Impact paramtr analysis in + N + pi+ N 1 Paul Hoyr Univrsity of Hlsinki INT Workshop Novmbr 14-18, 2016 Rlativistic lctron microscopy of hadron dynamics Gold atoms: 3D Th pion: 2D 8 ρπ(b) 6 ρ π (b) [fm -2 ] 4 2 1 www.york.ac.uk/nanocntr /facilitis/ftm/ 0-1 0 b x [fm] 1-1 0 b y [fm] Paul Hoyr INT 2016 Millr, Strikman, Wiss, arxiv 1011.1472
Procss dpndnc of transvrs siz 2 How dos th targt charg distribution dpnd on th proprtis of th final stat? Paul Hoyr INT 2016 PH and S. Kurki arxiv:1101.4810
Compar: Color Transparncy in A ρ X 3 Q 2 Paul Hoyr INT 2016 lf = 2ν/Δmh 2 formation lngth
Paul Hoyr INT 2016 Elctron microscopy of rlativistic chargs 4 Light quarks in hadrons mov with v c = 1 : How can w gt a sharp pictur using probs moving with th sam spd? In a fram whr ph z + (IMF or LF), th quark nrgy Eq = x Eh, hnc v q? = p q? xe h! 0 Th lctron can rsolv th transvrs positions of th quarks with arbitrary accuracy in hard collisions (Q ) In h X whr th lctron p z it scattrs from all targt quarks at qual Light-Front (LF) tim x + = t+z t=t z= t t=0 z=0 Th transvrs structur of h can b masurd at qual Light-Front tim x +
Paul Hoyr INT 2016 DIS: larg Q 2 rsolvs singl quarks 5 At low Q 2 th γ* may intract with diffrnt quarks in T and T*: N q T γ* γ* T* N Such contributions do not rflct th proprtis of a singl quark. Bj limit: q 0 = ν and Q 2, with x Bj = Q2 2m h N T Q 2 r At larg Q 2 th γ* is cohrnt on a singl quark Vrifid by scaling in Q 2 (up to log s) fixd Th quark can b at any r in th targt.
Impact paramtr distributions via th GPD s 6 Z d 2? (2 ) 2 ib? GPD(?) dtrmins th transvrs position b of th struck quark x N GPD N x Sopr (1977) Burkardt (2000) Dihl (2002) (P +, Δ /2) (P +, Δ /2) Δ f q/n (x, b) = n, i,k i=1 n dx i 4 d 2 b i 1 i x i 1 4 2 x i b i (2) (b b k ) (x x k ) n (x i, b i, i ) 2 Cntr of momntum at th origin LF wav function i Not: * b is conjugat to th (finit) nuclon momntum transfr Δ, not to th momntum transfr (Q ) in th hard collision q q * Paul Hoyr INT 2016 Th 2-dimnsional FT must b don in a fram whr Δ + = 0
Paul Hoyr INT 2016 In N N th lctron and nuclon momntum transfrs ar th sam: q = Δ Nuclon Form Factors γ* q = Δ 7 Th γ* coupls to a singl quark in th form factor (amplitud, not σ!) A 2-dim. FT ovr q will giv th distribution of th struck quark in b N F1, F2 N (P +, Δ /2) (P +, Δ /2) Δ Not: * Th γ*(q ) scattrs cohrntly ovr quarks within Δb 1/q, and thus masurs charg with this rsolution. * A FT ovr < q < givs th b-distribution with δ-function accuracy.
