Transversity Distributions and Todd Asymmetries in DrellYan Processes

Transversity Distributions and T-odd Asymmetries in Drell-Yan Processes Gary R. Goldstein Tufts University Leonard P. Gamberg Penn State-Berks Lehigh Valley College 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 1

Abstract Drell-Yan unpolarized processes display azimuthal asymmetries. One such asymmetry cos(2 ) is directly related to the leading twist transversity distribution h 1 (x, k. T). We use a model developed for semi-inclusive deep inelastic scattering that determines the “Sivers function” f 1 T (x, k. T) to predict the Drell-Yan asymmetry as a function of q 2, q. T and either x or x. F or a new variable, . The resulting predictions include a non-leading twist contribution from spin-averaged distributions that measurably effect lower energy results. 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 2

Outline • Transversity § § Short history Helicity flip, chirality, phases & k Quark distribution functions: T-even &T-odd Fragmentation functions: T-even &T-odd • SIDIS § Asymmetries: SSA & azimuthal § Rescattering & leading twist contributions • Drell-Yan § N & distributions § cos 2 asymmetry • Conclusions 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 3

Transversity - some history • 2 -body scattering amps - Exclusive hadronic § fa, b; c, d(s, t) with spin projections a, b; c, d • What spin frame leads to simplest description of theory or data? Amps to observables? § helicity has easy relativistic covariance - theory § states of S·p, e. g. |+1/2 , |-1/2 , etc. § transversity: eigenstates of S·(p 1 p 2) | 1/2 )T = {|+1/2 (i) |-1/2 }/√ 2 for spin 1/2, etc. Especially for relating to single spin asymmetries - only S·n Goldstein & Moravcsik, Ann. Phys. 1976 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 4

Transversity & simplicity • states of {S·(p 1 p 2)} or {S·(p 1 p 2)} are transversity normal to or parallel to scattering plane § Spin 1: | 1)T = {|+1> 2 |0> + |-1>}/2 | 0)T = {|+1> - |-1>}/ 2 § photon: | 1)T = {|+1> + |-1>}/ 2 linear polzn normal to plane | 0)T = {|+1> - |-1>}/ 2 linear polzn parallel to plane useful in photoproduction dynamics • Transversity amps in NN NN have phase simplicity (many observables!) • Goldstein & Moravcsik & Arash 1980’s 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 5

Phases & SSA • Single Spin Asymmetries (SSA) in 2 -body • Parity requires only <S·n> non-zero for any single spinning particle. Requires n p 1 p 2 some helicity flip p 2 p 1 or chirality flip for m=0 quarks & phase. <S·n> f*ab, cd[ ·n]dd’fab, cd’ Im[f*ab, c+ fab, c-] n requires some p 2 transverse to p 1 (at quark level? ) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 6

Azimuthal asymmetries - kinematics • Why similar to spin asymmetries? § Need plane established (P 1 P 2) transverse P § Need azimuthal angle relative to 1 st plane, i. e. 2 nd plane • via fragmentation or decay or pair production • How does orientation information get transferred from 1 st plane to 2 nd plane? Dynamical question. • SIDIS & Drell-Yan involve off-shell photons - like massive vector particles with & longitudinal polarization 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 7

SSA & Az. As dynamics: require loops & k • <S·n> f*AB, CD[ ·n]DD’f. AB, CD’ helicity basis or in transversity basis: {|f. AB, C(+ )|2 - |f. AB, C(- )|2} • Imaginary part or phase requires beyond tree level in any field theory § What is tree level in “effective” field theory? § Mixing PQCD & soft physics • Helicity or chirality flip requires a flipping interaction (m≠ 0, …) & non-zero transverse momentum of participants or k ’s 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 8

What reactions to look at? DIS vs. Drell-Yan vs. SIDIS for h 1(x)= q(x) & for T-odd Figures from R. Jaffe 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 9

