [TeV Figure 6.2: Upper panels: m eff distribution at LO (solid line) and at NLO in the high energy limit (dotted line), for the processes Z + 2 jet and γ + 2 jet at s = 14 TeV (with the event selection of eq. (4.1)), and at s = 33 and 100 TeV (with the event selection of eq. (4.3)). Lower panels: relative effect of the O(α) Sudakov corrections as defined in eq. (6.3). 52

59 6.2. Virtual O(α) Sudakov corrections to the R n γ ratio [ nb TeV d /HT [ nb TeV d /HT [ nb TeV d /HT Z + 3j s = 14 TeV LO NLO Virt. δ Virt /H T [TeV Z + 3j s = 33 TeV LO NLO Virt. δ Virt /H T [TeV Z + 3j s = 100 TeV LO NLO Virt. δ Virt /H T [TeV [ nb TeV d /HT [ nb TeV d /HT [ nb TeV d /HT γ + 3j s = 14 TeV LO NLO Virt. δ Virt /H T [TeV γ + 3j s = 33 TeV LO NLO Virt. δ Virt /H T [TeV γ + 3j s = 100 TeV LO NLO Virt. δ Virt /H T [TeV Figure 6.3: Upper panels: /H T distribution at LO (solid line) and at NLO in the high energy limit (dotted line), for the processes Z + 3 jet and γ + 3 jet at s = 14 TeV (with the event selection of eq. (4.2)), and at s = 33 and 100 TeV (with the event selection of eq. (4.4)). Lower panels: relative effect of the O(α) Sudakov corrections as defined in eq. (6.3). 53

60 6. Electroweak Sudakov corrections to the R n γ ratio [ nb TeV dp j T Z + 1j 14 TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt TeV δ Virt. 33 TeV δ Virt. 100 TeV δ Virt. [ nb TeV dp j T p j T [TeV 14 TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt γ + 1j p j T [TeV 14 TeV δ Virt. 33 TeV δ Virt. 100 TeV δ Virt. Figure 6.4: Upper panels: scaling of the jet p T distributions for Z + 1 jet and γ + 1 jet with the collider energy. Blue, green and red curves are the distributions at s = 14, 33 and 100 TeV, respectively. The effect of the O(α) virtual Sudakov corrections is shown in the lower panels: for the three energy setup the corrections overlap. 54

61 6.2. Virtual O(α) Sudakov corrections to the R n γ ratio [ nb TeV dmeff γ + 2j 14 TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt TeV δ Virt. 33 TeV δ Virt. 100 TeV δ Virt m eff [TeV [ nb TeV TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt. d /HT γ + 3j TeV δ Virt. 33 TeV δ Virt. 100 TeV δ Virt /H T [TeV Figure 6.5: Upper panels: scaling of the m eff and /H T distributions for γ + 2 and γ + 3 jet with the collider energy. Same notation, conventions and also conclusions as in fig

62 6. Electroweak Sudakov corrections to the R n γ ratio LO NLO Virt. R 1 γ V + 1j s = 14 TeV p j T [TeV δ Virt. LO NLO Virt. R 1 γ 0.22 R 1 γ V + 1j s = 33 TeV V + 1j s = 100 TeV p j T [TeV δ Virt p j T [TeV LO NLO Virt. δ Virt. Figure 6.6: LO (solid lines) and approximated NLO (dotted lines) distributions for the Rγ 1 ratio as a function of the jet p T at s = 14, 33 and 100 TeV. In the lower panels the relative effect of the virtual weak Sudakov corrections is shown. 56

63 6.2. Virtual O(α) Sudakov corrections to the R n γ ratio LO NLO Virt. R 2 γ V + 2j s = 14 TeV m eff [TeV δ Virt. LO NLO Virt. R 2 γ V + 2j s = 33 TeV R 2 γ V + 2j s = 100 TeV m eff [TeV δ Virt. LO NLO Virt m eff [TeV δ Virt. Figure 6.7: LO (solid lines) and approximated NLO (dotted lines) distributions for the Rγ 2 ratio as a function of m eff at s = 14 TeV with the event selection of eq. (4.1) and at s = 33, 100 TeV with the rescaled cuts of eq. (4.3). Lower panels: relative effect of the virtual weak Sudakov corrections. 57

64 6. Electroweak Sudakov corrections to the R n γ ratio R 3 γ V + 3j s = 14 TeV LO NLO Virt. δ Virt /H T [TeV LO NLO Virt. R 3 γ 0.26 R 3 γ V + 3j s = 33 TeV δ Virt /H T [TeV V + 3j s = 100 TeV /H T [TeV LO NLO Virt. δ Virt. Figure 6.8: LO (solid lines) and approximated NLO (dotted lines) distributions for the Rγ 3 ratio as a function of /H T at s = 14 TeV with the event selection of eq. (4.2) and at s = 33, 100 TeV with the rescaled cuts of eq. (4.4). Lower panels: relative effect of the virtual weak Sudakov corrections. 58

65 6.2. Virtual O(α) Sudakov corrections to the R n γ ratio R 1 γ TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt V + 1j p j T [TeV δ Virt. 14 TeV 33 TeV 100 TeV Figure 6.9: Comparison of the R 1 γ distributions as a function of the jet p T for the three different values of s = 14, 33 and 100 TeV TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt. R 2 γ V + 2j δ Virt. 14 TeV 33 TeV 100 TeV m eff [TeV Figure 6.10: LO (solid lines) and approximated NLO (dotted lines) distributions for the Rγ 2 ratio as a function of m eff at s = 14, 33 and 100 TeV once the same event selection of eq. (4.1) is imposed. 59

66 6. Electroweak Sudakov corrections to the R n γ ratio TeV LO 14 TeV NLO Virt. 33 TeV LO 33 TeV NLO Virt. 100 TeV LO 100 TeV NLO Virt. R 3 γ V + 3j δ Virt. 14 TeV 33 TeV 100 TeV /H T [TeV Figure 6.11: LO (solid lines) and approximated NLO (dotted lines) distributions for the Rγ 3 ratio as a function of /H T at s = 14, 33 and 100 TeV once the same event selection of eq. (4.2) is imposed. 60

