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Standard Model precision measurements Misure di precisione del modello standard

Lesson 2: Asymmetries at Z pole

Stefano Lacaprara

INFN Padova

Dottorato di ricerca in fisica

Universit` a di Padova, Dipartimento di Fisica e Astronomia

(2)

Outline

1 Introduction

2 Z-pole observables

3 Asymmetries

4 W mass and width

5 Top mass

6 Higgs mass and features

7 Global ElectroWeak fit

(3)

Outline

1 Introduction

2 Z-pole observables Standard Model Z lineshape

3 Asymmetries

4 W mass and width

5 Top mass

(4)

Outline

1 Introduction

2 Z-pole observables

3 Asymmetries

Forward-Backward Asymmetries Left-Right Asymmetries

Tau polarization Results

4 W mass and width

5 Top mass

6

Stefano Lacaprara (INFN Padova)

Higgs mass and features

SM precision measurements Padova 07/03/2019 4/54

(5)

Input to global EWK fit

(in parenthesis the order followed in these lessons)

2) (MZ (5) αhad

mt

b R0 c R0 0,b AFB

0,c AFB

Ab Ac (Tevt.) lept Θeff sin2

) FB lept(Q eff 2Θ sin

(SLD) Al

(LEP) Al

0,l AFB

lep R0

0 σhad ΓZ MZ ΓW MW MH

Global EW fit Indirect determination Measurement GfitterSM

Mar ’18

Higgs mass (5)

I

LHC

W mass and width (3)

I

LEP2, Tevatron, LHC Z-pole observables (1,2)

I

LEP1, SLD

I

M Z , Γ Z

I

σ 0 had

I

sin 2 θ lept eff

I

Asymmetries

I

BR R lep,b,c 0 = Γ had /Γ ``,b ¯ b,c ¯ c

Top mass (4)

(6)

Input to global EWK fit

(in parenthesis the order followed in these lessons)

Higgs mass (5)

I

LHC

W mass and width (3)

I

LEP2, Tevatron, LHC Z-pole observables (1,2)

I

LEP1, SLD

I

M Z , Γ Z

I

σ 0 had

I

sin 2 θ lept eff

I

Asymmetries

I

BR R lep,b,c 0 = Γ had /Γ ``,b ¯ b,c ¯ c

Top mass (4)

I

Tevatron, LHC other:

I

α s (M Z 2 ), ∆α had (M Z 2 )

(7)

Forward-Backward Asymmetries

(8)

Asymmetries [1]

The key features of the SM x-section for ee → f ¯ f production can be fully exploited by looking at the differential x-section d cos θ d σ as well as to effects related to the helicities of the fermions in the initial or final state.

Both are related to the parity violation present in the neutral current of the SM lagrangian.

Possible asymmetries

d σ

d cos θ : Forward-backward asymmetry A FB (LEP);

Helicity state: Left-Right asymmetry A LR (initial state: SLD, final state:LEP [τ ]);

Both: Left-Right Forward-Backward asymmetry A LRFB SLD;

We can differentiate the vector- and axial-vector coupling of the Z, including the measurement

of sin θ eff W

(9)

From Z-lineshape we got [2]

σ(s) = 12π Γ

e

M Γ

had2 Z

s

(s−M

z2

)

2

+M

Z2

Γ

2Z

: At peak: σ 0 = 12π M

2 Z

Γ

e

Γ

had

Γ

Z

Γ had = Σ q6=t Γ q ¯ q , and Γ Z = Γ ee + Γ µµ + Γ τ τ + Γ had + Γ inv

So we measure the SM prediction for the total coupling of Z to fermions, but not the actual, inner form of the V-A strucutre

−i g cos θ W γ µ

1 2



c V f − c A f γ 5



, c V f = T 3 − 2Q sin 2 θ W ) , c A f = T 3

fermion Q f T 3 c A f c V f

(ν e , ν µ , ν τ ) L 0 1/2 1/2 1/2 = 0.50 (e, µ, τ ) L -1 -1/2 -1/2 −1/2 + 2 sin 2 θ W = 0.03

(e, µ, τ ) R -1 0 0 +2 sin 2 θ W = 0.47

(u, c, t) L 2/3 1/2 1/2 1/2 + 4/3 sin 2 θ W = 0.19

(10)

At √

s = M Z the dominating term is the Z exchange.

