TMDs at small-x: an Operator Treatment

Transcription

1 TMDs at small-x: an Operator Treatment Yuri Kovchegov The Ohio State University work with Dan Pitonyak and Matt Sievert, arxiv: [nucl-th] and 5 other papers + papers in preparation

2 Outline Goal: understanding the small-x asymptotics of TMDS Helicity: helps evaluate the amount of proton spin coming from small x partons Transversity: small-x contribution to the proton tensor charge Quark Helicity: Quark helicity distribution at small x Small-x evolution equations for quark helicity Small-x asymptotics of quark helicity Gluon Helicity: Gluon helicity distribution at small x Small-x evolution equations for gluon helicity Small-x asymptotics of quark helicity TMDs Quark Transversity: Quark transversity TMD at small x Small-x evolution equation for quark transversity Small-x asymptotics of quark transversity TMDs

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4 Our Goal: calculating all PDFs and TMDs at Small x 4

5 How much spin is at small x? E. Aschenaur et al, arxiv: [hep-ph] Uncertainties are very large at small x! 5

6 Spin at small x The goal of this project is to provide theoretical understanding of helicity PDF s at very small x. Our work would provide guidance for future hpdf s parametrizations of the existing and new data (e.g., the data to be collected at EIC). Alternatively the data can be analyzed using our small-x evolution formalism. 6

7 <latexit sha_base64="hyy3okhventmn9wruvvfvtrbm=">aaachhicbvbns8nafnzu7/pv9ejlyreqseluisgevgoyfvordzbo3szsbuvogl+eo8+fe8efdeiwfbf+o9qdvgyvhzh5v3szfazd99mpjypte5nz5rn5+yxfitly+cmztxjdzbkvf9ghapfg+gqmkvm8pekl+exwp+v7fdddgpoomexkpenqlrfswilykk9t+zcvsuibaawtra/bbfwp9krkbjnrbhss34g9cj/sgdrsj/vbyqbpdwd4s7whqzihtslkux+nle+4qiapmu3pztaoqebbjl8r+7nhgwvdeswbliqacbmug+puynqmbrtbz9cgkg/jwqagnnliptmkhbmqncx//oaobb3gkkoleeupeidi4bu+g3bbhqnkhswukzfvavwdpuu4az7itwrs9+s8536p7bt73akeha7rmcarzi3uiedyqe5jiekqri5j4/kmbw4d86t8+q8fudlznbmhfyc8/efruid9a==</latexit> <latexit sha_base64="hyy3okhventmn9wruvvfvtrbm=">aaachhicbvbns8nafnzu7/pv9ejlyreqseluisgevgoyfvordzbo3szsbuvogl+eo8+fe8efdeiwfbf+o9qdvgyvhzh5v3szfazd99mpjypte5nz5rn5+yxfitly+cmztxjdzbkvf9ghapfg+gqmkvm8pekl+exwp+v7fdddgpoomexkpenqlrfswilykk9t+zcvsuibaawtra/bbfwp9krkbjnrbhss34g9cj/sgdrsj/vbyqbpdwd4s7whqzihtslkux+nle+4qiapmu3pztaoqebbjl8r+7nhgwvdeswbliqacbmug+puynqmbrtbz9cgkg/jwqagnnliptmkhbmqncx//oaobb3gkkoleeupeidi4bu+g3bbhqnkhswukzfvavwdpuu4az7itwrs9+s8536p7bt73akeha7rmcarzi3uiedyqe5jiekqri5j4/kmbw4d86t8+q8fudlznbmhfyc8/efruid9a==</latexit> <latexit sha_base64="hyy3okhventmn9wruvvfvtrbm=">aaachhicbvbns8nafnzu7/pv9ejlyreqseluisgevgoyfvordzbo3szsbuvogl+eo8+fe8efdeiwfbf+o9qdvgyvhzh5v3szfazd99mpjypte5nz5rn5+yxfitly+cmztxjdzbkvf9ghapfg+gqmkvm8pekl+exwp+v7fdddgpoomexkpenqlrfswilykk9t+zcvsuibaawtra/bbfwp9krkbjnrbhss34g9cj/sgdrsj/vbyqbpdwd4s7whqzihtslkux+nle+4qiapmu3pztaoqebbjl8r+7nhgwvdeswbliqacbmug+puynqmbrtbz9cgkg/jwqagnnliptmkhbmqncx//oaobb3gkkoleeupeidi4bu+g3bbhqnkhswukzfvavwdpuu4az7itwrs9+s8536p7bt73akeha7rmcarzi3uiedyqe5jiekqri5j4/kmbw4d86t8+q8fudlznbmhfyc8/efruid9a==</latexit> <latexit sha_base64="hyy3okhventmn9wruvvfvtrbm=">aaachhicbvbns8nafnzu7/pv9ejlyreqseluisgevgoyfvordzbo3szsbuvogl+eo8+fe8efdeiwfbf+o9qdvgyvhzh5v3szfazd99mpjypte5nz5rn5+yxfitly+cmztxjdzbkvf9ghapfg+gqmkvm8pekl+exwp+v7fdddgpoomexkpenqlrfswilykk9t+zcvsuibaawtra/bbfwp9krkbjnrbhss34g9cj/sgdrsj/vbyqbpdwd4s7whqzihtslkux+nle+4qiapmu3pztaoqebbjl8r+7nhgwvdeswbliqacbmug+puynqmbrtbz9cgkg/jwqagnnliptmkhbmqncx//oaobb3gkkoleeupeidi4bu+g3bbhqnkhswukzfvavwdpuu4az7itwrs9+s8536p7bt73akeha7rmcarzi3uiedyqe5jiekqri5j4/kmbw4d86t8+q8fudlznbmhfyc8/efruid9a==</latexit> Transversity and tensor charge Transversity at small x is also poorly known, resulting in poor knowledge of the tensor charge: Z q(q )= dx h (x, Q ) From the talk by M. Radici at DIS 8 + see his talk at this meeting Soffer bound 7