Nuclon Charg Distribution from N N 8 0 (b) = 1 2p + Z d 2 q (2 ) 2 iq b p +, 1 2 q, J + (0) p +, 1 2 q, = 0 dq 2 QJ 0(bQ)F 1 (Q 2 ) 0 (b) = n, i,k k n i=1 dx i 4 d 2 b i (1 i x i ) 1 4 (2) ( i x i b i ) (2) (b b k ) n (x i, b i, i ) 2 bx Complmntary to pdf s, but no factorization, hnc no univrsality. Nutron charg distribution vs. b + by Paul Hoyr INT 2016 Millr (2007) Carlson and Vandrhaghn (2008)
Paul Hoyr INT 2016 Byond lastic form factors 9 Th xprssion of ρ0(b) in trms of LC wav functions, 0 (b) = 1 2p + Z d 2 q (2 ) 2 iq b p +, 1 2 q, J + (0) p +, 1 2 q, is basd on th Fock xpansion of th initial and final stats: n 1 P + dx i d 2 k i, P, x + =0 = xi 16 3 16 3 (1 x i ) (2) ( n, i i=1 0 i i k i ) n (x i, k i, i) n; x i P +,x i P + k i, i x+ =0 Any stat f is dfind by its LF wav functions Th b-spac analysis applis similarly for any final stat f : hf J + (0) Ni By comparing th b-distributions for various stats f on larns about th raction dynamics. f n(x i, k i, i)
Paul Hoyr INT 2016 Byond lastic form factors 10 Th b-distribution of th struck quark in N f is givn by: A fn (b) = 1 4 n n i=1 1 0 dx i 4 d 2 b i (1 x i ) 2 ( i i x i b i ) ) n f (xi, b i ) n N (x i, b i ) k k 2 (b k b) Th xprssion is diagonal in th Fock stats n, and f n N n Th γ* both causs th transition N f and masurs th contributing Fock stats f = N* : Transition form factors: N N*
Comparison of N* transition form factors 11 N(1440)1/2 + b 2 ρ T b y (fm) Elctric Dipol N*(1440) ρ 0 ρ T charg dnsi<s b 2 ρ T N(1535)1/2 - b y (fm) Elctric Dipol N*(1535) ρ 0 ρ T CLAS data V. D. Burkrt arxiv:1610.00400 Paul Hoyr INT 2016
Paul Hoyr INT 2016 Connction to Color Transparncy 12 Brodsky, and Mullr (1988) Hard procsss ar xpctd to involv transvrsally compact Fock stats Masur thir siz via rscattring in nucli Exampl: A ρ X π π X CLAS Collaboration arxiv:1201.2735 Expct th transvrs siz of th qq cratd by th γ*(q 2 ) to dcras as Q 2 grows. A lf Tst by masuring th absorption of th th qq in th targt nuclus A. Choos kinmatics: Th γ* cohrnc lngth lc is short (< 1 fm) Th ρ formation lngth lf is long (> 1fm)
Evidnc for color Transparncy 13 lc (fm) A Paul Hoyr INT 2016 CLAS Collaboration arxiv:1201.2735
Hard scal from th final stat 14 In th CLAS xprimnt, th virtuality Q 2 of th γ* gav th hard scal, and σ(a) rflctd th siz of th qq which (latr) formd into th ρ. W may also considr A π π X at low Q 2, whr th rlativ k of th pions provids th hard scal. This is analogous to th E791 masurmnt of xclusiv dijt production π A jt jt A : Th rlativ k of th jts provids th hard scal. 500 GV π q k > 1.25 GV At low q th scattring is cohrnt on th nuclar targt A =Pt. E791 Collaboration hp-x/0010044 Us σ(a) to masur th siz of th qq which crats th jts.
π 500 GV σ(π A jt jt A) A α Fig. 4. Th Frmilab E791 di-jt yild from carbon and platinum as a function of th squar E791 Collaboration yilds shown ar for 1.5 appl k t appl 2.0 GV. Th curvs ar Mont Carlo simulations of th shows incohrnt dissociation, th background is shown by hp-x/0010044 th dot-dashd lin and th to Sourc: From Rf. [26]. α 1.5 E791 15 q q 2 < 0.015 GV 2 2/3 Data Thory Paul Hoyr INT 2016 Fig. 5. Th valus of obtaind from paramtrization of th E791 di-jt cross sction as = 0 A. Th data ar show quadratur sum of statistical and systmatic rrors and th k T bin siz. Th blu dashd lins ar th CT prdictions of Rf. [ α 1.5 indicats obsrvd inthat cohrnt th inlastic nuclus diffractiv pion nuclus is transparnt intractions. Typical to th virtualitis compact Q 2 = 4k 2 t qq Fock stats of th pion, slctd by k > 1.25 GV. ar also shown. incohrntly from nuclar targts, and th background. Th shaps of ths distributions ar calcu simulations as shownfig. in 5. Fig. Th 4 for valus transvrs of obtaind momntum, from paramtrization 1.5 appl k t of applth 2.0 E791 GV. di-jt Th cross pr-nuclo sction as production is paramtrizd quadratur as sum = of 0 statistical A, whr and 0 systmatic is th fr rrors cross andsction. th k T bin Th siz. valus Th blu of dashd th xpon lin xprimnt E791 ar shown obsrvdininfig. cohrnt 5 along inlastic withdiffractiv th CT prdictions pion nuclus of intractions. [16]. This is Typical to bvirtualitis compardq w 2
σ( N π N) 16 Rplac A : Lt th photon masur th transvrs siz, b(k ) π π N b k 500 GV q (12 GV) γ* q q 2 > 0 N q 2 < 0.015 GV 2 Not: * Th photon masurs th siz of th nuclon Fock stats at x + = 0 Th asymptotic πn stat mrgs from ths Fock stats as x + * Thr is no issu of cohrnc or formation lngths.
Comparison with th transition form factor 17 q 2 > 0 γ* q q 2 > 0 γ* q π(p1) N N*(p+q) N(p) b b k N(p2) p1 + p2 = p + q is not sufficint to fix p1(q) and p2(q) sparatly Th N and N* wav functions ar indpndnt of LF tim x + Th π N stat dvlops from th Fock stats masurd at x + = 0 Th N N* amplitud is ral, whras N π N has dynamical phass du to th πn intractions in th final stat.