Absorbing QED correction into effective coupling, and ignoring fermion masses, QCD ISR/FSR, . . .

d σ

d cos θ = 16πQ f (N c ) 2s |χ(s)| 2

(g Ve 2 + g Ae 2 )(g Vf 2 + g Af 2 )(1 + cos 2 θ) + 8g Ve g Ae g Vf g Af cos θ 

Where lineshape is

χ(s) = G F M W 2 8π √

2

s

s − M Z 2 + isΓ Z /M Z g Vf (g Af ) are effective vector (axial) coupling of Z to fermion f . At tree level, in S:

c Af = (T (3) ), c Vf = (T (3) − 2Q f sin 2 θ W )

(11)

A FB definition

Definition

Forward-Backward asymmetry as

A FB = σ F − σ B

σ F + σ B = N F − N B N F + N B

where F (B) is the direction of the final state fermion (e , µ , τ , b, c) with respect to the incoming e beam

σ F (B) = Z 1(0)

0(−1)

d σ

d cos θ d cos θ

Modulo asymmetries in the detector acceptances and efficiencies,

(12)

d σ

d cos θ = 16πQ

f

(N

c

) 2s |χ(s)|

2

(g

Ve2

+ g

Ae2

)(g

Vf2

+ g

Af2

)(1 + cos

2

θ) + 8g

Ve

g

Ae

g

Vf

g

Af

cos θ 

From

d cos θd σ

, and integrating over the angles σ

F

− σ

B

∝ R

1

0

cos θd cos θ − R

0

−1

cos θd cos θ = 1 σ

F

+ σ

B

∝ R

1

−1

(1 + cos

2

θ)d cos θ = 8/3

A

fFB

= 8g

Ve

g

Ae

g

Vf

g

Af

8/3(g

Ve2

+ g

Ae2

)(g

Vf2

+ g

Af2

)

= 3 4

 2g

Ve

g

Ae

g

Ve2

+ g

Ae2

  2g

Vf

g

Af

g

Vf2

+ g

Af2



= 3

4 A

e

A

f

(13)

Alternative

One can rewrite the x-section as:

d σ

d cos θ = 16πQ f (N c ) 2s |χ(s)| 2

(g Ve 2 + g Ae 2 )(g Vf 2 + g Af 2 )(1 + cos 2 θ) + 8g Ve g Ae g Vf g Af cos θ 



(1 + cos 2 θ) + 8 g Ve g Ae g Vf g Af

(g Ve 2 + g Ae 2 )(g Vf 2 + g Af 2 ) cos θ



∝ (1 + cos 2 θ) + 2A e A f cos θ 



(1 + cos 2 θ) + 8

3 A FB cos θ



N −N

(14)

d σ

d cos θ in µµ channel

On-peak (M Z ) and ±2GeV off-peak

As described before, there are two contributions:

I

∝ (1 + cos 2 θ) symmetical

I

∝ cos θ gives asymmetry the asymmetry is small

I

A FB ∝ g V f which is small for lepton. c V f ≈ 0.03 dependence of d cos θ d σ on √

s

I

at √

s = M Z 2 , resonance function is ≈ s−M s

2

Z

I

off-peak, the contribution from γ − Z interference term

does not vanish and so the actual shape changes

(15)

d σ

d cos θ in ee channel

Shape is different wrt to µµ due to presence of t-channel in ee final state.

most of asymmetry comes from t-channel

and s/t-channel interference

off-peak

(16)

A FB for ee channel wrt s

Interference contribution vanishes (only) on-peak;

it is dominant off-peak

I

A FB ∝ g Ae g Af

s(s−M

Z2

) (s−M

Z2

)

2

+M

Z2

Γ

2Z

t-channel is important (for ee), s-channel is small;

I

in particular on-peak, since:

F

A

FB

∝ g

Ve

g

Ae

g

Vf

g

Af

F

g

Vf

is small, expecially for leptons;

F

g

Ve,µ

∼ 0.038;

(17)

Corrections

As usual, the measured quantities suffers from large corrections: QED (ISR/FSR), self-energy, etc.

Use pseudo-observable A 0 FB (QED correction accounted for)

Other correction (QCD, top, Higgs, . . . ) still

there.

(18)

A f in Standard Model

A f depends on the vector- and axial-vector coupling of fermion to the Z.

