Electroweak Sudakov corrections to New Physics searches at the … New Physics searches. ELECTROWEAK SUDAKOV CORRECTIONS TO NEW PHYSICS SEARCHES AT THE LHC AND FUTURE HADRON COLLIDERS

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

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

Università

degli Studi

di Pavia

Dipartimento di

Fisica

Cover: Comparison between virtual and real electroweak

corrections to the process Z + 3 jets for the | �/HT | observable at√s = 100 TeV under a realistic event selection for the direct

New Physics searches.

ELECTROWEAK SUDAKOV CORRECTIONS TO NEWPHYSICS SEARCHES AT THE LHC AND FUTUREHADRON COLLIDERS

Mauro Chiesa

PhD Thesis - University of PaviaPrinted in Pavia, Italy, November, 2014ISBN: 978-88-95767-79-6

ELECTROWEAK SUDAKOV CORRECTIONS TO NEWPHYSICS SEARCHES AT THE LHC AND FUTURE

HADRON COLLIDERS

Mauro Chiesa

2014

Contents

1 Introduction 1

2 Sudakov logarithms as infrared limit of electroweak correc-tions 5

2.1 Infrared structure of one loop corrections for massless gaugetheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Infrared limit of EW corrections and Sudakov logarithms . . . . 9

3 The Denner-Pozzorini algorithm and its implementation in theALPGEN event generator 11

3.1 The Denner-Pozzorini algorithm: a short review . . . . . . . . . 11

3.1.1 DL contributions . . . . . . . . . . . . . . . . . . . . . . 13

3.1.2 SLC contributions . . . . . . . . . . . . . . . . . . . . . 14

3.2 Implementation of the algorithm in the ALPGEN generator . . . . 15

4 Virtual O(α) Sudakov corrections to New Physics searches atpp colliders 19

4.1 Direct search for New Physics in the channel /ET+ jets . . . . . . 19

4.2 Technical remarks on the computation . . . . . . . . . . . . . . 21

4.3 Phenomenological results . . . . . . . . . . . . . . . . . . . . . . 24

5 Real weak corrections to Z + 2 and Z + 3 jets production 33

5.1 Introduction: real weak corrections . . . . . . . . . . . . . . . . 33

5.2 Real weak corrections to Z + 2 and Z + 3 jets production: aparton level analysis . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 Preliminary results for the real weak corrections to Z + 3 jetsin the CKKW framework . . . . . . . . . . . . . . . . . . . . . . 36

6 Electroweak Sudakov corrections to the Rnγ ratio 47

6.1 Invisible Z+ jets background to New Physics searches in thechannel /ET+ jets . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Virtual O(α) Sudakov corrections to the Rnγ ratio . . . . . . . . 49

i

CONTENTS

7 Conclusions and future perspectives 61

List of publications 63

A Cross-checks and code validation 65A.1 EW Sudakov corrections to Z/γ + 1 jet . . . . . . . . . . . . . . 65A.2 EW Sudakov corrections to Z + 2 jets . . . . . . . . . . . . . . . 66A.3 Electroweak Sudakov corrections to di-jet production . . . . . . 68

B One loop renormalization counterterms 73B.1 On shell renormalization conditions at one loop . . . . . . . . . 73B.2 Unrenormalized self energies . . . . . . . . . . . . . . . . . . . . 74B.3 Definition of the A0, B0 and B1 functions . . . . . . . . . . . . . 78

C Collinear singularities in one loop radiative corrections 81C.1 Collinear singularities in O(α) QED corrections . . . . . . . . . 81

C.1.1 Collinear limit of real O(α) corrections . . . . . . . . . . 81C.1.2 Collinear limit of virtual O(α) corrections . . . . . . . . 84

C.2 Collinear limit of one loop EW Sudakov corrections . . . . . . . 86

D Sudakov corrections for external longitudinal gauge bosons inthe unitary gauge 87

Bibliography 90

ii

Chapter 1Introduction

It is somehow a paradigm that one loop electroweak corrections are smalleffects, 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 thegauge boson masses can be neglected if compared to the kinematical invariantsinvolved in the process, one loop electroweak corrections can be enhancedby large double and single logarithms of the invariants over the gauge bosonmasses. These large, energy growing, logarithms are called Sudakov logarithmsand are the leading part of the one loop O(α) corrections at high energies andin the asymptotic regime described above. The typical effect of the electroweakSudakov corrections is rather small on the integrated cross sections for standardevent selections, while can be of the order of tens of percent in the extremetails of the distributions for several interesting observables.

The presence of a Sudakov logarithmic structure has first been noticed inRef. [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 electroweakcomputation as an asymptotic limit, a complete survey of the existing Next-To-Leading (NLO) electroweak computations is probably beyond the scope ofthis introduction.

In Ref. [2] it has been pointed out for the first time that Sudakov logarithmsare 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 otherenergy scales, so that the weak gauge bosons are basically massless and MV

(V = W,Z) play the role of IR regulators like the infinitesimal mass param-eters which are usually introduced in order to regularize the IR singularitiesappearing in massless gauge theories, like QED.

The results of Ref. [2] have been the starting point of a quite rich andinteresting literature. The key points are the universality of the IR limit ofone loop corrections for massless theories and the well established knowledge ofthe IR behaviour of QED and QCD. Starting from Ref. [2] several works havebeen published which studied the Sudakov part of electroweak corrections in

1

1. Introduction

analogy with the infrared limit of QCD, trying to apply the standard techniquesdeveloped in QCD and pointing out the main differences basically related tothe electroweak spontaneous symmetry breaking and to the different nature ofthe 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 loopcorrections for massless gauge theories, in Refs. [3, 4, 5] a general algorithmhas been developed in order to compute the electroweak Sudakov corrections(for non mass suppressed processes) in a process independent way. Accordingto the algorithm, in the limit in which all the kinematical invariants are of thesame order and much bigger than the gauge boson masses (Sudakov limit),electroweak corrections depend only on the flavours and on the kinematics ofthe external particles of the LO process and can be written as the sum ofuniversal radiator functions which multiply tree level matrix elements. Thisalgorithm has been used in Refs. [6, 7, 8, 9, 10] to compute the EW Sudakovcorrections to Wγ production, WW scattering and V + 1 jet (V = Z, γ,W )production, respectively. The approach of Refs. [3, 4, 5] has been extended inRefs. [11, 12, 13, 14] to the Sudakov part of the two loop electroweak correc-tions.

In Refs. [15, 16, 17, 18, 19, 20, 21] the universality of the infrared limit ofradiative corrections has been used to write the evolution equations (IREE,infrared evolution equations) which describe the scaling of the Sudakov cor-rections 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 Sudakovcorrections has been addressed (an independent study of the exponentiationproperties 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 theAltarelli-Parisi splitting functions for the electroweak vertices has been foundtogether with the expression of the evolution equations for the electroweaksector of the Standard Model. An algorithm to describe weak corrections in aparton 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 correctionsand the gauge boson masses act as infrared cutoffs, these cutoffs are physicalparameters, so that in principle there is no need to include in the computa-tion of the one loop weak corrections the contribution of the extra radiationof 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 meansof arbitrary regulators, which are cancelled only once the contribution of theIR divergent real corrections is included in the calculation). Moreover, real

2

weak corrections usually are not included into EW NLO predictions becausethe additional gauge bosons decay, leading to final states that in principleare not degenerate with the LO ones. However, virtual weak corrections inthe Sudakov limit can be very large (of order -50% or larger), so that theeffect of the partially compensating real weak corrections may not necessar-ily be negligible. The impact of real EW corrections have been studied inRefs. [38, 39, 27, 26, 24, 23, 40, 41] in analogy with QED (i.e. without anycut on the additional gauge bosons) and it was found that the cancellationbetween real and virtual Sudakov logarithms may be only partial (these BlochNordsieck violation effects have been pointed out in Refs. [26, 24, 23]), whilein Refs. [42, 40, 43] the effect of real weak corrections under realistic eventselections has been considered.

This thesis is mainly focused on the implementation of the algorithm ofRefs. [3, 4, 5] (Denner-Pozzorini algorithm in the following) in the LO eventgenerator ALPGEN [44] and on the analysis of the phenomenological impact ofthe electroweak Sudakov corrections at the LHC and at future proton-protoncolliders. The basic features of the electroweak Sudakov corrections are de-scribed in chapter 2, pointing out the main similarities and differences withrespect to the infrared limit of one loop corrections in massless gauge theo-ries, such as QED and QCD. The Denner-Pozzorini algorithm and its imple-mentation 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 impactof the one loop electroweak corrections in the Sudakov limit to the processZ(→ νν) + n jets (with n ≤ 3), which is an irreducible Standard Model back-ground to the direct search for New Physics in the channel /ET+ jets at theLHC and at future hadron colliders. The effect of real weak corrections to Z+2and Z+3 jets production and the partial cancellation between real and virtualweak corrections in the Sudakov limit are studied in chapter 5. In chapter 6the electroweak Sudakov corrections to the process γ + n jets are consideredin order to compute the corrections to the Rn

γ = dσ(Z + n jets)/dσ(γ + n jets)ratio, which is the theoretical input for the partially data driven estimate ofthe Z(→ νν) + n jets background based on the experimental measurement ofthe differential distributions for γ + n jets events.

3

1. Introduction

4

Chapter 2Sudakov logarithms as infraredlimit 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, suchas QED and QCD, besides the ultraviolet (UV) divergences related to the be-haviour of the loop integrals in the q2 → ∞ limit (q being the loop momentum)and cancelled order by order in perturbation theory by means of the renormal-ization procedure, also infrared (IR) singularities arise related to the presenceof massless particles inside the loop diagrams.

In this section the basic features of the IR limit of one loop corrections formassless gauge theories are briefly recalled (basically following Ref. [45]), inorder to better explain the main differences and similarities with respect to theIR limit of one loop electroweak corrections. This section is mainly focusedon QED, where the IR singularities are usually regularized by means of massregularization, that is giving an unphysical mass to the massless particles (inthe case of QCD the standard regularization procedure used in the literatureis the dimensional regularization for both UV and IR divergences). Strictlyspeaking 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 IRdivergences related to the electron mass, the would-be singularities show up aslarge logarithms of m over the typical energy scale of the process and m canbe considered as an IR cutoff.

A process-independent analysis of the IR singularities appearing in oneloop 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− 2εdimensions reads:

TNµ1···µN

(p0, · · · , pN−1;m0, · · · ,mN−1) =(2πµ)4−D

iπ2

�dDq

qµ1 · · · qµN

N0 · · ·NN−1

, (2.1)

5

2. Sudakov logarithms as infrared limit of electroweak corrections

where the parameter µ has a mass dimension and serves to keep the dimensionof the integral constant as a function ofD, whileNk represents the denominatorof the loop propagator with momentum q + pk and internal mass mk, namely:

Nk = (q + pk)2 −m2

k + iε (k = 0, · · · , N − 1). (2.2)

Taking eq. (2.1) as a starting point, in Ref. [46] it has been proved that IRsingularities arise from the following two situations:

•

0

m2

m2

(m → 0),

(2.3)

soft singularities are related to the diagrams in which a massless particleis exchanged between two external on shell legs and originate from theintegration region where q → −pk (so that the momentum transfer inthe propagator k is zero);

•

0

0

m2 (m → 0),

(2.4)

collinear singularities are related to the diagrams in which an externalon shell particle splits into two internal massless lines and arise from theintegration region q → −pk + x(pk − pk+1) (x being an arbitrary realvariable), that is when the momentum flowing in the propagator k iscollinear to the external momentum (pk − pk+1).

For the case of QED, soft singularities are related to the diagrams involvingthe exchange of a photon between two external legs and to the electron selfenergy (which enters the electron wave function renormalization counterterm),while collinear singularities correspond to the splitting of an external electroninto an internal electron plus a photon. When regularized by means of massregularization, giving an infinitesimal mass λ to the photon, the IR singularitiesappear as logarithms of the IR cutoffs λ and m.

Taking eqs. (2.3)-(2.4) as a starting point, in Ref. [47] a general algorithmhas been developed for the separation of the soft/collinear structure of anarbitrary N point loop integral. In particular, in Ref. [47] the loop structurein the singular regions is decomposed at the integrand level as:

N−1�

k=0

1

Nk

� Asoftn

Nn−1NnNn+1

, (2.5)

N−1�

k=0

1

Nk

��

k �=n,n+1

Acollnk

NnNn+1Nk

, (2.6)

6

2.1. Infrared structure of one loop corrections for massless gauge theories

(where n is the index labelling the singular region considered) so that theIR part of the general N > 3 function reduces to a linear combination ofscalar Passarino-Veltman three point functions C0 [48, 49, 50]. In Ref. [47] theuniversal expression of the A coefficients is also given, together with a list ofthe IR singular C0 functions.

Once the IR limit of the loop integrals have been isolated the virtual oneloop matrix elements in the IR limit can be written in a process independentway 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 eikonalapproximation (that is neglecting the photon momentum and the masses in thenumerator of the propagators). First of all, the matrix element for the processi → f with the radiation of an additional (also off shell) photon becomes:

Mi→f+γµ (p, q) −→

q→0Mi→f

0 (p)�

l

2eQlpl, µ(q + ηlpl)2 −m2

l

(2.7)

where the photon leg with momentum q has been truncated, Mi→f0 (p) is the

matrix element for the LO process i → f , the factor ηk is set to 1 if the photonis radiated off an outgoing fermion line k (to −1 if k labels an incoming fermionline) and the index l spans all the external fermions with momentum pl andelectric chargeQl. Then the soft limit of one loop virtual corrections is obtainedconnecting each pair of terms appearing in eq. (2.7) with a photon propagator

−igµν

q2−λ2+iε(in the ξ = 1 gauge):

Mi→fVirt. soft =

α

4πMi→f

0

1

2

�

l,m

QlQmIlm, (2.8)

I lm =(2πµ)4−D

iπ2

�dDq

4(plpm)�q2 − λ2

��(q + ηlpl)2 −m2

l

��(q + ηmpm)2 −m2

m

� ,

(where the factor +iε in the loop denominators is understood). The loopintegrals in eq. (2.8) are the C0 functions entering eq. (2.5), while the diagonalterms m = l lead to the single soft logarithm associated with the electron wavefunction renormalization counterterm.

The soft limit of the virtual one loop QED corrections (2.8) is a functionof the unphysical parameter λ. However, also the process i → fγ when thephoton is not detected because it is either soft (i.e. below the detector energyresolution ΔE) or collinear with its emitter gives rise to O(α) corrections tothe LO process i → f . These real O(α) corrections in the soft limit can beobtained from eq. (2.7) considering the emission of a real photon and thusconnecting each pair of emitting legs in eq. (2.7) with the sum over the photon

7


polarizations and integrating over the soft photon phase space:

|Mi→fγReal soft|2 = e2|Mi→f

0 |2�

soft

d3�q

(2π)32q0

�

l,m

QlηlQmηm(plpm)

(qpm)(qpl)

=α

2π|Mi→f

0 |2�

l,m

� ΔE

λ

dq0q0

� 1

−1

dcosθqQlηlQmηm(plpm)

ElEm(1− βlcosθql)(1− βmcosθqm).

