Husimi distribution for nucleon tomography

Transcription

1 Husimi distribution for nucleon tomography Yoshitaka Hatta (Yukawa institute, Kyoto U.) Nucl.Phys. A940 (2015) 158 (arxiv: ) with Yoshikazu Hagiwara

2 Outline Wigner distribution in Quantum Mechanics and QCD Husimi distribution in QM and QCD 1-loop result Some speculations

3 3D tomography of the nucleon Partons inside the nucleon are characterized not only by the longitudinal momentum fraction x TMD PDF f(x; ~ k? ) GPD H(x; ~? ) F.T. eh(x; ~ b? ) Unintegrated distribution either in or space. Why not both a phase-space distribution for the nucleon?

4 5D tomography: Wigner distribution the mother distribution Wigner distribution TMD Z d ~ b? f(x; ~ k? ) W(x; ~ b? ; ~ k? ) Z Z d ~ k? GPD eh(x; ~ b? ) Z PDF d ~ b? R dx Form factor d ~ k? f(x) F( e ~ b? ) R dx charge Q Z d ~ b?

5 Wigner distribution in QM (1932) Phase space distribution in quantum mechanics f W (q; p) = = Z 1 1 Z 1 1 dxe ipx=~ hãjq x=2ihq + x=2jãi dxe ipx=~ hq + x=2j^½jq x=2i density matrix ^½ = jãihãj Moments of the Wigner distribtuion Z dq 2¼~ f W (q; p) = jhãjpij 2 ; Z dp 2¼~ f W (q; p) = jhãjqij 2 Eugene Wigner ( ) Expectation value of an operator h ^Oi = R dpdqf W (q; p)o(p; q)

6 Wigner distribution for the harmonic oscillator H = p2 2m + m!2 q 2 2 n=0 f W (q; p) = 2( 1) n e 2H=~! L n µ 4H ~! Laguerre polynomial

7 Wigner distribution for the harmonic oscillator H = p2 2m + m!2 q 2 2 f W (q; p) = 2( 1) n e 2H=~! L n µ 4H ~! n=0 n=1 Laguerre polynomial

8 Wigner distribution for the harmonic oscillator H = p2 2m + m!2 q 2 2 f W (q; p) = 2( 1) n e 2H=~! L n µ 4H ~! n=0 n=1 Laguerre polynomial n=4 n=2 Probability distribution? No way!

9 The uncertainty principle q p ~ The very notion of phase space distribution in quantum physics contradicts the uncertainty principle. Wigner distribution wildly oscillates and becomes negative. Incorporates (interesting) quantum interference effect, but no probabilistic interpretation 2

10 Wigner distribution in QCD Wigner distribution of quarks in the nucleon Ji (2003) Belitsky, Ji, Yuan (2003) W (x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 d 2? (2¼) 2 ei(xp + z ~ k? ~z? ) hp + 2 j¹q(b z 2 ) L q(b + z 2 )jp 2 i ~ b? integral TMD ~ k? integral GPD Connection to orbital angular momentum Lorce, Pasquini, (2011); YH (2011) ¹ = (0; 0; ~? ) momentum recoil (relativistic effect)

11 Model calculation light-cone quark models (no gluons included) Lorce, Pasquini, (2011)

12 Husimi distribution (1940) f H (q; p) = 1 ¼~ Z dq 0 dp 0 e m!(q0 q) 2 =~ (p 0 p) 2 =m!~ f W (q 0 ; p 0 ) Gaussian smearing of the Wigner distribution within the region of minimum uncertainty q = p ~=2m! p = p ~m!=2 q p = ~=2 Kodi Husimi ( )

13 Husimi distribution is positive f H (q; p) = h j^½j i = jhãj ij 2 0 Positive semi-definite! Coherent state aj i = j i j i = e j j2 =2 e ay j0i = m!q + ip p 2~m! Coherent state satisfies the minimum uncertainty relation q p = ~=2 Many applications in statistical physics, quantum optics, chaos, etc.

14 Husimi distribution for the harmonic oscillator n=4 f H (q; p) = 1 n! e H ~! µ n H ~! f W (q; p) = 2( 1) n e 2H=~! L n µ 4H ~! Localized around the classical orbit H = p2 2m + m!2 q 2 2 ¼ ~!(n )

15 Husimi distribution in QCD Hagiwara and YH (2015) Define H (x; ~ b? ; ~ k? ) 1 Z d 2 b 0 ¼?d 2 k?e `2 (~ b? ~ b 0? )2 `2( ~ k? ~ k 0? )2 W (x; ~ b 0?; ~ k?) 0 The parameter ` is arbitrary, but it is natural to take `. R hadron