67 Chapter 7 Conclusions and future perspectives At high energies and in extreme kinematical configurations the one loop virtual electroweak corrections are dominated by double and single Sudakov logarithms that involve the ratio of the kinematical invariants of the process under consideration over the gauge boson masses. As discussed in chapters 2-3 Sudakov logarithms correspond to the infrared limit of the weak corrections and, owing to the universality of the IR part of the virtual one loop corrections, in Refs. [3, 4, 5 a general algorithm has been developed in order to evaluate the Sudakov limit of the one loop EW corrections in a process independent way. The algorithm of Refs. [3, 4, 5 has been described in chapter 3 together with its implementation in the ALPGEN LO event generator. The electroweak Sudakov corrections to the processes Z + n jets (n 3) considered as Standard Model background to the direct search for New Physics at hadron colliders have been computed in chapter 4, showing that the size of the virtual weak corrections to the most relevant observables is of the order of tens of percent already at the LHC at 7 TeV and becomes even larger at higher centre of mass energies, which will allow to explore more and more extreme kinematical regions. In chapter 5 the effect of the real weak corrections to the processes Z+n jets (n 3) has been studied both in a realistic event selection and for fully inclusive setup. Real weak corrections partially compensate the large negative virtual contributions, the size of the cancellation being strongly dependent on the considered event selections. In chapter 6 the O(α) virtual EW Sudakov corrections to the production of a prompt photon in association with up to three jets have been computed in order to obtain the Sudakov corrections to the Rγ n ratio (n 3), which is the theoretical input for the data driven estimate of the invisible Z+ jets SM background to the direct NP searches in the channel /E T + jets based on the reference process γ+ jets. While the theoretical systematic uncertainties related to higher order QCD corrections, PDFs and scale choices largely cancel 61

68 7. Conclusions and future perspectives in the Rγ n ratio, the effect of the EW Sudakov corrections in the tails of the relevant distributions is of the order of 20% at the LHC at s = 7 TeV and increases at future pp colliders. Moreover, the Sudakov corrections modify the shape of the ratio, which is no longer flat, so that the EW corrections should be carefully considered in the extrapolation of the Z( νν)+ jets background from the measured γ+ jets events. The conclusion of the phenomenological studies of chapters 4-6 is twofold: on one hand, the results of chapters 4-6 show that the LHC reached the Sudakov limit for some of the typical event selections considered for the direct NP searches, and this limit will be deeply explored at future pp colliders with higher centre of mass energies. On the other hand, in chapters 4-6 it is shown how the size of the virtual weak Sudakov corrections in the extreme tails of several distributions can be of order of tens of percent at the LHC and can reach the value of 80, 85% at the hh-fccs: this means that the EW corrections should be included in theoretical predictions, in particular at the hh-fccs, but such large negative virtual one loop corrections suggest that also the partially compensating (and strongly process dependent) effect of the real weak contributions should be considered. Higher order EW corrections may also play a relevant role. The implementation of the Denner-Pozzorini algorithm in the ALPGEN event generator described in chapter 3 works only for external fermions and transverse gauge bosons: the next logical step is to improve the code including also the case of longitudinal gauge bosons and scalars following the ideas sketched in Appendix D. With this improvement, the algorithm could be used in order to study the high energy limit of the weak corrections to the processes HH+ jets and V V + jets (V, V = W, Z), which will play a crucial role (as signal and background, respectively) for the study of the Higgs self interactions, that is one of the main motivations for the proposed hh-fccs experiments. For the processes considered in chapters 4-6, the O(α) QED corrections to the leading O(αα Njets S ) Born contributions are a gauge invariant subset that can be computed separately from the weak contributions which are the subject of this thesis. However, this is no longer the case for the processes involving a W boson at the LO. In Refs. [3, 4, 5 the contribution of the photon loops is considered on the same footing as the one of the diagrams involving a W or Z exchange: this means that the radiator functions related to the photon loops depend on the QED-like IR cutoffs λ and m (with the notation of chapter 2), so that real QED corrections should be considered in order to get rid of the dependence on these two parameters. Future work will be devoted to the modification of the present version of the code to properly include also the effect of real QED corrections. The final and most ambitious goal of the project started in Ref. [73 is to provide a completely general tool to estimate the O(α) EW corrections in the Sudakov limit, not only for parton level analyses, but also in the context of the event generation in the CKKW framework. 62

69 List of publications 1. M. Chiesa, G. Montagna, L. Barzè, M. Moretti, O. Nicrosini, F. Piccinini and F. Tramontano, Electroweak Sudakov Corrections to New Physics Searches at the CERN LHC, Phys. Rev. Lett. 111, (2013) [arxiv: [hep-ph; 2. K. Mishra, T. Becher, L. Barzè, M. Chiesa, S. Dittmaier, X. Garcia i Tormo, A. Huss, T. Kasprzik et al. Electroweak Corrections at High Energies, arxiv: [hep-ph; 3. J.M. Campbell, K. Hatakeyama, J. Huston, F. Petriello, Jeppe R. Andersen et al., Working Group Report: Quantum Chromodynamics, arxiv: [hep-ph; 4. J. Butterworth, G. Dissertori, S. Dittmaier, D. de Florian, N. Glover, K. Hamilton, J. Huston and M. Kado et al., in Les Houches 2013: Physics at TeV Colliders: Standard Model Working Group Report, arxiv: [hep-ph, Electroweak Sudakov corrections to Z/γ+ jets at the LHC ; 5. M. Chiesa, Electroweak corrections in the Sudakov limit at the LHC, Nuovo Cim. C 037 (2014) 02, 143; 6. L. Barzè, M. Chiesa, G. Montagna, P. Nason, O. Nicrosini, F. Piccinini and V. Prosperi, W gamma production in hadronic collisions using the POWHEG+MiNLO method, arxiv: [hep-ph. 63

