Physique des Particules Avancées 2

Transcription

1 Physique des Particules Avancées Interactions Fortes et Interactions Faibles Leçon 6 Les collisions p p ( enseignant Alessandro Bravar Alessandro.Bravar@unige.ch tél.: 9610 bureau: EP 06 assistant Alexander Korzenev Korzenev@mail.cern.ch tél.: bureau: EP 1 9 avril 018

2 Hadron Hadron Collisions much higher c.o.m. energies than achievable with e + e - machines new particle searches at high mass scales underlying process: interaction of two partons (quarks, antiquarks, or gluons) 3 variables to describe the interaction: the parton momenta x 1 and x, and Q, which measures the hardness of the interaction [Q is not well defined, it depends on the process under study ] cfr. e-p scattering (DIS): variables x Bj and Q, and e + e - annihilation (fixed beam energies): 1 variable the scattering angle q

3 Understanding p p Cross-sections In QCD, hadron hadron collisions are described as an incoherent sum of elementary interactions between the hadron constituents: the quarks, the antiquarks, and the gluons. proton A proton B x a P a x b P b partonic cross section final state for a specific final state, we must add all possible amplitudes (i.e. Feynman diagrams) leading to that particular final state ( jets, top quark, W boson, Higss, ) f a/a (x a ) = probability to find a parton a in proton A, carrying a momentum fraction x a (, ) (, ) ( ( )) ˆ ABX dxadxb fa/ A xa Q fb/ B xb Q abx S Q The interacting partons carry only a fraction of the incident protons momenta. The incident protons (of fixed energy) can be viewed as beams of free partons with a large spectrum of energies. In the scattering between partons, therefore, the energy is not fixed with ŝ s. 3

4 Factorization Key point: FACTORIZATION! can be factorized in terms of: In general, the hard hadron reaction A + B C + D ABX dxadxb fa/ A( xa, mf ) fb/ B ( xb, mf ) 0 S mr 1 1. the elementary elastic parton parton cross section d / dt for the reaction a + b c + d. the parton density functions f a A (x a ) for finding a parton of momentum fraction x a and flavor a in the hadron A (same for hadron B), 3. the fragmentation function D cc (z) for parton c to fragment into the hadron C with a momentum fraction z. m F factorization scale (separates long and short distance physics, i.e. the Q of PDFs) m R is the renormalization scale of S (i.e. the Q at which of S is calculated) Finite corrections left behind are not universal and must be calculated for each process giving rise to order S (and higher) perturbative corrections. An all-orders cross section has no dependence on m F and m R. A residual depenence remains (to order S n+1 ) for a finite order S n calculation abx 4

5 At first order, in the perturbative expansion, we have several -body partonic processes that can contribute to the interaction: qq qq, qg qg, gg gg, etc. These processes are calculable from the corresponding Feynman diagrams using the QCD coupling constant Q 1 33 n lnq L f QCD L QCD ~ 00 MeV the additional constant C includes the contributions of higher order diagrams; the calculation of higher order terms is usually very lengthy and difficult (many Feynman diagrams involved). C Q is a momentum scale that characterizes the hardness of the reaction; in general we cannot define exactly the scale for each process as long as C is not determined (recall that in DIS Q was given by the virtuality Q < 0 of the exchanged photon, and in e + e - annihilation by the total c.o.m. energy Q > 0). Usually we employ one of the following definitions for Q : Q Q Q tˆ sˆ sˆ ˆˆˆ stu ˆ t uˆ ~ 4 p ~ ~ p 4 3 for cos q cm = 0 with m R = m F = Q and vary m between 1/ Q and Q T p T T 5

6 LO QCD Process d ˆ M1 3 4 dtˆ 16 sˆ sˆ S A strength at 90 0 in c.o.m. The coefficient 1 / (16 s ) is the phase space and flux factor. 6

