Lecture 11-12 October 5, 2017 Exploring SM QCD predictions Dissecting the QCD Lagrangian Ø Recap of derivation of QCD from SU c (3) Ø The quark part Ø The gluon part Observations about QCD predictions Ø Properties of quarks and gluons Ø Evolution of a s (Q 2 ) Ø Parton distribution functions Ø Carrying out QCD calculations Reading: Chapter 16 in the text book 1
Other experiments have great results too Rainer Weiss, Kip S. Thorne, Barry C. Barish Nobel 2017 Huge impact: GW150914 paper by LIGO (PRL,116, 061102, Jan 2016) collected 1840 citations to date A. Goshaw Physics 846 LIGO Virgo jointly observed gravitational wave signal from a binary black hole merger Event dates from Aug 14, 2017 at 10:30:43 UTC (GW170814); result released Sep 27: https://tds.virgo-gw.eu/gw170814 1.9 billion years ago Signal-to-noise ratio (SNR) +8 ms +14 ms GW170814: merger of 31 + 25 M BHs Time-frequency representation of strain data merging Detector data in colour; dark grey BBH merger template inspiral ringdown Addition of Virgo improves position uncertainty and polarisation information (tensor favoured) 2 7
Development of QCD theory 3
QCD from SU c (3) The prescription The QCD Lagrangian L QCD is obtained by postulating invariance under the SU c (3) transformation: 2! 0 =exp[ ig s a (x)t a ] = 4 q 3 r q g 5 = q b where the β a (x) are real functions of (ct,x,y,z) and and the T a = 1 with λ a the eight 3x3 Gell-Mann matrices. 2 a 2 4 q 1 q 2 q 3 3 5 Finding the invariance requires that the spin 1 boson fields G a µ be transformed by (a =1-8 for QCD): G µ a! G 0µ a = G µ a +(~c)@ µ a(x)+g s f abc b (x)g µ c For the case of QCD the f abc are the SU(3) group structure constants and g s = p 4 s is the strong interaction coupling constant. 4
QCD from SU c (3) The quark Lagrangian Start with the free field fermion Lagrangian for a quark of flavor q (one of u,d,c,s,t,b) and mass m q It is described by a Dirac spinor specified by a bold q j for a quark of color j. (color index j = 1,2,3 = r,g,b and 4-vector index µ = 0,1,2,3) L q free =[i(~c) q j µ@ µ q k -(m q c 2 ) q j q k ] jk As usual there is an implied sum over repeated indices.. Here I have anticipated notation arising in the QCD Lagrangian by introducing the δ jk which just insures each of the terms in the brokets [ ] connects quarks of the same color. 5
QCD from SU c (3) The quark Lagrangian The fermion part of the QCD Lagrangian is obtained by modifying the free particle Lagrangian L q. This is done by replacing: jk@ µ by D µ jk = jk@ µ + i g s ~c [T a] jk G µ a Again, g s = p 4 s = the strong interaction coupling and T a = a /2 with a = the eight Gell-Mann matrices of SU(3). The [T a ] jk are simply the jk component of the 3x3 T a matrices. For example [T 1 ] 11 = 0, [T 1 ] 12 =[T 1 ] 21 = 1 2. 6
QCD from SU c (3) The quark Lagrangian Therefore the fermion part of the QCD Lagrangian is: L q = i(~c) q j µ D µ jk q k -(m q c 2 ) q j q k jk =[i(~c) q j µ @ µ q k -(m q c 2 ) q j q k ] jk - g s [T a ] jk q j µ q k G µ a the free quark Lagrangian the quark-gluon interaction Analogous to QED (see L9, p29) define a quark QCD current: J a µ = g s [T a ] jk q j µ q k and L(q-g interaction) = - J a µ G µ a 7
The fermion-gluon coupling A. Goshaw Physics 846 The quark-gluon interaction mixes quarks of different colors. For example for a = 1 the interaction is simply: g s 2 [ q 1 µ q 2 + q 2 µ q 1 ]G µ 1 Using the color notation where j = 1,2,3 = red, green, blue the G µ 1 interaction is: g s 2 [ q r µ q g + q g µ q r ]G µ 1 q g q r q r q g g g r + g r g g s 2 µ The gluons are bi-colored = color-anticolor 8
QCD from SU c (3) The qluon Lagrangian Start with the free field Lagrangian for a boson of spin 1 and mass m described by a 4-vector G a µ : L G =- 1 4 F µ a F a µ + 1 2 (mc/~)2 G µ a G a µ (a = 1 to 8 for QCD): : Allowing for the possibility that the fields can be self-interacting the field tensor is: F µ a = @ µ G a @ G µ a g s f abc G µ b G c where as above g s = the strong interaction coupling of the fields and f abc = SU(3) structure constants. 9
QCD from SU c (3) The qluon Lagrangian As for the quark Lagrangian, demand invariance under the transformation (see page 3): G µ a! G 0µ a = G µ a +(~c)@ µ a(x)+g s f abc b (x)g µ c It is a non-trivial exercise to show that although F a µ = @ µ G a @ G µ a g s f abc G µ b G c is not invariant under this G µ a! G 0µ a transformation, the product F a µ Fµ a is invariant. (see Barger and Phillips page 38-39 for an outline of the proof). However the mass term in the Lagrangian proportional to G µ a G a µ is not invariant under G µ a! G 0µ a Therefore to preserve the invariance the gluon mass must be == 0. 10
QCD from SU c (3) The complete QCD Lagrangian Therefore the QCD Lagrangian for a quark of flavor q is: L QCD ==- 1 4 F µ a F a µ + i(~c) q j µ D µ jk q k -(m q c 2 ) q j q k jk with F µ a = @ µ G a @ G µ a g s f abc G µ b G c D µ jk = jk@ µ + i g s ~c [T a] jk G µ a There is one such Lagrangian for each quark flavor q = u,d,c,s,t,b. The complete QCD Lagrangian is P u,d,c,s,t,b Lq QCD Using the Euler-Lagrange equation you can obtain the field equations for each quark field q and gluon field G µ a (left for 11 a homework problem).