70

71 Appendix A Cross-checks and code validation A.1 EW Sudakov corrections to Z/γ + 1 jet In Refs. [8 and [9 the Denner-Pozzorini algorithm has been used in order to study the high energy limit of the virtual one loop electroweak corrections to the processes Z + 1 jet and γ + 1 jet, respectively, while in Refs. [66 and [9 the results of the high energy limit approximation have been compared to the predictions of the full NLO electroweak corrections, finding an agreement at the percent level already for p V T of order 100, 150 GeV (V = Z, γ). According to the algorithm, EW corrections in the Sudakov limit can be written in a factorized form as the sum of single and double logarithmic contributions, where each of the contributions is made up of a radiator function (containing the logarithmic structure of the corrections) times a tree level matrix element, that can be the one of the LO process or an SU(2) transformed of the Born matrix element. Since for Z + 1 jet and γ + 1 jet the matrix elements of the SU(2) correlated process W + 1 jet can be obtained from the ones of Z + 1 jet (or γ + 1 jet) through a ratio of couplings, in Refs. [8, 9 the analytic expression has been reported for the EW corrections in the Sudakov limit computed by means of the algorithm of Refs. [3, 4, 5. However, in the implementation of the Denner-Pozzorini algorithm in the ALPGEN event generator described in this thesis, all the matrix elements needed are computed numerically by means of the ALPHA algorithm [56, 57, 58. In fig. A.1 the results of the numerical implementation of the algorithm are compared to the ones of the analytic expressions of Refs. [8, 9 for the event selection: p j T > 100 GeV, y(j) < 2.5, y(v ) < 2.5 (V = γ, Z), (A.1) finding an agreement of the level of a few permille. To be precise, such a level of agreement is obtained once the formulas of Refs. [8, 9 have been 65

72 A. Cross-checks and code validation modified in order to drop the assumption M W M Z, so that all the terms like log(s/mw 2 )log(m Z 2/M W 2 ) are taken into account. [ pb GeV LO NLO Virt. A NLO Virt. B [ pb GeV LO NLO Virt. A NLO Virt. B dp j t γ + 1j δ Virt. A δ Virt. B dp j t Z + 1j δ Virt. A δ Virt. B Ratio A/B Ratio A/B Ratio Ratio p j t [GeV p j t [GeV Figure A.1: Upper panels: jet p T distribution at s = 14 TeV for γ + 1 jet (left plot) and Z +1 jet (right plot) at LO (solid blue line), at NLO obtained by means of the Denner-Pozzorini algorithm implemented in the ALPGEN generator (dotted red line, curve A) and at NLO obtained with the formulas of Refs. [9 and [8 (green dash-dotted line, curve B) for γ+1 jet and Z +1 jet, respectively. Central panels: relative effect of the virtual weak Sudakov corrections obtained from the curves A and B, respectively (δ EW ). Lower panels: ratio of the results of the two computations A and B. For both γ + 1 jet and Z + 1 jet the event selection of eq. (A.1) is considered. = NLO LO LO A.2 EW Sudakov corrections to Z + 2 jets Very recently, the full one loop EW corrections to the process Z + 2 jets have been computed by means of the RECOLA recursive amplitudes generator. While in Ref. [72 only the partonic subprocesses involving one fermion current have been considered, the effect of the O(α) corrections for the subprocesses with one and two fermion currents have been shown in Ref. [74. 66

73 A.2. EW Sudakov corrections to Z + 2 jets On one hand the code of Refs. [72, 74 is not yet publicly available, while on the other hand there are no analytic results in the literature for the electroweak corrections to Z + n jets with n > 1, so that the GOSAM package has been used in order to test the implementation of the Denner-Pozzorini algorithm in the ALPGEN generator for Z + 2 jets. GOSAM is a generator of one loop matrix elements originally developed for NLO QCD calculations [94, which has been recently extended to the one loop electroweak corrections [95, facing the issues of loops diagrams containing several mass scales and also unstable particles. However, it is worth noting that the present version of GOSAM still does not include the treatment of EW renormalization, which is necessary for the computation of the Sudakov corrections, since part of the single logarithmic structure comes from the running of the EW parameter and from the wave functions renormalization counterterms. In order to validate the present implementation of the algorithm of Refs. [3, 4, 5 using GOSAM, as a first step all the single logarithms coming from renormalization counterterms have been traced back and dropped. Then this modified version of the code has been used as follows: first phase space points have been generated using the ALPGEN routines and the LO and Sudakov NLO unrenormalized weights have been computed, then the phase space points are passed to a routine which calls the finite part of the unrenormalized exact NLO matrix element generated by GOSAM and at the end the distributions obtained with the three different weights have been compared. Figure A.2 shows some examples of this kind of comparisons between the predictions of the two different codes for the unrenormalized part of the one loop EW corrections to the partonic subprocesses considered under the event selection of eq. (4.1). Several comments on the procedure described in the previous paragraph are in order. First of all, it is worth noting that the results shown in fig. A.2 do not include renormalization, so that they are not physical and moreover they are gauge dependent. Nevertheless fig. A.2 is still a quite stringent technical test of the implementation of the Denner-Pozzorini algorithm in the ALPGEN generator, since both the calculations are performed in the same gauge (the GOSAM matrix element has been computed in the ξ = 1 gauge and the same gauge choice has been used in order to define the counterterms contributions in the ALPGEN routines) and the results nicely agree. Of course, the predictions of the two codes are not expected to be identical, as the GOSAM unrenormalized matrix element is exact, while in the ALPGEN routines only the logarithmic part of the corrections is computed: the two predictions should be the same only in the asymptotic regime, so that the percent level agreement shown in A.2 for an event selection which is only approximately in the Sudakov limit is really a non trivial validation of the implementation of the Denner-Pozzorini algorithm in the ALPGEN generator. An independent and physical (even if unfortunately only qualitative) test of the validity of the algorithm and of its implementation can be obtained from the comparison between the results of Ref. [72 and the predictions of the 67