(2.9)

The integrand in eq. (2.9) is again divergent in the soft limit q0 → 0 and in thecollinear limit θqj → 0 (in the case of photon radiation off a massless fermion,

that is βj =�1−m2

j/E2j → 1, j = l,m). While the finite electron mass

prevents from the collinear singularities, soft divergences are regularized by thelower bound of integration λ, that effectively corresponds to a mass parameterfor the emitted photon as the one introduced in eq. (2.8) (in eq. (2.9) theupper bound of integration is fixed by the degeneracy condition q0 ≤ ΔE andthe integral can be performed in four dimensions since no UV singularities arepresent in the real corrections). The integral in eq. (2.9) can be computedfollowing Ref. [48] and turns out that it develops the same kind of logarithmicstructure 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 massλ coming from eq. (2.8) and (2.9) are the same but with opposite sign, so thatthe sum of the real and virtual O(α) corrections is no longer a function ofthe arbitrary cutoff λ. This cancellation is an example of the Bloch-Nordsiecktheorem [51], which states that in the computation of radiative correctionsfor abelian gauge theories all the logarithms of the IR cutoffs cancel order byorder in perturbation theory in the sum of virtual and real contributions forinclusive observables. The Bloch-Nordsieck theorem also rules the cancellationof the collinear singularities in the m → 0 limit: a very short overview of thebasic features of the collinear limit of one loop QED corrections is given inAppendix C.

As regards the IR limit of the one loop QCD corrections, the behaviour ofthe loop integrals in the soft/collinear regions is the same as the one describedabove for the case of QED, while the factorization of the one loop correctionsis more involved since the radiation of additional real/virtual gluons changesthe original colour structure of the considered process, due to the fact thatthe 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 thevirtual corrections factorizes on the same LO matrix element, the O(αS) QCDcorrections in the IR limit factorize into radiator functions which multiplytree level matrix elements that are the LO amplitude and its colour correlatedamplitudes. However, the IR singularities again cancel in the sum of real andvirtual corrections for inclusive observables if, besides the sum over the finalstate colours, also the average over the colours of the initial state particles isconsidered, as stated by the KLN theorem [46], [52].

8

2.2. Infrared limit of EW corrections and Sudakov logarithms

2.2 Infrared limit of EW corrections and Su-

dakov logarithms

As pointed out in Refs. [1, 2], Sudakov logarithms correspond to the IR limitof the electroweak radiative corrections. In fact, when the considered energyscales are much larger than the weak gauge boson masses, MW and MZ canbe regarded as IR regulators for the loops involving the exchange of weakbosons and play the same role as the mass parameter λ, that was introducedin the previous section in order to regularize the QED IR singularities arisingfrom loop diagrams in which a soft/collinear photon is connected to externalon shell particles (with the following difference: in mass regularized QED it isassumed that λ < m in the intermediate steps of the calculations, while for EWcorrections MV > mf , mf being the mass of a light fermion and V = W,Z).

While the logarithmic structure of the EW Sudakov corrections shares thesame features as the one arising from the IR limit of QED or QCD in massregularization, since it is determined by the singular behaviour of the loopintegrals, the factorization properties of the Sudakov corrections are similarto the ones of the IR part of the QCD corrections, due to the non abelianstructure of the underlying gauge symmetry group SU(2)L×U(1)Y . For thesereasons, in the literature, Sudakov logarithms have been mainly studied inanalogy with the IR limit of QCD. There are, however, several differencesbetween the IR limit of EW and the one of QCD corrections, basically relatedto the spontaneous symmetry breaking of the EW theory and to the nature ofthe weak charges.

The spontaneous symmetry breaking of the EW theory implies that thegauge boson masses are physical parameters. In particular, this means that,even if from a formal point of view in the high energy limit MW and MZ actas IR regulators for the loop diagrams, they cannot be set to zero and theone loop virtual weak corrections are always finite (but potentially very largesince they are enhanced by large logarithms of the gauge boson masses overthe energy scale of the considered process).

As stressed in Refs. [3, 4, 5] , another less trivial consequence of the sponta-neous symmetry breaking of the EW theory is that also the scalar sector entersthe corrections both because of the correspondence between longitudinal gaugebosons and would-be Goldstone bosons and because the collinear factorizationis based on the BRS invariance of the full Standard Model.

Since the one loop virtual weak corrections are finite by themselves, theycan be computed separately from the O(α) contributions associated with thediagrams obtained from the ones of the LO process with the additional radi-ation of a real weak boson. Moreover, the additional gauge bosons decay sothat in principle their decay products lead to final states which are not degen-erate with the considered signature. For this reason the contribution of realweak corrections usually is not included in the computation of one loop EWcorrections.

9


In the high energy limit, where the size of the negative one loop virtualweak corrections can be of the order of several tens of percent, the positivecontribution of real weak corrections can lead to significant compensationsbetween real and virtual corrections. Of course, for realistic event selectionsthe size of the cancellation is strongly dependent on the specific observables andcuts considered, as will be discussed in chapter 5. However, also in the QED orQCD-like case of observables which are completely inclusive over the additionalradiation of real gauge bosons, the cancellation of the Sudakov logarithms inthe sum of virtual and real contributions may be only partial. This feature ofthe 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 EWtheory. The cancellation between real and virtual IR singularities for inclusiveobservables in QCD takes place according to the KLN theorem only after thatthe sum over the colours of the final state particles and the average over thecolours of the initial state ones is performed. This sum/average over the coloursis needed in QCD because the colour charges do not exist as free asymptoticstates, while this is not the case for the weak isospin that basically correspondsto the EW charge. As a result, Bloch Nordsieck violations arise from theincomplete sum/average over the weak isospins of the external particles of theconsidered process.

10

Chapter 3The Denner-Pozzorini algorithmand its implementation in theALPGEN event generator

3.1 The Denner-Pozzorini algorithm: a short

review

As already stated in the introduction, the main topic of this thesis is thephenomenological study of the impact of the one loop electroweak correctionsin the Sudakov limit to the Z and γ plus jets production processes whichare of great interest for the direct search for New Physics at the LHC andat future pp colliders. These corrections have been computed by means ofthe Denner-Pozzorini algorithm which has been implemented in the ALPGEN

generator [44]. In this section the basic formulas proved in Refs. [3, 4, 5] arevery briefly summarized in order to better explain the technical aspects of theimplementation of the algorithm.

The starting point is the fact that electroweak Sudakov logarithms arethe infrared limit of virtual one loop EW corrections and the IR structure ofone loop corrections is universal, depending only on the flavour and on thekinematics of the particles of the considered process at the LO. Formally, theasymptotic IR limit is defined by the requirement that all the kinematicalinvariants involved in the process under consideration are of the same orderand much larger than the W and Z boson masses (Sudakov limit). Actually,there are also single Sudakov logarithms related to the one loop renormal-ization: they also are universal, since they are ruled by the renormalizationgroup equations, and are naturally encoded in the usual expression of the EWrenormalization counterterms (collected in Appendix B) that can be found forexample in Ref. [53].

Concerning the IR structure of virtual one loop corrections for masslessgauge theories, as already discussed in the previous chapter, in Ref. [46] it has

11

3. The Denner-Pozzorini algorithm and its implementation in the ALPGEN eventgenerator

been proved that infrared singularities arise from two classes of diagrams: fromthose diagrams in which a massless propagator is connected to two externalon shell legs and from the ones in which an on shell external leg splits intotwo internal massless lines. Once the singularities are regularized giving aninfinitesimal mass to the massless propagators, they appear as logarithms ofthese masses. Of course, this is a gauge dependent statement which rules forthe ξ = 1 gauge: all the intermediate steps of the calculation are performedassuming the ξ = 1 gauge, while the final result is gauge independent. In theSudakov limit all the particles of the Standard Model are basically masslesscompared to the other energy scales involved, so that the W and the Z bosonscan be considered as almost massless and their masses play the role of (phys-ical) IR cutoffs. Only internal bosonic propagators lead to the Sudakov loga-rithms, as can be easily understood by direct inspection of the SM vertices andneglecting all the mass dependent ones (since they would give contributionslike M 2log(s/M 2), in the limit M 2 → 0). This kind of argument, however,requires some care when external longitudinally polarized gauge bosons areconsidered: for this reason, in Refs. [3, 4, 5] longitudinal gauge bosons are re-placed with the would-be Goldstone bosons by means of the Goldstone bosonequivalence 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) partand the remaining single logarithms associated with the renormalization coun-terterms. The DL part has the form:

δDLMNLOi1···in =

N�

k=1

�

l>k

δDLkl MLO

i1···jk···jl···in , (3.1)

where the indexes k and l span all the electroweak charged external legs ofthe LO order process, N is the number of these legs, δDL

kl is a radiator func-tion which depends on the flavours of the particles ik and il and containsthe double logarithm of their invariant mass over the gauge boson masses,while MLO

i1···jk···jl···in is the tree level matrix element involving the external legsi1 · · · jk · · · jl · · · in (jk = ik and jl = il in the case of the exchange of a photonor a Z boson between the two legs k and l, otherwise jk and jl are the SU(2)transformed of ik and il, respectively).

The single logarithmic part of the corrections coming from the unrenormal-ized one loop matrix element is:

δSLCMNLOi1···in =

N�

k=1

δSLCk MLOi1···jk···in , (3.2)

where the notation is the same as in eq. (3.1), the radiator functions containingthe single logarithms factorize on each external leg separately and MLO

i1···jk···indiffers from the matrix element of the LO process only for the possible mixingbetween neutral gauge bosons or scalars.

12

3.1. The Denner-Pozzorini algorithm: a short review

The single logarithms associated with the fields renormalization are ob-tained from the fields wave function renormalization counterterms by settingthe renormalization scale to the one of the kinematical invariants of the pro-cess in order to extract the mass singular part of these counterterms. In thecase of external longitudinally polarized gauge bosons replaced by the corre-sponding would-be Goldstone bosons, the one loop corrections to the GBETare also included, as described in Refs. [3, 4, 5]. Once added the SLC part,these contributions can be written as

δSLMNLOi1···in =

N�

k=1

δSLk MLOi1···jk···in . (3.3)

The remaining single logarithms come form the running of the dimension-less parameters (the renormalization of the mass parameters leads only to masssuppressed contributions):

δPRMNLOi1···in = δe

δMLOi1···inδe

+ δcWδMLO

i1···inδcW

+ δht

δMLOi1···in

δht

+ δheffH

δMLOi1···in

δhH

, (3.4)

where ht = mt/MW , hH = M2H/M

2W and cW = MW/MZ . As for the fields wave

function renormalization, the logarithmic structure of eq. (3.4) is obtained fromthe expression of the counterterms by setting the renormalization scale to theone of the kinematical invariants of the process (in δheff also the Higgs tadpolecontribution has to be included [54]).

The next two subsections summarize the most important steps of thederivation of eqs. (3.1) and (3.2).

3.1.1 DL contributions

The DL contributions arise from the one loop diagrams in which a soft-collineargauge boson is exchanged between two external legs. Using the graphicalnotation of Refs. [3, 4, 5], eq. (3.1) reads:

δDLM(i1 · · · in) =�

V a=γ,Z,W±

�

k

�

l > k

�

jl, jk

V a

ik

il

jk

jl

,

(3.5)

where the blob represents the sum of all the tree level diagrams for the externalflavour string (i1 · · · jk · · · jl · · · in) with amputated off shell legs jl and jk. jland jk are equal to il and ik if V a = Z or γ, otherwise they are the SU(2)transformed of il and ik, respectively.

As for the case of QED, the soft-collinear limit of each of the diagramsin eq. (3.5) can be computed in the so called eikonal approximation (i.e. ne-glecting the masses and the loop momentum in the numerators), while the ex-pressions for the vertices V aikjk and V ailjl can be worked out using the Dirac

13


equation for external fermions, or the transversity condition together with theGBET for external transverse gauge bosons. Once all the diagrams in eq. (3.5)have been computed in the eikonal approximation, eq. (3.5) becomes:

δDLM(i1 · · · in) =N�

k=1

�

l<k

�

V a=γ,Z,W±

IVa

jk,ikIV

a

jl,ilM0(i1 · · · jk · · · jl · · · in)δV

a

k,l ,

(3.6)where IV

a

jm,im is the gauge group generator corresponding to the V aimjm ver-tex (m = k, l), M0(i1 · · · jk · · · jl · · · in) is the matrix element for the processinvolving the external particles (i1 · · · jk · · · jl · · · in) and δV

a

k,l is the radiatorfunction associated to the exchange of a V a boson between the two exter-nal particles k and l. Of course, the external legs for the NLO process arestill (i1 · · · ik · · · il · · · in), while in the r.h.s. of eq. (3.6) the tree level matrixelements are evaluated for the flavour string (i1 · · · jk · · · jl · · · in), but the re-placement of the external wave functions for the fields ik and il with the onesfor jk and jl leads only to mass suppressed effects (once the GBET is used forthe longitudinal gauge bosons).

In eq. (3.6) the loop contributions have been factorized as:

δVa

k,l = −ie2�

d4q

(2π)44(pkpl)

(q2 −M2V a)[(pk + q)2 −m2

jk][(pl − q)2 −m2

jl], (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 resultcan also be found in Ref. [47]).

3.1.2 SLC contributions

The SLC contributions are associated to the collinear splitting ik → jkVa,

where ik is an external on shell leg and the gauge boson V a becomes collinearto the internal line jk. With the notation of Refs. [3, 4, 5], the SLC terms arisefrom the contributions of the form:

V a

jkik ,

(3.8)

where the blob is again the sum of all the tree level-like diagrams with exter-nal legs (i1 · · · jk · · · in, V a) and jk, V

a are off shell. Each of the diagrams ineq. (3.8) can be written as:

I = −i(4π)2µ4−D

�dDq

(2π)DN(ik, jk, V

a; q)

(q2 −M2V a + iε)[(p− q)2 −M2

jk+ iε]

, (3.9)

where N(ik, jk, Va; q) represents a contribution to the blob in eq. (3.8).

14

3.2. Implementation of the algorithm in the ALPGEN generator

Since part of the collinear singularities appearing in eq. (3.9) have alreadybeen computed in eqs. (3.6)-(3.7), in order to isolate the SLC part of thecorrections the results of eqs. (3.6)-(3.7) must be subtracted from eq. (3.9),obtaining:

δSLCM(i1 · · · ik · · · in) =

=�

V a=A,Z,W±

�

jk

V a

jkik

trunc.

−�

l �= k

�

jl

V a

ik

il

jk

jl

eik.

coll.

=�

jk

jkδSLC(jk, ik)M0(i1 · · · jk · · · in),

(3.10)

where the last line is obtained by means of the collinear Ward identities provedin Ref. [4] after that the collinear logarithmic part of the integral in eq. (3.9)has been factorized out (as described in Appendix C), M0(i1 · · · jk · · · in) is thematrix element for the flavour string (i1 · · · jk · · · in) (with jk �= ik in the caseof mixing between neutral gauge bosons or scalars), while δSLC(ik, jk) is thesingle logarithmic radiator function corresponding to the external leg ik.

The proof of the collinear Ward identities of Ref. [4] relies on the BRSsymmetry of the spontaneously broken electroweak theory. Since the details ofthe proofs are different for external scalars, fermions and gauge bosons, theyare 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 theSudakov limit can be written in a factorized form as the sum of contributionsthat consist of radiator functions multiplied by tree level matrix elements. Theradiator functions are universal due to the universality of the infrared struc-ture of the virtual one loop corrections, while the tree level matrix elementsneeded are the ones of the considered LO process and its SU(2) correlatedamplitudes. The algorithm is thus well suited for the implementation in a LOevent generator such as ALPGEN which provides automatically all the matrixelements needed.

ALPGEN is basically a collection of process-specific packages for several hardprocesses interfaced to a process-independent section of code. In the followingchapters only the vbjet package will be considered: it deals with the pro-duction of nW + mZ + jγ + lH + k jets with n + m + j + l + k ≤ 8 andk ≤ 3. The process-dependent packages generate the phase space points to-gether with the flavour and the helicity configurations, while the common part

15


of the code uses these variables as input to compute the corresponding treelevel matrix element through a call to the matrix routine (which implementsthe ALPHA algorithm [56, 57, 58]) and takes care of the integration proce-dure. ALPHA is based on the Dyson-Schwinger method to recursively defineone-particle off-shell Green’s functions, which are computed numerically. Inorder to implement the Denner-Pozzorini algorithm in the ALPGEN generator,the program work-flow is left unchanged, simply the LO matrix element inthe integral is multiplied by a factor 1 + 2Re δEW . The computation of thecorrection δEW is briefly described in the following paragraphs.