8 Quark Helicity at Small x (flavor-singlet case) Yu.K., M. Sievert, arxiv:55.76 [hep-ph] Yu.K., D. Pitonyak, M. Sievert, arxiv: [hep-ph], arxiv:6.697 [hep-ph], arxiv:6.688 [hep-ph], arxiv: [hep-ph] + in preparation 8

9 Observables We want to calculate quark helicity PDF and TMD at small x. Leading Twist TMDs Nucleon Spin Quark Spin Quark Polarization Un-Polarized (U) Longitudinally Polarized (L) Transversely Polarized (T) Nucleon Polarization U L T ƒ = ƒ T = g L = g T = Helicity h = h L = h = Boer-Mulders Transversity Sivers h T = 9

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sha_base64="j4qmtan9npfjjukfx6ltplv5a8=">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</latexit> Quark Helicity TMD We start with the definition of the quark helicity TMD with a futurepointing Wilson line staple. g q L (x, k T )= ( ) 3 X S L S L At small-x, in anticipation of the shock-wave formalism, we rewrite the quark helicity TMD as (in A - = gauge for the + moving proton) g q L (x, k T )= p+ ( ) 3 Z where the fundamental light-cone Wilson line is Z d rdr d d d d e ik ( ) e ik r hp, S L () U[,r] (r) p, S L i r + = D ( ) V [, ] V [, ] ( )E V x [b,a ]=Pexp 8 >< >: ig a bz dx 9 >= A + (x,x) >;

11 <latexit sha_base64="4jwozwkz4lze4rfjpkabkgvwpm=">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</latexit> <latexit sha_base64="4jwozwkz4lze4rfjpkabkgvwpm=">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</latexit> <latexit sha_base64="4jwozwkz4lze4rfjpkabkgvwpm=">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</latexit> <latexit sha_base64="4jwozwkz4lze4rfjpkabkgvwpm=">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</latexit> Quark Helicity TMD at Small x At high energy/small-x the proton is a shock wave, and we have the following contributions to the SIDIS quark helicity TMD: z A B C x D E F g q L (x, k T )= p+ ( ) 3 Z d d d d e ik ( ) + 5 D ( ) V [, ] V [, ] ( )E

12 Quark Helicity TMD at Small x A B C D E F Clearly diagrams C and D do not transfer spin information from the target, since Wilson lines do not know about polarization. Diagram F is energy-suppressed, since the gluon should have no time to be emitted and absorbed inside the shock wave. Diagrams of the types A and E++ can be shown to cancel each other. We are left with the diagram B.

