Hadron Structure Jianwei Qiu Brookhaven National Laboratory Stony Brook University The plan for my three lectures q The Goal: To understand the hadron structure in terms of QCD and its hadronic matrix elements of quark-gluon field operators, to connect these matrix elements to physical observables, and to calculate them from QCD (lattice QCD, inspired models, …) q The outline: See also lectures by Shepard on Hadrons, partons (quarks and gluons), “Hadron Spectroscopy”, and and probes of hadron structure lectures by Deshpande on One lecture “Electron-Ion Collider” and Parton Distribution Functions (PDFs) and lectures by Gandolfi on “Nuclear Structure” Transverse Momentum Dependent PDFs (TMDs) and lectures by Aschenauer on One lecture “Accelerators & detectors” Generalized PDFs (GPDs) and multi-parton correlation functions One lecture New particles, new ideas, and new theories q Early proliferation of new hadrons – “particle explosion”:
… and many more! Hadrons have internal structure! q Nucleons cannot be point-like spin-1/2 Dirac particles: 1933: Proton’s magnetic moment e~ µp = gp Otto Stern 2m ✓ p ◆ Nobel Prize 1943 g =2.792847356(23) = 2! p 6 e~ µ = 1.913 = 0! 1960: Elastic e-p scattering n 2m 6 ✓ p ◆ Robert Hofstadter Nobel Prize 1961
Proton Proton EM charge radius!
Neutron Form factors Electric charge distribution New particles, new ideas, and new theories q Early proliferation of new particles – “particle explosion”:
… and many more!
Quark Model
Proton Neutron Nobel Prize, 1969 The naïve Quark Model q Flavor SU(3) – assumption: Physical states for , neglecting any mass difference, are represented by 3-eigenstates of the fund’l rep’n of flavor SU(3) q Generators for the fund’l rep’n of SU(3) – 3x3 matrices:
with Gell-Mann matrices q Good quantum numbers to label the states:
simultaneously diagonalized Isospin: , Hypercharge: q Basis vectors – Eigenstates: The naïve Quark Model q Quark states:
Spin: ½ Baryon #: B = ⅓ Strangeness: S = Y – B Electric charge:
q Antiquark states: The naïve Quark Model q Mesons = quark-antiquark flavor states: ² Group theory says:
There are three states with : ² Physical meson states (L=0, S=0): 1 1 ⇡0 = (uu¯ dd¯) ⌘ = (uu¯ + dd¯ 2ss¯) p 8 p 2 6 (⌘ , ⌘ ) (⌘, ⌘0) 1 8 1 ! ⌘1 = (uu¯ + dd¯+ ss¯) p3 q Baryon states = 3 quark states:
Proton Neutron The naïve Quark Model q A complete example: Proton q Normalization: q Charge:
q Spin:
q Magnetic moment:
µ 1 µu 2/3 n = 0.68497945(58) µn = [4µd µu] = 2 µ ⇡ 1/3 µp Exp 3 d ✓ ◆ Deep inelastic scattering (DIS) q Modern Rutherford experiment – DIS (SLAC 1968) e(p)+h(P ) e0(p0)+X ! ² Localized probe: 2 2 2 Q = (p p0) 1fm 1 1fm Q ⌧ ² Two variables: 2 2 Q =4EE0 sin (✓/2) Q2 xB = 2mN ⌫ ⌫ = E E0
Discovery of spin ½ quarks, and partonic structure!
The birth of QCD (1973) Nobel Prize, 1990 – Quark Model + Yang-Mill gauge theory Quantum Chromo-dynamics (QCD)
= A quantum field theory of quarks and gluons = q Fields: Quark fields: spin-½ Dirac fermion (like electron) Color triplet: Flavor:
Gluon fields: spin-1 vector field (like photon) Color octet: q QCD Lagrangian density:
q QED – force to hold atoms together: f 1 ( ,A)= [(i@ eA ) µ m ] f [@ A @ A ]2 LQED µ µ f 4 µ ⌫ ⌫ µ Xf q QCD Color confinement: Gluons are dark, No free quarks or gluons ever been detected! QCD Asymptotic Freedom q Interaction strength:
μ2 and μ1 not independent
Discovery of QCD Asymptotic Freedom
Collider phenomenology Nobel Prize, 2004 – Controllable perturbative QCD calculations New particles, new ideas, and new theories q Proliferation of new particles – “November Revolution”:
November Revolution!