The algorithm of Refs. [3, 4, 5] has been implemented in the vbjet packageof the ALPGEN generator basically following eqs. (3.1), (3.3) and (3.4). First ofall the double logarithmic part of the corrections is computed.

• For each pair of external EW charged legs the radiator contribution as-sociated with the exchange of either a Z boson or a photon is computedas 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 alsoallowed, the code works as follows:

– the original LO flavour string (i1 · · · ik · · · il · · · in) is mapped in theSU(2) transformed one (i1 · · · jk · · · jl · · · in);

– the kinematics of the event is rescaled in order to put the two trans-formed legs k, l on their mass shell preserving the transverse and thelongitudinal momenta of all the final state particles. More precisely,the kinematics is modified only if one of the two legs k, l is a finalstate vector boson V (V = Z, γ,W ), since the vbjet package onlydeals with light (massless) quarks: when m(ja) �= m(ia) (a = k, l),the energy of the leg a is rescaled according to the mass-shell condi-tionm2(ja) = E2(a)−|�p(a)|2, this of course changes the total energyin the final state, so that the energy and the longitudinal momen-tum of the initial state particles are fixed by the four-momentumconservation.

– The tree level matrix element for the SU(2) transformed flavourstring (i1 · · · jk · · · jl · · · in) is evaluated numerically by means of theALPHA algorithm through a call to the matrix routine.

While needed in order to provide on shell particles as input for the matrixroutine, the transformation of the kinematics has basically negligibleimpact in the applications discussed in chapters 4, 5 and 6 for the eventselections considered.

The SL part of the correction contains both the SLC terms defined insubsection 3.1.2 and the logarithmic part of the wave function renormalizationcounterterms. The SL contribution is computed in analogy with the DL one:for each EW charged external leg k, the corresponding radiator function is

16

3.2. Implementation of the algorithm in the ALPGEN generator

computed together with the transformed tree level matrix element needed (inthe case of the mixing between neutral gauge bosons or scalars) after the usualtransformation 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 arebasically the same (2pkpl � r for each pair of external legs k and l), and thescale r in the argument of the single logarithms of eq. (3.3) is naturally fixedto the one of the kinematical invariants. For the realistic applications of thealgorithm, however, even if the invariants should be of the same order, they arenot identical, so that in the implementation of the algorithm the scale of the SLpart is chosen event by event as the mean of the largest and the smallest of theinvariants: clearly it is an arbitrary choice, however different scale choices leadto 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 theones considered in chapters 4, 5 and 6.

While the SLC part of the correction and the logarithmic structure of thewave function renormalization counterterms share the same factorization prop-erties and can be cast in the form (3.3), in general this is not the case for thesingle logarithms coming from parameter renormalization. However, in Ap-pendix E of Ref. [5] has been shown that for processes involving the productionof an arbitrary number of transverse vector bosons in fermion anti-fermion an-nihilation also the logarithms related to the parameter renormalization can beincluded in eq. (3.3): the argument of Ref. [5] can be easily extended to theprocesses V +N jets considered in chapters 4, 5 and 6 for the most relevant par-tonic subprocesses of order α

Njets

S α. Concerning the present implementation ofthe Denner-Pozzorini algorithm in the ALPGEN generator another remark is inorder: the matrix elements provided by ALPHA for the processes Z +N jetsand γ + N jets (N = 1, 2, 3) considered in the phenomenological studies of

the following chapters contain both contributions of order αNjets

S α and of order

αNjets−2S α3 (for N > 1) which cannot be disentangled. Of course the parameter

renormalization for the two classes of processes is different: in the present ver-sion of the code the contribution of the parameter renormalization is correctlyimplemented only for the dominant order α

Njets

S α subprocesses (the impact ofthis approximation will be discussed in subsection 4.2).

As discussed in chapter 4, for the phenomenological results shown in thisthesis the Sudakov corrections for the longitudinally polarized gauge bosonshave been neglected. However, also these contributions can be included asdescribed in Ref. [7]. Since the tree level matrix elements provided by ALPHAare computed in the unitary gauge, the amplitudes involving external would-beGoldstone bosons are not directly available. However the Sudakov correctionsfor the case of longitudinally polarized gauge bosons can be obtained by usingthe tree level matrix elements for external longitudinal gauge bosons with theformulas of Refs. [3, 4, 5] for the radiator functions associated with externalwould-be Goldstone bosons, once these formulas have been modified in orderto account for the phase transformation which relates the tree level amplitudes

17


for the longitudinal gauge bosons and the ones for the corresponding would-beGoldstone bosons (see for example Appendix D).

18

Chapter 4Virtual O(α) Sudakovcorrections to New Physicssearches at pp colliders

4.1 Direct search for New Physics in the chan-

nel /ET+ jets

One of the main tasks of the LHC proton-proton collider is the search forPhysics beyond the Standard Model. Among the possible extensions of the SM,Supersymmetry (SUSY) is probably one of the most theoretically explored andappealing, since it addresses the main issues of the SM like the mass hierarchyand the naturalness problem. SUSY relates fermionic and bosonic degrees offreedom postulating for each SM particle a supersymmetric partner with thesame quantum numbers but opposite spin-statistics. According to R-parityconserving SUSY models, SUSY particles should be produced in pairs anddecay into SM particles plus light stable supersymmetric particles (LSP). Ifthe LSP are weakly interacting and neutral, like the neutralino, they alsoprovide a possible candidate for Dark Matter. According to these theories, atthe LHC coloured SUSY particles (squarks and gluinos) should be producedin pairs, interact strongly producing jets, and decay into LSP which escapedetection, leading to signatures with missing transverse energy /ET and severaljets. The amount of /ET and the pT scale of the jets should be of hundreds ofGeV, in order to fulfil the lower limits on the SUSY particles masses set bythe previous results of the direct searches for New Physics (NP) performed atLEP and TEVATRON.

Both the ATLAS [59, 60] and CMS [61, 62] Collaborations published theirresults on the direct search for NP in the channel /ET + multi-jets. Theseanalyses require at least two or three jets at high pT and a large amount ofmissing momentum. Additional cuts on the total transverse momentum of thejets HT =

�j p

jT and on the angular separation between the jet �pT s and the

19

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 /ET + multi-jets ismade up of pure QCD multi-jets events (with large missing momentum comingfrom leptonic decays of heavy-flavour hadrons inside the jets or jet energymismeasurement), W + jets events (when the W boson decays leptonicallyand the charged lepton is not detected), Z + jets events (when the Z bosondecays into neutrinos) and t t events. Among these processes, Z + multi-jetswith Z → νν turns out to be the most important irreducible SM backgroundfor the event selections with up to three jets.

The background processes V+jets (V = W,Z) have been widely studiedin 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 partonshower generator SHERPA can be found in Refs. [63, 43] and [64], respectively.Fixed-order (NLO) QCD predictions for the production of a vector boson inassociation with 5 jets at hadron colliders are presented in Ref. [65]. Leadingand 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 EWand QCD corrections to the same processes have been computed using thesoft and collinear effective theory in [32]. The exact NLO EW calculation forV = 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]. TheNLO EW calculation for Z(νν)+ 2 jets, for the partonic subprocesses withone fermion current only (i.e. including only gluon-gluon contributions toZ + 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 limitincluding all partonic subprocesses have been computed using the Denner-Pozzorini algorithm [3, 4, 5] implemented in the ALPGEN multiparticle eventgenerator [44]. Recently exact O(α) corrections for the process Z+2 includingalso four quark subprocesses have been presented in Ref. [74].

In this chapter the results of Refs. [73] and [75, 76, 77] are collected, showingthe effect of one loop virtual weak corrections in the Sudakov limit for theprocesses Z + n jets (with n = 1, 2, 3) as SM background to the direct searchfor NP at the LHC and future proton-proton colliders. In the following theresults for Z+2 jets are given for a set of cuts which mimics the ATLAS eventselection of Ref. [59], namely:

meff > 1 TeV /ET/meff > 0.3

pj1T > 130 GeV pj2T > 40 GeV |ηj| < 2.8

Δφ(�pjT , /�pT ) > 0.4 ΔR(j1,j2) > 0.4 , (4.1)

where the effective mass is defined as meff =�

i pT i + /ET and j1, j2 are thehardest and next-to-hardest jet, respectively. For Z +3 jets the CMS baseline

20

4.2. Technical remarks on the computation

selection of Ref. [61] is considered:

HT > 500 GeV |/�HT | > 200 GeV

pjT > 50 GeV |ηj| < 2.5 ΔR(ji,jk) > 0.5

Δφ(�pj1,j2T , /�HT ) > 0.5 Δφ(�pj3T , /�HT ) > 0.3, (4.2)

where HT =�

i pT i,�/HT = −�i �pT i and j1, j2, j3, are the hardest, the next-to-

hardest and the remaining jet, respectively. Finally, for the numerical resultsat

√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 forthe jets are kept fixed (as they are essentially determined by the geometry of

the detector), the pjT and meff (or | �/HT | and HT ) cuts are rescaled as the centreof mass energy grows form 14 to 33 or 100 TeV. The rescaled event selectionadopted for Z + 2 jets is:

meff > 2(7) TeV , (√s = 33(100) TeV) ,

pj1T > 260(910) GeV , (√s = 33(100) TeV) ,

pj2T > 80(280) GeV, (√s = 33(100) TeV) ,

|ηj| < 2.8, ΔR(j1,j2) > 0.4, Δφ(�pjT , /�pT ) > 0.4, (4.3)

while the rescaled cuts for Z + 3 jets are:

HT > 1(3.5) TeV , (√s = 33(100) TeV)

|/�HT | > 0.4(1.4) TeV , (√s = 33(100) TeV)

pjT > 100(350) GeV , (√s = 33(100) TeV) ,

|ηj| < 2.5, Δφ(�pj1,j2T , /�HT ) > 0.5, Δφ(�pj3T , /�HT ) > 0.3. (4.4)

4.2 Technical remarks on the computation

In this chapter the electroweak Sudakov corrections to the processes Z+2 andZ + 3 jets are computed using the Denner-Pozzorini algorithm [3, 4, 5]. Theunderlying hypothesis of the algorithm is that all the kinematical invariantsof the electroweak charged legs are of the same order and larger than the Wmass. Figure 4.1 (left plot) shows the maximum invariant mass distributionsfor 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 invariantmasses between electroweak charged particles at the parton level: most of theevents are characterized by at least one invariant mass above few hundredsGeV. For the event selections considered, the approximation of Refs. [3, 4, 5]is still expected to hold, since the radiator contributions depending on largekinematical invariants are reliable, whereas those depending on small kine-matical invariants (which are at least of order MW , as shown in the plot onthe right of fig. 4.1) may lead to unreliable contributions, which, however, are

21


10−9

10−7

10−5

10−3

10−1

100 500 1000 1500 2000

dσ

dm

MAX

�pb

GeV

�

mMAX [GeV]

ATLAS 7 TeV

ATLAS 14 TeV

CMS 7 TeV

CMS 14 TeV

10−9

10−7

10−5

10−3

10−1

100 500 1000 1500 2000

dσ

dm

min

�pb

GeV

�

mmin [GeV]

ATLAS 7 TeV

ATLAS 14 TeV

CMS 7 TeV

CMS 14 TeV

Figure 4.1: Maximum (left plot) and minimum (right plot) invariant massdistributions for Z + 2 and Z + 3 jets under the ATLAS and CMS cuts ofeqs. (4.1)-(4.2)

numerically below the stated accuracy, since the arguments of the involvedlogarithms are of order one. Figure 4.1 can be considered as a first assessmentof the applicability of the method: a detailed description of the validation ofthe calculation can be found in Appendix A.

Figures 4.2-4.5 show the relative importance of the partonic subprocessesat the leading order. While the subprocesses with one fermion current (curveA) are the main part of the cross section, the relative importance of the sub-processes with four identical quarks (curve B) and with two pairs of identicalquarks in the same isodoublet (e.g. ud → udZ(g), curve C) in the tails of theconsidered distributions is of order 45, 50%. The contribution of the other fourquark subprocesses is small even if their sum is not negligible.

The Denner-Pozzorini algorithm has been implemented in the vbjet.f rou-tine: this means that the LO subprocesses considered are both of order α2

Sαand of order α3 for Z+2 jets (α3

Sα and αSα3 for Z+3 jets). While the double

and the single collinear logarithmic part of the Sudakov corrections are thesame for the two classes of subprocesses, the single logarithms coming fromthe running of the electroweak parameters are different: in the present imple-mentation of the algorithm it is assumed that the dominant subprocesses areof order α

Njets

S α. At 14 TeV the size of the purely weak LO subprocesses in thetails of the distributions can be conservatively estimated to be of order 8, 10%by comparing the left and the right plots of figs. 4.2-4.5 (in the plots on the

right only the order αNjets

S α contributions have been considered): in principle

22

4.2. Technical remarks on the computation

0.0

0.2

0.4

0.6

0.8

1.0

1000 1500 2000 2400

dσi

dm

eff/dσLO

dm

eff

meff [GeV]

√s = 7 TeV

ABC

0.0

0.2

0.4

0.6

0.8

1.0

1000 1500 2000 2400

dσi

dm

eff/dσLO

dm

eff

meff [GeV]

√s = 7 TeV

QCD only

ABC

Figure 4.2: Relative importance of the most relevant partonic subprocessesfor 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 identicalquarks (e.g. uu → uuZ) and with two pairs of identical quarks in the sameisodoublet (e.g. ud → udZ), respectively. The plots on the right are obtainedconsidering only the QCD LO contributions.

the corrections to this part of the cross sections are overestimated, howeverin 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 byfar the most relevant one, so that the overall result is still within the statedaccuracy.

For the leading order αNjets

S α partonic subprocesses the Z boson can onlybe emitted by a quark line. As a result, the contribution of longitudinallypolarized Z bosons is suppressed at high energy (fig. 4.6), where the longi-tudinal gauge bosons behave like the would-be Goldstone bosons accordingto the Goldstone boson equivalence theorem. For this reason, the Sudakovcorrections to the longitudinally polarized Z bosons are not included in thecomputation.

As a conclusive remark, on one hand QED corrections are a gauge invariantsubset of the one loop electroweak corrections to Z + 2 and Z + 3 jets for theO (α

Njets

S α) partonic subprocesses, while on the other hand they turn outto be rather small for fully inclusive setup (see for instance the results ofRefs. [10, 67]). Since this chapter is mainly focused on the effect of weakcorrections, following Ref. [8] the QED contribution is not included in thenumerical results.

23


0.0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000

dσi

dm

eff/dσLO

dm

eff

meff [GeV]

√s = 14 TeV

ABC

0.0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000

dσi

dm

eff/dσLO

dm

eff

meff [GeV]

√s = 14 TeV

QCD only

ABC

Figure 4.3: Relative importance of the most relevant partonic subprocessesfor Z + 2 jets under the cuts of eq. (4.1) at

√s = 14 TeV. Same notation and

conventions as in fig. 4.2.

4.3 Phenomenological results

In this section the results of Refs. [73, 75, 76] are collected for the virtualelectroweak Sudakov corrections to the processes Z+2 and Z+3 jets consideredas Standard Model background to the direct search for New Physics in thechannel /ET+ jets at hadron colliders. For the numerical results the defaultALPGEN input parameters and PDFs sets have been used, namely:

• CTEQ6l (PDFs),

• GF = 1.16639 × 10−5 GeV−2, MW = 80.419 GeV, MZ = 91.188 GeV(electroweak input parameters),

• µ2 = M2Z +

�jet p

2T jet (factorization and renormalization scale).