13 Quark Helicity TMD at Small x Dominance of diagram B can also be obtained by applying crossing symmetry to the SIDIS process (KS 5): x y x y c.c. z = z z c.c. x y = the answer Compare the last line to the diagram B: reflecting the cc amplitude into the amplitude reduces the above diagram to the one on the right. ζ k w k ξ 3

14 <latexit sha_base64="e5ukzxphkdcitw/wylsyjnuohq=">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</latexit> <latexit sha_base64="e5ukzxphkdcitw/wylsyjnuohq=">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</latexit> <latexit sha_base64="e5ukzxphkdcitw/wylsyjnuohq=">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</latexit> <latexit sha_base64="e5ukzxphkdcitw/wylsyjnuohq=">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</latexit> Quark Helicity TMD at Small x Evaluating diagram B we arrive at g q L (x, k T )= 4N c ( ) 6 Z d d wd ye ik ( y) where G wz is the polarized dipole amplitude (defined on the next slide). Here s is the cms energy squared, L is some IR cutoff, underlining denotes transverse vectors, z = smallest longitudinal momentum fraction of the dipole momentum out of those carried by the quark and the antiquark The same result was previously obtained starting with the SIDIS process (KPS 5) instead of the operator definition of quark helicity TMD: we have thus shown that the two approaches are consistent at small x. Z /s dz z w w y w y w G w, (zs) 4

15 Polarized Dipole All flavor singlet small-x helicity observables depend on one object, polarized dipole amplitude : G (z) DD h i tr V V pol N c +tr h iee V pol V (z) V x P exp 4ig Z unpolarized quark 3 dx + A (x +,,x) 5 polarized quark: eikonal propagation, non-eikonal spin-dependent interaction Double brackets denote an object with energy suppression scaled out: DD EE O (z) zs D O E (z) 5

16 Polarized fundamental Wilson line To complete the definition of the polarized dipole amplitude, we need to construct the definition of the polarized Wilson line V pol, which is the leading helicity-dependent contribution for the quark scattering amplitude on a longitudinally-polarized target proton. p p k p p + k σ σ σ a λ λ b σ k k p x x At the leading order we can either exchange one non-eikonal t-channel gluon (with quark-gluon vertices denoted by blobs above) to transfer polarization between the projectile and the target, or two t-channel quarks, as shown above. 6

17 <latexit sha_base64="jxogv6kfiegehcie7jzgsw7gu4=">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</latexit> <latexit 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sha_base64="qoivdvlmcc9yiiexshya/btd8=">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</latexit> <latexit sha_base64="qoivdvlmcc9yiiexshya/btd8=">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</latexit> <latexit sha_base64="qoivdvlmcc9yiiexshya/btd8=">aaacehicbvhlatwwfjxdvzp9znous7lgpqqmceohwqtsntnlpojahlnrisz4jisphk4kh4g/jvfvduskq8mqobdilgsm59+hk5a4noews/pf/t4ydnnw88hl6+er9fpptfenomxokgpi4xojrhkc8onybeyqtmbdvplr/+vlppjsv5a+zqllckoxkbafeocodxstixuzm9xfynuugyhljmmgyssge8almutkhjgflbaxasmbfwwazafebsymflbnrqnadlrln3keobtacpwwlnhs9dcgb7bxjweox7pef/tiwwes+gjp4nq4cqbbuuahcddghdys4fxok9ouzjpqcbarfqm9gsztgvrbvgrroaeuyyjgdkprmx3ydxae7jsmhqjq7sca/dthsanqsxcz5+dvq/5p+qdhfcwy5rbvdjlbvdqctax9fidnilejvg4qqrh7k9aluyqat6ubcyg8/+wh4oxwggbt8punemxtrxbaae9r3sorefofhdmzrhfp3drwp3q5344p/d+/a/w9jecd+qf8wti87t+</latexit> Polarized fundamental Wilson line p p k p p + k σ σ σ a λ λ b σ k k In the end one arrives at V pol x = igp+ s g p + s Z Z dx The first term on the right (the gluon exchange contribution) was known before (KPS 7), the second term (quark exchange) is new. We have employed an adjoint light-cone Wilson line p dx V x [+,x ] F (x,x) V x [x, ] Z x b dx V x [+,x ] t apple (x,x) Ux ba [x,x ] U x [b,a ]=P exp 6 4ig + 5 a bz x x dx (x,x) t a V x [x, ]. 3 A + (x + 7 =,x,x) 5 7