7 The Role of Structure Functions Given that quarks and gluons are not free inside the hadrons, we have to consider also 1. the probability of finding a parton of given momentum inside the interacting hadrons (the parton probability distribution function f(x) ) and. the probability that the outgoing parton produced fragments into a hadron h (fragmentation function D(z) ). These probabilities cannot be obtained from a perturbative QCD calculation; in principle we should be able to obtain them from a full calculation starting from the QCD Lagrangian (confinement problem), however, at present they are parameters that we extract from experiment. The factorization hypothesis allows us to use the same structure and fragmentation functions measured in DIS, e + e - annihilation, and Drell-Yan process to predict the cross sections in different processes. The validity of this hypothesis has to be verified each time by an exact calculation of the perturbative series. The separation between short- and long-distance physics is controlled by the factorization scale m F, which can be chosen in several ways. A parton emitted with a small p T < m F is considered to be part of the hadron structure and is absorbed into the parton distribution function. A parton emitted at large p T is part of the short-distance sub process. 7

8 What Can We Calculate Scattering processes at high energy hadron colliders can be classified as either HARD or SOFT. Quantum Chromodynamics (QCD) is the underlying theory for all such processes, but the approach (and the level of understanding) is very different for the two cases. For HARD processes, e.g. W boson or high-e T jet production, the rates and event properties can be predicted with high precision using QCD perturbation theory. For SOFT processes, e.g. total cross sections or diffractive processes the rates and properties are dominated by non-perturbative QCD effects, which are much less understood and not calculable. SppS 8

9 The hadron hadron interactions has the form A + B jet1 + jet + jet3 + jet4 The generated hadrons will be collimated in jets around the direction of the outgoing partons emerging from the elementary reaction (sub process) a + b c + d jet1 and jet are the fragments of partons c and d jet3 and jet4 are the fragments of the spectators, which did not participate in the interaction. Since the initial state (pp system) is colorless, there must be some soft gluon exchange Between the jets to assure that also the final state is colorless. At high energies even more high p T jets are expected to be produced. They originate from the radiation of hard gluons from the quarks (higher order QCD diagrams). 9

10 Feynman Diagrams for Di-jet Production not back-to-back because x 1 x 10

11 Hard vs. Soft Processes In order for the parturbative QCD calculation to be valid, it is important that the process is a short-distance interaction, so that S (Q ) is small enough for the perturbative expansion to be valid; in other words p T (or Q ) must be large. uncertainty principle: large energy short time scale ignore interactions with spectator partons The average p T of hadrons produced in hadronic collisions is <p T > ~ 0.4 GeV! Perturbative QCD analyses, however, work for p T > 3 GeV (Q > 10 GeV ). Large p T processes are rare processes; the cross sections is smaller by several orders of magnitude w.r.t. the dominant production. Low p T physics referred to as soft physics, is not calculable within QCD, and only phenomenological approaches are possible. Typical cross sections are of the order of millibarns, while large p T cross sections are of the order of nanobarns. Soft interaction cross sections are usually described by the phenomenological expression pph d T ~ F dx dp F T p 1 x exp( 4 p ) T where x F = p* L / s is the Feynman variable, which describes the longitudinal phase space, and p T the transverse momentum of the outgoing hadron. 11

12 p + p o + X, x F ~ 0 (q CM ~0 o ) hard soft 1

13 Typical Hadronic Cross Sections s X s Y s 1 GeV 1 TeV 13

14 Hadronic Total Cross Sections All hadronic cross sections as function of s can be described by a universal expression s X s Y s with same exponents and different coefficients 14

15 The ATALS LHC 15

16 The CMS LHC 16

17 The CMS LHC 17

18 Proton Proton Collisions at LHC Measurements of structure functions not only provide a powerful test of QCD, the parton distribution functions are essential for the calculation of cross sections at pp and pp colliders. Example: Higgs production at the Large Hadron Collider LHC The LHC collides 7 TeV protons on 7 TeV protons However underlying collisions are between partons Higgs production the LHC dominated by gluon-gluon fusion p 7 TeV p t t t H 0 Cross section depends on gluon PDFs 1 1 ( pp HX ) ~ dx d x g( x ) g( x ) ( gg H) Uncertainty in gluon PDFs lead to a ±5 % uncertainty in Higgs production cross section 7 TeV Prior to HERA data uncertainty was ±5 % 18