Since no cuts are applied to the decay products of the Z, the Z boson isproduced on shell. As in Refs. [73, 75, 76], in this section only a purely parton-level analysis has been considered: this means, in particular, that the jetsare represented by tree-level partons, while the missing transverse momentumcorresponds to the transverse momentum of the Z boson.

Figure 4.7 shows the effect of weak Sudakov corrections on the effectivemass distribution (meff =

�Njetsj=1 pT j + /ET ) for Z + 2 jets under the ATLAS

cuts of eq. (4.1). As can be seen, the size of virtual weak corrections on thetotal cross section is very small, while in the tails of the meff distribution atthe LHC at

√s = 7 TeV is of order −20, −25% and it grows up to −40, −45%

at√s = 14 TeV.

24

4.3. Phenomenological results

0.0

0.2

0.4

0.6

0.8

1.0

200 600 1000

dσi

d|� /H

T|/

dσLO

d|� /H

T|

| �/HT | [GeV]

√s = 7 TeV

ABC

0.0

0.2

0.4

0.6

0.8

1.0

200 600 1000

dσi

d|� /H

T|/

dσLO

d|� /H

T|

| �/HT | [GeV]

√s = 7 TeV

QCD only

ABC




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 | �/HT | distribution (which is the most interesting region for the direct NPsearches) at

√s = 7 and 14 TeV is of order −25% and −45%, respectively.

Recently, several proposals have been presented for future proton-protoncolliders that will operate after the second run and the high luminosity runof 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 ofTLEP [79]. The main tasks of these future hadron colliders (hh-FCCs) will bethe study of the Higgs self-interaction and, if no NP signals will appear in thenext run of the LHC, the direct search for New Physics. In this latter scenarioit seems natural that the event selections used for the NP searches will bemodified in order to look at higher NP mass scales. For this reason, in figs. 4.9and 4.10 the selection cuts of eqs. (4.3)-(4.4) have been considered: while theacceptance and the angular separation cuts are the same of eqs. (4.1)- (4.2) the

pT , meff , HT and | �/HT | 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 oforder −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 | �/HT | distributions under the cuts of eq. (4.4) isshown in fig. 4.10: again the corrections in the tails of the distributions growas the centre of mass energy increases and reach the value of −60% at 33 TeV

25


0.0

0.2

0.4

0.6

0.8

1.0

200 700 1200 1700 2200

dσi

d|� /H

T|/

dσLO

d|� /H

T|

| �/HT | [GeV]

√s = 14 TeV

ABC

0.0

0.2

0.4

0.6

0.8

1.0

200 700 1200 1700 2200

dσi

d|� /H

T|/

dσLO

d|� /H

T|

| �/HT | [GeV]

√s = 14 TeV

QCD only

ABC




and −80% at 100 TeV. In principle, at 100 TeV even larger values of meff and

| �/HT | can be reached, however the PDFs extrapolation in these regions becomesrather uncertain, so that in the present analysis 20 TeV and 8 TeV have been

chosen as maximum values of meff and | �/HT |, 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 themeff or | �/HT | distribution the correctionremains essentially the same at 14, 33 or 100 TeV, as shown in figs. 4.11-4.12where 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. 4.11-4.12 with figs. 4.7-4.9 and with figs. 4.8-4.10. In the tails of the distributions the size of the Sudakov corrections growswith the collider energy moving from 14 to 33 and 100 TeV simply becauseas the centre of mass energy increases, more and more extreme kinematicalconfigurations are involved and for these configurations the corrections areexpected to be very large. This behaviour has been found for other processesstudied in Refs. [75, 76], such as di-jet, di-boson and inclusive single vectorboson production.

In conclusion, in Refs. [73, 75, 76] the one loop weak Sudakov correctionshave been computed to the process Z + n jets (with n ≤ 3), which is an irre-ducible Standard Model background to the direct search for New Physics at theLHC in the signatures with multi-jets and missing transverse momentum. Atthe LHC the effect of virtual weak corrections in the event selections consideredis large and it becomes even larger at future proton-proton colliders (where thehigher energy allows to look at more and more extreme kinematical regions).

26


10−7

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total LOlong. Z

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dσ

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T|

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GeV

�

| �/HT | [GeV]

Z + 3j

CMS 14 TeV

total LOlong. Z

Figure 4.6: LO meff (|/�HT |) 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 heavygauge boson radiation and the higher-order electroweak contributions (beyondone-loop) require further investigation.

27


10−6

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dm

eff

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GeV

�

√s = 7 TeV

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LONLO Virt.

δ Virt.

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�

√s = 14 TeV

-0.45

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meff [GeV]

LONLO Virt.

δ Virt.

Figure 4.7: Weak corrections to Z+2 jets in the ATLAS setup of eq. (4.1) atthe LHC centre of mass energies of 7 and 14 TeV. The upper panels show theeffective mass distribution at LO (solid blue line) and at NLO including onlyvirtual one loop weak corrections (dotted red line). The lower panels show

the relative effect (δEW = dσNLO−dσLO

dσLO ) of virtual weak corrections (dotted redline).

28


10−6

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GeV

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LONLO Virt.

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200 700 1200 1700 2200

δ EW

| �/HT | [GeV]

LONLO Virt.

δ Virt.

Figure 4.8: Weak corrections to Z + 3 jets in the CMS setup of eq. (4.2) atthe LHC centre of mass energies of 7 and 14 TeV, with the same notation and

conventions as in fig. 4.7. The change in the slope of the distributions at | �/HT |around 500 GeV is an effect of the HT cut of eq. (4.2)

29


10−9

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δ EW

meff [TeV]

LONLO Virt.

δ Virt.

Figure 4.9: Weak corrections to Z + 2 jets in the rescaled ATLAS setup ofeq. (4.3) at the hh-FCC energies of 33 and 100 TeV, with the same notationand conventions as in fig. 4.7.

30


10−7

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T|

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√s = 100 TeV

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δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

Figure 4.10: Weak corrections to Z + 3 jets in the rescaled CMS setup ofeq. (4.4) at the hh-FCC energies of 33 and 100 TeV, with the same notationand conventions as in fig. 4.7.

31


10−7

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meff [TeV]

14 TeV LO14 TeV NLO Virt.



14 TeV δ Virt.33 TeV δ Virt.

100 TeV δ Virt.

Figure 4.11: Comparison between the Sudakov corrections to Z+2 jets at theLHC

√s = 14 TeV (blue curve) and at FCC energies of 33 (green curve) and

100 TeV (red curve) for the meff distribution under the cuts of eq. (4.1). In thelower panel are shown the relative electroweak Sudakov corrections defined infig. 4.7: the corrections for the three energy setup overlap.

10−7

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Z + 3j

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| �/HT | [TeV]





100 TeV δ Virt.

Figure 4.12: Same plot as in fig. 4.11 but for Z + 3 jets under the cutsof eq. (4.2). As in fig. 4.11, the corrections for the three energy setup arebasically the same.

32

Chapter 5Real weak corrections to Z + 2and Z + 3 jets production

5.1 Introduction: real weak corrections

In the computation of one loop virtual electroweak corrections, the diagramsinvolving virtual photons can lead to infrared and/or collinear singularities thathave to be regularized either by dimensional regularization or by introducing anunphysical mass parameter for the photon. This way virtual O(α) correctionsbecome functions of the arbitrary infrared regulators, and the dependence onthe infrared regulators only cancels when also real photonic corrections areincluded (i.e. all the diagrams obtained from the LO ones with the emission ofan additional real photon). The extra emission of additional W and Z bosonsusually is not included in the computation of real O(α) corrections for tworeasons: first of all the gauge bosons decay, so that in principle they lead tofinal states which are not degenerate with the LO ones; the second reason isthat real purely weak corrections are always finite, in the sense that they do notdepend on unphysical parameters (in fact, even in the Sudakov limit, when thecorrections are dominated by the Sudakov logarithms which are the IR limit ofthe 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 realgauge bosons may lead to significant positive contributions.

The issue of real weak boson emission has been addressed by several au-thors. For example, Refs. [38, 39, 27, 26, 24, 23, 40, 41] studied the effect ofreal weak corrections in analogy with QED or QCD: all the diagrams obtainedfrom the LO ones with the emission of an additional W or Z boson are consid-ered as real corrections, the additional gauge boson is produced on-shell andintegrated over the full phase space. The result is a significant cancellationbetween real and virtual corrections, which however may be incomplete due tothe incomplete average on the isospin of the initial state particles (this effect

33

5. Real weak corrections to Z + 2 and Z + 3 jets production

ZW (→ νlνljj) + jj ZZ(→ νlνljj) + jj WW (→ νlljj) + jjZW (→ νlνlνll) + jj ZW (→ νllll) + jj ZZ(→ νlνlll) + jjZZ(→ νlνlνlνl) + jj WW (→ νlνlll) + jj ZW (→ νlljj) + jj

ZW (→ νlνljj) ZW (→ νlljj) ZZ(→ νlνljj)WW (→ νlljj) ZW (→ νlljj) + j ZW (→ νlνljj) + j

ZZ(→ νlνljj) + j WW (→ νlljj) + j

Table 5.1: Vector boson radiation processes contributing to the consideredsignatures. In brackets vector boson decay channels are specified, while outsidethe brackets j stands for a matrix element QCD parton. The above processesare for the Z +2 jet final state, whereas for three jet final states the processesare the same ones plus an additional QCD parton.

is known as Bloch Nordsieck violation [26, 24, 23]). For example, in e+e− col-lisions the cancellation would require also the processes with e+νe, e

−νe andνeνe as initial states, while in pp collisions the cancellation would take place ifu and d type quarks were weighted by the same PDFs.

The approach of Refs. [42, 40, 43] is more phenomenological: the additionalgauge bosons decay and are included in the real corrections only when thefinal states are degenerate with the LO ones. Again the cancellation betweenreal and virtual weak corrections is only partial and moreover it is stronglydependent on the event selection considered.

Following the approach of Refs. [42, 40, 43], in Ref. [73] a first estimate ofthe impact of real weak corrections to the processes Z +2 and Z +3 jets withthe event selection of eqs. (4.1) and (4.2) has been computed. In the followingsections 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(α2αnS)

with n ≤ 2 for Z+2 jets (n ≤ 3 for Z+3 jets) has been considered as real weakradiation. From a purely perturbative point of view, only the processes withn = 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 lowerpanel of Table 5.1 contribute to the same experimental signature and moreoverthey are the most relevant ones among the real EW radiation contributions.In Ref. [73] jets coming from vector bosons decay are distinguished from theother jets (called matrix element jets) and the latter are always required withinthe acceptance cuts in order to avoid infrared QCD singularities: this can beconsidered as a LO prediction of the real contributions, which provides at leasta 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

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 therewere exactly two matrix element jets generated with pT > 40 GeV, |η| < 2.8and ΔR separation larger than 0.4, this is not the case for the real correctionswhen one of the two gauge bosons decays into quarks. As a first step, all thefinal state partons closer than 0.4 in ΔR are combined into a single jet. Thenthe jets and the final state charged leptons (if present) are divided into twocategories: the tagged jets (or charged leptons) and the untagged ones, wherethe tagged jets are defined by the requirement pjT > 40 GeV and |η| < 2.8, whilethe charged leptons are tagged if plT > 20 GeV and |ηl| < 2.4. Finally, in orderto 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 atagged jets and a tagged lepton are closer than ΔR = 0.2 the jet is removedfrom the tagged ones and the event is discarded if it does not contain exactlytwo tagged jets. At the very end, if a tagged charged lepton is closer to atagged jet than ΔR = 0.4, the lepton is removed from the tagged ones and theevent is discarded if any tagged charged lepton survives. The effective massis then computed as the sum of the pT s of the tagged jets together with themissing 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) isconsidered. In Ref. [73] a much more simplified event reconstruction procedurehas been considered: as a first step, charged leptons and final state partonscloser than ΔR = 0.2 are merged into a single jet and if any charged leptonwith plT > 10 GeV and |ηl| < 2.5 survives the event is discarded, then the jetsare merged together if ΔRjj ≤ 0.5 and the event is considered only if contains

exactly three jets within the cuts of eq. (4.2). After this selection, | �/HT | isdefined as minus the sum of the �pT s of these three jets.

Figures 5.1-5.2 and 5.3-5.4 show the effect of the real weak corrections toZ + 2 jets (under the cuts of eq. (4.1)) and Z + 3 jets (under the cuts ofeq. (4.2)), respectively, when the processes of Tab. 5.1 are considered as realcorrections and the additional experimental-like cuts described in the previoustwo paragraphs are imposed. As can be seen, the requirement that the finalstates of the real correction processes are degenerate with the signal leadsto rather small real contributions (ranging from 5% in the tail of the meff

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 | �/HT | distributions range from 6% to 20% moving from 7 to100 TeV), so that the cancellation between real and virtual weak corrections isquite 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 weakcorrections turns out to be small once realistic selection cuts are imposed onthe decay products of the vector bosons. This picture changes completely if amore 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


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 standardcomputation of one loop electroweak corrections). With this definition of thereal contributions only matrix element jets (generated within the acceptance)are involved and the missing momentum is computed as minus the sum ofthe �pT s of the jets. Figures 5.5-5.6 and 5.7-5.8 show the effect of real weakcorrections to Z +2 and Z +3 jets, respectively, according to this latter pointof 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 thevirtual corrections for the LHC centre of mass energies of 7 and 14 TeV, whileat the FCC energies of 33 and 100 TeV real corrections turn out to be largerthan the virtual ones, leading to an overcompensation of the virtual Sudakovcorrections already found in Ref. [41].

5.3 Preliminary results for the real weak cor-

rections to Z+3 jets in the CKKW frame-

work

When the real weak corrections to Z + n jets (n = 2, 3) are defined as inTab. 5.1, the contribution of those subprocesses in which the additional gaugeboson decays in a quark- antiquark pair can lead to QCD infrared singularities.Considering as an example the process ZW+3 jets with Z → νν andW → qq�,that is a real correction to Z + 3 jets, if the W boson is boosted enough itsdecay products will likely be merged into a single parton level jet : if this jet istagged, then one of the three ME partons can become unresolved, thus leadingto QCD infrared divergences.

Of course, in section 5.2, the kind of behaviour described above is avoidedby means of the generation cut imposed on the pT of the ME partons, howeversome criticism may arise concerning the IR safety of the separation between MEjets and jets coming from vector boson decays. One possible way to overcomethis point is to generate the ZV + n jets samples (with V = Z,W , n = 0, 1, 2and 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 generatorin the MLM framework [82]. At variance with the parton level analysis ofsection 5.2, this requires to generate the event samples ZV + n partons thatare passed to a shower Monte Carlo: after the showering and hadronizationprocess, each of the samples can lead to an arbitrary number of jets and theproper matching condition of the samples (which avoids possible double count-ing) is obtained by the requirement that the hardest jets correspond to the MEpartons of the original event sample considered. The sum of the showered andmatched results for ZV +n partons samples gives the prediction for ZV+ jetswithout sharp generation cuts on the pT of the jets.

36

5.3. Preliminary results for the real weak corrections to Z + 3 jets in the CKKWframework

In this section, the process Z(νν)W (qq�)+ jets is considered as a case ofstudy for the real weak corrections to Z+3 jets in the CKKW framework: thisprocess is the most relevant among the ones in Tab. 5.1 and allows to investigatethe potential IR dependence of the parton level analysis of section 5.2, aspointed out in the previous two paragraphs. In order to study the relativeeffect of the real weak corrections corresponding to the Z(νν)W (qq�)+ jetsevents, also the LO prediction for Z+3 jets have been computed in the CKKWframework 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 com-pared to the ones of the hadron level analysis obtained in the CKKW frame-work for the LHC centre of mass energy of 14 TeV. For the latter results, anevent selection as close as possible to the experimental one has been consid-ered, namely: in addition to the angular separation cuts of eq. (4.2), HT is

defined as the sum of the tagged jets pT s (pjT > 50 GeV, |ηj| < 2.5), | �/HT | isminus the vector sum of the transverse momenta of the jets with pT > 30 GeVand |η| < 5, the events containing any charged lepton with plT > 10 GeV and|ηl| < 2.4 are discarded (in fig. 5.9 these leptons can only come from hadrondecays), jets are reconstructed by means of the anti-kt clustering algorithm [83](implemented in the FASTJET code [84, 85]) with a distance parameter of 0.5and 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, sincethey are based on quite different theoretical ingredients, however the resultsbasically agree. Of course, this is still a preliminary result, as all the classes ofprocesses in Tab. 5.1 should be computed in the CKKW framework, howeverthere is no particular reason to believe that the results will change, since all ofthem share the same kind of potential QCD IR sensitivity.