Brodsky, Hwang & Schmidt provided non-trivial model calculation Final state corrections to tree-level DIS-> f 1 T (x, p 2) & SSA G&G: contributes to h 1(x) & f 1 T (x, p 2) = ± h 1 (x, p 2) Need SIDIS or D-Y to make functions experimentally accessible in asymmetry or polarization Brodsky, Hwang, Schmidt PLB 2002 Collins PLB 2002; Ji & Yuan PLB 2002 Goldstein & Gamberg ICHEP 2002 Gamberg, Goldstein, Oganessyan PRD 2003 &hep-ph 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 10

h 1 • h 1 (x, p 2) is “T-odd” distribution probability of finding quark with non-zero transversity in unpolarized hadron (it is Peven) • Vanishes at tree level in T-conserving models, as in spectator diquark model e. g. N quark+diquark where q is struck quark (like ordinary decay amps - final state interactions are essential - no T violation) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 11

SIDIS kinematics In spectator model yellow inclusive blob becomes diquark - scalar for simplicity 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 12

Brodsky, Hwang, Schmidt rescattering 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 13

Interpreting rescattering 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 14

Model calculation 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 15

Calculating h 1 (x, k. T) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 16

Distribution definitions fj/A(0) (x) is integral over k. T of j/A with gauge link added to insure gauge invariance 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 17

Expanding distributions Feynman rules obtained with intermediate states inserted 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 18

Ingredients for h 1 NPB 194(1982)445 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 19

Integration results Spin independent tree level: Transversity T-odd GPD: ( . . k. j factor on both sides) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 20

Regularization • SSA’s & asymmetries involve moments of distribution & fragmentation functions e. g. h 1 (1)(x) = ∫ d 2 k k 2 h 1 (x, k 2) which would diverge without k 2 damping 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 21

Transverse momentum hadronic tensor 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 22

h 1 (x, k ) calculation with Gaussian h 1/(m 2 -k 2)=(1 -x)/ (k. T 2) result of p->q+diq kinematics h 1 (x, k )= f 1 T (x, k ) (sign+) in diquark model Gamberg, Goldstein, Oganessyan PRD 2003 & hep-ph 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 23

Drell-Yan coordinates p 1 l’ p 2 l 9/5/2021 y lepton CM frame defines plane tilted at rel. t. hadron plane of p 1 &p 2 x Coordinates? z is direction of q z in initial frame or x is direction along q. T from initial cm boost (Collins-Soper frame) or … SIR 2005: T-Odd & D-Y G. R. Goldstein 24

Drell-Yan Cross Sections see early papers ‘ 70’s Collins&Soper 1977 effects of transverse momenta -> , , non-zero D. Boer PRD 60 & D. Boer, S. J. Brodsky & D. S. Hwang hep-ph/0211110 Unpolarized pair of hadrons l + l’ + X involves transversity at leading twist 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 25

D-Y angular dependent How are angular asymmetries calculated? is related to T-odd distributions at leading twist (D. Boer). 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 26

Az. As: h 1 (1) • H 1 (1) cos 2 Both distribution & fragmentation calculated in spectator models with gaussian k π π +h. c. 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 27

From SIDIS to Drell-Yan - analogous calculations Beam (π, p, …) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 28

Azimuthal asymmetry Integrate over all quark transverse momenta. +p l+l- X is in process; p+anti-p is calculated for s at fixed target Fermilab. x direction is QT direction Notation of Boer, Mulders, Teryaev & Boer, Brodsky, Hwang 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 29

QT dependences General form: expectation of a hadronic tensor with distributions from quark model of incoming particles • Asymmetry must vanish as QT 0 ; no 1 st plane orientation in forward limit of initial state. • What is role of quark spin? • In lepton rest frame or q+ q (CM) “fat” photon produced. • Whether q & q polarized or not, photon’s spin tensor (T & L) is fixed by QED. • Unpolarized q+ q defines a plane via QT & tensor behaves ~ (QT 2 / Q 2)2. • Transversely polarized q+ q have ST 1 ST 2 tensor structure to combine with k. T & p. T (2 planes) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 30