19 Inclusive Jet Production g g g g g q g q q q q q sub-process fraction contributing to p + p jet + X (qq, qg, gg scattering) at Tevatron (s ~ TeV) 19

20 Test QCD predictions by looking at production of pairs of high energy jets pp jet + jet + X p p Measure cross-section in terms of transverse energy pseudorapidity q = 6-90 o q = o at low E T cross-section is dominated by low x partons (gg scattering) E T [GeV] at high E T cross-section is dominated by high x partons (qq scattering) 0

21 Inclusive Jet -section dp T dy p T transverse momentum y rapidity 1

22 The Kinematics parton + parton parton + parton q 1 q p 1 p In the c.o.m. of the interacting partons, there is only one independent variable in addtion to ^s: the angle q or p T = p cosq. (this is the usual -body process encountered already several times) Consider for instance the qg qg scattering: q p1 q q x 1 x q1 g q q g q p PP c.o.m. g g qg c.o.m. s PA PB Pcm s s q1 x1pa x1 q xpb x

23 Consider the difference between parton momenta We obtain a first measurable variable P1 x x z z xf 1 Pcm that can be determined from the longitudinal momenta P 1z, P z of the outgoing jets. A second measurable variable can be derived from the parton c.o.m. energy ^s P cm 1 sˆ q q 4q q 4x x P x x s 1 1 sˆ p p M M1 sˆ xx 1 s s where M 1 is the invariant mass of the outgoing di-jet. s q q x x P P 1 1 1z z only a fraction of the c.o.m. energy is available for the partonic sub process Finally, from the kinematics of the outgoing jets x F x 1 x 1 x x x 1, 1 x F xf 4 we can determine experimentally x 1 and x. 3

24 The Differential Cross Section The invariant cross section for the production of two jets h A + h B jet 1 + jet + X can be factorized as E E 1 6 d dp dp 1 Q Q dx dx 1 f A ABJ1 J X sˆ d ˆ( S ( mr)) 4 ( x1, mf ) fb( x, mf ) d ( q1 q p1 p) dtˆ The two kinematical variables d 3 p 1 d 3 p will disappear after integration (d function). x 1 and x can be determined from the jet kinematics. We have to add all possible -body parton interactions Q 1 Q P 1 P and convolute with the corresponding parton distribution functions. Since we are not detecting a specific hadron in the final state, there are no fragmentation functions involved. The interaction dynamics is contained in the sub process cross section M if the sub processes are distinguishable, like qq qq and qq gg dˆ ˆ dt M M if the sub processes are not distinguishable ( interference terms!). 4

25 To arrive at the cross section, the sub process cross section must be convoluted with the probability of finding a parton Q 1 or Q inside the hadrons h A and h B : f A (x 1 )dx 1 and f B (x )dx. This convolution can change completely the relative weight of various sub processes. At low x, gluons dominate; at large x valence quarks dominate. The factorization and renormalization scales usually are taken equal to the p T of the outgoing jets: m F = m R = p T At leading order, changing S (i.e. m R ) by 10% can vary the cross section by 50%! In general, leading order calculations are not satisfactory, but we have to start somewhere Several processes have been already calculated beyond the first order in S, Considering all diagrams, real or virtual, for the production of 3 or 4 partons the cross sections are written as ˆ C LO S C NLO 3 S C NNLO 4 S L To study the inclusive production of a single particle C, we introduce the fragmentation function D(z): E C 3 d dp C Q Q dx dx 1 f A ( x1 ) fb( x AB sˆ d ˆ ) dtˆ C X D C q ( z) 5

26 Some Typical QCD Processes (1) 6

27 Some Typical QCD Processes () 7

28 Top Pair Production qq annihilation top anti-top production in proton anti-proton collisions at the Tevatorn gg fusion at Tevatron pp collisions E CM ~ TeV ~ 10 pb at LHC pp collisions E CM ~ 14 TeV ~ 400 pb 8