As a conclusive remark, also the virtual weak corrections to Z + 3 jetscould be computed in the CKKW framework, generating the event samples forZ + n partons (n ≤ 3) according to the approximated NLO O(α) normaliza-tion. However, as can be seen from fig. 5.10, once the cuts on the pZT and onthe angular separation between the Z and the final state partons are imposed,the Sudakov corrections for the three samples are basically the same regard-less of the parton multiplicities (and also of the details of the event selectionconsidered). As a consequence, the results of chapter 4 for the virtual weakSudakov corrections to the processes Z+2 and Z+3 jets will still hold for theanalysis at the level of fully showered, matched and hadronized events.

37


10−6

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LONLO Virt.NLO Real

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Figure 5.1: Upper panels: meff distribution for Z + 2 jets under the eventselection of eq. (4.1) at LO (solid blue line), at NLO including only virtualSudakov corrections (red dotted line) and at NLO including only the realcorrections (green dash-dotted line) at the LHC centre of mass energy of 7 and

14 TeV. Lower panels: relative effect (δEW = dσNLO−dσLO

dσLO ) of the virtual (reddotted line) and virtual+real corrections (green dash-dotted line); the size ofreal weak corrections is the difference between the two curves. The processesconsidered as real corrections are defined in Tab. 5.1 under the realistic eventselection described in the text.

38


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LONLO Virt.NLO Real


Figure 5.2: Virtual and real corrections to Z + 2 jets for the meff observableat the FCC

√s of 33 and 100 TeV under the cuts of eq. (4.1) with the same

notation and conventions as in fig. 5.1. The processes considered as real cor-rections are defined in Tab. 5.1 under the realistic event selection described inthe text.

39


10−6

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Figure 5.3: Virtual and real corrections to Z + 3 jets for the | �/HT | observablefor 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. 5.1. The processes consideredas real corrections are defined in Tab. 5.1 (with an additional matrix elementjet) under the realistic event selection described in the text.

40


10−7

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| �/HT | [TeV]

LONLO Virt.NLO Real


Figure 5.4: Virtual and real corrections to Z + 3 jets for the | �/HT | observablefor

√s = 33 and 100 TeV under the cuts of eq. (4.2) with the same notation

and conventions as in fig. 5.1. The processes considered as real corrections aredefined in Tab. 5.1 (with an additional matrix element jet) under the realisticevent selection described in the text.

41


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LONLO Virt.NLO Real


Figure 5.5: Comparison between virtual and real weak corrections to Z+2 jetsfor the meff observable when real corrections are defined as the contribution ofthe processes Z + (V ) + 2 jets (V = W , Z) and no cuts are imposed on theadditional 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 asin fig. 5.1.

42


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LONLO Virt.NLO Real


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LONLO Virt.NLO Real


Figure 5.6: Comparison between virtual and real weak corrections to Z+2 jetsfor the meff observable when real corrections are defined as the contribution ofthe processes Z + (V ) + 2 jets (V = W , Z) and no cuts are imposed on theadditional 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 conventionsas in fig. 5.1.

43


10−6

10−5

10−4

10−3dσ

d|� /H

T|

�pb

GeV

�

√s = 7 TeV

Z + 3j

-0.25

-0.15

-0.05

0.05

200 400 600 800 1000

δ EW

| �/HT | [GeV]

LONLO Virt.NLO Real


10−7

10−6

10−5

10−4

10−3

10−2

dσ

d|� /H

T|

�pb

GeV

�

√s = 14 TeV

Z + 3j

-0.40

-0.20

0.00

0.20

200 700 1200 1700 2200

δ EW

| �/HT | [GeV]

LONLO Virt.NLO Real


Figure 5.7: Comparison between virtual and real weak corrections to Z+3 jets

for the | �/HT | observable when real corrections are defined as the contributionof the processes Z + (V ) + 3 jets (V = W , Z) and no cuts are imposed on theadditional 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 asin fig. 5.1.

44


10−7

10−6

10−5

10−4

10−3

10−2

10−1dσ

d|� /H

T|

�nb

TeV

�

√s = 33 TeV

Z + 3j

-0.60

-0.40

-0.20

0.00

0.20

1 2 3

δ EW

| �/HT | [TeV]

LONLO Virt.NLO Real


10−7

10−5

10−3

10−1

dσ

d|� /H

T|

�nb

TeV

�

√s = 100 TeV

Z + 3j

-0.75

-0.50

-0.25

0.00

0.25

0.50

1 2 3 4 5 6 7

δ EW

| �/HT | [TeV]

LONLO Virt.NLO Real


Figure 5.8: Comparison between virtual and real weak corrections to Z+3 jets

for the | �/HT | observable when real corrections are defined as the contributionof the processes Z + (V ) + 3 jets (V = W , Z) and no cuts are imposed on theadditional 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 conventionsas in fig. 5.1.

45


0.03

0.05

0.07

0.09

200 600 1000

δReal

| �/HT | [GeV]

√s = 14 TeV

ZW + jets

δRealpart.

δRealCKKW

Figure 5.9: Relative contribution dσReal

dσLO of the real weak corrections related tothe process Z(νν)W (qq�)+ jets at the LHC at 14 TeV. δReal

part. is the correctioncorresponding to the sum of the results of the parton level analysis of section 5.2for the subprocesses Z(νν)W (qq�)+n partons (n = 1, 2, 3), while δReal

CKKW is thecorrection obtained in the CKKW framework.

-0.45

-0.35

-0.25

-0.15

-0.05

0.5 1 1.5 2

δEW

Virt.

pZT [TeV]

√s = 14 TeV

Z + 1j

Z + 2j ATLAS

Z + 3j CMS

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 greenline). For Z + 1 jet no cuts are imposed on the final state particles, whilefor Z + 2 and Z + 3 jets the event selection of eq. (4.1) and (4.2) have beenconsidered, respectively.

46

Chapter 6Electroweak Sudakovcorrections to the Rn

γ ratio

6.1 Invisible Z+ jets background to New Physics

searches in the channel /ET+ jets

The Standard Model process Z + n jets, with Z → νν, is an irreducible SMbackground to the direct search for New Physics (NP) in pp collisions in thechannel /ET+ jets. In chapter 4 the effect of the one loop virtual weak Sudakovcorrections to the processes Z +2 and Z +3 jets has been studied consideringtwo different sets of observables and cuts which mimic the ATLAS and CMSevent selections of Refs. [59, 60] and [61, 62], respectively. For the relevantobservables, the corrections have been found to be very large in the mostinteresting regions for the direct NP searches (of order −50% at the LHC at14 TeV and even larger at future pp colliders).

Because New Physics signals could manifest themselves as a mild deviationwith respect to the large SM background, precise theoretical predictions for theprocesses under consideration are needed. In order to reduce the theoreticalsystematic uncertainties, the experimental procedure for the irreducible back-ground determination relies on data driven methods. For instance, the mostconceptually straightforward estimate of the cross section for Z(→ νν)+n jetscould be done through the measured cross section for Z(→ l+l−)+n jets times

the B(Z→νν)B(Z→l+l−)

ratio, which is known with high precision from LEP1 data. Withthe exception of the ratio of branching ratios, the method is free of theoreticalsystematics. However, due to the low production rate of Z(→ l+l−)+n jets, inparticular in the signal regions, this method results to be affected by large sta-tistical uncertainty. Other possible choices of reference processes areW+n jetsand γ + n jets (Refs. [86, 87, 88, 89, 62]). In both cases the statistics is not a

47

6. Electroweak Sudakov corrections to the Rnγ ratio

limitation and the required theoretical input is the ratio

RnV =

�dσ(Z(→ νν) + n jets)

dX

�/

�dσ(V + n jets)

dX

�, (6.1)

where X is the observable under consideration and V = W , γ. For each ofthe reference processes, additional sources of uncertainties are related to themisidentification of either the W or the prompt γ, such as tt background (forthe case of W+ jets) or the photon isolation procedure (for γ+ jets).

The estimate of the invisible Z+ jets background to the direct NP searchesin the channel /ET+ jets is obtained from the measured cross section for theprocess γ+ jets corrected by the Rn

γ ratio. In particular, the photon is requiredto be detected, but it is considered as invisible: the photon �pT is added tothe total /ET and the selection cuts are only applied to the resulting missingmomentum. The same strategy is applied to the W+ jets control sample, fromwhich an independent estimate of the Z+ jets background is obtained. Alsothe Z(→ µ+µ−)+ jets sample is used in order to quantify the error on theestimate of the Z(→ νν)+ jets background, from the analysis of the doubleratio RTH/REXP as a function of the jet multiplicities, where RTH and REXP

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 theoreticaluncertainties related to the ratio in eq. (6.1). In particular, in Ref. [90] a studyof the cancellation in Rn

W of systematic theoretical uncertainties originatingfrom higher-order QCD corrections, including scale variations and choice ofPDFs, has been presented. A first detailed analysis of the impact of higher-order QCD corrections, PDFs and scale choices to R1,2,3

γ has been shown inRef. [91]. More recently, the level of theoretical uncertainty induced by QCDhigher-order corrections in the knowledge of R2

γ and R3γ, relying on the compar-

ison of full NLO QCD calculations with parton shower simulations matched toLO matrix elements, has been discussed in Refs. [92] and [43], respectively. Allthese studies point out that many theoretical systematics related to pQCD andPDFs largely cancel in the ratio and the corresponding theoretical uncertaintyin the Rn

γ ratio can be safely estimated to be within 10%.

What is not expected to cancel in the ratio is the contribution of higher-order electroweak corrections, which are different for the processes Z + n jetsand γ+n jets and can be enhanced by Sudakov logarithms in the typical signalregions for the NP searches. For R1

γ the higher-order EW corrections have beencalculated in Refs. [9, 41], where it is shown that for a vector boson transversemomentum of 2 TeV they are of the order of 20%, while the EW correctionsto R1

W have been computed in Ref. [67] and turn out to be moderate (of theorder of 8% for pWT = 2 TeV).

In view of the forthcoming run II of the LHC, it is therefore necessary toquantitatively estimate the effects of EW corrections to Rn

γ , in particular forn ≥ 2. In this chapter the results of Ref. [93] for the one loop virtual weak

48

6.2. Virtual O(α) Sudakov corrections to the Rnγ ratio

corrections to the Rnγ 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

Rnγ ratio

The one loop virtual weak corrections to the Rnγ ratio are obtained from the

virtual O(α) Sudakov corrections to the processes Z + n jets and γ + n jets as

RnγNLO = Rn

γ LO

�1 + δEW (Z)− δEW (γ)

�, (6.2)

where

δEW (V ) =dσNLO(V + n jets)/dX

dσLO(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 resultsat the LHC at 14 TeV the event selection of eqs. (4.1) and (4.2) is consideredfor 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 imposedfor V + 2 jets (eq. (4.4) for V + 3 jets). For n = 1 no cut on the two finalstate particles is imposed. As described in the previous section, for the processγ+n jets the photon �pT is considered as missing momentum (and actually it isequal to the total /ET in the present parton level analysis). With respect to thecalculation of chapter 4, the only difference is the choice of the renormalizationand factorization scale which has been defined as µ2 =

�jet p

2T 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 meff and | �/HT | 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 6.4-6.5 show the scaling of the corrections to γ + n jets (n = 1, 2and 3) with the collider energy when the same event selection is consideredfor 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 Rnγ ratio, as can be seen from figs. 6.6, 6.7 and 6.8, where

the Sudakov corrections to R1γ, R

2γ and R3

γ 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 Rnγ 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. Evenmore important: while the ratio tends to be flat at the LO in the high HT

49


regions, the Sudakov corrections modify the shape of the Rnγ distributions and

thus should be taken into account in the extrapolation of the Z(→ νν)+ jetscross section from the measured cross section for γ+ jets events.

50


10−1110−910−710−510−310−1101

dσ

dpj T

�nb

TeV

�

Z + 1j

-0.50

-0.30

-0.10

1 2 3 4

δ EW

pjT [TeV]

√s = 14 Tev

LONLO Virt.

δ Virt.

10−1010−810−610−410−2

1102

dσ

dpj T

�nb

TeV

�

γ + 1j

-0.30

-0.20

-0.10

1 2 3 4

δ EW

pjT [TeV]

√s = 14 TeV

LONLO Virt.

δ Virt.

10−1110−910−710−510−310−1101

dσ

dpj T

�nb

TeV

�

Z + 1j

-0.70

-0.50

-0.30

-0.10

1 2 3 4 5 6 7

δ EW

pjT [TeV]

√s = 33 TeV

LONLO Virt.

δ Virt.10−1010−810−610−410−2

1102

dσ

dpj T

�nb

TeV

�

γ + 1j

-0.40

-0.30

-0.20

-0.10

1 2 3 4 5 6 7

δ EW

pjT [TeV]

√s = 33 TeV

LONLO Virt.

δ Virt.

10−9

10−7

10−5

10−3

10−1

101

dσ

dpj T

�nb

TeV

�

Z + 1j

-0.80

-0.60

-0.40

-0.20

1 2 3 4 5 6 7 8 9

δ EW

pjT [TeV]

√s = 100 TeV

LONLO Virt.

δ Virt.10−8

10−6

10−4

10−2

1

102

dσ

dpj T

�nb

TeV

�

γ + 1j

-0.40

-0.30

-0.20

-0.10

1 2 3 4 5 6 7 8 9

δ EW

pjT [TeV]

√s = 100 TeV

LONLO Virt.

δ Virt.

Figure 6.1: Upper panels: jet pT distribution at LO (solid line) and at NLOin the high energy limit (dotted line), for the processes Z +1 jet and γ +1 jetat

√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 ineq. (6.3).

51


10−710−610−510−410−310−2

dσ

dm

eff

�nb

TeV

�

Z + 2j√s = 14 TeV

-0.40

-0.30

-0.20

-0.10

1 2 3 4

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

10−710−610−510−410−310−2

dσ

dm

eff

�nb

TeV

�

γ + 2j√s = 14 TeV

-0.30

-0.20

-0.10

1 2 3 4

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

10−910−810−710−610−510−410−3

dσ

dm

eff

�nb

TeV

�

Z + 2j√s = 33 TeV

-0.60

-0.40

-0.20

2 4 6 8 10 12

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

10−910−810−710−610−510−410−310−2

dσ

dm

eff

�nb

TeV

�

γ + 2j√s = 33 TeV

-0.40

-0.30

-0.20

-0.10

2 4 6 8 10 12

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

10−9

10−8

10−7

10−6

10−5

dσ

dm

eff

�nb

TeV

�

Z + 2j√s = 100 TeV

-0.90

-0.80

-0.70

-0.60

-0.50

7 9 11 13 15 17 19

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

10−8

10−7

10−6

10−5

10−4

dσ

dm

eff

�nb

TeV

�

γ + 2j√s = 100 TeV

-0.50

-0.40

-0.30

7 9 11 13 15 17 19

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

Figure 6.2: Upper panels: meff distribution at LO (solid line) and at NLO inthe high energy limit (dotted line), for the processes Z + 2 jet and γ + 2 jetat

√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 effectof the O(α) Sudakov corrections as defined in eq. (6.3).