Observations about QCD predictions 12
Properties of quarks A. Goshaw Physics 846 u Quarks carry single colors r, g or b ; anti-quarks carry single anti-colors r, ḡ or b. 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 J P = 1 -, Q = 0 and color = 0. Gluons can change quark colors but not flavors (wait for the weak interaction to do this). q g q r g g r q g γ q g QCD QED 13
Properties of gluons A. Goshaw Physics 846 u Gluons are bi-colored and members of the octet representation of SU(3): 3x3 = 8 + 1. The structure of the 8 G aµ can be obtained following the example of G 1 µ on page 7 (or look it up in group theory). G µ 1 =(rḡ + g r)/p 2 G µ 2 =-i(rḡ g r)/p 2 G µ 3 =(r r + gḡ)/p 2 G µ 4 =(r b + b r)/ p 2 G µ 5 =-i(r b b r)/ p 2 G µ 6 =(g b + bḡ)/ p 2 G µ 7 =-i(g b bḡ)/ p 2 G µ 8 =(r r + gḡ 2b b)/ p 6 Gluons can self interact via either a triplet and quartic vertex. g s g 2 s 14
Properties of gluons A. Goshaw Physics 846 One of the first experimental confirmation of QCD was the observation of 3 jet events from e + e - collisions. The 3 rd jet was gluon radiation of one of the quarks. JADE s = 35 GeV LEP s = 91 GeV s Measurement of 15
Running of α s (Q 2 ) A. Goshaw Physics 846 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, a s 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. α s (Q 2 ) 0.3 0.2 0.1 τ decays (N 3 LO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) e.w. precision fits (N 3 LO) ( ) pp > jets (NLO) pp > tt (NNLO) QCD α s (M z ) = 0.1181 ± 0.0011 1 10 100 Q [GeV] April 2016 1000 A useful paramaterization is: 1 s (Q 2 ) = 33 2n f 12 ln[ Q 2 2 QCD where n f = the number of quark flavors in the loops and QCD is a parameter fit from data. ] 16
CP violation in QCD A. Goshaw Physics 846 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 3 rd one is C = charge conjugation, defined to be an operator changing particles to anti-particles. The eigenvalues of these operators are conserved via a multiplicative conservation law if the underlying theory obeys the symmetry à conservation of parity, etc. The product of all 3, TCP, is conserved under very basic theory assumptions (it is difficult to dream up theories that violate TCP). We will return to this in more detail in study of the weak interaction. 17
CP violation in QCD A. Goshaw Physics 846 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 SU c (3) invariance. This term can violate CP invariance big time. It is built out of the gluon field tensor F µ a = @ µ G a @ G µ a g s f abc G µ b G c already used for - 1 4 F µ a F a µ 18
CP violation in QCD A. Goshaw Physics 846 The additional CP violating term is: where F µ a = 1 2 µ 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. F a 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. Experimental limits on the electric dipole moment, interpreted in terms of the above QCD violating parameter, require that QCD apple 10 11 QCD 32 g 2 2 s F a µ F a µ The un-naturally small value of this QCD CP violation suggests looking for a reason that it is exactly zero. 19
CP violation in QCD A. Goshaw Physics 846 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. 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. 20
Testing QCD theory 21
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 α s (Q 2 ) 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 α s (Q 2 ) ~ 0.1, requiring higher order terms in calculations using Feynman diagrams. This often involves evaluating literally 10 s of thousand diagrams. 22
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 u +ū! b + b 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. 23
Parton distribution functions A proton with some high momentum p carries along with it a collection of partons. Let p i = the momentum carried by the ith parton and x i = p i /p. The x i 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). 24
Parton distribution functions 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. These PDF s must obey sum rules. For a proton: 25
Parton distribution functions A. Goshaw Physics 846 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 Q o à u(x;q o ), then QCD theory predicts the PDF at any other value of Q à u(x;q). 26
Parton distribution functions A. Goshaw Physics 846 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: 27
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 α s at some fixed Q 2 scale Ø Calculate from QCD theory the evolution of α s to the Q 2 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). 28
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 Q 2 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%. 29
LHC data from 7 TeV proton proton collisions producing quark/gluon fragmentation into a high energy jet A. Goshaw Physics 846 30
LHC data from 7 TeV proton proton collisions Distribution of the invariant mass of di-jets. A. Goshaw Physics 846 31
LHC data from 8 TeV proton proton collisions QED plus QCD test from the production of a photon A. Goshaw Physics 846 32
LHC data from 7 TeV proton proton collisions Production of a pair of beauty quarks, measured as a pair of beauty hadrons A. Goshaw Physics 846 33
LHC data from 7 TeV proton proton collisions Invariant mass of the jets containing beauty hadrons A. Goshaw Physics 846 34
End Lecture 11 (+12) Next Lecture: More QED plus QCD examples Why they are not enough. 35