lecture 11-12 - duke universitywebhome.phy.duke.edu/~goshaw/lecture11-12_2017_v3.pdf · 2017. 10....

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A. Goshaw Physics 846

Lecture 11-12 October 5 , 2017

Exploring SM QCD predictions Dissecting the QCD Lagrangian Ø  Recap of derivation of QCD from SUc(3) Ø The quark part Ø The gluon part

Observations about QCD predictions Ø  Properties of quarks and gluons Ø  Evolution of as(Q2) Ø  Parton distribution functions Ø  Carrying out QCD calculations

Reading: Chapter 16 in the text book

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Other experiments have great results too

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LIGO–Virgo jointly observed gravitational wave signal from a binary black hole mergerEvent dates from Aug 14, 2017 at 10:30:43 UTC (GW170814); result released Sep 27: https://tds.virgo-gw.eu/GW170814

+8 ms +14 msSignal-to-noise ratio (SNR)

Huge impact: GW150914 paper by LIGO (PRL,116, 061102, Jan 2016) collected 1840 citations to date

Time-frequency representation of strain data

Addition of Virgo improves position uncertainty and polarisation information (tensor favoured)

inspiral

merging

ringdown

1.9 billion years ago …

GW170814:merger of 31 + 25 M⨀ BHs

Detector data in colour; dark grey BBH merger template

Rainer Weiss, Kip S. Thorne, Barry C. Barish

Nobel 2017

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Development of QCD theory

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QCD from SUc(3) The prescription

●  The QCD Lagrangian LQCD is obtained by postulating invariance under the SUc(3) transformation:

!

0= exp [�igs�a(x)Ta]

=

2

4qrqgqb

3

5 =

2

4q1q2q3

3

5

●  Finding the invariance requires that the spin 1 boson fields Gaµ

be transformed by (a =1-8 for QCD): For the case of QCD the fabc are the SU(3) group structure constants and is the strong interaction coupling constant.

where the βa(x) are real functions of (ct,x,y,z) and and the with λa the eight 3x3 Gell-Mann matrices. Ta = 1

2�a

gs =p4⇡↵s

G

µa ! G

0µa = G

µa + (~c)@µ

�a(x) + gsfabc�b(x)Gµc

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●  Start with the free field fermion Lagrangian for a quark of flavor q (one of u,d,c,s,t,b) and mass mq It is described by a Dirac spinor specified by a “bold” qj for a quark of color j . (color index j = 1,2,3 = r,g,b and 4-vector index µ = 0,1,2,3) As usual there is an implied sum over repeated indices..

●  Here I have anticipated notation arising in the QCD Lagrangian by introducing the δjk which just insures each of the terms in the brokets [ ] connects quarks of the same color.

QCD from SUc(3) The quark Lagrangian

Lqfree = [i(~c)qj�µ@µqk - (mqc2)qjqk] �jk

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● 

● 

The fermion part of the QCD Lagrangian is obtained by modifying the

free particle Lagrangian Lq. This is done by replacing:

● 

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The [Ta]jk are simply the jk component of the 3x3 Ta matrices.

For example [T1]11 = 0, [T1]12 = [T1]21 =

12 .

Again, gs =

p4⇡↵s = the strong interaction coupling

and Ta = �a/2 with �a = the eight Gell-Mann matrices of SU(3).


�jk@µ by Dµjk = �jk@µ + i gs~c [Ta]jkGµ

a

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●  Therefore the fermion part of the QCD Lagrangian is:

 the free quark Lagrangian  the quark-gluon interaction


= [i(~c)qj�µ@µqk - (mqc2)qjqk ]�jk - gs[Ta]jk qj�µqk Gµa

●  Analogous to QED (see L9, p29) define a quark QCD current:

Jaµ = gs[Ta]jkqj�µqk

and L(q-g interaction) = - Jaµ Gµ

a

Lq = i(~c)qj�µDµjkqk - (mqc2)qjqk�jk

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The fermion-gluon coupling

●  Using the color notation where j = 1,2,3 = red, green, blue the G1

µ interaction is:

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 qr  qg

 g r g

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 qg  qr

 g g r  

 +

●  The quark-gluon interaction mixes quarks of different colors. For example for “a” = 1 the interaction is simply:

� gs2 [q1�µq2 + q2�µq1]G

µ1

� gs2 [qr�µqg + qg�µqr]G

µ1

� gs2 �µ

●  The gluons are bi-colored = color-anticolor

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●  Start with the free field Lagrangian for a boson of spin 1 and mass m described by a 4-vector Ga

µ :

(a = 1 to 8 for QCD): : LG = - 14F

µ⌫a F a

µ⌫ + 12 (mc/~)2 Gµ

a Gaµ

●  Allowing for the possibility that the fields can be self-interacting the field tensor is: where as above gs = the strong interaction coupling of the fields and fabc = SU(3) structure constants.