A f = 2g Vf g Af

g Vf 2 + g Af 2 = 2 g Vf /g Af

1 + (g Vf /g Af ) 2

In particular, it is strictly null if no axial/vector coupling exists for the neutral current.

Coupling SM: c Vf = (T 3 − 2Q f sin 2 θ W ) and c Af = (T 3 )

Using effective SM quantities, the radiative correction are re-absorbed sin θ W ,eff f = (1 + ∆ρ) sin θ f W with ∆ρ = α(M π

Z

) M m

2t2

Z

α(M

Z

) ln m M

H22 Z

g Vf /g Af = 1 − 2 2Q f

T f 3 sin θ f W ,eff

So A f is directly related to sin θ f W ,eff

(19)

A f for different fermions

The actual A f dependence on sin θ W ,eff f varies with the fermion charge and weak-isospin

(20)

A FB results

A ` FB : testing lepton universality

A 0,e FB =0.0145 ± 0.025 A 0,µ FB =0.0169 ± 0.013 A 0,τ FB =0.0188 ± 0.017 A 0,` FB =0.0171 ± 0.010

A 0,e FB suffers from uncertainties related to

t-channel, as well as to s-t interference.

(21)
(22)

A FB for quarks

Need to tag also the flavour and distinguish bc from ¯ b ¯ c, not only bc from q Done only for A c ¯ FB c and A b ¯ FB b with different techniques:

lifetime tagging;

I

cτ for b is ∼ 450µm (in lab ×γ)

I

cτ for c is ∼ 150µm (in lab ×γ)

I

detect displaced secondary vertex by impact parameter or decay lenght measurements Exclusive decays:

I

leptonic decay;

I

decay with D mesons;

additional techniques:

I

jet charge;

I

vertex charge;

I

kaons;

Correction to asymmetries from QCD (gluon radiation from final state quarks), in addition to QED

correction and LEP energy scale.

(23)

lifetime tagging

d 0 significance L significance and NN

b ↔ c separation via M vtx

b-tagging performance

(24)

Lepton tagging

A lepton inside a jet is a stong indication of a b or c flavour.

To distinguish from b to ¯ b use semileptonic decay of b quarks.

b → ` ; ≈ 10%

c → ` + ; ≈ 13%

b → c → ` + cascade decay ;

Distinguish between direct and cascade decay using lepton p T and p

lower purity and efficiency wrt lifetime tagger

(25)

D-Meson Tags

Tagging via reconstruction of charmed meson and barions in jet:

direct evidence of a jet originated by an heavy quark.

low B ∼ O(%)

b → c fragmentation is small

distinguish c from b decay by charmed mesons momentum;

I

D 0 → K π +

I

D + → K π + π +

I

D s → K + K π +

I

Λ + c → pK π +

(26)

Jet charge, vertex charge and kaons

Consider ee → q ¯ q → jet + jet events.

The average charge of all particles within a jet retains some info about initial q charge.

Q h = Σ i q i p i k k Σ i p i k k

where p i k k is the p parallel to trust axis of the event (corresponding to jets’ directions)

Likewise, the sum can run over all the charged particles from a secondary (displaced) vertex.

Tag efficieny is rather small: data driven estimation from fraction of same-sign double tag in pure sample of q quarks.

Kaons: deduce the q flavour via decay chain b → c → s and c → s

(27)

A FB for b and c

A 0,c FB = 0.0707 ± 0.0035 A 0,b FB = 0.0992 ± 0.0016

(28)

A FB for b and c vs √ s

Note larger error off-peak: contribution from s/t interference term

(29)

Inclusive Hadronic Charge Asymmetry

Do not try to distinguish between quark flavour, but rather measure an “average” asymmetry based on charge asymmetries.

hQ FB had i = hQ F − Q B i = X

q=udcsb

R f A f FB δ q

δ q = hQ q − Q q ¯ i jet charge diff for q ¯ q

hQ FB had i can be interpreted as a function of sin 2 θ lept eff (in A f FB )

δ q from Z → bb/cc and MC for udc

(large syst uncert)

(30)

Left-Right Asymmetries

The forward-backward asymmetry is due to the presence of axial and vector couplig of fermions in the neutral current, introducing a dependence on cos θ which can be measured.

Z coupling to left and right fermions are different as well, since L and R fermions belongs to different multiplet in the SM, with different quantum numbers.