74 A. Cross-checks and code validation routines for the Sudakov corrections included in ALPGEN. The event selection considered in Ref. [72 does not consider particularly tight cuts: p j T > 25 GeV, y(j) < 4.5, ΔR jj 0.4 (j = j 1, j 2 ), (A.2) (where j 1 and j 2 are the leading and the remaining jet, respectively), so that the kinematical configurations involved can hardly be considered as Sudakov regions. Figure A.3 shows the predictions of the algorithm of Refs. [3, 4, 5 for the leading jet p T distribution at s = 8 TeV for the process Z + 2 jets with the event selection of eq. (A.2) when only the partonic subprocesses with one fermion current are considered, as in Ref. [72. As can be seen, the results in fig. A.3 are very similar to the ones of Ref. [72, where the full one loop EW corrections to Z + 2 gluons turned out to be of order 15% for p j1 T and 10% for p j1 T = 500 GeV. = 1 TeV The corrections shown in fig. A.3 for Z +2 gluons with the cuts of eq. (A.2) are sensibly smaller than the ones obtained in chapter 4 for Z + 2 jets under the event selection of eq. (4.1): this is mainly due to the fact that in eq. (A.2) there are no cuts on the transverse momentum of the Z boson, so that the most relevant kinematical regions are the ones in which the two jets are back to back while the Z is soft. As a result, the larger invariants (and thus the leading part of the correction) are related to a di-jet like kinematics and the EW corrections to the di-jet production at hadron colliders turn out to be rather small, as described in the next section. Of course, since in fig. A.3 only the partonic subprocesses with one fermion current are included, the number of EW charged legs is in general smaller than the one considered for the case of Z +2 jets studied in chapter 4. However, the impact of the different number of EW charged particles should be very small, as can be seen from the stability of the corrections as a function of the final state parton multiplicity (as soon as the Z boson is required to be hard), as shown in chapter 5. A.3 Electroweak Sudakov corrections to di-jet production The one loop electroweak corrections to the production of two jets in hadron collisions have been computed for the first time in Ref. [96 and an independent calculation has been recently performed in Ref. [97. In particular, in Ref. [97 it is shown the impact of EW corrections for the leading jet p T distribution at the LHC energies of 7, 8 and 14 TeV under the event selection: p j T > 25 GeV, y(j) < 2.5, ΔR jj > 0.6 (j = j 1, j 2 ), (A.3) where j 1 and j 2 are the hardest and the next-to-hardest jets, respectively. As in the case of the results of fig. A.3, the kinematical configurations selected by the cuts of eq. (A.3) are not that extreme and probably they are 68

75 A.3. Electroweak Sudakov corrections to di-jet production still far from the Sudakov limit. Despite of this, the electroweak Sudakov corrections to the dijet production have been computed by means of the Denner- Pozzorini algorithm implemented in the ALPGEN event generator and the results are show in fig. A.4 for the LHC at s = 14 TeV under the event selection of eq. (A.3). As can be seen, the corrections turn out to be rather small, of the same order as the ones shown in fig. A.3. Moreover, the predictions in fig. A.3 at large p T basically agree with the ones of Ref. [97, where the one loop EW corrections have been found to be of order 5% and 12% for the leading jet p T of the order of 1.5 and 3 TeV, respectively. 69

76 A. Cross-checks and code validation δ Virt. A δ Virt. B δ Virt. A δ Virt. B s = 14 TeV uu Zgg m eff [GeV 0.00 δ Virt. A δ Virt. B uu Zcc s = 14 TeV m eff [GeV 0.00 δ Virt. A δ Virt. B s = 14 TeV uu Zuu m eff [GeV s = 14 TeV ud Zud p Z T [GeV Figure A.2: Virtual EW corrections to some of the partonic subprocesses involved in the production of a Z boson in association with two jets under the cuts of eq. (4.1) at s = 14 TeV. In particular, the last two plots refer to the most relevant classes of partonic subprocesses with two fermion currents, as shown in chapter 4. The results obtained with the finite part of the unrenormalized one loop matrix element generated by GOSAM are compared to the predictions of the Denner-Pozzorini algorithm implemented in the ALPGEN event generator (where the logarithmic structure coming from parameters and wave function renormalization has been dropped, as described in the text). Even if these plots have no physical meaning, they represent a non trivial technical test of the validity of the algorithm and of its implementation. 70

77 A.3. Electroweak Sudakov corrections to di-jet production 0.00 δ Virt s = 8 TeV p j1 T [GeV Figure A.3: Relative effect of the virtual weak Sudakov corrections to the leading jet p T distribution for the process Z + 2 jets (where only the partonic subprocesses with one fermion current have been included) under the cuts of eq. (A.2) at s = 8 TeV. To be compared with fig. 4 of Ref. [ δ EW δ EW pp jj, s = 14 TeV p j1 T [GeV Figure A.4: Relative effect of the virtual weak Sudakov corrections to the leading jet p T distribution for the dijet production process at the LHC centre of mass energy of 14 TeV under the cuts of eq. (A.3). To be compared with fig. 12 of Ref. [97. 71

78 72 A. Cross-checks and code validation

79 Appendix B One loop renormalization counterterms B.1 On shell renormalization conditions at one loop In the on shell scheme the form of the renormalization counterterms is fixed by the following requirements: the renormalized masses are the poles of the propagators at one loop; the renormalized fields are normalized to one; the renormalized electric charge is obtained from the eeγ vertex in the Thomson limit; the renormalized Higgs tadpole is set to zero. With the above conditions the one loop EW counterterms for fermions, gauge bosons and EW couplings read: δm f = m f 2 Re ( Σ f,l ii ( ) = m 2 f δz f,l ii δz f,r ii ReΣ f,l ii m 2 ( ( ) f ReΣ f,l p 2 ii p 2 + = ( ) f,r ReΣii m 2 f m 2 ( ( ) f ReΣ f,l p 2 ii p 2 + ( ) ( ) ( ) ) m 2 f + Σ f,r ii m 2 f + 2Σ f,s ii m 2 f, ( f,r ReΣ ) ( ii p 2 f,s + 2 ReΣ )) ii p 2 ( f,r ReΣ ) ( ii p 2 f,s + 2 ReΣ )) ii p 2 p 2 =m 2 f p 2 =m 2 f,, (B.1) 73

80 B. One loop renormalization counterterms δm 2 W = ReΣ W T δm 2 Z = ReΣ ZZ T δz AA = ( ) k 2 ΣAA T k 2 ( M 2 W ), δzw k 2 ReΣ W T ( k 2 ) k 2 =M 2 W ( ) M 2 Z, δzzz = ( ReΣ ) ZZ k2 T k 2, k 2 =MZ 2 k 2 =0 ΣAZ T (MZ δz AZ = 2 Re 2), δz MZ 2 ZA = 2 ΣAZ T (0), MZ 2 δmh 2 = ReΣ ( ) H MH 2, δzh = k ReΣ ( H k 2) 2,, k 2 =M 2 H, (B.2) δe e = 1 2 δz AA s W 1 c W 2 δz ZA = 1 2 ( ) δm 2 W δc W c W = 1 2 δs W s W M 2 W δm 2 Z M 2 Z = c2 W δc W = 1 s 2 W c W 2 δt = T H. c 2 W s 2 W ( ) k 2 ΣAA T k 2 s W Σ AZ T (0), c W M 2 k 2 Z =0 = 1 ( 2 Re Σ W T (MW 2 ) ΣZZ T (MZ 2) ), MW 2 MZ 2 ( Σ W Re T (MW 2 ) T (MZ 2) ), M 2 W ΣZZ MZ 2 (B.3) Where the notation Re means that the real part is taken only for the scalar functions contained in the unrenormalized self energies. B.2 Unrenormalized self energies ( Σ ) { AA T k 2 = α 4 4π 3 f,i N f C Q2 f + 2m 2 f,ib 0 (0, m f,i, m f,i ) + k2 3 [ ( k 2 + 2m 2 f,i) B0 (k, m f,i, m f,i ) + ( 3k 2 + 4M 2 W ) B0 (k, M W, M W ) 4M 2 W B 0 (0, M W, M W ) } (B.4) 74