52


10−710−610−510−410−310−2

dσ

d|� /H

T|

�nb

TeV

�

Z + 3j√s = 14 TeV

-0.40

-0.30

-0.20

-0.10

0.5 1 1.5 2

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

10−610−510−410−310−210−1

dσ

d|� /H

T|

�nb

TeV

�

γ + 3j√s = 14 TeV

-0.20

-0.10

0.00

0.5 1 1.5 2

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

10−7

10−6

10−5

10−4

10−3

dσ

d|� /H

T|

�nb

TeV

�

Z + 3j√s = 33 TeV

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

1 2 3 4

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

10−710−610−510−410−310−2

dσ

d|� /H

T|

�nb

TeV

�

γ + 3j√s = 33 TeV

-0.30

-0.20

-0.10

1 2 3 4

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

10−8

10−7

10−6

10−5

10−4

dσ

d|� /H

T|

�nb

TeV

�

Z + 3j√s = 100 TeV

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

1.5 3 4.5 6 7.5

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

10−7

10−6

10−5

10−4

dσ

d|� /H

T|

�nb

TeV

�

γ + 3j√s = 100 TeV

-0.40

-0.30

-0.20

1.5 3 4.5 6 7.5

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

Figure 6.3: Upper panels: | �/HT | distribution at LO (solid line) and at NLO inthe high energy limit (dotted line), for the processes Z + 3 jet and γ + 3 jetat

√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 effectof the O(α) Sudakov corrections as defined in eq. (6.3).

53


10−11

10−9

10−7

10−5

10−3

10−1

101dσ

dpj T

�nb

TeV

�

Z + 1j

-0.50

-0.30

-0.10

1 2 3 4

δ EW

pjT [TeV]





100 TeV δ Virt.

10−10

10−8

10−6

10−4

10−2

1

102

dσ

dpj T

�nb

TeV

�

γ + 1j

-0.30

-0.20

-0.10

1 2 3 4

δ EW

pjT [TeV]





100 TeV δ Virt.

Figure 6.4: Upper panels: scaling of the jet pT distributions for Z + 1 jetand γ + 1 jet with the collider energy. Blue, green and red curves are thedistributions 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 energysetup the corrections overlap.

54


10−7

10−6

10−5

10−4

10−3

10−2

10−1

1

101dσ

dm

eff

�nb

TeV

�

γ + 2j

-0.30

-0.10

1 2 3 4

δ EW

meff [TeV]





100 TeV δ Virt.

10−6

10−5

10−4

10−3

10−2

10−1

1

101

dσ

d|� /H

T|

�nb

TeV

�

γ + 3j

-0.20

-0.10

0.5 1 1.5 2

δ EW

| �/HT | [TeV]





100 TeV δ Virt.

Figure 6.5: Upper panels: scaling of the meff and | �/HT | distributions for γ + 2and γ + 3 jet with the collider energy. Same notation, conventions and alsoconclusions as in fig. 6.4.

55


0.18

0.22

0.26

0.30

R1 γ

V + 1j√s = 14 TeV

-0.25-0.15-0.05

1 2 3 4

δ EW

pjT [TeV]

LONLO Virt.

δ Virt.

0.14

0.18

0.22

0.26

0.30

R1 γ

V + 1j√s = 33 TeV

-0.35-0.25-0.15-0.05

1 3 5 7

δ EW

pjT [TeV]

LONLO Virt.

δ Virt.

0.18

0.22

0.26

0.30

0.34

R1 γ

V + 1j√s = 100 TeV

-0.35

-0.15

0.05

0.5 2 4 6 8 10

δ EW

pjT [TeV]

LONLO Virt.

δ Virt.

Figure 6.6: LO (solid lines) and approximated NLO (dotted lines) distribu-tions for the R1

γ ratio as a function of the jet pT at√s = 14, 33 and 100 TeV.

In the lower panels the relative effect of the virtual weak Sudakov correctionsis shown.

56


0.24

0.26

0.28

0.30

0.32

R2 γ

V + 2j

√s = 14 TeV

-0.20

-0.10

0.00

1 2 3 4

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

0.22

0.26

0.30

0.34

R2 γ

V + 2j√s = 33 TeV

-0.30-0.20-0.10

2 4 6 8 10 12

δ EW

meff [TeV]

LONLO Virt.

δ Virt.

0.18

0.22

0.26

0.30

0.34

0.38

R2 γ

V + 2j√s = 100 TeV

-0.40-0.30-0.20

7 9 11 13 15 17 19

δ EW

meff [TeV]

LONLO Virt.

δ Virt.


γ ratio as a function of meff 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


0.22

0.24

0.26

0.28

0.30

0.32

R3 γ

V + 3j√s = 14 TeV

-0.20

-0.10

0.00

0.5 1 1.5 2

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

0.22

0.26

0.30

0.34

R3 γ

V + 3j√s = 33 TeV

-0.30-0.20-0.100.00

1 2 3 4

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

0.18

0.22

0.26

0.30

0.34

0.38

R3 γ

V + 3j√s = 100 TeV

-0.40-0.30-0.20-0.10

1.5 3 4.5 6 7.5

δ EW

| �/HT | [TeV]

LONLO Virt.

δ Virt.

Figure 6.8: LO (solid lines) and approximated NLO (dotted lines) distribu-

tions for the R3γ ratio as a function of | �/HT | 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


0.18

0.22

0.26

0.30

0.34

0.38

0.42

R1 γ

V + 1j

-0.30

-0.20

-0.10

0.00

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

δ EW

pjT [TeV]




δ Virt. 14 TeV33 TeV100 TeV

Figure 6.9: Comparison of the R1γ distributions as a function of the jet pT for

the three different values of√s = 14, 33 and 100 TeV.

0.25

0.29

0.33

0.37

0.41

R2 γ

V + 2j

-0.20

-0.10

0.00

1 1.5 2 2.5 3 3.5 4 4.5

δ EW

meff [TeV]






γ ratio as a function of meff at√s = 14, 33 and 100 TeV once

the same event selection of eq. (4.1) is imposed.

59


0.22

0.26

0.30

0.34

0.38

0.42

R3 γ

V + 3j

-0.20

-0.10

0.00

0.5 1 1.5 2

δ EW

| �/HT | [TeV]





Figure 6.11: LO (solid lines) and approximated NLO (dotted lines) distribu-

tions for the R3γ ratio as a function of | �/HT | at

√s = 14, 33 and 100 TeV once

the same event selection of eq. (4.2) is imposed.

60

Chapter 7Conclusions and futureperspectives

At high energies and in extreme kinematical configurations the one loop vir-tual electroweak corrections are dominated by double and single Sudakov loga-rithms that involve the ratio of the kinematical invariants of the process underconsideration over the gauge boson masses. As discussed in chapters 2-3 Su-dakov 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, inRefs. [3, 4, 5] a general algorithm has been developed in order to evaluate theSudakov 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 withits 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 Physicsat hadron colliders have been computed in chapter 4, showing that the size ofthe virtual weak corrections to the most relevant observables is of the order oftens of percent already at the LHC at 7 TeV and becomes even larger at highercentre of mass energies, which will allow to explore more and more extremekinematical 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 fullyinclusive setup. Real weak corrections partially compensate the large negativevirtual contributions, the size of the cancellation being strongly dependent onthe considered event selections.

In chapter 6 the O(α) virtual EW Sudakov corrections to the productionof a prompt photon in association with up to three jets have been computedin order to obtain the Sudakov corrections to the Rn

γ ratio (n ≤ 3), whichis the theoretical input for the data driven estimate of the invisible Z+ jetsSM background to the direct NP searches in the channel /ET+ jets based onthe reference process γ+ jets. While the theoretical systematic uncertaintiesrelated to higher order QCD corrections, PDFs and scale choices largely cancel

61

7. Conclusions and future perspectives

in the Rnγ 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 theshape of the ratio, which is no longer flat, so that the EW corrections shouldbe carefully considered in the extrapolation of the Z(→ νν)+ jets backgroundfrom 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 Su-dakov limit for some of the typical event selections considered for the directNP searches, and this limit will be deeply explored at future pp colliders withhigher centre of mass energies. On the other hand, in chapters 4-6 it is shownhow the size of the virtual weak Sudakov corrections in the extreme tails of sev-eral distributions can be of order of tens of percent at the LHC and can reachthe value of −80, −85% at the hh-FCCs: this means that the EW correctionsshould be included in theoretical predictions, in particular at the hh-FCCs, butsuch large negative virtual one loop corrections suggest that also the partiallycompensating (and strongly process dependent) effect of the real weak contri-butions should be considered. Higher order EW corrections may also play arelevant role.

The implementation of the Denner-Pozzorini algorithm in the ALPGEN eventgenerator described in chapter 3 works only for external fermions and trans-verse gauge bosons: the next logical step is to improve the code including alsothe case of longitudinal gauge bosons and scalars following the ideas sketched inAppendix D. With this improvement, the algorithm could be used in order tostudy the high energy limit of the weak corrections to the processes HH+ jetsand V V �+ jets (V , V � = W,Z), which will play a crucial role (as signal andbackground, respectively) for the study of the Higgs self interactions, that isone of the main motivations for the proposed hh-FCCs experiments.

For the processes considered in chapters 4-6, the O(α) QED corrections tothe leading O(ααN jets

S ) Born contributions are a gauge invariant subset thatcan be computed separately from the weak contributions which are the subjectof this thesis. However, this is no longer the case for the processes involving aW boson at the LO. In Refs. [3, 4, 5] the contribution of the photon loops isconsidered on the same footing as the one of the diagrams involving a W or Zexchange: this means that the radiator functions related to the photon loopsdepend 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 thedependence on these two parameters. Future work will be devoted to themodification of the present version of the code to properly include also theeffect of real QED corrections.

The final and most ambitious goal of the project started in Ref. [73] is toprovide a completely general tool to estimate the O(α) EW corrections in theSudakov limit, not only for parton level analyses, but also in the context ofthe event generation in the CKKW framework.

62

List of publications

1. M. Chiesa, G. Montagna, L. Barze, M. Moretti, O. Nicrosini, F. Piccininiand F. Tramontano, Electroweak Sudakov Corrections to New PhysicsSearches at the CERN LHC, Phys. Rev. Lett. 111, 121801 (2013)[arXiv:1305.6837 [hep-ph]];

2. K. Mishra, T. Becher, L. Barze, M. Chiesa, S. Dittmaier, X. Garcia i Tormo,A. Huss, T. Kasprzik et al. Electroweak Corrections at High Energies,arXiv:1308.1430 [hep-ph];

3. J.M. Campbell, K. Hatakeyama, J. Huston, F. Petriello, Jeppe R. An-dersen et al., Working Group Report: Quantum Chromodynamics,arXiv:1310.5189 [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:1405.1067 [hep-ph], Electroweak Sudakov corrections to Z/γ+ jetsat the LHC ;

5. M. Chiesa, Electroweak corrections in the Sudakov limit at the LHC,Nuovo Cim. C 037 (2014) 02, 143;

6. L. Barze, M. Chiesa, G. Montagna, P. Nason, O. Nicrosini, F. Piccininiand V. Prosperi, W gamma production in hadronic collisions using thePOWHEG+MiNLO method, arXiv:1408.5766 [hep-ph].

63

Appendix ACross-checks and codevalidation

A.1 EW Sudakov corrections to Z/γ + 1 jet

In Refs. [8] and [9] the Denner-Pozzorini algorithm has been used in order tostudy the high energy limit of the virtual one loop electroweak corrections tothe 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 thepredictions of the full NLO electroweak corrections, finding an agreement atthe percent level already for pVT of order 100, 150 GeV (V = Z, γ).

According to the algorithm, EW corrections in the Sudakov limit can bewritten in a factorized form as the sum of single and double logarithmic con-tributions, where each of the contributions is made up of a radiator function(containing the logarithmic structure of the corrections) times a tree level ma-trix element, that can be the one of the LO process or an SU(2) transformedof the Born matrix element. Since for Z + 1 jet and γ + 1 jet the matrixelements of the SU(2) correlated process W + 1 jet can be obtained from theones of Z + 1 jet (or γ + 1 jet) through a ratio of couplings, in Refs. [8, 9] theanalytic expression has been reported for the EW corrections in the Sudakovlimit computed by means of the algorithm of Refs. [3, 4, 5].

However, in the implementation of the Denner-Pozzorini algorithm in theALPGEN event generator described in this thesis, all the matrix elements neededare computed numerically by means of the ALPHA algorithm [56, 57, 58].In fig. A.1 the results of the numerical implementation of the algorithm arecompared to the ones of the analytic expressions of Refs. [8, 9] for the eventselection:

pjT > 100GeV, |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 alevel of agreement is obtained once the formulas of Refs. [8, 9] have been

65

A. Cross-checks and code validation

modified in order to drop the assumption MW � MZ , so that all the terms likelog(s/M 2

W )log(M 2Z/M

2W ) are taken into account.

10−7

10−5

10−3

10−1

101

dσ

dpj t

�pb

GeV

�

γ + 1j

-0.20

-0.15

-0.10

-0.05

δ EW

0.990

0.995

1.000

1.005

100 500 1000 1500 2000

Ratio

pjt [GeV]

LONLO Virt. ANLO Virt. B

δ Virt. Aδ Virt. B

Ratio A/B

10−7

10−5

10−3

10−1

101

dσ

dpj t

�pb

GeV

�

Z + 1j

-0.40

-0.30

-0.20

-0.10δ E

W

0.990

0.995

1.000

1.005

100 500 1000 1500 2000

Ratio

pjt [GeV]

LONLO Virt. ANLO Virt. B


Ratio A/B

Figure A.1: Upper panels: jet pT 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 bymeans 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 obtainedfrom the curves A and B, respectively (δEW = dσNLO−dσLO

dσLO ). Lower panels:ratio of the results of the two computations A and B. For both γ + 1 jet andZ + 1 jet the event selection of eq. (A.1) is considered.

A.2 EW Sudakov corrections to Z + 2 jets

Very recently, the full one loop EW corrections to the process Z + 2 jets havebeen computed by means of the RECOLA recursive amplitudes generator.While in Ref. [72] only the partonic subprocesses involving one fermion currenthave been considered, the effect of the O(α) corrections for the subprocesseswith one and two fermion currents have been shown in Ref. [74].

66

A.2. EW Sudakov corrections to Z + 2 jets

On one hand the code of Refs. [72, 74] is not yet publicly available, while onthe other hand there are no analytic results in the literature for the electroweakcorrections to Z + n jets with n > 1, so that the GOSAM package has been usedin order to test the implementation of the Denner-Pozzorini algorithm in theALPGEN generator for Z + 2 jets.

GOSAM is a generator of one loop matrix elements originally developed forNLO QCD calculations [94], which has been recently extended to the one loopelectroweak corrections [95], facing the issues of loops diagrams containing sev-eral mass scales and also unstable particles. However, it is worth noting thatthe present version of GOSAM still does not include the treatment of EW renor-malization, which is necessary for the computation of the Sudakov corrections,since part of the single logarithmic structure comes from the running of theEW 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 renormal-ization counterterms have been traced back and dropped. Then this modifiedversion of the code has been used as follows: first phase space points have beengenerated using the ALPGEN routines and the LO and Sudakov NLO unrenor-malized weights have been computed, then the phase space points are passedto a routine which calls the finite part of the unrenormalized exact NLO matrixelement generated by GOSAM and at the end the distributions obtained with thethree different weights have been compared. Figure A.2 shows some examplesof this kind of comparisons between the predictions of the two different codesfor the unrenormalized part of the one loop EW corrections to the partonicsubprocesses considered under the event selection of eq. (4.1).