Quark Model QCD … andEW many more!
H0 Completion of SM? New particles, new ideas, and new theories q Proliferation of new particles – “November Revolution”:
November Revolution!
Quark Model QCD … andEW many more!
X, … H0 Y, … Completion of SM? Z, … Pentaquark, … A new particle explosion? QCD and hadron internal structure q Our understanding of the proton evolves
1970s 1980s/2000s Now Hadron is a strongly interacting, relativistic bound state of quarks and gluons q QCD bound states: ² Neither quarks nor gluons appear in isolation! ² Understanding such systems completely is still beyond the capability of the best minds in the world q The great intellectual challenge: Probe nucleon structure without “seeing” quarks and gluons? What holds hadron together – the glue? q Understanding the glue that binds us all – the Next QCD Frontier!
q Gluons are weird particles! ² Massless, yet, responsible for nearly all visible mass “Mass without mass!”
Bhagwat & Tandy/Roberts et al What holds it together – the glue? q Understanding the glue that binds us all – the Next QCD Frontier!
q Gluons are weird particles! ² Massless, yet, responsible for nearly all visible mass ² Carry color charge, responsible for color confinement and strong force What holds it together – the glue? q Understanding the glue that binds us all – the Next QCD Frontier!
q Gluons are weird particles! ² Massless, yet, responsible for nearly all visible mass ² Carry color charge, responsible for color confinement and strong force but, also for asymptotic freedom
QCD perturbation theory What holds it together – the glue? q Understanding the glue that binds us all – the Next QCD Frontier!
q Gluons are weird particles! ² Massless, yet, responsible for nearly all visible mass ² Carry color charge, responsible for color confinement and strong force but, also for asymptotic freedom, as well as the abundance of glue What holds it together – the glue? q Understanding the glue that binds us all – the Next QCD Frontier!
q Gluons are wired particles! ² Massless, yet, responsible for nearly all visible mass ² Carry color charge, responsible for color confinement and strong force but, also for asymptotic freedom, as well as the abundance of glue
Without gluons, there would be NO nucleons, NO atomic nuclei… NO visible world! See A. Deshpande’s talk on EIC Hadron properties in QCD q Mass – intrinsic to a particle:
= Energy of the particle when it is at the rest ² QCD energy-momentum tensor in terms of quarks and gluons
² Proton mass: X. Ji, PRL (1995) GeV Rest frame⇠ q Spin – intrinsic to a particle: = Angular momentum of the particle when it is at the rest ² QCD angular momentum density in terms of energy-momentum tensor
² Proton spin:
We do NOT know the state, P, S , in terms of quarks and gluons! | i Hadron properties in QCD q Hadron mass from Lattice QCD calculation:
Input
A major success of QCD – is the right theory for the Strong Force! How does QCD generate this? The role of quarks vs that of gluons? Hadron properties in QCD q Role of quarks and gluons – sum rules: ² Invariant hadron mass (in any frame): p T µ⌫ p pµp⌫ m2 p T ↵ p h | | i/ /h | ↵ | i (g) T ↵ = F µ⌫,aF a + m (1 + ) ↵ 2g µ⌫ q m q q q=u,d,s X 3 2 QCD trace anomaly (g)= (11 2n /3) g /(4⇡) + ... f (g) p F 2 p 2g h | | i Kharzeev @ Temple workshop At the chiral limit, the entire proton mass is from gluons! ² Hadron mass in the rest frame – decomposition (sum rule):
where Hq = †( iD ↵) q – quark energy q · q X 9↵s 2 2 Hm = qmq q – quark mass Ha = (E B ) q 16⇡ 1X 2 2 Hg = (E + B ) – gluon energy – trace anomaly 2 Hadron properties in QCD q Role of quarks and gluons – sum rules: ² Partonic angular momenta, the contributions to hadron spin: 1 S(µ)= P, S Jˆz(µ) P, S = J (µ)+J (µ) ⇥ | f | ⇤ 2 q g Xf where – quark angular mom.