81 B.2. Unrenormalized self energies Σ AZ T { 2 3 ( ) k 2 = α N f C 4π ( Q f) ( g + f + ) [ k 2 g f 3 f,i ( k 2 + 2mf) 2 B0 (k, m f, m f ) + 2m 2 fb 0 (0, m f, m f ) { [( 1 9c 2 W + 1 ) k 2 + ( 12c 2 W + 4 ) MW 2 3s W c W 2 ( 12c 2 W 2 ) M 2 W B 0 (0, M W, M W ) + k2 3 }} B 0 (k, M W, M W ) (B.5) Σ ZZ T ( ) { { k 2 = α 2 ((g ) N f + 2 ( ) ) [ C f + g 2 f ( k 2 + 2m 4π 3 f,i) 2 B0 (k, m f,i, m f,i ) f,i } + 2m 2 f,ib 0 (0, m f,i, m f,i ) + k2 3 + m 2 3 4s f,ib 2 0 (k, m f,i, m f,i ) W c2 W { 1 + ( 24c 4 6s 2 W c2 W 8c 2 W + 2 ) MW 2 B 0 (0, M W, M W ) + ( 4c 2 W 1 ) k 2 W 3 ( + 18c 4 W + 2c 2 W 1 ) k 2 B 0 (k, M W, M W ) 2 ( ) + 24c 4 W + 16c 2 W 10 MW 2 B 0 (k, M W, M W ) } s 2 W c2 W (M 2 H M 2 Z )2 k 2 { (2M 2 H 10M 2 Z k 2) B 0 (k, M Z, M H ) ( B 0 (k, M Z, M H ) B 0 (0, M Z, M H ) 2M 2 ZB 0 (0, M Z, M Z ) 2M 2 HB 0 (0, M H, M H ) 2k2 3 ) }} (B.6) where g + f = s W Q f, g f c = I3,f W s2 W Q f W s W c W 75 (B.7)

82 B. One loop renormalization counterterms Σ W T ( k 2 ) = α 4π { i 1 3s 2 W [ ( ) k 2 m2 l,i B 0 (k, 0, m l,i ) + m 2 2 l,ib 0 (0, m l,i, m l,i ) + m4 l,i 2k (B 2 0 (k, 0, m l,i ) B 0 (0, 0, m l,i )) + k2 3 + ( 1 V s 2 ij [ 2 k 2 m2 u,i + m 2 d,j i,j W 2 ) B 0 (k, m u,i, m d,j ) + k2 3 + m 2 u,ib 0 (0, m u,i, m u,i ) + m 2 d,jb 0 (0, m d,j, m d,j ) ( m 2 u,i m 2 2 d,j) + (B 2k 2 0 (k, m u,i, m d,j ) B 0 (0, m u,i, m d,j )) { (2M 2 W + 5k 2) B 0 (k, M W, λ) 2M 2 W B 0 (0, M W, M W ) } M W 4 (B k 2 0 (k, M W, λ) B 0 (0, M W, λ)) + k2 3 {[ + 1 (40c 2 12s 2 W 1 ) ( k c 2 W W c 2 W ( 16c 2 W + 2 ) [ MW 2 B 0 (0, M W, M W ) + MZB 2 0 (0, M Z, M Z ) ) MW 2 B 0 (k, M W, M Z ) + ( 4c 2 W 1 ) 2 3 k2 ( 8c 2 W + 1 ) ( )} (MW 2 M Z 2)2 B 0 (k, M W, M Z ) B 0 (0, M W, M Z ) k 2 { + 1 (2M 2 12s 2 H 10MW 2 k 2) B 0 (k, M W, M H ) W ( (M H 2 M W 2 )2 B (k, M k 2 W, M H ) B 0 (0, M W, M H ) }} 2M 2 W B 0 (0, M W, M W ) 2M 2 HB 0 (0, M H, M H ) ) 2k2 3 (B.8) 76

83 B.2. Unrenormalized self energies Σ ( { H k 2) = α N f m 2 f,i C 4π 2s 2 f,i W M [2A W 2 0 (m f,i ) + (4m f,i k 2 )B 0 (k 2, m f,i, m f,i ) ( ) 1 6MW 2 2k 2 + M H 4 B 0 (k 2, M W, M W ) 2s 2 W 1 2s 2 W 1 ( 4s 2 W c2 W 3 + M 2 H ( 2M 2 W ) 2M 2 W A 0 (M W ) + 3M 2 W 6M 2 Z 2k 2 + M 4 H 2M 2 Z ) s 2 W B 0 (k 2, M Z, M Z ) ( ) M H 2 A 4s 2 W c2 W 2MZ 2 0 (M Z ) + 3M Z 2 2s 2 W c2 W 3 [ 3 M H 4 B 8s 2 W MW 2 0 (k 2, M H, M H ) + M H 2 A MW 2 0 (M H ) } (B.9) ( Σ ) f,l i,j k 2 = α 4π { δ ij Q 2 f [ 2B 1 (p 2, m f,i, λ) δ ij (g f [2B )2 1 (p 2, m f,i, M Z ) + 1 m 2 [ f,i + δ ij 4s 2 W M B W 2 1 (p 2, m f,i, M Z ) + B 1 (p 2, m f,i, M H ) [ ( ) 2 + m2 f,k + 1 2s 2 W k V ik V kj M 2 W B 1 (p 2, m f,k, M W ) + 1 } (B.10) ( Σ ) f,r i,j k 2 = α 4π { δ ij Q 2 f [ 2B 1 (p 2, m f,i, λ) δ ij (g + f [2B )2 1 (p 2, m f,i, M Z ) + 1 m 2 [ f,i + δ ij 4s 2 W M B W 2 1 (p 2, m f,i, M Z ) + B 1 (p 2, m f,i, M H ) } + 1 m f,i m f,j V 2s 2 W MW 2 ik V kj B 1(p 2, m f,k, M W ) k (B.11) 77