This feature can be measured studing the Z production with polarized e ± beam, or by looking at the polarization of the decay product of the Z .

I

SLD produces Z with polarized e ± beams;

F

Most precise determination of A

e

I

LEP measure the polarization of τ in Z → τ τ decay, studing subsequent τ decays;

(31)

Differential x-section for polarized beam

For unpolarized beam

d σ

d cos θ ∝ (1 + cos 2 θ) + A e A f 2 cos θ

Differential cross section (born-level) in case of initial- or final-state fermion helicity are:

d σ Ll

d cos θ ∝ g Le 2 g lf 2 (1 + cos θ) 2 + f (A e,f ) cos θ d σ Rr

d cos θ ∝ g Re 2 g rf 2 (1 + cos θ) 2 + . . . d σ Lr

d cos θ ∝ g Le 2 g rf 2 (1 − cos θ) 2 + . . . d σ Rl

d cos θ ∝ g Re 2 g lf 2 (1 − cos θ) 2 + . . .

(32)

For e with partial polarization P e and e + unpolarized (SLC), summing over final-state polarization

d σ f ¯ f d cos θ = 3

8 σ f ¯ tot f (1 − P e A e )(1 + cos 2 θ) + 2(A e − P e )A f cos θ  P e is positive for right-handed beam helicity, negative for left.

Measurements (see also next slides)

if we look at total x-section for left/right-polarized e beam:

I

σ tot L/R sensitive to (1 + cos 2 ) term:

I

A LR sensitive to A e

if we look at forward/backward x-section for left/right-polarized e beam:

I

σ F /B L/R sensitive to cos θ term:

I

A LRFB sensitive to A f

A FB sensitive to A e A f

(33)

Measure the x-section for e L e + → f ¯ f (σ f L ) and e R e + → f ¯ f (σ R f ).

I

Note: only e beam is polarized;

I

the polarization is not complete P e ∼ 0.8

I

P e > 0 for Left, P e < 0 for Right electrons;

I

Final state is inclusive, no flavour determation needed;

A LR = σ L f − σ R f σ L f + σ R f From A LR get A e from (1 + cos 2 θ) term integration:

σ L/R f = Z +1

−1

d σ L

d cos θ d cos θ ∝ 1 ± P e A e

N f − N f 1

(34)

It is also possible to combine the two techniques: FB and LR.

A LRFB = (σ F − σ B ) L − (σ F − σ B ) R

F + σ B ) L + (σ F + σ B ) R

The integration to get F and B drop the (1 + cos 2 θ) term and keep the cos θ one.

σ F /B ∝ (A e ∓ P e )A f , (σ F − σ B ) L/R ∝ (P e )A f

A LRFB = (N F − N B ) L − (N F − N B ) R (N F + N B ) L + (N F + N B ) R

1 h|P e |i = 3

4 A f

Possible to measure directly A f for identified final state (f = e, µ, τ, b, c)

In both case, the precise knowledge of P e is critical

(35)

SLC: beam polarization [3]

First e ± linear collider e source

I

polarized by shining circ. polarized light on a ArGa photo-cathode (P e ∼ 22%)

(36)

SLC: beam polarization [3]

linear acceleration;

I

e + produced by e on a target, and collected, squized and accelerated;

two damping rings to reduce size and energy spread of e ± bunches;

final arch to single interaction region 1 GeV lost for sync. rad.

I

spin further rotated, and collision with longitudinal pol.

repetition rate 120 Hz (45 o 90 kHz at LEP)

Polarization randomly rotated every pulse (120 Hz) by changing the laser circ pol

(37)

P e measured continously via a downstream Compton polarimeter [4, 5]

Compton scattering is senstitive to P e and P γ (σ(j = 3/2) > σ(j = 1/2))

maximum for “head-on” γ − e collisions

High intensity laser (25MW), lead to 1000 compton interaction per pulse

Compton-scattered e are bent by a magnet and

analyzed by multichannel cherenkov detector

Compton-scattered γ are analyzed by a quarz-fiber

calorimeter for cross-check

(38)

Electron polarization durin data-taking

σ P

e

∼ 0.5% (initial 2.7%)

(39)

Energy calibration

√ s at SLC was measured and monitored downstream of the beams;

precise vertical bending dipole to bend the beam;

two horizontal bending magnets to produce synchrotron radiation before and after dipole;

syncrotron light monitor to measure the bending;

Expected precision 20 MeV

Absolute calibration from Z peak mass and

comparing with LEP results

(40)

SLDetector

Beam pipe (and beam) smaller than LEP, closer vertex detector;

Slow repetition rate, use slow but high resolution CCD arrays;

Central Drift Chamber LAr calorimeter

Event selection as at LEP,

better b/c tagging.