Fµ⌫a = @µG⌫

a � @⌫Gµa � gsfabcG

µbG

⌫c

QCD from SUc(3) The qluon Lagrangian

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QCD from SUc(3) The qluon Lagrangian

●  As for the quark Lagrangian, demand invariance under the transformation (see page 3):

●  It is a non-trivial exercise to show that although

is not invariant under this transformation, the product is invariant. (see Barger and Phillips page 38-39 for an outline of the proof).

Fµ⌫a = @µG⌫


µbG

⌫c

Gµa ! G0µ

a

Fµ⌫a F a

µ⌫

●  However the mass term in the Lagrangian proportional to is not invariant under Therefore to preserve the invariance the gluon mass must be == 0.

Gµa Ga

µ

Gµa ! G0µ

a

G

µa ! G

0µa = G

µa + (~c)@µ

�a(x) + gsfabc�b(x)Gµc

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QCD from SUc(3) The complete QCD Lagrangian

●  Therefore the QCD Lagrangian for a quark of flavor q is:

with

Dµjk = �jk@µ + i gs~c [Ta]jkGµ

a

Fµ⌫a = @µG⌫


µbG

⌫c

●  Using the Euler-Lagrange equation you can obtain the field equations for each quark field q and gluon field Ga

µ (left for a homework problem).

●  There is one such Lagrangian for each quark flavor q = u,d,c,s,t,b . The complete QCD Lagrangian is

Pu,d,c,s,t,b L

qQCD

LQCD = = - 14Fµ⌫a F a

µ⌫ + i(~c)qj�µDµjkqk - (mqc2)qjqk�jk

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Observations about QCD predictions

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Properties of quarks

u  Quarks carry single colors r, g or b ; anti-quarks carry single anti-colors . In group language these are the 3 and 3 triplets of SU(3). For example a charm quark with spin ½ , Q= +2/3 , color = red has an anti-charm partner with spin ½ , Q= -2/3 , color = red The J/psi is a (c c) meson with JP = 1- , Q = 0 and color = 0.

r, g or

¯b

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 qg  qr

 g g r  

Gluons can change quark colors but not flavors (wait for the weak interaction to do this).

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 qg  qg  g g r

 γ

QCD QED

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Gµ1 = (rg + gr)/

p2

Gµ2 = -i (rg � gr)/

p2

Gµ3 = (rr + gg)/

p2

Gµ4 = (rb+ br)/

p2

Gµ5 = -i (rb� br)/

p2

Gµ6 = (gb+ bg)/

p2

Gµ7 = -i (gb� bg)/

p2

Gµ8 = (rr + gg � 2bb)/

p6

u  Gluons are bi-colored and members of the octet representation of SU(3): 3x3 = 8 + 1 . The structure of the 8 Gaµ can be obtained following the example of G1

µ on page 7 (or look it up in group theory).

Properties of gluons

Gluons can self interact via either a triplet and quartic vertex.

⇡ gs ⇡ g2s

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Nuclear and Particle Physics Franz Muheim 15

Evidence for GluonsEvidence for Gluons

Quarks radiate Gluons2nd order diagram

Experimental SignatureGluons confined, fragmentshadronises into jetÎ3-jet events

Measurement of αSWhen including gluon radiationadditional factor √αS in matrix elementadds term with factor αS in cross section

e.g R(q2 = (25 GeV)2) ≈ 3.85 > 11/3 → αS = 0.15

gqqee →−+

JADE √s = 35 GeV

( )( ) ⎟

⎠⎞

⎜⎝⎛ +=

→→

= ∑−+−+

−+

πα

µµσσ S2 13

hadronsq qQee

eeR

LEP √s = 91 GeV






e.g R(q2 = (25 GeV)2) ≈ 3.85 > 11/3 → αS = 0.15

gqqee →−+

JADE √s = 35 GeV

( )( ) ⎟

⎠⎞

⎜⎝⎛ +=

→→

= ∑−+−+

−+

πα

µµσσ S2 13

hadronsq qQee

eeR

LEP √s = 91 GeV






e.g R(q2 = (25 GeV)2) ≈ 3.85 > 11/3 → αS = 0.15

gqqee →−+

JADE √s = 35 GeV

( )( ) ⎟

⎠⎞

⎜⎝⎛ +=

→→

= ∑−+−+

−+

πα

µµσσ S2 13

hadronsq qQee

eeR

LEP √s = 91 GeV

●  One of the first experimental confirmation of QCD was the observation of 3 jet events from e+ e- collisions. The 3rd jet was gluon radiation of one of the quarks.