– gluon angular mom.
² Spin decomposition (sum rule) – an incomplete story: 1 1 = ⌃ + G +(L + L ) Jaffe-Manohar, 90 2 2 q g Ji, 96, … Proton Spin Different from QM! Quark helicity: 30% – Best known ⇠ Quark helicity: 20%(with RHIC data) – Start to know ⇠ Orbital Angular Momentum (OAM): Not uniquely defined – Little known Gauge field is tied together with the motion of fermion – Dμ! Hadron properties in QCD q Role of quarks and gluons – sum rules: ² Jaffe-Manohar’s quark OAM density: 3 3 = † ~x ( i@~) Lq q ⇥ q ² Ji’s quark OAM density:h i 3 Hatta, Yoshida, Burkardt, 3 Meissner, Metz, Schlegel, L = † ~x ( iD~ ) q q ⇥ q … ² Difference – generatedh by a “torque”i of color Lorentz force 2 + 3 3 dy d yT 1 q Lq 3 P 0 q(0) dz (0,z ) L / (2⇡) h | 2 y Z 3ijZ i +j ✏ y F (z ) (z ,y) (y) P + ⇥ T | iy =0 i,j=1,2 X ⇥ ⇤ “Chromodynamic torque” Sum rules are NOT unique – None of these matrix elements are physical!!! q Value of the decomposition – a good sum rule: Every term of the sum rule is “independently measurable” – can be related to a physical observable with controllable approximation! Hadron structure in QCD q What do we need to know for the structure? ² In theory: P, S ( , ,Aµ) P, S – Hadronic matrix elements h |O | i with all possible operators: ( , ,Aµ) O ² In fact: None of these matrix elements is a direct physical observable in QCD – color confinement! ² In practice: Accessible hadron structure = hadron matrix elements of quarks and gluons, which 1) can be related to physical cross sections of hadrons and leptons with controllable approximation; and/or 2) can be calculated in lattice QCD q Single-parton structure “seen” by a short-distance probe: ² 5D structure: 2 1) d bT f(x, kT ,µ) – TMDs: 2D confined motion! kT Z 2) d2k – GPDs: xp T F (x, bT ,µ) 2D spatial imaging! Z bT 2 2 3) d kT d bT f(x, µ) – PDFs: Number density! Z Hadron structure in QCD q What do we need to know for the structure? ² In theory: P, S ( , ,Aµ) P, S – Hadronic matrix elements h |O | i with all possible operators: ( , ,Aµ) O ² In fact: None of these matrix elements is a direct physical observable in QCD – color confinement! ² In practice: Accessible hadron structure = hadron matrix elements of quarks and gluons, which 1) can be related to physical cross sections of hadrons and leptons with controllable approximation; and/or 2) can be calculated in lattice QCD q Multi-parton correlations: 2 p, ~s k (Q, ~s) + + + – Expansion / t 1/Q ··· ⇠
Quantum interference 3-parton matrix element – not a probability! Hadron structure in QCD
We need the probe!
How to connect QCD quarks and gluons to observed hadrons and leptons?