Several comments on the procedure described in the previous paragraph arein order. First of all, it is worth noting that the results shown in fig. A.2 donot include renormalization, so that they are not physical and moreover theyare gauge dependent. Nevertheless fig. A.2 is still a quite stringent technicaltest of the implementation of the Denner-Pozzorini algorithm in the ALPGEN

generator, since both the calculations are performed in the same gauge (theGOSAM matrix element has been computed in the ξ = 1 gauge and the samegauge choice has been used in order to define the counterterms contributionsin the ALPGEN routines) and the results nicely agree. Of course, the predictionsof the two codes are not expected to be identical, as the GOSAM unrenormalizedmatrix element is exact, while in the ALPGEN routines only the logarithmic partof the corrections is computed: the two predictions should be the same only inthe asymptotic regime, so that the percent level agreement shown in A.2 foran event selection which is only approximately in the Sudakov limit is really anon trivial validation of the implementation of the Denner-Pozzorini algorithmin the ALPGEN generator.

An independent and physical (even if unfortunately only qualitative) testof the validity of the algorithm and of its implementation can be obtainedfrom the comparison between the results of Ref. [72] and the predictions of the

67


routines for the Sudakov corrections included in ALPGEN. The event selectionconsidered in Ref. [72] does not consider particularly tight cuts:

pjT > 25GeV, |y(j)| < 4.5, ΔRjj ≥ 0.4 (j = j1, j2), (A.2)

(where j1 and j2 are the leading and the remaining jet, respectively), so thatthe kinematical configurations involved can hardly be considered as Sudakovregions. Figure A.3 shows the predictions of the algorithm of Refs. [3, 4, 5] forthe leading jet pT 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 onefermion current are considered, as in Ref. [72]. As can be seen, the results infig. A.3 are very similar to the ones of Ref. [72], where the full one loop EWcorrections to Z + 2 gluons turned out to be of order −15% for pj1T = 1 TeVand −10% for pj1T = 500 GeV.

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 underthe 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 themost relevant kinematical regions are the ones in which the two jets are backto back while the Z is soft. As a result, the larger invariants (and thus theleading part of the correction) are related to a di-jet like kinematics and theEW corrections to the di-jet production at hadron colliders turn out to berather small, as described in the next section. Of course, since in fig. A.3 onlythe partonic subprocesses with one fermion current are included, the numberof EW charged legs is in general smaller than the one considered for the caseof Z+2 jets studied in chapter 4. However, the impact of the different numberof EW charged particles should be very small, as can be seen from the stabilityof the corrections as a function of the final state parton multiplicity (as soonas 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 hadroncollisions have been computed for the first time in Ref. [96] and an independentcalculation has been recently performed in Ref. [97]. In particular, in Ref. [97]it is shown the impact of EW corrections for the leading jet pT distribution atthe LHC energies of 7, 8 and 14 TeV under the event selection:

pjT > 25GeV, |y(j)| < 2.5, ΔRjj > 0.6 (j = j1, j2), (A.3)

where j1 and j2 are the hardest and the next-to-hardest jets, respectively.As in the case of the results of fig. A.3, the kinematical configurations se-

lected by the cuts of eq. (A.3) are not that extreme and probably they are

68

A.3. Electroweak Sudakov corrections to di-jet production

still far from the Sudakov limit. Despite of this, the electroweak Sudakov cor-rections to the dijet production have been computed by means of the Denner-Pozzorini algorithm implemented in the ALPGEN event generator and the resultsare 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 thesame order as the ones shown in fig. A.3. Moreover, the predictions in fig. A.3at large pT basically agree with the ones of Ref. [97], where the one loop EWcorrections have been found to be of order −5% and −12% for the leading jetpT of the order of 1.5 and 3 TeV, respectively.

69


-0.12

-0.08

-0.04

1000 1400 1800

δ EW

meff [GeV]

√s = 14 TeV

uu → Zgg


-0.16

-0.12

-0.08

-0.04

1000 1400 1800 2200

δ EW

meff [GeV]

√s = 14 TeV

uu → Zcc


-0.40

-0.30

-0.20

-0.10

0.00

1000 3000 5000 7000

δ EW

meff [GeV]

√s = 14 TeV

uu → Zuu


-0.40

-0.30

-0.20

-0.10

0.00

300 900 1500 2100

δ EW

pZT [GeV]

√s = 14 TeV

ud → Zud


Figure A.2: Virtual EW corrections to some of the partonic subprocesses in-volved in the production of a Z boson in association with two jets under thecuts 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 un-renormalized one loop matrix element generated by GOSAM are compared tothe predictions of the Denner-Pozzorini algorithm implemented in the ALPGENevent generator (where the logarithmic structure coming from parameters andwave function renormalization has been dropped, as described in the text).Even if these plots have no physical meaning, they represent a non trivialtechnical test of the validity of the algorithm and of its implementation.

70

A.3. Electroweak Sudakov corrections to di-jet production

-0.15

-0.10

-0.05

0.00

0 200 400 600 800 1000

δ EW

pj1T [GeV]

√s = 8 TeV

δ Virt.

Figure A.3: Relative effect of the virtual weak Sudakov corrections to theleading jet pT distribution for the process Z + 2 jets (where only the partonicsubprocesses with one fermion current have been included) under the cuts ofeq. (A.2) at

√s = 8 TeV. To be compared with fig. 4 of Ref. [72].

-0.15

-0.12

-0.09

-0.06

-0.03

0.00

500 1000 1500 2000 2500 3000

δEW

pj1T [GeV]

pp → jj,√s = 14 TeV

δEW

Figure A.4: Relative effect of the virtual weak Sudakov corrections to theleading jet pT distribution for the dijet production process at the LHC centreof mass energy of 14 TeV under the cuts of eq. (A.3). To be compared withfig. 12 of Ref. [97].

71


72

Appendix BOne loop renormalizationcounterterms

B.1 On shell renormalization conditions at one

loop

In the on shell scheme the form of the renormalization counterterms is fixedby 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 theThomson limit;

• the renormalized Higgs tadpole is set to zero.

With the above conditions the one loop EW counterterms for fermions, gaugebosons and EW couplings read:

δmf =mf

2Re�Σf,L

ii

�m2

f

�+ Σf,R

ii

�m2

f

�+ 2Σf,S

ii

�m2

f

��,

δZf,Lii = −ReΣf,L

ii

�m2

f

�

−m2f

∂

∂p2

�ReΣf,L

ii

�p2�+ ReΣf,R

ii

�p2�+ 2ReΣf,S

ii

�p2��p2=m2

f

,

δZf,Rii = −ReΣf,R

ii

�m2

f

�

−m2f

∂

∂p2

�ReΣf,L

ii

�p2�+ ReΣf,R

ii

�p2�+ 2ReΣf,S

ii

�p2��p2=m2

f

, (B.1)

73

B. One loop renormalization counterterms

δM 2W = ReΣW

T

�M2

W

�, δZW − ∂

∂k2ReΣW

T

�k2��k2=M2

W

,

δM 2Z = ReΣZZ

T

�M2

Z

�, δZZZ = − ∂

∂k2ReΣZZ

T

�k2��k2=M2

Z

,

δZAA = − ∂

∂k2ΣAA

T

�k2��k2=0

,

δZAZ = −2ReΣAZ

T (M 2Z)

M2Z

, δZZA = 2ΣAZ

T (0)

M2Z

,

δM 2H = ReΣH

�M2

H

�, δZH = − ∂

∂k2ReΣH

�k2��k2=M2

H

, (B.2)

δe

e= −1

2δZAA − sW

cW

1

2δZZA =

1

2

∂

∂k2ΣAA

T

�k2��k2=0

− sWcW

ΣAZT (0)

M2Z

,

δcWcW

=1

2

�δM 2

W

M2W

− δM 2Z

M2Z

�=

1

2Re

�ΣW

T (M 2W )

M2W

− ΣZZT (M 2

Z)

M2Z

�,

δsWsW

= −c2Ws2W

δcWcW

= −1

2

c2Ws2W

Re

�ΣW

T (M 2W )

M2W

− ΣZZT (M 2

Z)

M2Z

�,

δt = −TH . (B.3)

Where the notation Re means that the real part is taken only for the scalarfunctions contained in the unrenormalized self energies.

B.2 Unrenormalized self energies

ΣAAT

�k2�= − α

4π

�4

3

�

f,i

N fCQ

2f

�−�k2 + 2m2

f,i

�B0 (k,mf,i,mf,i)

+ 2m2f,iB0 (0,mf,i,mf,i) +

k2

3

�

+�3k2 + 4M 2

W

�B0 (k,MW ,MW )− 4M 2

WB0 (0,MW ,MW )

�(B.4)

74

B.2. Unrenormalized self energies

ΣAZT

�k2�= − α

4π

�2

3

�

f,i

N fC (−Qf )

�g+f + g−f

� �k2

3

−�k2 + 2m2

f

�B0 (k,mf ,mf ) + 2m2

fB0 (0,mf ,mf )�

− 1

3sW cW

��9c2W +

1

2

�k2 +

�12c2W + 4

�M2

W

�B0 (k,MW ,MW )

−�12c2W − 2

�M2

WB0 (0,MW ,MW ) +k2

3

��(B.5)

ΣZZT

�k2�= − α

4π

�2

3

�

f,i

N fC

��g+f�2

+�g−f�2� �−

�k2 + 2m2

f,i

�B0 (k,mf,i,mf,i)

+ 2m2f,iB0 (0,mf,i,mf,i) +

k2

3

�+

3

4s2W c2Wm2

f,iB0 (k,mf,i,mf,i)

�

+1

6s2W c2W

�−�24c4W − 8c2W + 2

�M2

WB0 (0,MW ,MW ) +�4c2W − 1

� k2

3

+

�18c4W + 2c2W − 1

2

�k2B0 (k,MW ,MW )

+�24c4W + 16c2W − 10

�M2

WB0 (k,MW ,MW )

�

+1

12s2W c2W

��2M 2

H − 10M 2Z − k2

�B0 (k,MZ ,MH)

− (M 2H −M2

Z)2

k2

�B0 (k,MZ ,MH)− B0 (0,MZ ,MH)

�

− 2M 2ZB0 (0,MZ ,MZ)− 2M 2

HB0 (0,MH ,MH)−2k2

3

��(B.6)

where

g+f = −sWcW

Qf , g−f =I3,fW − s2WQf

sW cW(B.7)

75


ΣWT

�k2�= − α

4π

��

i

1

3s2W

�−�k2 −

m2l,i

2

�B0 (k, 0,ml,i) +m2

l,iB0 (0,ml,i,ml,i)

+m4

l,i

2k2(B0 (k, 0,ml,i)− B0 (0, 0,ml,i)) +

k2

3

�

+�

i,j

1

s2W|Vij|2

�−�k2 −

m2u,i +m2

d,j

2

�B0 (k,mu,i,md,j) +

k2

3

+m2u,iB0 (0,mu,i,mu,i) +m2

d,jB0 (0,md,j ,md,j)

+

�m2

u,i −m2d,j

�2

2k2(B0 (k,mu,i,md,j)− B0 (0,mu,i,md,j))

�

+2

3

��2M 2

W + 5k2�B0 (k,MW ,λ)− 2M 2

WB0 (0,MW ,MW )

− M4W

k2(B0 (k,MW ,λ)− B0 (0,MW ,λ)) +

k2

3

�

+1

12s2W

��40c2W − 1

�k2 +

�16c2W + 54− 10

c2W

�M2

W

�B0 (k,MW ,MZ)

−�16c2W + 2

� �M2

WB0 (0,MW ,MW ) +M2ZB0 (0,MZ ,MZ)

�

+�4c2W − 1

� 23k2

−�8c2W + 1

� (M 2W −M2

Z)2

k2

�B0 (k,MW ,MZ)− B0 (0,MW ,MZ)

��

+1

12s2W

��2M 2

H − 10M 2W − k2

�B0 (k,MW ,MH)

− (M 2H −M2

W )2

k2

�B (k,MW ,MH)− B0 (0,MW ,MH)

�− 2k2

3

− 2M 2WB0 (0,MW ,MW )− 2M 2

HB0 (0,MH ,MH)

��(B.8)

76

B.2. Unrenormalized self energies

ΣH�k2�= − α

4π

��

f,i

N fC

m2f,i

2s2WM2W

[2A0(mf,i) + (4mf,i − k2)B0(k2,mf,i,mf,i)]

− 1

2s2W

�6M 2

W − 2k2 +M4

H

2M 2W

�B0(k

2,MW ,MW )

− 1

2s2W

�3 +

M2H

2M 2W

�A0(MW ) +

3M 2W

s2W

− 1

4s2W c2W

�6M 2

Z − 2k2 +M4

H

2M 2Z

�B0(k

2,MZ ,MZ)

− 1

4s2W c2W

�3 +

M2H

2M 2Z

�A0(MZ) +

3M 2Z

2s2W c2W

− 3

8s2W

�3M4

H

M2W

B0(k2,MH ,MH) +

M2H

M2W

A0(MH)��

(B.9)

Σf,Li,j

�k2�= − α

4π

�δijQ

2f

�2B1(p

2,mf,i,λ) + 1�

+ δij(g−f )

2�2B1(p

2,mf,i,MZ) + 1�

+ δijm2

f,i

4s2WM2W

�B1(p

2,mf,i,MZ) + B1(p2,mf,i,MH)

�

+1

2s2W

�

k

VikV†kj

��2 +

m2f �,k

M2W

�B1(p

2,mf �,k,MW ) + 1��

(B.10)

Σf,Ri,j

�k2�= − α

4π

�δijQ

2f

�2B1(p

2,mf,i,λ) + 1�

+ δij(g+f )

2�2B1(p

2,mf,i,MZ) + 1�

+ δijm2

f,i

4s2WM2W

�B1(p

2,mf,i,MZ) + B1(p2,mf,i,MH)

�

+1

2s2W

mf,imf,j

M2W

�

k

VikV†kjB1(p

2,mf �,k,MW )

�(B.11)

77


Σf,Si,j

�k2�= − α

4π

�δijQ

2f

�4B0(p

2,mf,i,λ)− 2�

+ δijg+f g

−f

�4B0(p

2,mf,i,MZ)− 2�

+ δijm2

f,i

4s2WM2W

�B0(p

2,mf,i,MZ)− B0(p2,mf,i,MH)

�

+1

2s2W

�

k

VikV†kj

m2f �,k

M2W

B0(p2,mf �,k,MW )

�(B.12)

In eqs.(B.4)-(B.12) an infinitesimal mass parameter λ has been introducedfor the photon in order to regularize the QED infrared singularities, N f

C is thenumber of colours for the fermion f , while in eqs. (B.10)-(B.12) f � stands forthe isospin partner of f .

B.3 Definition of the A0, B0 and B1 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 arecollected.

The scalar integral A0 is defined as:

A0(m) =(2πµ)4−D

iπ2

�dDq

1

q2 −m2 + iε. (B.13)

Eq. (B.13) can be computed in the standard way by performing the Wickrotation, obtaining:

A0(m) = m2(Δ− logm2

µ2+ 1) +O(D − 4), (B.14)

where (γE is the Euler’s constant)

Δ =2

4−D− γE + log4π. (B.15)

The definition of the scalar function B0 is:

B0(p,m0,m1) =(2πµ)4−D

iπ2

�dDq

1�q2 −m2

0 + iε��(q + p)2 −m2

1 + iε� .