84 B. One loop renormalization counterterms Σ f,s i,j ( ) { k 2 = α [ δ ij Q 2 f 4B 0 (p 2, m f,i, λ) 2 4π [ + δ ij g + f g f 4B 0 (p 2, m f,i, M Z ) 2 m 2 [ f,i + δ ij 4s 2 W M B W 2 0 (p 2, m f,i, M Z ) B 0 (p 2, m f,i, M H ) } + 1 V 2s 2 ik V m 2 f,k kj B W MW 2 0 (p 2, m f,k, M W ) k (B.12) In eqs.(b.4)-(b.12) an infinitesimal mass parameter λ has been introduced for the photon in order to regularize the QED infrared singularities, N f C is the number of colours for the fermion f, while in eqs. (B.10)-(B.12) f stands for the isospin partner of f. B.3 Definition of the A 0, B 0 and B 1 functions In this subsection the definitions of the one and two point scalar functions [49, 50, 48 appearing in the expression of the unrenormalized self energies are collected. The scalar integral A 0 is defined as: A 0 (m) = (2πµ)4 D iπ 2 d D 1 q q 2 m 2 + iε. (B.13) Eq. (B.13) can be computed in the standard way by performing the Wick rotation, obtaining: where (γ E is the Euler s constant) A 0 (m) = m 2 (Δ log m2 + 1) + O(D 4), (B.14) µ 2 The definition of the scalar function B 0 is: B 0 (p, m 0, m 1 ) = (2πµ)4 D iπ 2 Δ = 2 4 D γ E + log4π. (B.15) d D 1 q [ [. q 2 m iε (q + p) 2 m iε 78 (B.16)

85 B.3. Definition of the A 0, B 0 and B 1 functions After introducing the Feynman parametrization and performing the Wick rotation, eq. (B.16) becomes: B 0 (p, m 0,m 1 ) = Δ 1 0 dxlog that can be solved in a general way as: [ p 2 x 2 x(p 2 m m 2 1) + m 2 1 iε + O(D 4), µ 2 (B.17) B 0 (p, m 0,m 1 ) = Δ + 2 log m 0m 1 µ 2 + m2 0 m 2 1 p 2 log m 1 m 0 m 0m 1 p 2 (1 r r) logr + O(D 4), (B.18) where r and 1 r are obtained from the condition: x 2 + m2 0 + m 2 1 p 2 iε x + 1 = (x + r)(x + 1 ). (B.19) m 0 m 1 r The scalar coefficient B 1 comes from the Passarino-Veltman reduction of the two point tensor integral B µ : (2πµ) 4 D d D q q [ [ µ iπ 2 q 2 m iε (q + p) 2 m iε = B µ (p, m 0, m 1 ) = p µ B 1 (p, m 0, m 1 ). (B.20) Even if from eq. (B.20) it is possible to write the following general expression for the B 1 coefficient: B 1 (p, m 0, m 1 ) = m2 1 m 2 ( ) 0 B 2p 2 0 (p, m 0, m 1 ) B 0 (0, m 0, m 1 ) 1 2 B 0(p, m 0, m 1 ), (B.21) for many applications it is convenient to evaluate the B 1 coefficient directly from: B 1 (p, m 0,m 1 ) = Δ dx(x 1)log [ p 2 x 2 x(p 2 m m 2 1) + m 2 1 iε, (B.22) µ 2 where the O(D 4) terms have been omitted since they are not relevant at one loop level. 79

86 B. One loop renormalization counterterms 80

87 Appendix C Collinear singularities in one loop radiative corrections C.1 Collinear singularities in O(α) QED corrections For the sake of brevity, in chapter 2 only the infrared and infrared-collinear limit of one loop QED corrections has been considered. This Appendix is focused on the collinear limit of both real and virtual one loop QED corrections following the treatment of Ref. [45. C.1.1 Collinear limit of real O(α) corrections The matrix element for the process i f(eγ) involving the radiation of a photon off a final state electron q p p, (C.1) can be written as: M i f(eγ) λ,α (p, q) = eu α (p )γ µ /p + /q (p + q) 2 m 2 ɛ µ(q, λ)a i f(e) (p), (C.2) (where A i f(e) (p) represents the sum of the LO diagrams for the process i f(e) involving an off-shell final state electron with momentum p and helicity α, while λ is the photon polarization). As pointed out in chapter 2, the propagator in eq. (C.2) is divergent when the emitted photon becomes collinear with the external momentum p and the electron mass acts as IR cutoff. If the electron mass in eq. (C.2) is set to zero, 81

88 C. Collinear singularities in one loop radiative corrections the collinear singularity corresponds to the kinematical configurations in which q T 0 (where q T is the transverse momentum of the photon with respect to the emitting electron). In the collinear limit, the following parametrization for the momenta p, q and p can be introduced: ( p µ = E + ( q µ = (1 x)e + q T 2 ) 2x(1 x)e, 0, 0, E q T 2 4x(1 x)e, 0, 0, (1 x)e + q T 2 (1 2x) 4x(1 x)e q T 2 4x(1 x)e, 0, 0, xe q T 2 (1 2x) 4x(1 x)e ( xe + ) + q µ T ) q µ T p µ =p µ q µ = ) q µ T (0, = q T cosφ, q T sinφ, 0, (C.3) where (up to O( q T 4 ) terms): p 2 = q T 2 x(1 x), q2 = 0, p 2 = 0. (C.4) Using eq. (C.3) and assuming to replace the /p term with the polarization sum /p β u β(p)u β (p) also for the internal off-shell electron, eq. (C.2) becomes: M i f(eγ) λ,α (p, q) = e β = β u α (p )γ µ u β (p)u β (p)ɛ µ(q, λ)a i f(e) x(1 x) (p) q T 2 V α,β,λ (p, p, k) x(1 x) M i f(e) q T 2 β (p) (C.5) (where the regular terms in the limit q T 0 have been omitted). The vertex part in eq. (C.5) can be computed from the explicit expression of the polarization vectors for the photon and the Dirac spinors for the internal and external electrons corresponding to the momenta in eq. (C.3) up to O( q T 3 ) contributions: ( u (p) = 0, 0, 0, T 2E), ) T u + (p) =( 2E, 0, 0, 0, u (p ) = (0, 0, q T e iφ, T 2xE), 2xE ( u + (p q T e iφ T ) = 2xE,, 0, 0), 2xE ɛ µ ±(k) = 1 ( qt e iφ 2 2(1 x)e, 1, i, q T e iφ ). (C.6) 2(1 x)e 82