18 Polarized adjoint Wilson line Quarks mix with gluons. Therefore, we need to construct the adjoint polarized Wilson line --- the leading helicity-dependent part of the gluon scattering amplitude on the longitudinally polarized target. p ν µ p + k b p k b p a a c λ a ρ λ b λ µ σ σ λ k k p x x The calculation is similar to the quark scattering case. It yields (U pol x ) ab = igp+ s + g p + s Z dx Z + Z x dx U x [+,x ] F (x + =,x,x) U x [x, ] ab aa dx U x [+,x ] (x,x) t a V x [x,x ] + 5 t b (x,x) U b b x [x, ]+c.c. 8

19 <latexit sha_base64="fffohepr7kf7tbzyohxelomq=">aaacc3icbvdlssnafj3uv6vqes3q4tqquoibdirrdwce+oihmpmyyetmdmraujejb/ixouibvbd/6nzaivg9cojxzl/fe48wmsmvznzuyxfpesw/wlhb39jcmrd3wjjkbcznhlfiddwkcaocnbvvjhriqvdomdlhuctv3lhkqrvkjmdgh6nmauiyullyzwl9okx8zdlpbgspy3qe8hvxyo4tb8usbfmgl+jxzgsibdwzu/en6ek5bwhrmssmtbsxjsjbtfjiwlvussgoeh6pouphyfrdrp9jcx3nekd4ni6oikttwfeykkpryfnu4mkrriew8i/udexwcocnlcaiix7nfqckgiuakgohtqbbiiqfltfcveacysvjq+gq7dnx/5lwkcv6ryl9vs7syliw/qbgugqoqqcgazoagzuwsn4bi/gg/fkvbpvs9ackc3sgl8w3r8aaqxq==</latexit> <latexit sha_base64="fffohepr7kf7tbzyohxelomq=">aaacc3icbvdlssnafj3uv6vqes3q4tqquoibdirrdwce+oihmpmyyetmdmraujejb/ixouibvbd/6nzaivg9cojxzl/fe48wmsmvznzuyxfpesw/wlhb39jcmrd3wjjkbcznhlfiddwkcaocnbvvjhriqvdomdlhuctv3lhkqrvkjmdgh6nmauiyullyzwl9okx8zdlpbgspy3qe8hvxyo4tb8usbfmgl+jxzgsibdwzu/en6ek5bwhrmssmtbsxjsjbtfjiwlvussgoeh6pouphyfrdrp9jcx3nekd4ni6oikttwfeykkpryfnu4mkrriew8i/udexwcocnlcaiix7nfqckgiuakgohtqbbiiqfltfcveacysvjq+gq7dnx/5lwkcv6ryl9vs7syliw/qbgugqoqqcgazoagzuwsn4bi/gg/fkvbpvs9ackc3sgl8w3r8aaqxq==</latexit> <latexit sha_base64="fffohepr7kf7tbzyohxelomq=">aaacc3icbvdlssnafj3uv6vqes3q4tqquoibdirrdwce+oihmpmyyetmdmraujejb/ixouibvbd/6nzaivg9cojxzl/fe48wmsmvznzuyxfpesw/wlhb39jcmrd3wjjkbcznhlfiddwkcaocnbvvjhriqvdomdlhuctv3lhkqrvkjmdgh6nmauiyullyzwl9okx8zdlpbgspy3qe8hvxyo4tb8usbfmgl+jxzgsibdwzu/en6ek5bwhrmssmtbsxjsjbtfjiwlvussgoeh6pouphyfrdrp9jcx3nekd4ni6oikttwfeykkpryfnu4mkrriew8i/udexwcocnlcaiix7nfqckgiuakgohtqbbiiqfltfcveacysvjq+gq7dnx/5lwkcv6ryl9vs7syliw/qbgugqoqqcgazoagzuwsn4bi/gg/fkvbpvs9ackc3sgl8w3r8aaqxq==</latexit> <latexit sha_base64="fffohepr7kf7tbzyohxelomq=">aaacc3icbvdlssnafj3uv6vqes3q4tqquoibdirrdwce+oihmpmyyetmdmraujejb/ixouibvbd/6nzaivg9cojxzl/fe48wmsmvznzuyxfpesw/wlhb39jcmrd3wjjkbcznhlfiddwkcaocnbvvjhriqvdomdlhuctv3lhkqrvkjmdgh6nmauiyullyzwl9okx8zdlpbgspy3qe8hvxyo4tb8usbfmgl+jxzgsibdwzu/en6ek5bwhrmssmtbsxjsjbtfjiwlvussgoeh6pouphyfrdrp9jcx3nekd4ni6oikttwfeykkpryfnu4mkrriew8i/udexwcocnlcaiix7nfqckgiuakgohtqbbiiqfltfcveacysvjq+gq7dnx/5lwkcv6ryl9vs7syliw/qbgugqoqqcgazoagzuwsn4bi/gg/fkvbpvs9ackc3sgl8w3r8aaqxq==</latexit> Small-x Evolution at large N c At large N c the evolution is gluon-driven. We will evolve a gluon dipole, remembering that at large N c the relation between the adjoint and fundamental longitudinally-polarized gluon dipoles is G adj (z) =4G (z) (Note that the factor is 4, not like in the unpolarized dipole case.) 9