16 Moments of the Husimi distribution The b-moment of the QCD Husimi distribution does not reduce to the TMD. Z d 2 b? H (x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 e i(xp + z ~ k? ~z? ) e z2? 4`2 hp j¹q( z=2) L q(z=2)jp i?? cf. Gaussian ansatz in TMD phenomenology Double moments are the same as in the Wigner case R d 2 b? d 2 k? H + (x; ~ b? ; ~ k? ) = f(x) (PDF) R d 2 b? d 2 k? ( ~ b? ~ k? )H + (x; ~ b? ; ~ k? ) = L can (canonical OAM)

17 Positivity? In the A + = 0 gauge Z H» d 2? e i~? ~ b? ` 2 2? 4 hp + =2jq y + ±(K+ (1 x)p + )e `2( ~ K? + ~ k? ) 2 q + jp =2i good component Positive definite if it were not for the momentum recoil (relativistic effect) However, the Gaussian factor suppresses

18 1-loop Wigner distribution for an on-shell quark W(x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 d 2? (2¼) 2 ei(xp + z ~ k? ~z? ) hp + 2 j¹q(b z 2 ) + L q(b + z 2 )jp 2 i Zeroth order W[x; ~ b? ; ~ k? ] = ±(x 1)± (2) ( ~ b? )± (2) ( ~ k? ) =) H(x; ~ b? ; ~ k? ) = ±(1 x) e b2? =`2 `2k 2? ¼ 2 ~ b? = ~ k? = 0!? Violates the uncertainty principle

19 First order in s W + [x; ~ b? ; ~ k? ] = Z sc F d 2? 2¼ 2 (2¼) 2 e i ~? ~ b? ³ 2 k? 2 2? 4 (1 x) P qq (x) + m 2 (1 x) 3 (q+ 2 + m 2 (1 x) 2 )(q 2 + m 2 (1 x) 2 ) splitting function P qq (x) = 1 + x2 1 x ~q = ~ k? ~? 2 (1 x)

20 First order in s Bad convergence, sensitive to max? Divergent when ~ b? = 0, oscillates in. b? W + [x; ~ b? ; ~ k? ] = Z sc F d 2? 2¼ 2 (2¼) 2 e i ~? ~ b? ³ 2 k? 2 2? 4 (1 x) P qq (x) + m 2 (1 x) 3 Negative when (q+ 2 + m 2 (1 x) 2 )(q 2 + m 2 (1 x) 2 ) j ~ k? j < (1 x) j~? j 2 splitting function P qq (x) = 1 + x2 1 x ~q = ~ k? ~? 2 (1 x)

21 Mukherjee, Nair, Ojha,

22 One-loop Husimi distribution Smearing in j ~ k? ~ k 0? j» 1=` Integration region limited to j ~? j < 2=` H + [x; ~ b? ; ~ k? ] = `2 sc F 2¼ 3 Z Z d 2 k?e 0 `2( ~ k? ~ k 0? d 2 )2? (2¼) 2 cos(~? ~ b? )e ` 2 ³ 2 (k? 0 )2 2? 4 (1 x) P qq (x) + m 2 (1 x) 3 ((q 0 +) 2 + m 2 (1 x) 2 )((q 0 ) 2 + m 2 (1 x) 2 ) 4 2? Smearing region larger than the negative region

23 Numerical result Before (Wigner) After (Husimi) x = 0:5; m 2 = 0:1 GeV 2 ; ` = 1 GeV 1

24 Before (Wigner) After (Husimi)

25 Entropy? Since the Husimi distribution is positive, one can define entropy (Husimi-Wehrl entropy) S R dqdp 2¼¹h f H ln f H cf. von Neumann entropy S = tr^½ ln ^½ Nonvanishing even for a pure state. A measure of complexity (chaoticity) of the nucleon wavefunction.

26 Relation to Color Glass Condensate? At small-x, the gluons can be treated as a coherent classical field McLerran, Venugopalan (1993) Husimi distribution is the coherent state expectation value. Any relation between the two?

27 A tantalizing hint The b-moment of the Husimi distribution is not exactly TMD. Z d 2 b? H (x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 e i(xp + z ~ k? ~z? ) e z2? 4`2 hp jf +¹ ( z=2)l F ¹ + (z=2)jp i At low-x, identify ` $ 1 Q s (x) saturation scale e z2? =4`2! e Q2 s z2? =4 dipole S-matrix

28 A tantalizing hint The b-moment of the Husimi distribution is not exactly TMD. Z d 2 b? H (x; ~ b? ; ~ k? ) = Z dz d 2 z? 16¼ 3 e i(xp + z ~ k? ~z? ) e z2? 4`2 hp jf +¹ ( z=2)l F ¹ + (z=2)jp i At low-x, identify ` $ 1 Q s (x) saturation scale e z2? =4`2! e Q2 s z2? =4 dipole S-matrix What is computed within CGC/quasi-classical approximation could be interpreted as the Husimi distribution.

29 Conclusions Wigner distribution badly behaved. Nowhere near what one would naively expect for a phase space distribution. Husimi distribution much better behaved. Can be interpreted as a probability distribution of quarks and gluons. Classical description of the nucleon (nucleus).