89 C.1. Collinear singularities in O(α) QED corrections Using eq. (C.6) the vertex term reads: V α,β,λ (p, p, k) =u α (p )γ µ u β (p)ɛ µ(q, λ) 2x qt =e x(1 x) (δ α,λ + xδ α,λ )δ α,β e iλφ + O( q T 3 ), (C.7) Vα,β,λ(p, p, k)v α,β,λ(p, p, k) = 2e 2 q T x 2 x(1 x) 1 x δ α,βδ α,β + O( q T 4 ). λ=± (C.8) In order to compute the cross section for the process i f(eγ), the matrix element squared obtained form eqs. (C.5) and (C.8) have to be integrated over the final state photon and electron phase space. In the collinear limit k = (1 x)p, leading to an extra 1/x factor, since from: p p q 1 x, as can be seen d 3 q d 3 p d 3 q d 3 p = d 4 pδ 4 (p p q) (2π) 3 2q 0 (2π) 3 2p 0 (2π) 3 2q 0 (2π) 3 2p 0 d 3 q d 3 p p = (C.9) (2π) 3 2q 0 (2π) 3 2p 0 p q p0= p q + p. As a result, the cross section for the process i f(eγ) in the collinear limit reads: i f(eγ) coll α = 2e 2 d 3 q x(1 x) 1 + x 2 [ 1 (2π) 3 2q 0 q T 2 1 x x i f(e) α + O( q T ). (C.10) Using the following identity for the photon phase space in the collinear limit: equation (C.10) becomes: d 3 q (2π) 3 2q 0 = dxdφ q T 2 4(2π) 3 (1 x) + O( q T 2 ), i f(eγ) α coll = α 2π log ( Q 2 m 2 ) i f(e) α 1 0 (C.11) dx 1 + x2 1 x. (C.12) Concerning eq. (C.12) several comments are in order. First of all, i f(eγ) α is part of the real O(α) corrections to the process i f: eq. (C.12) shows that real corrections in the collinear limit factorize in tree level cross sections multiplied by a factor, which is the usual (unregularized) Altarelli-Parisi splitting [98. The same kind of factorization also holds for the case of initial state radiation, with the difference that the radiation changes the centre of mass energy of the hard process i f. It should be noticed that eq. (C.12) is divergent in the x 1 limit, that corresponds to the soft limit of real O(α) corrections computed in chapter 2 in the eikonal approximation. As a conclusive remark, while in eq. (C.12) the electron mass has been chosen as the lower bound of the q T integration, since in eq. (C.2) m is the IR regulator for the collinear singularity, the upper bound of integration is somehow arbitrary, even if it should be small enough in order not to spoil the validity of the collinear approximation. 83 coll

90 C.1.2 C. Collinear singularities in one loop radiative corrections Collinear limit of virtual O(α) corrections The collinear singularities of the one loop virtual QED corrections correspond to the diagrams in which an on shell external fermion emits a virtual photon and arise from the integration region where the photon momentum becomes collinear to the one of the external fermion: p p q q. (C.13) The set of diagrams in (C.13) can be written as: µ 4 D d D q i(/p /q) ( ) ig µν (2π) D Ai(eγ) 0 ν (p q, q) iei (p q) 2 m 2 l γ µ q 2 λ u(p), 2 (C.14) where A i(eγ) 0 ν represents the sum of the diagrams for the process i(e) 0 involving an additional photon, all the external legs have been considered as incoming, the factor I l is Q l or +Q l in the case of an incoming electron or positron, respectively, and in all the denominators a +iε term is always understood. Since also the soft limit of the virtual O(α) corrections is involved in eq. (C.14), in order to separate the collinear part of the virtual one loop QED corrections, the soft contribution computed in the eikonal approximation in chapter 2 should be subtracted, namely: µ 4 D ei l d D q (2π) D [q 2 λ 2 [ (p l q) 2 m 2 l { A i(eγ) 0 µ (p l q, q)i(/p l /q)γ µ u(p) m l iei m ( 4p l p m ) } M (p m + q) 2 m 2 0 (p l, p m ), m (C.15) where the second term within the brackets can be simplified in the collinear limit q xp (m 2 0) and using the total charge conservation l m I m = I l, obtaining: M 1 l, coll = µ 4 D lim q xp M i(eγ) 0 [γµ { d D q (2π) D M i(eγ) 0 [γµ (p(1 x), xp) ei [ l [q 2 λ 2 (p l q) 2 m 2 l ( 2 x 1 ) q µ 2e x M 0(p l, p m ) }, (C.16) being the Green function for the process i(eγ) 0, where the photon leg has been truncated. In eq. (C.16) only the 1PI one loop diagrams are included, since the electron self energy diagrams are collected in the wave function renormalization counterterms and will be considered separately. 84

91 C.1. Collinear singularities in O(α) QED corrections Using the Ward identity: q µ M i(eγ) 0 [γµ (p q, q) = ei l M i(e) 0 (p), (C.17) equation (C.16) reads: M 1 l, coll = µ 4 D M 0 (p l, p m ) d D q 2i(eI l ) [ 2. (C.18) (2π) [q D 2 λ 2 (p l q) 2 m 2 l As can be seen from eq. (C.18), the one loop QED corrections in the collinear limit q xp factorize on the external leg l, so that the leg index will be suppressed in the following. Moreover, as the soft singularities have been subtracted, the λ parameter can be set to zero. Introducing the Sudakov parametrization [99: q µ =xp µ + zn µ + q µ T ( n µ p ) = p 0, p 0 p q µ T =(0, q T ) n q T =p q T = 0, (C.19) with: d D q = p n dx dz d D 2 q T, (C.20) equation (C.16) can be written as (Il 2 = Q 2 l = 1): d M 1 coll = M 0 (p l, p m )µ 4 D D 2 q T 2e 2 i dx dz (2π) D 2 4p nx(x 1)[z z 0 [z z 1, (C.21) where: z 0 = q T 2 x 2 m 2 + iε 2xp n z 1 = q T 2 (x 1) 2 m 2 + m 2 + iε. (C.22) 2p n(x 1) The two poles of the integrand in eq. (C.21) have opposite imaginary parts for 0 < x < 1, so that the integration over the z variable can be performed closing the contour of integration in the upper or in the lower half complex plane: M 1 coll = α d 2π M 0(p l, p m )4πµ 4 D D 2 q T 1 dx dz (2π) D 2 q T 2 + x 2 m 2 = α 1 ( 4πµ 2 ) ε 2π M 0(p l, p m )Γ(ε) dx m 2 x 2 0 = α µ2 log 2π m M 0(p 2 l, p m ) + collinear finite terms. 85 (C.23)