(41)

Results

A 0 LR = 0.1514 ± 0.0022 syst. uncert:

I

δP e /P e ∼ 0.5%

I

interference correction (0.39%)

I

s (0.15%)

Additional asymmetries (arc spin trasport, e + polarization, lumi) negligible

A 0 LR = 2(1 − 4 sin 2 θ lept eff )

1 + (1 − 4 sin 2 θ lept eff )

sin 2 θ lept eff = 0.23097 ± 0.00027

at LEP

(42)

A LRFB

Computed for ` = e, µ, τ

I

15k Z → ee, 12k → µµ, 11k → τ τ

I

via likelihood fit to d cos θ d

I

uncertainties dominated by statistics

I

then radiative corrections, polarization, background larger for τ due to τ decay ID

smaller for e: from A LR and A LRFB

A e =0.1544 ± 0.0060 A µ =0.142 ± 0.015 A τ =0.136 ± 0.025

A ` = 0.1513 ± 0.0021, sin 2 θ eff lept = 0.23098 ± 0.00026

(43)

τ polarization

At LEP e ± not polarized, but it is possible to look at the polarization of final state in Z → τ τ decays, by using the τ → ν τ + X decay.

P τ = σ + − σ σ + + σ −

where σ + is the x-sec to produce positive helicity τ etc.

Must take care that τ have mass, so helicity state do not correspond to chirality state (the one involved directly in the SM coupling).

I

However at LEP, p τ ≈ M Z /2  m τ , so helicity ≈ chirality

P τ = − A τ (1 + cos 2 θ τ

) + 2A e cos θ τ

(1 + cos 2 θ ) + 8/3A τ cos θ

(44)

Experimental method

Consider τ → πν decay.

It is purely V-A charged weak current, and obviously conserve angular momentum. In the τ rest frame, the neutrino (LH) is emitted opposite to τ spin, and so the π is emitted

backward, namely along the τ spin.

If the τ has positive helicity (spin k to momentum) then the π is emitted in the p τ direction (left configuration), and so, in the LAB (LEP CM), has larger momentum.

If helicity is negative (spin antik), π is emitted in opposite direction and has smaller momentum.

Bottomline: if τ has positive (negative) helicity, the π momentum is higher (lower)

(45)

(a) Measure the direction of π in the τ rest frame cos θ π

1 Γ

d Γ

d cos θ pi = 1

2 (1 + P τ cos θ π )

boosted in the LAB (x π = E π /E τ = E π /E beam ) 1

Γ d Γ

dx pi = 1 + P τ (2x π − 1)

Similar, but more complex, and less sensitive, for

(b) τ → ρν

(c) τ → a 1 ν

(46)
(47)

P τ results

A τ =0.1439 ± 0.0043 A e =0.1498 ± 0.0049

A ` =0.1465 ± 0.0033

sin 2 θ eff lept =0.23159 ± 0.00041

(48)

Results on axial and vector coupling

g V −A f scale from σ ff ∝ g Vf 2 + g Af 2 g V −A f ratio from A f = 2 1+(g g

Vf

/g

Af

Vf

/g

Af

)

2

A 0 FB = 3 4 A e A f A 0 LR =A e A 0 LRFB = 3

4 A f

P τ = − A τ

(49)

Results on axial and vector coupling

g V −A f scale from σ ff ∝ g Vf 2 + g Af 2 g V −A f ratio from A f = 2 1+(g g

Vf

/g

Af

Vf

/g

Af

)

2

A 0 FB = 3 4 A e A f A 0 LR =A e

A 0 LRFB = 3

4 A f

P τ = − A τ

(50)

Results on axial and vector coupling

g V −A f scale from σ ff ∝ g Vf 2 + g Af 2 g V −A f ratio from A f = 2 1+(g g

Vf

/g

Af

Vf

/g

Af

)

2

A 0 FB = 3 4 A e A f A 0 LR =A e

A 0 LRFB = 3

4 A f

P τ = − A τ

(51)