Paul Hoyr INT 2016 FT of γ* matrix lmnt in momntum spac 18 In th fram: p = (p +,p, 1 2 q) w hav q = (0 +,q, q) p f = (p +,p + q, 1 2 q) d 2 q (2 ) 2 iq b 1 2p + f(p f ) J + (0) N(p) = A fn (b) p N l + q f pf A fn (b) = 1 4 n 1 n i=1 0 dx i 4 d 2 b i (1 x i ) 2 ( i i x i b i ) ) n f (xi, b i ) n N (x i, b i ) k k 2 (b k b)
Paul Hoyr INT 2016 Exampl: f = π (p 1 ) N(p 2 ) In ordr to conform with th Lorntz covarianc of LF stats, at any pf : 1 N(p + f, p f ; f dx d 2 k ) f 0 x(1 x) 16 3 (x, k) (p 1 )N(p 2 ) whr Ψ f (x,k) dfins th final stat in trms of th rlativ variabls x, k : 19 p + 1 = xp+ f p 1 = xp f + k p + 2 =(1 x)p+ f p 2 =(1 x)p f k With x, k bing indpndnt of pf, our choic of Ψ f (x,k) dfins th pion and nuclon momnta p1, p2 at all photon momnta q.
Paul Hoyr INT 2016 QED illustration: + γ* + µ + µ 20 Th targt is a virtual photon γ*(p). Th Fourir transform of A µµ, 1, 2 = 1 µ (p1 2p +, 1)µ + (p 2, 2) J + (0) (p, )i givs, dnoting m = mµ and M 2 = m 2 x(1-x) p 2, A µµ,+1 (b; x, k) = + 1 2 + 1 2 p m appl p Mb K0 x(1 x) 1 x 2 (1 x) 2 xp i k b 1 x K0 Mb x x 2 xp +i k b x This agrs with th gnral xprssion in trms of th LF wav functions. γ*(p) γ*(q) µ + (p1) + µ (p2)
+ γ* + µ + µ xampl (1) Avrag Impact paramtr vs. x: m and k dpndnc 0.4 0.3 0.2 0.1 b 3 hb 2 i k = 5.0 1.5 p 2 = 0 0 m = 1 0.1 0.2 0.3 0.4 0.5 m = 2 x 161010_impact.nb
+ γ* + µ + µ xampl (2) Avrag Impact paramtr vs. x: p 2 dpndnc 0.6 0.4 0.2 b 3 hb 2 i m = 1 k = 0 0.1 0.2 0.3 0.4 0.5 p 2 = +3m 2 p 2 = 0 p 2 = 3m 2 x 161010_impact.nb
Paul Hoyr INT 2016 Fourir transform of th cross sction 23 Th γ*+n f amplituds hav dynamical phass (rsonancs,...). Calculating thir Fourir transforms rquirs an amplitud analysis. On can also Fourir transform th masurd cross sction itslf. Thn th b-distribution rflcts th diffrnc btwn th impact paramtrs of th photon vrtx in th amplitud and its complx conjugat: d 2 q (2 ) 2 iq b 1 2p + f(p f ) J + (0) N(p) 2 = d 2 b q A fn (b q ) A fn(b q b) A narrowing of AfN(bq) as a function of th final stat f will b rflctd in th convolution.
+ γ* + µ + µ xampl (3) Avrag Impact paramtr from cross sction 2.0 1.5 1.0 0.5 hbi S m = 1, p 2 = 0 m = 2, p 2 = 0 k = 0 m = 1, p 2 = 5m 2 0.1 0.2 0.3 0.4 x Non-flip amplitud, b 4 momnt, normalizd to S(b=0), to powr 1/6. 161010_impact.nb
Exampl: γ (*) + D p + n at 90 Th 90 brak-up cross sction at q 2 =0 agrs with dimnsional scaling for Eγ > 1 GV. σ(γd pn) E 22 E 22 CM dσ(γd pn) / kb GV dt C. Bochna t al, PRL 81 (1998) 4576 20 25 Dos this man that only compact configurations of th dutron, with R < 0.2 fm, contribut to this procss? E (GV) γ If so, xpct no q 2 -dpndnc for q 2 < 1 GV 2. With lctroproduction data R could b masurd:
Paul Hoyr INT 2016 Many othr procsss within rach 26 Multiparticl final stats Havir flavors γ*n ππn, γ*n KΛ, K*Λ γ*n DΛc, Nuclar targts
Paul Hoyr INT 2016 Summary 27 Th q -dpndnc of th virtual photon masurs th charg distribution in transvrs spac. Th charg dnsity is masurd at an instant of Light-Front tim x + = t + z Unlik pdf s, no lading twist limit is implid: All Q 2 ar usful Th dnsity can b dtrmind for any initial and final stat: γ*a f Comparisons of b-distributions in diffrnt procsss giv insights into th scattring dynamics in transvrs spac. Modl indpndnt analysis Rady to b trid out with Jlab data!