Fundamentals of QCD factorization and evolution (probing scale) QCD factorization – approximation q Creation of an identified hadron – a factorizable example:
“Long-lived” parton states
Non-perturbative, but, universal kˆ2 =0
⇡ k2 ↵ h i Perturbative! + O Q2 ✓ ◆ Factorization: factorized into a product of “probabilities” ! Effective quark mass q Running quark mass:
Quark mass depend on the renormalization scale! q QCD running quark mass:
q Choice of renormalization scale: for small logarithms in the perturbative coefficients q Light quark mass:
QCD perturbation theory (Q>>ΛQCD) is effectively a massless theory Infrared and collinear divergences q Consider a general diagram with a “unobserved gluon”:
for a massless theory
² Singularity Infrared (IR) divergence
²
Collinear (CO) divergence
IR and CO divergences are generic problems of a massless perturbation theory Pinch singularity and pinch surface q “Square” of the diagram with a “unobserved gluon”: “Cut-line” – final-state – in a “cut-diagram” notation 1 1 (p k, Q) d4k (k2) / T (p k)2 + i✏ (p k)2 i✏ + Z 1 1 (l, Q) dl2 / T l2 + i✏ l2 i✏ Z )1 Complex conjugate Amplitude of the Amplitude
Pinch singularities Pinch surfaces “perturbatively”
= “surfaces” in k, k’, … determined by (p-k)2=0, (p-k-k’)2=0, … “perturbatively” Hard collisions with identified hadron(s)
q Creation of an identified hadron: Non-perturbative!
Pinch in k2
q Identified initial hadron:
Perturbative! Perturbative! Pinch in k2
Non-perturbative!
q Initial + created identified hadron(s):
Pinch in both k2 and k’2
Cross section with identified hadron(s) is NOT perturbatively calculable Hard collisions with identified hadron(s)
q Creation of an identified hadron: Non-perturbative!
Pinch in k2
q Identified initial hadron:
Perturbative! Perturbative! Pinch in k2
Non-perturbative!
q Initial + created identified hadron(s):
Pinch in both k2 and k’2
Dynamics at a HARD scale is linked by partons almost on Mass-Shell Hard collisions with identified hadron(s)
q Creation of an identified hadron: Non-perturbative!
Pinch in k2
q Identified initial hadron:
Perturbative! Perturbative! Pinch in k2
Non-perturbative!
q Initial + created identified hadron(s):
Pinch in both k2 and k’2
Quantum interference between dynamics at the HARD and hadronic scales is powerly suppressed! Example: Inclusive lepton-hadron DIS q Scattering amplitude:
,';,qukieuk ' μ Μ−(λλ σ) = λ' ( )⎣⎦⎡⎤ γµλ( ) μ ⎛⎞i * g µµ ' ⎜⎟2 (− ) μ’ ⎝⎠q μ’ * em XeJµ ' (0,) pσ q Cross section:
2 X 3 3 2 ⎡⎤ X DIS 11⎛⎞ dli dk ' 4 4 ⎛⎞ dq,';, 2'πδ lkpk+−− σλλσ=⎜⎟∑∑ Μ( ) ⎢⎥∏ 33( ) ⎜⎟∑ i 22s X λλσ,', i 1 22EE 22' ⎝⎠i=1 ⎝⎠ ⎣⎦⎢⎥= ( ππ) i ( ) 2 dσ DIS 11⎛⎞ E ' Lkµν k,'Wq, p 32= ⎜⎟ ( ) µν ( ) dk' 2 s⎝⎠Q q Leptonic tensor: e2 – known from QED Lkkµν(, ')=+ kkkkkkgµ''ν νµ−⋅ ' µν 2π 2 ( ) DIS structure functions q Hadronic tensor: 1 Wqp(, ,SSS )= dz4† eiq⋅ z p , JzJ () (0), p µν4π ∫ µν q Symmetries:
² Parity invariance (EM current) WWµν= νµ sysmetric for spin avg. ² Time-reversal invariance * WWµν= µν real ² Current conservation µν qWµν== qWµν 0 ⎛⎞qq 1 ⎛⎞⎛⎞pq⋅⋅ pq Wgµν FxQ,,22 pqpq FxQ µν=−−⎜⎟µν22212( BB) +⎜⎟⎜⎟µµ− ν − ν ( ) ⎝ qp⎠ ⋅qq⎝⎠⎝ q⎠ ⎡⎤Q2 = q2 µνρσ S 22( pq..)SSσσ−( q) p +iM q σ gx ,Q gxQ , 2 pε ρ ⎢⎥1 ( BB) + 2 2 ( ) Q ⎢⎥p.q ( pq. ) x = ⎣ ⎦ B 2p q q Structure functions – infrared sensitive: ·
FxQ,,2222 FxQ ,, gxQ ,, g xQ , No QCD parton dynamics 1212( BBBB) ( ) ( ) ( ) used in above derivation! Long-lived parton states q Feynman diagram representation of the hadronic tensor:
W µν ∝ … q Perturbative pinched poles:
4 ⎛⎞⎛⎞11 1 dk H(,)Q k T( k , ) ⇒ ∞ perturbatively ∫ ⎜⎟⎜⎟22 ⎝⎠⎝⎠k +iεεk −i r0 q Perturbative factorization: kk22+ Nonperturbative matrix element k µµ=+xpnkT µµ + 2xpn⋅ T
dx 2 2 2 ⎛⎞⎛⎞11 1 dk H(Q ,k = 0) dk T,)( k ∫ T ∫ ⎜⎟⎜⎟22 x ⎝⎠⎝⎠k +iiεεk − r0 Short-distance Collinear factorization – further approximation
2 q Collinear approximation, if Q ∼ xp ⋅ n ≫ kT , k
– Lowest order:
⎛ k 2 ⎞ +O T +UVCT ≈ ⎜⎟2 ⊗ ⎝⎠Q Scheme dependence
Same as an elastic x-section Parton’s transverse momentum is integrated into parton distributions, and provides a scale of power corrections 2 q DIS limit: ν,,Qx→∞ whilefixed B Feynman’s parton model and Bjorken scaling Spin-½ parton! q Corrections: Parton distribution functions (PDFs) q PDFs as matrix elements of two parton fields: – combine the amplitude & its complex-conjugate
(µ2) ZO can be a hadron, or a nucleus, or a parton state!
But, it is NOT gauge invariant!
– need a gauge link: (µ2) ZO
– corresponding diagram in momentum space: d4k (x k+/p+) 2 (2⇡)4 + UVCT(µ ) Z μ-dependence
Universality – process independence – predictive power Gauge link – 1st order in coupling “g” q Longitudinal gluon:
q Left diagram: + 2 + + p dy1 ix p (y y ) a a p i((x x1 xB) p +(Q /2xBp ) n) dx e 1 1 n A (y ) ( igt ) · · · 1 2⇡ · 1 M p+ (x x x )Q2/x + i✏ Z Z 1 B B dy 1 + 1 1 a a ix1p (y1 y ) = g n A (y1 )t dx1 e = ig dy1 n A(y1 ) 2⇡ · x + i✏ M y · M Z Z 1 Z q Right diagram: + 2 + + p dy1 ix p y a i((x + x1 xB) p +(Q /2xBp ) n) a p dx e 1 1 n A (y ) · · (+igt ) · 1 2⇡ · 1 (x + x x )Q2/x i✏ p+ M Z Z 1 B B dy 1 + 1 1 a a ix1p y1 = g n A (y1 )t dx1 e = ig dy1 n A(y1 ) 2⇡ · x1 i✏ M · M Z Z Z0 q Total contribution: 1 1 O(g)-term of ig dy1 n A(y1 ) LO 0 y · M the gauge link! Z Z QCD high order corrections q NLO partonic diagram to structure functions:
2 −Q 2 2 dk k1 0 ∝ 1 Dominated ∫ k 2 0 1 by tAB → ∞ Diagram has both long- and short-distance physics q Factorization, separation of short- from long-distance: QCD high order corrections
dk4 q QCD corrections: pinch singularities in ∫ i
+ + + … q Logarithmic contributions into parton distributions:
⊗ + +…+UVCT
2 2 2 ⎛⎞xQB 2 ⎛⎞ΛQCD FxQ2 (,B )= Cf , ,αs ⊗ϕf x ,µF + O⎜⎟ ∑ ⎜⎟xQ2 ( ) ⎜⎟2 f ⎝⎠µF ⎝⎠ 2 q Factorization scale: µ F To separate the collinear from non-collinear contribution Recall: renormalization scale to separate local from non-local contribution Dependence on factorization scale q Physical cross sections should not depend on the factorization scale 2 d 2 µF 2 FxQ2 (,B )0= dµF 2 2 2 2 F2(xB,Q )= Cf (xB/x, Q /µF , ↵s) f (x, µF ) Xf Evolution (differential-integral) equation for PDFs ' d ! x Q2 $* ! x Q2 $ d )µ 2 C # B , ,α &,⊗ϕ x,µ 2 + C # B , ,α & ⊗ µ 2 ϕ x,µ 2 = 0 ∑ F d 2 f # x 2 s & f ( F ) ∑ f # x 2 s & F d 2 f ( F ) f () µF " µF %+, f " µF % µF q PDFs and coefficient functions share the same logarithms
22 22 PDFs: log(µµFF0QCD) or log( µΛ ) 22 22 Coefficient functions: log(QQµµF ) or log( ) DGLAP evolution equation: ∂ x 22(,xx2 )P ⎛⎞, (', ) µFF2 ϕiijjµ= ∑ / ⎜ αϕ sF⎟ ⊗ µ ∂µF j ⎝⎠x' Calculation of evolution kernels q Evolution kernels are process independent ² Parton distribution functions are universal ² Could be derived in many different ways q Extract from calculating parton PDFs’ scale dependence
P P P P P P P P pp1 p k 1 k k k + p + p kk 2 p-k 2 p-k
Collins, Qiu, 1989
Change “Gain” “Loss” ² Same is true for gluon evolution, and mixing flavor terms q One can also extract the kernels from the CO divergence of partonic cross sections Scaling and scaling violation
Q2-dependence is a prediction of pQCD calculation PDFs of a spin-averaged proton q Modern sets of PDFs @NNLO with uncertainties:
Consistently fit almost all data with Q > 2GeV Summary of lecture one q QCD has been extremely successful in interpreting and predicting high < 1/10 fm energy experimental data! q But, we still do not know much about hadron structure – work just started! q Cross sections with large momentum transfer(s) and identified hadron(s) are the source of structure information q TMDs and GPDs, accessible by high energy scattering, encode important information on hadron’s 3D structure – distributions as well as motions of quarks and gluons q QCD factorization is necessary for any controllable “probe” for hadron’s quark-gluon structure! Thank you! Backup slides How to calculate the perturbative parts?