20 Small-x Evolution at large N c We need to sum the following diagrams (box denotes the polarized Wilson lines ): inhomogeneous term I I k k k k x x x x II x x II k k k k x x eikonal k k other eikonal diagrams x x

21 Large-N c Evolution In the strict DLA limit (S=) and at large N c we get (here G is an auxiliary function we call the neighbour dipole amplitude ) (KPS 5) G(x,z)= G () (x,z)+ s N c The initial conditions are given by the Born-level graphs Z z x s (x,x,z )= () (x,x,z )+ s N c dz z Z z x s Z x z s dz z dx x (x,x,z )+3G(x,z ) min n o x,x z z Z z s dx 3 x 3 (x,x 3,z )+3G(x 3,z ) x x z z x x z z x x z z () (x,x,z)=g () (x,z) x x z z x x z z G () (x,z)= sc F N c h C F ln zs i ln(zsx )

22 Resummation Parameter For helicity evolution the resummation parameter is different from BFKL, BK or JIMWLK, which resum powers of leading logarithms (LLA) s ln(/x) Helicity evolution resummation parameter is double-logarithmic (DLA): s ln x The second logarithm of x arises due to transverse momentum (or transverse coordinate) integration being logarithmic both in the UV and IR. This was known before: Kirschner and Lipatov 83; Kirschner 84; Bartels, Ermolaev, Ryskin 95, 96; Griffiths and Ross 99; Itakura et al 3; Bartels and Lublinsky 3.

23 Quark Helicity at Small x These equations can be solved both numerically and analytically. (KPS 6-7) s = r s N c ln x = r s N c zs ln The small-x asymptotics of quark helicity is (at large N c ) q(x, Q ) x q h with q h = 4 p 3 r s N c.3 r s N c

24 Small-x Evolution at large N c &N f At large N c &N f there is no simple relation between the adjoint and fundamental polarized dipole amplitudes. We need to construct coupled evolution equations mixing them with each other. Here s the adjoint dipole evolution: inhomogeneous term I I II x x II k k k k k k k k x x x x x x III III eikonal k k k k k k other eikonal diagrams x x x x x x 4

25 Small-x Evolution at large N c &N f At large N c &N f there is no simple relation between the adjoint and fundamental polarized dipole amplitudes. We need to construct coupled evolution equations mixing them with each other. Here s the fundamental dipole evolution: inhomogeneous term I I II x x II k k k k k k k k x x x x x x III eikonal k k k k other eikonal diagrams x x x x 5