29 W & Z Production W p T distribution (s ~ 1.8 GeV) W & Z cross sections 9

30 Direct g Production This QCD Compton diagram gives direct access to the gluons inside the hadrons direct g production q g By detecting in coincidence an isolated high-p T photon and the away side jet originated by the fragmenting quark, one can fully reconstruct the sub process kinematics 1 1 ( pp g jet) ~ dx d x [ g( x ) q( x ) g( x ) q( x )] ( qg qg )

31 The Drell Yan Process The Drell Yan process is the reaction h A + h B g* + X l + l - The mass of the lepton pair l + l - (e + e -, m + m -, + - ) M ll 0 since the photon is virtual (Q > 0!) One assumes that the leptons are produced directly by the electroweak interaction qq l + l -. dx ˆ adxb fa/ A( xa ) fb/ B( xb ) ABl l abl l To annihilate, the quark and anti-quark must have same flavor and color. Very clean, golden process in QCD, because 1. dominated by quarks in initial state. no quarks or gluons in the final state (small QCD corrections) 3. experimentally leptons are easier to measure 4. most important and precise SM test at LHC The process is similar to the annihilation e + e - qq and to the DIS eq eq. The quarks inside the hadrons are described by the hadron structure functions f(x,q ). 31

32 The Drell-Yan Cross Section We begin with the partonic sub process (QED) 1 4 ˆ( qq l l ) e N 3ˆ s c q To embed the partonic cross section in the hadronic process, we rewrite it as a differential cross section d / dq for the production of a lepton pair with invariant mass M ll = Q = ^s d ˆ 1 4 dq N 3Q 4 c e d Q sˆ q The hadronic cross section can be obtained by adding incoherently the contributions of different quarks and convoluting the partonic d / dq with the structure functions f q (x) d pp l l X dq q 1, q, 4 q 3 3 q 3Q d ˆ dx dx f x Q f x Q e Q s The factors 1/3 come from averaging over initial state colors and the factor 3 from summing over qq same color combinations. If the quarks carried no color, the cross section would be 3 times larger (another evidence for color!). 3

33 The kinematics is similar to the di-jet production in hadron hadron collisions. We replace the outgoing jet 1 and jet with the lepton pair l + and l - and can determine the c.o.m. kinematics: x x x p p M Next, we can replace ^s = x 1 x s F 1 M 1 1 x 1 s x s sˆ d pp l l X s 4 eq dx1 dx f q x fq x d x1x dq Nc 3Q q Q s // sˆ x cm 1 x and in terms of x 1 and x d pp l l X 1 4 dx dx N 3x x s 1 c 1 q e f x f x f x f x A B A B q q 1 q q 1 q or in terms of x F and M 1 d pp l l X 1 4 x1x 4 dm1dxf Nc 3M1 x1 x q e f x f x f x f x A B A B q q 1 q q 1 q The Drell-Yan process is extremely useful, because it allows us to access the anti-quark distributions directly. 33

34 In lowest QCD order with no gluon emission we expect a scaling result: although the cross section is a function of s and the lepton pair mass Q, the quantity Q 4 d/dq is a function of the ratio s/q only. (NB the LO diagram does not depend on any scale). Several experiments verified this scaling invariance with a precision of 10 0%. The angular distribution of the leptons in the lepton pair c.o.m. follows well the (1 + cos q) distribution (yet another proof that the quarks have spin 1/). At NLO we have to consider also gluon radiation from the interacting quarks. This will introduce logarithmic scaling violations similar to those observed in DIS. 34

35 LO vs. Higher Order QCD Leading Order QCD + fastest option, often the only available large scale uncertainty reflecting large theory uncertainty no control on renormalization small scale dependence can be misleading Next to Leading Order QCD + reduce dependence on unphysical scales + establish normalization and shape of the cross section + large NLO corrections higher order terms are important calculations often difficult or not existing for all sub processes kinematical distributions of outgoing jets (particles) not very intuitive Collider processes know at NNLO today Drell Yan Higgs 3-jets in e + e - annihilation 35