q Use DIS structure function F2 as an example: ⎛⎞xQ2 ⎛⎞Λ2 FxQ(,2 )= CB , ,α ⊗ϕ x ,µ2 + O QCD 2hB∑ qf⎜⎟2 s fh/ ( ) ⎜⎟2 qf, ⎝⎠xQµ ⎝⎠ ² Apply the factorized formula to parton states: hq→ 2 Feynman 2 ⎛⎞xQ 2 Feynman FxQ(, )= CB , ,α ⊗ϕ x ,µ diagrams 2qB∑ qf⎜⎟2 s f / q ( ) diagrams q, f ⎝⎠x µ
² Express both SFs and PDFs in terms of powers of αs:
th (00) 22( ) 22(0) 0 order: FxQCxxQ2qB(, )= q ( B /, /µ )⊗ϕqq/ ( x ,µ) (00) ( ) (0) CxFxqq()= 2 () ϕqq/ (xx) =δδ qq (1−)
th (11) 2 ( ) 2 22(0) 1 order: FxQCxxQ2qB(, )= q ( B /, /µ )⊗ϕqq/ ( x ,µ) (0) 2 22(1) +CxxQqB ( / , /µ )⊗ϕqq/ ( x , µ) (110) 22222( ) ( ) (1) CxQqq(, /µ )= FxQF22 (, )−⊗q (, xQ )ϕq/q ( x ,µ) PDFs of a parton q Change the state without changing the operator:
– given by Feynman diagrams q Lowest order quark distribution: ² From the operator definition:
q Leading order in αs quark distribution:
2 ² Expand to (gs) – logarithmic divergent:
UV and CO divergence Partonic cross sections q Projection operators for SFs:
⎛⎞qq 1 ⎛⎞⎛⎞pq⋅⋅ pq Wgµν FxQpqpqFxQ,,22 µν=−−⎜⎟µν22212( ) +⎜⎟⎜⎟µµ− ν − ν ( ) ⎝⎠qpqqq⋅ ⎝⎠⎝⎠
2 2214⎛⎞µνx µν FxQ1(, )=⎜⎟− g+2 pp Wµν (, xQ ) 2 ⎝⎠Q 2 22⎛⎞µν12x µν FxQ2 (, )= x⎜⎟− g+2 pp Wµν (, xQ ) ⎝⎠Q 1 q 0th order: FxxgW(0) ()==µν(0) xgµν 2,qqµν 4π e2 1 xgµν q Tr⎡⎤ p p q 2 ( p q )2 =( ) γγγ⋅⋅µ( ++) γν πδ( ) 42π ⎣⎦⎢⎥ 2 =exq δ (1− x ) (0) 2 Cxexxqq()=δ (1− ) How does dimensional regularization work? q Complex n-dimensional space: Im(n)
(1) Start from here: (2) Calculate IRS quantities UV renormalization here Theory cannot be
renormalized! a renormalized theory
(3) Take εè 0 for IRS quantities only
4 6 Re(n)
UV-finite, IR divergent
UV-finite, IR-finite
NLO coefficient function – complete example
(110) 22222( ) ( ) (1) CxQqq(, /µ )= FxQF22 (, )−⊗q (, xQ )ϕq/q ( x ,µ) µν q Projection operators in n-dimension: ggµν = n≡−42ε
2 ⎛⎞µν4x µν (1(32)−εε) Fxg2 =⎜⎟−+− 2 ppWµν ⎝⎠Q q Feynman diagrams:
1 ( ) + Real Wµν ,q + +
Virtual + + c.c. + + c.c. + UV CT q Calculation: µν(11) µν ( ) −gWµν,,qq and ppWµν Contribution from the trace of Wμν q Lowest order in n-dimension:
µν (0) 2 −gWµν ,qq= e(1−−εδ ) (1 x ) q NLO virtual contribution:
µν (1)V 2 −gWµν ,qq= e(1−−εδ ) (1 x ) ε α ⎡⎤4(1)(1)131πεεµ 2 Γ+Γ2 − *⎛⎞s C ⎡⎤ 4 ⎜⎟−F ⎢⎥22⎢⎥++ ⎝⎠πεεε⎣⎦Q Γ−(1 2 )⎣⎦ 2 q NLO real contribution: 2 ε 1 R α ⎡⎤4(1)πεµ Γ+ gWµν ( ) e2 (1 )C ⎛⎞s −µν ,qq=−−ε F ⎜⎟⎢⎥2 ⎝⎠2(12)πε⎣⎦Q Γ− ⎧⎫12112−⎡εεε⎛⎞⎛⎞x ⎤ − *⎨⎬−−⎢⎥ 1 x +⎜⎟⎜⎟ + + ⎩⎭εεεε⎣⎦⎝⎠⎝⎠1122(12)(1)12−−x −−x − q The “+” distribution: 1+ε ⎛⎞11 1 ⎛l nx (1)− ⎞ 2 ⎜⎟=−δεε(1 −xO ) + + ⎜ ⎟ + ( ) ⎝⎠1(1)1−−−xxxε + ⎝ ⎠+ 11fx() fx ()− f (1) ∫∫dx≡ dx+l n(1− z ) f (1) zz(1−−xx )+ 1 q One loop contribution to the trace of Wμν:
2 µν 1 2 ⎛⎞αs ⎧ 1 ⎛⎞Q −gW( ) = e(1−ε ) −P ( x )+P ( xn )l µν ,qq ⎜⎟⎨ qq qq ⎜⎟2 −γ E ⎝⎠2(4)πε⎩ ⎝⎠ µπe 2 ⎡−2 ⎛⎞⎛⎞l nx(1 ) 3 1 1 + x ++CF ⎢(1()xn)⎜⎟⎜⎟ − −l x ⎣ ⎝⎠⎝⎠1211−−−x + xx+ ⎛⎞9 π 2 ⎤⎪⎫ +3(1)−−xx⎜⎟ +δ − ⎥⎬ ⎝⎠23 ⎦⎭⎪ q Splitting function: ⎡⎤13+ x2 Pqq ()xx=+CF ⎢⎥δ (1− ) ⎣⎦(1− x )+ 2 μ ν q One loop contribution to p p Wμν: 2 µν (1)V µν 1 R 2 α Q ppW = 0 ppW( ) = eC s µν ,q µν ,qqF 24π x q One loop contribution to F2 of a quark: 2 1 22αs ⎧⎛⎞1 −γ ⎛⎞Q FxQex( ) (, )P ()1 xll n (4 eE )P ()xn 2q =q ⎨⎜⎟−qq( +ε π ) + qq ⎜⎟2 2π⎩⎝⎠ε CO ⎝⎠µ
22 ⎡⎤2 ⎛⎞⎛⎞l nx(1− ) 3 1 1+ x ⎛⎞ 9 π ⎪⎫ ++CF (1xnxxx ) −−l ( )++ 3 2−+δ (1− ) ⎬ ⎢⎥⎜⎟⎜⎟1211xxx ⎜⎟ 23 ⎣⎦⎝⎠⎝⎠−−−++ ⎝⎠ ⎭⎪ ⇒∞ as ε → 0 q One loop contribution to quark PDF of a quark:
(1) 2 ⎛⎞αs ⎧⎫⎛⎞11 ⎛ ⎞ ϕqq/ (,xxµ )=+⎜⎟Pqq ()⎨⎬⎜⎟ ⎜− ⎟ + UV-CT ⎝⎠2π ⎩⎭⎝⎠ε UVCO ⎝ε ⎠ – in the dimensional regularization
Different UV-CT = different factorization scheme! q Common UV-CT terms:
αs ⎛⎞1 ² MS scheme: UV-CTMS =− Pqq (x )⎜⎟ 2π ⎝⎠ε UV
αs ⎛⎞1 −γ ² MS scheme: UV-CTP (xn ) 1l (4e E ) MS =−qq ⎜⎟ ( +επ ) 2π ⎝⎠ε UV (1) 22 ² DIS scheme: choose a UV-CT, such that CxQq (, /µ )DIS = 0 q One loop coefficient function:
(110) 22222( ) ( ) (1) CxQqq(, /µ )= FxQF22 (, )−⊗q (, xQ )ϕq/q ( x ,µ)
⎧ 2 (1) 22 2αs ⎪ ⎛⎞Q CxQqq(, /µ )= ex⎨Pqq () xnl ⎜⎟2 2π⎜⎟µ ⎩⎪ ⎝⎠MS 22 ⎡⎤2 ⎛⎞⎛⎞l nx(1− ) 3 1 1+ x ⎛⎞ 9 π ⎪⎫ ++CF (1xnxxx ) −−l ( )++ 3 2−+δ (1− ) ⎬ ⎢⎥⎜⎟⎜⎟1211xxx ⎜⎟ 23 ⎣⎦⎝⎠⎝⎠−−−++ ⎝⎠ ⎭⎪