Applications of Light-Front Dynamics in Hadron Physics

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1 Applications of Light-Front Dynamics in Hadron Physics Chueng-Ryong Ji North Carolina State University 1. What is light-front dynamics (LFD)? 2. Why is LFD useful in hadron physics? 3. Any first principle QCD predictions? 4. Any use of effective field theory? 5. Any treacherous points? 6. Any relevance to JLab experiments? June 5-7, 2013 Questions

2 Outline(tentative) Tutorials of LFD (5 th, 9-10am) Interpolating QFT (5 th, 1-2pm) Perturbative QCD Predictions (6 th,9-10am) Chiral Effective Field Theory (6 th, 1-2pm) Anomaly and Zero Modes(7 th, 10:15-11:15am) Application to DVCS and GPDs (7 th,1-2pm)

3 LFD Tutorial Simultaneity is broken in relativity.

4 Lorentz Transformation

5 Minkowski Space

6 The Boost Problem t z

7 LFD Tutorial t z

8 LFD Tutorial t z

9 LFD Tutorial t z

10 LFD Tutorial t z

11 The Boost Problem Number of quanta can change under boost. t t z z

12 LFD Tutorial Number of quanta can change under boost. t t z z

13 LFD Tutorial Number of quanta can change under boost. t t z z

14 LFD Tutorial Number of quanta can change under boost. t t z z

15 LFD Tutorial Number of quanta can change under boost. t t z z

16 The Boost Effect

17 Dirac s Proposition 1949

19 Poincaré Generators 3 boosts 3 rotations K = (K 1, K 2, K 3 ) 3 space translations 1 time translation J = (J 1, J 2, J 3 ) P = (P 1, P 2, P 3 ) E = P 0 Poincaré Algebra: 10x9/2=45

20 Poincaré Algebra: Covariant Form [p m, p n ] = 0 [p m,j rs ] = i(g mr p s - g ms p r ) [J mn,j rs ] = i(g ms J nr + g nr J ms - g mr J ns - g ns J mr ) p n = é ë p 0 p 1 p 2 p 3 ù û = ò d 3 x T 0n J mn = é 0 K 1 K 2 K 3 ù -K 1 0 J 3 -J 2 -K 2 -J 3 0 J 1 ë -K 3 J 2 -J 1 0 û = ò d 3 x M 0mn T mn = k L ( m f k ) n f k - g mn L å ; T mn = 0 m M mnl = T mn x l -T ml x n ; L = r p

21 6 How many generators leave the time surface invariant? Stability Group 7 6

22 6 Is the form change just formal? or Does it offer any physical consequences? 7 6

23 Physical Meaning of Stability Group Equal-time Wavefunction Time-ordered Scattering Amplitudes Invariant under Stability Group Elements Kinematic Transformations

24 Feynman Diagram: Invariant under all Poincaré generators Individual Time-Ordered Diagrams: Invariant under stability group Kinematic vs. Dynamic Generators

25 Return of Prodigal Son Longitudinal Boost from dynamic to kinematic

26 Interpolation between Instant and Front Forms K. Hornbostel, PRD45, 3781 (1992) C.Ji and C. Mitchell, PRD64, (2001) C.Ji and A. Suzuki, PRD87, (2013)

27 Interpolating Hadronic Wavefunction x 0 x ˆ+ x + Invariant under kinematic transformations (P, J) (P +, P^, J 3, E^, K 3 ) 0 d p / 4

28 d = 0 p 0 = p 0 -p 3 = p 3 0 < d < p /4 pˆ + = p 0 cosd - p 3 sind pˆ = - p0 sind + p 3 cosd d = p /4 p + = p - p - = p +

29 Σ(a)+Σ(b)=1/(s-m 2 ) ; s=2 GeV, m=1gev J-shape peak & valley : P z = - s(1- C) 2C ; C = cos(2d)