26 <latexit 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<latexit 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<latexit 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<latexit sha_base64="t+yldpukgpdtafjuvom7a5rwkak=">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</latexit> Small-x Evolution at large N c &N f The resulting equations are Q (zs) =Q () (zs)+ sn c Z z s + s N c 4 G adj () (z) =Gadj (z) + s N c adj, (z )= s N f adj (), (z )+ s N c s N f, (z )= (), (z )+ sn c dz z Z z /s Zx /(z s) Z z dz z max{,/x }/s Z z /s Z z dz z max{,/x }/s Z z s + s N c 4 Z z /s dz z Z z /s dz z dx x x Z z/z /(z s) dz z x Z z/z /(z s) dz z x Z z /z /(z s) ( dx x Zx /(z s) dx x adj, (z )+ Gadj (z )+Q (z ), (z ) Q (z ), dx x ; (z ), min{x Z,x z /z } dx 3 x 3 min{x Z,x z /z } dz z /(z s) x Z z/z /(z s) dx 3 x 3 /(z s) 3;3 (z ), dx 3 x 3 ( Q 3 (z ). h i adj, (z )+3G adj (z ) dx 3 x 3 h i adj,3 (z )+3G adj 3 (z ) adj 3,3 (z )+ Gadj 3 (z )+Q 3 (z ),3 (z ) These are yet to be solved. 6

27 Gluon Helicity at Small x Yu.K., D. Pitonyak, M. Sievert, arxiv: [nucl-th]

28 Dipole Gluon Helicity TMD Now let us repeat the calculation for gluon helicity TMDs. We start with the definition of the gluon dipole helicity TMD: g G (x, k T )= is L xp + Z d d ( ) 3 D i E e ixp + ik P, S L ij T hf tr +i () U [+] [, ] F +j ( ) U [ ] [, ] P, S L + = U^[+] t Here U [+] and U [-] are future and past Wilson line staples (hence the name `dipole TMD, F. Dominguez et al looks like a dipole scattering on a proton): j z i U^[ ] proton

29 Dipole Gluon Helicity TMD At small x, the definition of dipole gluon helicity TMD can be massaged into g Gdip (x, k T )= 8iN c S L g ( ) 3 Here we obtain a new operator, which is a transverse vector (written here in A - = gauge): G i (z) 4N c Note that Z dx k i? ij T D tr can be thought of as a transverse curl acting on Z d x e ik x k i? ij T applez d b G j Q (zs = x ) h i E V [, ]V [,x ]( ig) Ãi (x,x) V [x, ] +c.c. (z) G i (z) i t and not just on à i (x,x) -- different z from the polarized dipole amplitude! proton

30 Dipole TMD vs dipole amplitude Note that the operator for the dipole gluon helicity TMD G i (z) 4N c Z dx D h i E tr V [, ]V [,x ]( ig) Ãi (x,x) V [x, ] +c.c. (z) is different from the polarized dipole amplitude G (z) 4N c Z dx D h i E tr V [, ]V [,x ]( ig) r Ã(x,x) V [x, ] +c.c. (z) We conclude that the dipole gluon helicity TMD does not depend on the polarized dipole amplitude! (Hence the dipole name may not even be valid for such TMDs.) This is different from the unpolarized gluon TMD case.

31 Evolution Equation To construct evolution equation for the operator G " governing the gluon helicity TMD we resum similar (to the quark case) diagrams: k k c.c. i i i x x other LLA diagrams i

32 Large-N c Evolution: Equations This results in the following evolution equations: G i (zs) =G i () (zs)+ sn c s N c + sn c Z z s Z z x s dz z Z dz z Z z s dz z Z d x ln x Zx z s dx x h G i (z s) d x ln x ij T (x ) j? x ij T (x ) j? x h i gen, (z s)+g (z s) h i gen, (z s)+ gen, (z s) i i, (z s) i (z s)=g i () (z s)+ sn c s N c + sn c Z z s Z z x s dz z Z dz z Z z s dz z Z d x 3 ln x 3 apple min x,x Z z s z z d x 3 ln x 3 ij T (x 3) j? x 3 dx 3 x 3 h G i 3(z s) ij T (x 3) j? x 3 h i gen 3, 3 (z s)+g 3 (z s) h i gen 3, 3 (z s)+ gen 3, 3 (z s) i i, 3(z s).