Properties of gluons

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Running of αs(Q2)

9. Quantum chromodynamics 39

They are well within the uncertainty of the overall world average quoted above. Note,however, that the average excluding the lattice result is no longer as close to the valueobtained from lattice alone as was the case in the 2013 Review, but is now smaller byalmost one standard deviation of its assigned uncertainty.

Notwithstanding the many open issues still present within each of the sub-fieldssummarised in this Review, the wealth of available results provides a rather precise andreasonably stable world average value of αs(M2

Z), as well as a clear signature and proof ofthe energy dependence of αs, in full agreement with the QCD prediction of AsymptoticFreedom. This is demonstrated in Fig. 9.3, where results of αs(Q2) obtained at discreteenergy scales Q, now also including those based just on NLO QCD, are summarized.Thanks to the results from the Tevatron and from the LHC, the energy scales at whichαs is determined now extend up to more than 1 TeV♦.

QCD αs(Mz) = 0.1181 ± 0.0011

pp –> jetse.w. precision fits (N3LO)

0.1

0.2

0.3

αs (Q2)

1 10 100Q [GeV]

Heavy Quarkonia (NLO)e+e– jets & shapes (res. NNLO)

DIS jets (NLO)

April 2016

τ decays (N3LO)

1000

(NLOpp –> tt (NNLO)

)(–)

Figure 9.3: Summary of measurements of αs as a function of the energy scale Q.The respective degree of QCD perturbation theory used in the extraction of αs isindicated in brackets (NLO: next-to-leading order; NNLO: next-to-next-to leadingorder; res. NNLO: NNLO matched with resummed next-to-leading logs; N3LO:next-to-NNLO).

♦ We note, however, that in many such studies, like those based on exclusive states ofjet multiplicities, the relevant energy scale of the measurement is not uniquely defined.For instance, in studies of the ratio of 3- to 2-jet cross sections at the LHC, the relevantscale was taken to be the average of the transverse momenta of the two leading jets [381],but could alternatively have been chosen to be the transverse momentum of the 3rd jet.

January 6, 2017 18:42

●  Just as for QED, the QCD coupling strength evolves with the energy used to probe the vertex. Due to contributions from gluon self-interaction loops, as decreases with probe energy Q à asymptotic freedom. This allows perturbative calculations to be done with high energy probes, but not for bound state hadrons.

●  A useful paramaterization is:

where nf = the number of quark flavors in the loops and is a parameter fit from data.

1↵s(Q2) = 33�2nf

12⇡ ln[ Q2

⇤2QCD

]

⇤QCD

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CP violation in QCD

●  I mentioned in L2 the discrete symmetry operators: Ø  P = parity (x,y,z) à (-x,-y,-z) Ø  T = time reversal t à -t Ø  A 3rd one is C = charge conjugation, defined to be an operator changing particles to anti-particles.

●  The product of all 3, TCP, is conserved under very basic theory assumptions (it is difficult to dream up theories that violate TCP).

●  The eigenvalues of these operators are conserved via a multiplicative conservation law if the underlying theory obeys the symmetry à “conservation of parity”, etc.

●  We will return to this in more detail in study of the weak interaction.

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CP violation in QCD

●  Experimentally, studies of processes mediated by QED and QCD find no violation of P, C and T symmetry. ●  The QED and QCD Lagrangians we developed have exact P, C and T symmetry.

●  However there is an additional term that can be added to the QCD Lagrangian that passes muster under the requirement of SUc(3) invariance. This term can violate CP invariance big time. It is built out of the gluon field tensor already used for

Fµ⌫a = @µG⌫


µbG

⌫c

- 14Fµ⌫a F a

µ⌫

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CP violation in QCD

●  The additional CP violating term is:

●  This term would introduce a large violation of CP invariance in the strong interaction where none is observed experimentally. In particular it predicts that the neutron should have an electric dipole moment.

where

˜F aµ⌫ =

12✏µ⌫⇢� F a⇢�

and ✏µ⌫⇢� is a totally antisymmetric tensor

(=+1 even permutation of 0,1,2,3, -1 if odd, 0 otherwise)

⇥QCD is an unconstrained dimensionless paramater.

●  Experimental limits on the electric dipole moment, interpreted in terms of the above QCD violating parameter, require that

⇥QCD

32⇡2 g2s Fµ⌫a F a

µ⌫

|⇥QCD| 10�11

●  The un-naturally small value of this QCD CP violation suggests looking for a reason that it is exactly zero.

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●  There have been many experimental searches for axions with no observed signals. However it remains an option for making a contribution to the mystery of dark matter.