Non-leading contributions Spin dependent leading part ~ QT 2 for small QT 2 / Q 2 Non-leading, spin independent part ~ extra QT 2 Collins & Soper ‘ 77 defined tensor A 2 = B = (2 k. Tx p. Tx k. T p. T) / M 2 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 31

convolutions There will be the tensor B and azimuthal dependence, crucial for transmitting plane orientation information. Integrate numerically to obtain convoluted functions depending on x, mee, QT (and s). Note x = mee 2/xs for s >> mee 2 >>QT 2. Convolutions of h’s have extra factors of S at appropriate scale compared to f’s. But f’s in numerator have QT 2 relative to h’s. 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 32

Drell-Yan kinematics Asymmetry is function of 3 variables: x, √q 2 =m , q. T Want to obtain integrated over 2 variables. How to do this while keeping “symmetry” x 1, x 2 Using x. F treats x 1, x 2 symmetrically, but different range vs. q. Use = x. F /2(1 - ) from -1/2 to +1/2 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 33

Az. As function Insert convolutions into asymmetry expression: Obtained for range of x, mee, QT (and s). Choose kinematic ranges of Conway, et al. (FNAL fixed target π p) applied to p p. Sum over their (limited) ranges to obtain (x), (mee 2), (QT). 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 34

(QT 2) leading h 1 contribution Calculated for same s 50 Gev 2 - lower kinematic range than Conway, et al. E 615. Antiproton beam vs. π Data for π-p at s=500 Ge. V 2 E 615 Very similar to Boer, Brodsky, Hwang But gaussian supressed f f part is at most 10% At higher s 500 Gev 2 with comparable range curve decreases a bit f f part can be 10 - 15% of this for some values of 3 variables. 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 35

(m ) leading h 1 contribution (s=50 Ge. V 2) Data for π-p at s=500 Ge. V 2 E 615 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 36

q vs. x x = 0. 9 x = 0. 8 x = 0. 7 x = 0. 6 x = 0. 5 x = 0. 4 x = 0. 3 x = 0. 2 x = 0. 1 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 37

( ) s=50 Ge. V 2 ( ) Blue -leading Red - with non-leading 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 38

versus x 1 s=50 Ge. V 2 Leading twist only Including non-leading 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 39

Summary& Conclusions • Transversity is important for full understanding of hadron spin composition. Accessed via SIDIS & Drell-Yan with SSA’s & azimuthal asymmetries. Require flips & loops. Probing applicability of models of factorized soft-hadronic & PQCD. • TMD’s are important distributions for accessing spin & transversity content of hadrons. • BHS rescattering is mechanism for generating TMD’s at leading twist that can be measured via SSA’s & Az. As’s. • quark-diquark (S=0) model with gaussian regulators allows simple calculations to demonstrate existence of interesting TMD’s and thus SSA’s & Az. As’s. 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 40

Summary (cont’d) • Example considered: “T-odd” contribution to cos 2 in D-Y compared to “T-even” non-leading spin independent piece. • Does data support “T-odd” TMD? Large effect in π+p at hi s makes this very plausible. Need Az. As data on anti-p+p. • Improvements: § S=1 diquark is I=1 uu flavor p->d+diq § better starting model (2 -body constraints are limiting) • Questions: § How do “T-odd” TMD’s evolve? § Are Sudakov effects important for low q. T in Az. As’s? • Many workers, much work to be done. Everyone here + experiments! • Transversity has arrived! 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 41

Fitting f 1(x) 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 42

cos 2 asymmetry in SIDIS Ignoring 1/Q 2 T-even contribution Boer & Mulders PRD 1998 Gamberg, Goldstein, Oganessyan PRD 2003 & hep-ph 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 43

(x) for s = 50 Ge. V 2 Including non-leading Leading twist only 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 44

(QT 2) leading h 1 contribution preliminary Calculated for same s 500 Gev 2 kinematic range as Conway, et al. E 615 for beam they get larger asymmetry that grows. Very similar to Boer, Brodsky, Hwang f f part is only 1 or 2% of this. At lower s 50 Gev 2 with comparable range curve increases f f part can be 10 - 15% of this for some values of 3 variables. 9/5/2021 SIR 2005: T-Odd & D-Y G. R. Goldstein 45