92 C. Collinear singularities in one loop radiative corrections In eq. (C.23) the collinear finite terms also include the UV poles, which are however cancelled by means of the renormalization procedure. Without the subtraction of the soft contributions, the computation of eq. (C.14) would lead to the result: M 1 coll + soft = α 2π M 0(p l, p m )log µ2 m dx x 1 x, (C.24) which is divergent in the soft limit x 1. Adding eq. (C.24) to the collinear limit of the wave function renormalization counterterms δz f, with coll δz f = α µ2 log 2π m dx(1 x), (C.25) it turns out that the part of the IR limit of the virtual QED corrections that can be factorized on a single external leg is: Virt. coll = α 2π log ( Q 2 m 2 ) LO 1 0 dx 1 + x2 1 x (C.26) and cancels the corresponding real contribution of eq. (C.12) for the photon radiation off a final state electron. This is not the case for the initial state radiation contributions, where the photon radiation changes the centre of mass energy of the hard process, so that real and virtual corrections factorize on tree level cross sections computed in different phase space points. C.2 Collinear limit of one loop EW Sudakov corrections As already discussed in chapter 3, single collinear Sudakov logarithms are related to the diagrams of the form (C.1). At variance with the QED case, the emitted gauge boson can also be a Z or a W, while the external on shell leg can be a fermion, a scalar or another gauge boson. The collinear limit of the one loop virtual weak corrections is computed following the same steps described in section C.1.1. First of all the soft contributions included in the double logarithmic part of the correction are subtracted. Then the collinear limit is evaluated: the total charge conservation used in the QED case, for the EW corrections is replaced by the approximate SU(2) U(1) invariance (which is fulfilled up to mass suppressed terms). The proof of the collinear factorization in QED is obtained by means of the Ward identity (C.17), that no longer holds for the EW interactions: in Refs. [3, 4, 5 the factorization of the collinear limit of the one loop weak corrections is obtained using the collinear Ward identities, proved in [4, 5, that are a consequence of the BRS invariance of the Standard Model. As a final step, the loop integrals are computed with the same techniques of section C

93 Appendix D Sudakov corrections for external longitudinal gauge bosons in the unitary gauge In Refs. [3, 4, 5 the electroweak corrections for the processes involving external longitudinal gauge bosons are computed by means of the Goldstone boson equivalence theorem. In chapters 4, 5 and 6, the O(α) corrections for the case of longitudinally polarized Z bosons have been neglected, since the corresponding LO contributions are strongly suppressed for the event selections considered. Moreover, the tree level matrix elements provided by ALPGEN are computed in the unitary gauge, so that the procedure of Refs. [3, 4, 5 for the longitudinal gauge bosons cannot be followed directly. It is however possible to improve the present implementation of the Denner- Pozzorini algorithm in the ALPGEN event generator in a way very similar to the one described in Ref. [7. At LO the formula for the GBET is: m 0 = M φ1 φm 0 i 1 Q Va, M V 1 L V L m a=1 (D.1) where m is the number of longitudinal gauge bosons V L (V L = Z L, W L ), Q Va is the electric charge of Va L and φ a is the would-be Goldstone boson corresponding to Va L. At NLO eq. (D.1) becomes: M V 1 L V L m NLO = M φ1 φm NLO 87 m a=1 i 1 Q Va (1 + δa Va ), (D.2)

94 D. Sudakov corrections for external longitudinal gauge bosons in the unitary gauge where δa Z L = ΣZZ L (M Z 2) im ZΣ Zχ (MZ 2) + δm Z + 1 MZ 2 M Z 2 δz ZZ (D.3) δa W L = ΣW L W (MW 2 ) + M W Σ W φ (MW 2 ) + δm W + 1 MW 2 M W 2 δz W W. (D.4) For the computation of one loop EW corrections in the Sudakov limit, the term proportional to δa in eq. (D.2) can be easily included into the single logarithmic part of the correction: δ C GBETM V L 1 V L m NLO = M V L 1 V L m 0 m δa Va, a=1 (D.5) where the leading order expression for the GBET has been used, since the δa terms are of O(α). For the remaining part of eq. (D.2), one can think to use the original approach of Refs. [3, 4, 5: N NLO (φ) = δk,l DL M i1 j l j k i N 0 (φ) + δm i1 i N l=1 k>l N l=1 δl SL M i1 j l i N 0 (φ), (D.6) where M i1 i N (φ) stands for M φ1 φm and the radiator functions are the ones derived for the would-be Goldstone bosons. Only at the very end, when the corrections of eq. (D.6) have been evaluated, those tree level matrix elements which involve external would-be Goldstone bosons can be mapped back to the physical fields using the LO expression for the GBET, while the ones involving external Higgs bosons are computed directly. In general the overall procedure leads to a non null phase shift, since in eq. (D.6) the SU(2) transformed of the original φ a fields are considered, however, this phase shift is uniquely determined by the flavour of the φ a fields and its SU(2) transformed, so it can be included in the general expression for the radiator functions involving external would-be Goldstone bosons. 88

95 Acknowledgements I am very grateful to Guido Montagna, Mauro Moretti, Oreste Nicrosini and Fulvio Piccinini for all the things I had the opportunity to learn during my PhD. I would like to thank Luca Barzè, Paolo Nason, Valeria Prosperi and Francesco Tramontano for the very fruitful collaboration. Thanks also to Stefano Boselli, Carlo Carloni Calame and Homero Martinez. The work presented in this theses was supported in part by the Research Executive Agency (REA) of the European Union under Grant Agreement No. PITN-GA (LHCPhenoNet) and by the Italian Ministry of University and Research under the PRIN Project No. 2010YJ2NYW. I would like to thank the INFN for the financial support. 89

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