30 g mn = é ù ë û é ò d 3 x T 0m = P m = ë P 0 P 1 P 2 P 3 ù û é ò d 3 x (T 0m x n -T 0m x n ) = J mn = ë T mn = L ( m f k ) n f k - g mn L å ; k é P ˆm = ë g ˆ m ˆ n = 0 K 1 K 2 K 3 -K 1 0 J 3 -J 2 -K 2 -J 3 0 J 1 -K 3 J 2 -J 1 0 é cos2d 0 0 sin2d ù ë sin2d 0 0 -cos2d û P ˆ+ = cosd P 0 +sind P 3 P ˆ1 = P 1 P ˆ2 = P 2 P ˆ- = sind P 0 - cosd P 3 m T mn = 0 ù û J ˆ m ˆ n = ù = ò dx ˆ- d 2 x^t û ˆ+ ˆm é 0 E ˆ 1 E ˆ 2 -K 3 ù -E ˆ 1 0 J 3 -F ˆ 1 -E ˆ 2 -J 3 0 -F ˆ 2 ë K 3 F ˆ 1 F ˆ 2 0 û E ˆ 1 = J 2 sind + K 1 cosd E ˆ 2 = K 2 cosd - J 1 sind F ˆ 1 = K 1 sind - J 2 cosd F ˆ 2 = J 1 cosd + K 2 sind

31 Interpolating Poincaré Algebra [P ˆm, P ˆn ] = 0 [P ˆm, J ˆr ˆ s ] = i(g ˆm ˆr P ˆ s - g ˆm ˆ s P ˆr ) [J ˆm ˆn, J ˆr ˆ s ] = i(g ˆm ˆ s J ˆn ˆr + g ˆn ˆr J ˆm ˆ s - g ˆm ˆr J ˆn ˆ s - g ˆn ˆ s J ˆm ˆr ) e.g. [P ˆ+, J ˆ+ˆ- ]= i(g ˆ+ˆ+ P ˆ- - g ˆ+ˆ- P ˆ+ ) [P ˆ+,-K 3 ]= i(p ˆ- cos2d - P ˆ+ sin2d) Return of Prodigal Son [K 3, P + ] = -ip + Exp -iw K 3 ( ) x + > µ x + > One more kinematic generator appears only in the front form. Maximum number (7) of members in the stability group.

32 Kinematic Operators (Members of Stability Group) Exp( -iwà ˆ i ) x + ˆ > µ x + ˆ > [ ˆÀ i, P ˆ+ ] = 0 À ˆ i = F ˆ i cos2d - E ˆ i sin2d d = 0 -J 2 J 1 À ˆ 1 = -J 2 cosd - K 1 sind ˆ À 2 = J 1 cosd - K 2 sind (J 3, P 1, P 2, Pˆ-) d = p /4 -E 1 = -(J 2 + K 1 )/ 2 E 2 = (J 1 - K 2 )/ 2

33 particle at rest p 0 = M, p 1 = p 2 = p 3 = 0 (pˆ + = Mcosd, pˆ = Msind) - same p 0 Under ˆ p 0 + p 3 À i transformation same remain at rest P 0 = M; p 3 = 0 can move 2 P 0 = M + p ^ 2M ; p3 = - p ^ 2M 2 (p 0 ) 2 - (p 3 ) 2 = (M + p 2 ^ 2M )2 - (- p 2 ^ 2M )2 = M 2 + p 2 ^ = 2p + p - > 0 Rational Energy-Momentum Dispersion Relation Vacuum gets simpler in LFD.

36 Angular Momentum [J i,j j ] = ie ijk J k,[j i,m] = 0 { } T = Exp -i(b ˆ 1 À 1 + b ˆ 2 À 2 ) [ ˆ Á i, ˆ Á j ] = ie ijk ˆ Á k,[ ˆ Á i,m] = 0 T n >= p,n > ˆ Á i p,n >= TJ i n > Á ˆ i = TJ i T +

37 Á ˆ ì 3 = J 3 pˆ + ˆ - z (p ^ À ˆ ü í î ^) ý þ / M sind ì Á ˆ ï æ z ^ = J ^ + p ^ cosd J 3 + ˆ (p ^ À ˆ ö æ í ^) ç è pˆ + M sind - (ˆ z p ^)sind ç K 3 + î ï - ø è pˆ p ˆ ö ü ^ E ^ ï + M sind ý - ø þï ì Á ^ = í z ˆ (p - î Á 3 = J 3 + ˆ z (E ^ p ^)/ p + E ^ - p + F ^+ p ^K 3 ) - p ^ p + ü ( p + J 3 + z ˆ E ^ p ) ^ ý þ / M Á 3 = W + p + W ˆm = 1 2 e ˆm ˆnâ ˆb p ˆn Jâ ˆb [Á 3, Stability Group Members] = 0

38 Interpolating Spinors ˆÁ 3 û (1) (1) CR = (+1)û CR

39 Interpolating Spinors ˆÁ 3 û (2) (2) CR = (-1)û CR

40 Interpolating Helicity Amplitude ˆM(l 1,l 2,l 3, l 4 ) = û( ˆp 2, l 2 )g ˆmû( ˆp 1, l 1 )û( ˆp 3, l 3 )g ˆmû( ˆp 4, l 4 )