●  In 1977 Peccei and Quinn (Phys. Rev. Lett. 38, 1440) proposed a very elegant solution to this “strong CP problem”. They postulated the existence of an additional U(1) symmetry for QCD, that would require the CP violating term to be zero at the expensive of introducing a so-called Goldstone boson that acquires a small mass when the symmetry is broken.

●  This particle is called the axion. It is a neutral, scalar, weakly interacting particle and could have a long lifetime. Therefore it is a candidate for making a contribution to dark matter.

CP violation in QCD

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Testing QCD theory

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Testing QCD with experiment

●  The QCD Lagrangian gives an exact prescription for making predictions for any process governed by the strong interaction.

●  However making the QCD theory vs experiment comparisons are not easy. 1. The coupling strength αs(Q2) at low energies is so large ( à ~ 1) that perturbative calculations can not be done. This requires development on new techniques such as space-time lattice based calculations to make approximate predictions. Even at high energies αs(Q2) ~ 0.1, requiring higher order terms in calculations using Feynman diagrams. This often involves evaluating literally 10’s of thousand diagrams.

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Testing QCD with experiment

2. The predictions from QCD are at the quark/gluon (parton) level and there are no free particle beams quarks/gluons. The best you can do is throw a container of them stored inside a hadron to make random collisions of the partons.

●  For example consider the production of two beauty mesons in high energy p p collisions: p + p à B+ + B- + X

●  The parton-level process here can be calculated from QCD theory. In order to compare with experiment the the flux of u quarks from the proton must be known.

u+ u ! b+ b

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Parton distribution functions

●  A proton with some high momentum p carries along with it a collection of partons.

●  Let pi = the momentum carried by the ith parton and xi = pi/p .

●  The xi have distributions that depends on the parent hadron and the particular parton ( u, d, g, s, …). These distributions are called parton distribtution functions (PDF’s).

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●  These PDF’s must obey sum rules. For a proton:

●  Let u(x) = the number distribution function of u quarks. That is u(x) dx = the number of u quarks with x between x and x+dx. Similarly for d(x), g(x), s(x), u (x), d (x), s (x) . Seven PDF’s ignoring small sea contributions of c, b, t quarks.

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●  The distributions in x of a particular parton also depend on the energy Q with which the parton enters a collision. Therefore in fact the distributions are of the form u(x;Q).

●  When measured at some scale Qo à u(x;Qo), then QCD theory predicts the PDF at any other value of Q à u(x;Q).

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●  In a particular collision, a parton can carry any fraction of the protons momentum from ~ 0 to ~ 1. ●  But averaged over many collisions, the average fraction of momentum carried by partons of a given flavor can be calculated from the PDF’s. For example for the proton (at some modest Q):

●  The “missing” momentum fraction of ~ 0.55 is carried by gluons:

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Experimental tests of QCD theory

●  Therefore in making predictions that can be used to test QCD theory there are several steps:

Ø  Measure from experiment αs at some fixed Q2 scale Ø  Calculate from QCD theory the evolution of αs to the Q2 scale appropriate for your measurement and the parton level cross section

Ø  Use measured Parton Distribution Functions (PDF) to predict the parton momentum spectrum for the collision you are studying.

Ø  Convolute the parton level cross section with the parton flux from the PDF, and integrate over the parton flux. Ø  Introduce fragmentation functions for the specific final state hadrons (e,g, b quark -> B meson).

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Experimental tests of QCD theory

●  The precision of QCD tests are limited by various factors:

1. The strong coupling strength is known to ~ 2% and must be evolved to a Q2 characteristic of the process studied.

2. Parton distributions are not well known at high x values, limiting the precision of some cross section predictions.

3. Higher order perturbative corrections can be large. Typically at high energies there are on the 30% corrections in going from the lowest order (LO) to the next term in the perturbative expansion.

●  The bottom line is that most QCD predictions are tested experimentally to ~ few to 10% compared to EWK predictions that are often tested to better than 0.1%.

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LHC data from 7 TeV proton proton collisions producingquark/gluon fragmentation into a high energy jet

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LHC data from 7 TeV proton proton collisionsDistribution of the invariant mass of di-jets.

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LHC data from 8 TeV proton proton collisionsQED plus QCD test from the production of a photon

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LHC data from 7 TeV proton proton collisionsProduction of a pair of beauty quarks, measured as a pair of beauty hadrons

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LHC data from 7 TeV proton proton collisionsInvariant mass of the jets containing beauty hadrons

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End Lecture 11 (+12) Next Lecture:

More QED plus QCD examples Why they are not enough.

lecture 11-12 - duke universitywebhome.phy.duke.edu/~goshaw/lecture11-12_2017_v3.pdf · 2017. 10....

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