33 Large-N c Evolution: Equations Here gen, (z s)= (x x ),(z s)+ (x x ) G (z s) is an object which we know from the quark helicity evolution, as the latter gives us G and G. Note that our evolution equations mix the gluon (G i ) and quark (G) small-x helicity evolution operators: G i (zs) =G i () (zs)+ sn c s N c + sn c Z z s Z z x s dz z Z dz z Z z s dz z Z d x ln x Zx z s dx x h G i (z s) d x ln x ij T (x ) j? x ij T (x ) j? x h i gen, (z s)+g (z s) h i gen, (z s)+ gen, (z s) i i, (z s)

34 Initial Conditions Initial conditions for this evolution are given by the lowest order t-channel gluon exchanges: i i b b Z d b G i () (zs) = Z d b i (), (zs) = sc F N c ij x j ln x Note that these initial conditions have no ln s, unlike the initial conditions for the quark evolution: Z d b G () (zs) = Z d b (), (zs) = sc F N c ln(zsx )

35 Large-N c Evolution Equations: Solution These equations can be solved in the asymptotic high-energy region yielding the small-x gluon helicity intercept h G = 3 r r 4 p s N c 3.88 s N c We obtain the small-x asymptotics of the gluon helicity distributions: G(x, Q ) g Gdip L (x, k T ) 3 4 p 3 x p s Nc 35

36 Quark Transversity at Small x Yu.K., M. Sievert, in preparation (and hence preliminary)

37 Observables We want to calculate quark transversity TMD at small x: Leading Twist TMDs Nucleon Spin Quark Spin Quark Polarization Un-Polarized (U) Longitudinally Polarized (L) Transversely Polarized (T) Nucleon Polarization U L T ƒ = ƒ T = g L = g T = Helicity h = h L = h = Boer-Mulders Transversity Sivers h T = 37

38 <latexit sha_base64="5mtxqtk9kmtcblx77qyrrq+erg=">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</latexit> <latexit sha_base64="5mtxqtk9kmtcblx77qyrrq+erg=">aaada3icbvlljtmwfhxcayivdqwqmysqpfytvvjaseeawyynidamzhqjnict7xqpmzuctpjdb8chswimswndh3+ckesozxcnj8bn33bsfo8ofv9rzfjvupctxrl7bu966cfpw7tvt/bthkiskzroaiuyereqxwvmvwldpjlrpjisono9brkh39guveshetnzmyjwar8zinrlgr3nfvlpjjmjif7noaq3acjhrqbzyxhjpvua5gpxlrx4ahjqandozv8rpuaucl43ml3hhpec57uosxtdxewalmtyhk8tt8wcwyhfwaazxqklrqkxqgg+qdehvbdpr+gecisjwr3vv6daue6culwkzkqteqs5qrh+ifsrispkuzltb9cmflqnqfbtqfrihz39h9x3ash4fghnleovybvjdb64clwg9bbtryg7v84zmirsfrtqzsa+l6uz4zizalgzqsxiuwersictsmscluznrnwmijy8qwz6r9rfsu6swjffqkssrjanc9v5p9ylpx8wmt/ncs5rub8latqd6kjazcwjwmwsifry+69al8q6quavkt/pnbvgiorkpfg/rvnnyoxjv7keh6chqih89rwfodtpeesdj85n56vzzf3kfng/uz+pa7tao6hf8l9+qfcmozr</latexit> <latexit sha_base64="5mtxqtk9kmtcblx77qyrrq+erg=">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</latexit> <latexit sha_base64="5mtxqtk9kmtcblx77qyrrq+erg=">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</latexit> Quark Transversity Operator Analysis of quark transversity is similar to quark helicity: we start with the operator definition (for proton spin in the x-direction) h q T (x, k T )+ k x M h? q T (x, k T )= ( ) 3 Z d rdr e ik r hp, S x =+ () U[,r] 5 + Diagram analysis again shows that the B-type diagrams dominate: (r) p, S x =+i r + = ζ k w k ξ 38

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sha_base64="xdnte5kgq9rsx/exvqhdhrmu7i=">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</latexit> <latexit sha_base64="xdnte5kgq9rsx/exvqhdhrmu7i=">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</latexit> Quark Transversity Operator ζ Calculating the B-graph contribution we get h q T (x, k T )+ k x M h? q T (x, k T )= x 8N c ( ) 4 Z d x d x Z /s dz z k x k x + k k m it (zs) k w k Z d k ( ) ei(k +k) x ξ k k apple k + k We have defined a transversely polarized dipole operator T (zs) (zs) N c i D h tr V V pol,t Transverse-spin dependent interaction is suppressed by two (!) powers of energy, hence we rescaled T by s. i tr h ie V pol,t V S x =+ 39