(B.16)

78

B.3. Definition of the A0, B0 and B1 functions

After introducing the Feynman parametrization and performing the Wick ro-tation, eq. (B.16) becomes:

B0(p,m0,m1) = Δ

−� 1

0

dxlog

�p2x2 − x(p2 −m2

0 +m21) +m2

1 − iε

µ2

�+O(D − 4),

(B.17)

that can be solved in a general way as:

B0(p,m0,m1) = Δ+ 2− logm0m1

µ2

+m2

0 −m21

p2log

m1

m0

− m0m1

p2�1r− r�logr +O(D − 4), (B.18)

where r and 1rare obtained from the condition:

x2 +m2

0 +m21 − p2 − iε

m0m1

x+ 1 = (x+ r)(x+1

r). (B.19)

The scalar coefficient B1 comes from the Passarino-Veltman reduction ofthe two point tensor integral Bµ:

(2πµ)4−D

iπ2

�dDq

qµ�q2 −m2

0 + iε��(q + p)2 −m2

1 + iε�

= Bµ(p,m0,m1) = pµB1(p,m0,m1). (B.20)

Even if from eq. (B.20) it is possible to write the following general expressionfor the B1 coefficient:

B1(p,m0,m1) =m2

1 −m20

2p2

�B0(p,m0,m1)− B0(0,m0,m1)

�

− 1

2B0(p,m0,m1), (B.21)

for many applications it is convenient to evaluate the B1 coefficient directlyfrom:

B1(p,m0,m1) = −Δ

2

−� 1

0

dx(x− 1)log

�p2x2 − x(p2 −m2

0 +m21) +m2

1 − iε

µ2

�, (B.22)

where the O(D − 4) terms have been omitted since they are not relevant atone loop level.

79


80

Appendix CCollinear singularities in oneloop radiative corrections

C.1 Collinear singularities in O(α) QED cor-

rections

For the sake of brevity, in chapter 2 only the infrared and infrared-collinearlimit of one loop QED corrections has been considered. This Appendix isfocused on the collinear limit of both real and virtual one loop QED correctionsfollowing 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 aphoton off a final state electron

q

p′p,

(C.1)

can be written as:

Mi→f(eγ)λ,α (p�, q) =− euα(p

�)γµ /p� + /q

(p� + q)2 −m2�∗µ(q,λ)A

i→f(e)(p), (C.2)

(where Ai→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 helicityα, while λ is the photon polarization).

As pointed out in chapter 2, the propagator in eq. (C.2) is divergent whenthe emitted photon becomes collinear with the external momentum p� and theelectron mass acts as IR cutoff. If the electron mass in eq. (C.2) is set to zero,

81

C. Collinear singularities in one loop radiative corrections

the collinear singularity corresponds to the kinematical configurations in which|�qT | → 0 (where �qT is the transverse momentum of the photon with respect tothe emitting electron).

In the collinear limit, the following parametrization for the momenta p, qand p� can be introduced:

pµ =�E +

|�qT |22x(1− x)E

, 0, 0, E�

qµ =�(1− x)E +

|�qT |24x(1− x)E

, 0, 0, (1− x)E +|�qT |2(1− 2x)

4x(1− x)E

�+ qµT

p�µ =pµ − qµ =�xE +

|�qT |24x(1− x)E

, 0, 0, xE − |�qT |2(1− 2x)

4x(1− x)E

�− qµT

qµT =�0, |�qT |cosφ, |�qT |sinφ, 0

�, (C.3)

where (up to O(|�qT |4) terms):

p2 =|�qT |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 uβ(p)uβ(p) also for the internal off-shell electron, eq. (C.2) becomes:

Mi→f(eγ)λ,α (p�, q) =− e

�

β

uα(p�)γµuβ(p)uβ(p)�

∗µ(q,λ)A

i→f(e)(p)x(1− x)

|�qT |2

=−�

β

Vα,β,λ(p, p�, k)

x(1− x)

|�qT |2Mi→f(e)

β (p) (C.5)

(where the regular terms in the limit |�qT | → 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 inter-nal and external electrons corresponding to the momenta in eq. (C.3) up toO(|�qT |3) contributions:

u−(p) =�0, 0, 0,

√2E�T

,

u+(p) =�√

2E, 0, 0, 0�T

,

u−(p�) =

�0, 0,

|�qT |e−iφ

√2xE

,√2xE

�T,

u+(p�) =

�√2xE,− |�qT |eiφ√

2xE, 0, 0

�T,

�µ±(k) =1√2

� |�qT |e∓iφ

2(1− x)E, 1,∓i,− |�qT |e∓iφ

2(1− x)E

�. (C.6)

82

C.1. Collinear singularities in O(α) QED corrections

Using eq. (C.6) the vertex term reads:

Vα,β,λ(p, p�, k) =uα(p

�)γµuβ(p)�∗µ(q,λ)

=e

√2x|�qT |

x(1− x)(δα,λ + xδ−α,λ)δα,βe

−iλφ +O(|�qT |3), (C.7)

�

λ=±V ∗α�,β,λ(p, p

�, k)Vα��,β,λ(p, p�, k) = 2e2

|�qT |2x(1− x)

1 + x2

1− xδα�,βδα��,β +O(|�qT |4).

(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 integratedover the final state photon and electron phase space. In the collinear limitk = (1 − x)p, leading to an extra 1/x factor, since |�p|

|�p−�q| → 1x, as can be seen

from:�

d3�q

(2π)32q0

�d3�p�

(2π)32p�0=

�d3�q

(2π)32q0

�d3�p�

(2π)32p�0

�d4 pδ4(p− p� − q)

=

�d3�q

(2π)32q0

�d3�p

(2π)32p0

|�p||�p− �q|

��p0=|�p−�q|+|�p|

. (C.9)

As a result, the cross section for the process i → f(eγ) in the collinear limitreads:

dσi→f(eγ)α�

��coll

= 2e2�

d3�q

(2π)32q0

x(1− x)

|�qT |21 + x2

1− x

�1xdσ

i→f(e)α� +O(|�qT |)

�. (C.10)

Using the following identity for the photon phase space in the collinear limit:

d3�q

(2π)32q0=

dxdφ|�qT |24(2π)3(1− x)

+O(|�qT |2) , (C.11)

equation (C.10) becomes:

dσi→f(eγ)α�

��coll

=α

2πlog�Q2

m2

�dσ

i→f(e)α�

� 1

0

dx1 + x2

1− x. (C.12)

Concerning eq. (C.12) several comments are in order. First of all, dσi→f(eγ)α�

��coll

is part of the real O(α) corrections to the process i → f : eq. (C.12) showsthat real corrections in the collinear limit factorize in tree level cross sectionsmultiplied by a factor, which is the usual (unregularized) Altarelli-Parisi split-ting [98]. The same kind of factorization also holds for the case of initial stateradiation, with the difference that the radiation changes the centre of massenergy of the hard process i → f . It should be noticed that eq. (C.12) isdivergent 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 conclu-sive remark, while in eq. (C.12) the electron mass has been chosen as the lowerbound of the |�qT | integration, since in eq. (C.2) m is the IR regulator for thecollinear singularity, the upper bound of integration is somehow arbitrary, evenif it should be small enough in order not to spoil the validity of the collinearapproximation.

83


C.1.2 Collinear limit of virtual O(α) corrections

The collinear singularities of the one loop virtual QED corrections correspondto the diagrams in which an on shell external fermion emits a virtual photonand arise from the integration region where the photon momentum becomescollinear to the one of the external fermion:

q

p− qp

.

(C.13)

The set of diagrams in (C.13) can be written as:

µ4−D

�dDq

(2π)DAi(eγ)→0

ν (p− q, q)i(/p− /q)

(p− q)2 −m2

�ieIl

�γµ

−igµν

q2 − λ2u(p), (C.14)

where Ai(eγ)→0ν represents the sum of the diagrams for the process i(e) → 0

involving an additional photon, all the external legs have been considered asincoming, the factor Il is −Ql or +Ql in the case of an incoming electronor positron, respectively, and in all the denominators a +iε term is alwaysunderstood.

Since also the soft limit of the virtual O(α) corrections is involved ineq. (C.14), in order to separate the collinear part of the virtual one loop QEDcorrections, the soft contribution computed in the eikonal approximation inchapter 2 should be subtracted, namely:

µ4−D

�dDq

(2π)DeIl�

q2 − λ2��(pl − q)2 −m2

l

�×

�Ai(eγ)→0

µ (pl − q, q)i(/pl − /q)γµu(p)−�

m�=l

ieIm(−4plpm)

(pm + q)2 −m2m

M0(pl, pm)�,

(C.15)

where the second term within the brackets can be simplified in the collinearlimit q → xp (m2 → 0) and using the total charge conservation

�l �=m Im = −Il,

obtaining:

M1l, coll = µ4−D

�dDq

(2π)D−eIl�

q2 − λ2��(pl − q)2 −m2

l

�×

limq→xp

�Mi(eγ)→0

[γ]µ (p(1− x), xp)�2x− 1�qµ − 2e

xM0(pl, pm)

�, (C.16)

Mi(eγ)→0[γ]µ 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 areincluded, since the electron self energy diagrams are collected in the wavefunction renormalization counterterms and will be considered separately.

84

C.1. Collinear singularities in O(α) QED corrections

Using the Ward identity:

qµMi(eγ)→0[γ]µ (p− q, q) = eIlMi(e)→0(p), (C.17)

equation (C.16) reads:

M1l, coll = −µ4−DM0(pl, pm)

�dDq

(2π)D2i(eIl)

2

�q2 − λ2

��(pl − q)2 −m2

l

� . (C.18)

As can be seen from eq. (C.18), the one loop QED corrections in the collinearlimit q → xp factorize on the external leg l, so that the leg index will besuppressed in the following. Moreover, as the soft singularities have beensubtracted, the λ parameter can be set to zero.

Introducing the Sudakov parametrization [99]:

qµ =xpµ + znµ + qµT

nµ =�p0,−p0

�p

|�p|�

qµT =(0, �qT )

n · qT =p · qT = 0, (C.19)

with:

dDq = p · n�

dx

�dz

�dD−2�qT , (C.20)

equation (C.16) can be written as (I2l = Q2l = 1):

M1coll = M0(pl, pm)µ

4−D

�dx

�dD−2�qT(2π)D−2

�dz

−2e2i

4p · nx(x− 1)[z − z0][z − z1],

(C.21)where:

z0 =|�qT |2 − x2m2 + iε

2xp · n

z1 =|�qT |2 − (x− 1)2m2 +m2 + iε

2p · n(x− 1). (C.22)

The two poles of the integrand in eq. (C.21) have opposite imaginary parts for0 < x < 1, so that the integration over the z variable can be performed closingthe contour of integration in the upper or in the lower half complex plane:

M1coll =

α

2πM0(pl, pm)4πµ

4−D

�dx

�dD−2qT(2π)D−2

�dz

1

|�qT |2 + x2m2

=α

2πM0(pl, pm)Γ(ε)

� 1

0

dx� 4πµ2

m2x2

�ε

=α

2πlog

µ2

m2M0(pl, pm) + collinear finite terms. (C.23)

85


In eq. (C.23) the collinear finite terms also include the UV poles, which arehowever cancelled by means of the renormalization procedure.

Without the subtraction of the soft contributions, the computation ofeq. (C.14) would lead to the result:

M1coll+ soft = − α

2πM0(pl, pm)log

µ2

m2

� 1

0

dxx

1− x, (C.24)

which is divergent in the soft limit x → 1. Adding eq. (C.24) to the collinearlimit of the wave function renormalization counterterms δZf , with

δZf

��coll

= − α

2πlog

µ2

m2

� 1

0

dx(1− x), (C.25)

it turns out that the part of the IR limit of the virtual QED corrections thatcan be factorized on a single external leg is:

dσVirt.

��coll

= − α

2πlog�Q2

m2

�dσLO

� 1

0

dx1 + x2

1− x(C.26)

and cancels the corresponding real contribution of eq. (C.12) for the photonradiation off a final state electron. This is not the case for the initial stateradiation contributions, where the photon radiation changes the centre of massenergy of the hard process, so that real and virtual corrections factorize on treelevel 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 arerelated to the diagrams of the form (C.1). At variance with the QED case, theemitted gauge boson can also be a Z or a W , while the external on shell legcan be a fermion, a scalar or another gauge boson.

The collinear limit of the one loop virtual weak corrections is computedfollowing the same steps described in section C.1.1. First of all the soft con-tributions included in the double logarithmic part of the correction are sub-tracted. Then the collinear limit is evaluated: the total charge conservationused in the QED case, for the EW corrections is replaced by the approximateSU(2)×U(1) invariance (which is fulfilled up to mass suppressed terms). Theproof of the collinear factorization in QED is obtained by means of the Wardidentity (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 ob-tained using the collinear Ward identities, proved in [4, 5], that are a conse-quence of the BRS invariance of the Standard Model. As a final step, the loopintegrals are computed with the same techniques of section C.1.1.

86

Appendix DSudakov corrections for externallongitudinal gauge bosons in theunitary gauge

In Refs. [3, 4, 5] the electroweak corrections for the processes involving externallongitudinal gauge bosons are computed by means of the Goldstone bosonequivalence theorem. In chapters 4, 5 and 6, theO(α) corrections for the case oflongitudinally polarized Z bosons have been neglected, since the correspondingLO contributions are strongly suppressed for the event selections considered.Moreover, the tree level matrix elements provided by ALPGEN are computed inthe unitary gauge, so that the procedure of Refs. [3, 4, 5] for the longitudinalgauge 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 theone described in Ref. [7].

At LO the formula for the GBET is:

MV L1 ···V L

m ···0 = Mφ1···φm···

0

m�

a=1

i1−QVa , (D.1)

wherem is the number of longitudinal gauge bosons V L (V L = ZL,WL), QVa isthe electric charge of V L

a and φa is the would-be Goldstone boson correspondingto V L

a .

At NLO eq. (D.1) becomes:

MV L1 ···V L

m ···NLO = Mφ1···φm···

NLO

m�

a=1

i1−QVa

�1 + δAVa

�, (D.2)

87

D. Sudakov corrections for external longitudinal gauge bosons in the unitary gauge

where

δAZL =− ΣZZL (M 2

Z)− iMZΣZχ(M 2

Z)

M2Z

+δMZ

MZ

+1

2δZZZ (D.3)

δAWL =− ΣWWL (M 2

W ) +MWΣWφ(M 2W )

M2W

+δMW

MW

+1

2δZWW . (D.4)

For the computation of one loop EW corrections in the Sudakov limit, theterm proportional to δA in eq. (D.2) can be easily included into the singlelogarithmic part of the correction:

δCGBETMV L1 ···V L

m ···NLO = MV L

1 ···V Lm ···

0

m�

a=1

δAVa , (D.5)

where the leading order expression for the GBET has been used, since the δAterms are of O(α).

For the remaining part of eq. (D.2), one can think to use the originalapproach of Refs. [3, 4, 5]:

δMi1···iNNLO (φ) =

N�

l=1

�

k>l

δDLk,l Mi1···jl···jk···iN

0 (φ) +N�

l=1

δSLl Mi1···jl···iN0 (φ), (D.6)

where Mi1···iN (φ) stands for Mφ1···φm··· and the radiator functions are the onesderived for the would-be Goldstone bosons. Only at the very end, when thecorrections of eq. (D.6) have been evaluated, those tree level matrix elementswhich involve external would-be Goldstone bosons can be mapped back to thephysical fields using the LO expression for the GBET, while the ones involvingexternal Higgs bosons are computed directly. In general the overall procedureleads to a non null phase shift, since in eq. (D.6) the SU(2) transformed of theoriginal φa fields are considered, however, this phase shift is uniquely deter-mined by the flavour of the φa fields and its SU(2) transformed, so it can beincluded in the general expression for the radiator functions involving externalwould-be Goldstone bosons.

88

Acknowledgements

• I am very grateful to Guido Montagna, Mauro Moretti, Oreste Nicrosiniand Fulvio Piccinini for all the things I had the opportunity to learnduring my PhD.

• I would like to thank Luca Barze, Paolo Nason, Valeria Prosperi andFrancesco Tramontano for the very fruitful collaboration.

• Thanks also to Stefano Boselli, Carlo Carloni Calame and Homero Mar-tinez.

• The work presented in this theses was supported in part by the ResearchExecutive Agency (REA) of the European Union under Grant AgreementNo. PITN-GA-2010-264564 (LHCPhenoNet) and by the Italian Ministryof 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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