C and B quarks coupling

(52)

A q vs A `

vert band: A ` , horiz: A q , diagonal: A 0,q FB

(53)

sin 2 θ lept eff

(54)

Input to global EWK fit

(in parenthesis the order followed in these lessons)

3

− −21 0 1 2 3

tot

O) / σ

indirect

(O

2) (MZ αs

2) (MZ (5) αhad

mt

b R0 c R0 0,b AFB

0,c AFB

Ab Ac (Tevt.) lept Θeff sin2

) FB lept(Q eff 2Θ sin

(SLD) Al

(LEP) Al

0,l AFB

lep R0

0 σhad ΓZ MZ ΓW MW MH

Global EW fit Indirect determination Measurement GfitterSM

Mar ’18

Higgs mass (4)

I

LHC

W mass and width (2)

I

LEP2, Tevatron, LHC Z-pole observables (1)

I

LEP1, SLD

I

M Z , Γ Z

I

σ 0 had

I

sin 2 θ lept eff

I

Asymmetries

I

BR R lep,b,c 0 = Γ had /Γ ``,b ¯ b,c ¯ c Top mass (3)

I

Tevatron, LHC other:

I

α s (M Z 2 ), ∆α had (M Z 2 )

(55)

Input to global EWK fit

(in parenthesis the order followed in these lessons)

Higgs mass (4)

I

LHC

W mass and width (2)

I

LEP2, Tevatron, LHC Z-pole observables (1)

I

LEP1, SLD

I

M Z , Γ Z

I

σ 0 had

I

sin 2 θ lept eff

I

Asymmetries

I

BR R lep,b,c 0 = Γ had /Γ ``,b ¯ b,c ¯ c

Top mass (3)

(56)

ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., “Precision electroweak measurements on the Z resonance,” Phys.Rept. 427 (2006) 257–454, arXiv:hep-ex/0509008 [hep-ex].

F. Halzen and A. D. Martin, Quarks and Leptons: an introductory course in modern particle physics.

Wiley, New York, USA, 1984.

G. A. Loew, “The slac linear collider and a few ideas on future linear colliders,” Proceedings of the 1984 Linear Accelerator Conference (1984) . https://inis.iaea.org/search/search.aspx?orig_q=RN:16018252.

G. Shapiro, S. Bethke, O. Chamberlain, R. Fuzesy, M. Kowitt, D. Pripstein, B. Schumm, H. Steiner, M. Zolotorev, P. Rowson, D. Blockus, H. Ogren, M. Settles, M. Fero, A. Lath, D. Calloway, R. Elia, E. Hughes, T. Junk, and G. Zapalac, “The compton polarimeter at the slc,”.

https://escholarship.org/uc/item/78t2f6wp.

SLD Collaboration, M. J. Fero, “The Compton polarimeter for SLC,” in Frontiers of high energy spin physics. Proceedings, 10th International Symposium, 35th Yamada Conference, SPIN’92, Nagoya, Japan, November 9-14, 1992, pp. 899–904.

1992.

http://www-public.slac.stanford.edu/sciDoc/docMeta.aspx?slacPubNumber=SLAC-PUB- 6026.

(57)

Outline

1 Introduction

2 Z-pole observables

3 Asymmetries

4 W mass and width Motivation At LEP II M W at Tevatron M W at LHC

5 Top mass

(58)

Outline

1 Introduction

2 Z-pole observables

3 Asymmetries

4 W mass and width

5 Top mass

6 Higgs mass and features

7 Global ElectroWeak fit

(59)

Outline

1 Introduction

2 Z-pole observables

3 Asymmetries

4 W mass and width

5 Top mass

6 Higgs mass and features

(60)

Outline

1 Introduction

2 Z-pole observables

3 Asymmetries

4 W mass and width

5 Top mass

6 Higgs mass and features

7 Global ElectroWeak fit

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Since we are comparing two polynomials, it follows that the identity holds for all x

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.. We will show a more

(American Mathematical Monthly, Vol.118, January 2011). Proposed by Kieren MacMillan (Canada) and Jonathan

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.. Let π ∈

Now we modify the last row of M n by adding a proper linear combination of the other rows. This operation does not change the determinant of M n , but it will help us to

Recall that a variety is k-unirational, or unirational over k, if it is unirational and if moreover the rational dominant map from the projective space to the variety is defined