44 Jacob-Wick Helicity vs. Light-Front Helicity Invariant under kinematic transformations Related by a rotation

45 Canonical Quantization Simple 1+1 dim QFT: L = 1 2 mf m f m2 f 2 L = C é ë 2 ( ˆ+f) 2 -( ˆ-f) 2 ù û + S( ˆ+f)( ˆ-f)- 1 2 m2 f 2 C = cos(2d), S = sin(2d) p(x) = L ( 0 f) = 0 f p(x) = C ˆ+f + S ˆ-f [p(x),f(y)] x 0 =y 0 = -id(x 3 - y 3 ) [p(x),f(y)] x ˆ+ =y ˆ+ = -id(x ˆ- - y ˆ- )

46 T mn = å k Pˆ+ Energy = ò ˆ- dx (p ˆ+f - L) = 1 2 Periodic Boundary Condition: f(x ˆ+ = 0, x ˆ- ) = w n = L ( m f k ) n f k - g mn L é ë P 0 = ò dx 3 (p 0 f - L) ù û f(x ˆ+, x ˆ- - ) = f(x ˆ+, x ˆ- + ) ò dx ˆ- C{( ˆ+f) 2 + ( ˆ-f) 2 }+ m 2 f 2 å n=- 2 æ n 2 +C m ö ç è p ø Pˆ+ = æ p ö ç è ø å n 1 é a n e -i 4pw n ë æ np ö ç è ø æw n - S n ö ç a + n a n è C ø x ˆ- + a n + e i æ np ö ç è ø [a m, a n + ] = d mn x ˆ- ù û

47 Symmetry Breaking f f = f +n L L = L - m 2 n f m2 n 2 Pˆ+ Pˆ+ m3 = Pˆ+ + (a C 1/4 0 + a 0 ( ) 1/2 n + )

48 Nontrivial Vacuum State Translation in scalar field: 0 > W > f f = f +n æ + ö W > = expçi ò dx ˆ- n p(x ˆ- ) 0 > è ø p(x ˆ+ = 0, x ˆ- ) = -i - å n=- æp ö ç è ø w é n 4p a ne -i ë æ np ö ç è ø é W >= exp -(C 1/2 m ) n 2 ù expé-(c 1/2 m ) 1/2 + ë na 0 ù û ë 2 û 0 > Condensation of Zero-Modes x ˆ- - a n + e i æ np ö ç è ø x ˆ- ù û

49 Vacuum Energy Pˆ+ W>= E W W> a e aa+ 0 >=a e aa+ 0 > Pˆ+ é mn W > C a ( + 1/2 0a 0 + m3 ) 1/2 n (a C 1/4 0 + a 0 ë + ) = (-m 2 n 2 ) expé-(c 1/2 m ) 1/2 + ë na 0 ù û 0 > + ò - E W = -m 2 n 2 = (- 1 2 m2 n 2 )dx ˆ- Independent of interpolation angle! ù exp é -(C1/2 m ) 1/2 + na 0 ù ë û 0 > û

50 Recovery of Trivial Vacuum in LFD é W >= exp -(C 1/2 m ) n 2 ë 2 ù expé-(c 1/2 m ) 1/2 + ë na 0 û ù û 0 > W> 0 > as C 0 However, E W and < W f(x) W >= -n are still independent of interpolation angle!

51 What is going on? < W f(x) W > æ + =< 0 exp é ë(c 1/2 m ) 1/2 n(a a 0 ) ù a 0 + a 0 ûç è 2(C 1/2 m ) 1/2 = -n ö expé ë-(c 1/2 m ) 1/2 n(a a 0 ) ù û ø 0 > Complication is transferred from vacuum to operator.

52 Summary LFD is not just formal but consequential in the analysis of physical observables. Longitudinal boost joins stability group in LFD. LF helicity amplitudes are independent of all references frames that are related by front-form boosts. Energy-momentum dispersion relation becomes rational and vacuum gets simpler in LFD.

Angular Momentum in Light-Front Dynamics

Angular Momentum in Light-Front Dynamics Chueng-Ryong Ji North Carolina State University INT Worksho INT-1-49W Orbital Angular Momentum in QCD Seattle, WA, Feb. 7, 01 ν Poincaré Algebra [ µ, ν ] 0 [ µ,j