Transverse Momentum Distributions: Matches and Mismatches

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1 Transverse Momentum Distributions: Matches and Mismatches Ahmad Idilbi ECT* M.G. Echevarría, Ahmad Idilbi, Ignazio Scimemi. [arxiv:.947] MGE, AI, Andreas Schäfer, IS. [arxiv: 08.8] MGE, AI, IS. JHEP 07 (0) 00. [arxiv:.4996] QCD Evolution 03 JLab

2 Hadronic ME with TMD: Definition and their evolution! Outline New Definition TMDPDF: Generalizes the one introduced last year. (Echevarria-Idilbi-Scimemi: EIS (ICE in German!)) Role of the Soft Function, Rapidity Space and Rapidity Divergencies Properties: Universality, Anomalous Dimension, TMDPDF->PDF Collins ( JCC ) Definition and Comparison with EIS Q -resummation, Evolution, Phenomenology - See M. Echevarria Talk Conclusions and Outlook

3 Transverse Dependence The next-to-simple hadronic matrix element: F.ThP (x,x? )[x, 0] P i F naive n (0,y, ~y? )= X hp, n W n (0,y, ~y? ) n/ W n n P, i W n (x) = P exp apple ig Z 0 We would also need transverse gauge links to maintain gauge invariance [Ji andyuan, Belisky, Ji and Yuan 0, A.I and I.C, EIS ] ds n A n (x s n) 3

4 i(p/ k/ ) (p k) i! k i, = p i( p/ k/ ) ( p k) i! k i, = p 4

5 i(p/ k/ ) (p k) i! k i, = p i( p/ k/ ) ( p k) i! k i, = p F naive n = ( x) sc F 4 3 L T L T ln ( x) Q apple " UV ln Q 3 " UV ( x)ln( x) P q/q ln µ L T P q/q It is ill-defined!! Mixed UV/Rapidity Divergences!! Transverse Dependence: Non-Cancelation Among Real And Virtual Contributions 4 L T =ln µ b 4e E

6 Remarks In full QCD there are no mixed divergences (UV or Long-Distance div.) When we are restricted to certain Regions (Soft or Collinear) the mixed divergences show up. They cancel only when, after factorization, all contributions appearing are added up The Soft and Collinear Limits: Problems!!

7 DY q T -Spectrum M = H(Q /µ ) Z d 4 ye iq y J n (0,y,~y? ) S(0, 0,~y? ) J n (y, 0,~y? ) P q l q T Q P X l 0 J n (0,y,~y? )= J n (y, 0,~y? )= X hn (P, ) n(0,y,~y? ) n/ X hn ( P, ) n n/ S(0, 0,~y? )=h0 Tr T Sn T S T n n N (P, )i zb subtracted n(y, 0,~y? ) N ( P, )i zb subtracted (0, 0,~y? )T S T n S T n 0i 6 [Ji, Ma and Yuan 003]

) In QCD there are no mixed divergences, so they are cancelled in the combination of collinear,

8 Definition of TMDPDF (st attempt...) The hadronic tensor is: Z M = H(Q /µ ) d 4 ye iq y J n (0,y,~y? ) S(0, 0,~y? ) J n (y, 0,~y? ) In QCD there are no mixed divergences, so they are cancelled in the combination of collinear, anti-collinear and soft. Thus we can try to define the TMDPDF by symmetry as: F trial n (x, ~ k? )= Z dr d ~r? ( ) 3 e i( r xp ~r? ~k? ) J n (0,r,~r? ) p S(0, 0,~r? ) 7

9 Results at One Loop: Soft Function h.c. S (, ) = sc F apple " UV " UV ln µ Q L T L T ln µ Q 6 8 L T =ln µ b 4e E

10 Results at One Loop: Pure Collinear n ( ) = sc F ( x) apple " UV ln " UV µ 3 " UV 4 L T 3 L T L T ln µ ( x)ln( x) P q/q ln µ L T P q/q We subtract the soft contamination (zero-bin) for each diagram!! 9 L T =ln µ b 4e E

11 Results at One Loop: TMDPDF F trial n = sc F apple ( x) " UV " UV ln Q µ 3 " UV F trial n = p n S L T 3 L T L T ln Q µ ( x) L T P q/q P q/q ln µ 4 ( x) ( x)[ ln( x)] [EIS, JHEP 0] 0

12 Results at One Loop: TMDPDF F trial n = sc F apple ( x) " UV " UV ln Q µ 3 " UV F trial n = p n S L T 3 L T L T ln Q µ ( x) L T P q/q P q/q ln µ 4 ( x) ( x)[ ln( x)] F trial n Breakdown: The Two Sectors Are Treated Independently!! = sc F apple ( x n ) " UV " UV ln Q µ 3 " UV L T 3 L T L T ln Q µ ( x n) L T P q/q P q/q ln µ 4 ( x n) ( x n )[ ln( x n )] [EIS, JHEP 0] There are mixed UV/nUV divergences!! The definition I provided has a catch... 0

Only can be distinguished by their relative rapidities: Modes mix under boosts, so we

13 Factorization of Modes (/) p =(,,?) k n Q(,, )! y 0 k n Q(,, )! y 0 k s Q(,, )! y 0 q T Q k n k n k s q T Soft and Collinear modes have the same invariant mass. Only can be distinguished by their relative rapidities: Modes mix under boosts, so we need rapidity cuts. y = ln k k Rapidity divergence when k goes to 0 We need a lower rapidity cutoff

14 Factorization of Modes (/) We need to impose rapidity cutoffs to separate the modes: H(Q ) n ( n ) S( n, n ) n ( n ) A is collinear B is soft C is anti-collinear y = ln k k Sensible TMDPDF: We will identify positive & negative rapidity quanta with each TMDPDF!! The Complete Soft function has (or might not!) to be split.

15 Splitting of the Soft Function (/3) The soft function can be split in two pieces. On one hand: M = H(Q /µ ) n ( ) n ( ) S(, ) ln M =lnh ln n ln n ln S Pure-collinear ln ln n = R n x n, s,l?, ln µ n = R n x n, s,l?, ln ln S = R s s,l?, ln 3 µ Q µ

16 Splitting of the Soft Function (/3) The soft function can be split in two pieces. On one hand: M = H(Q /µ ) n ( ) n ( ) S(, ) Pure-collinear ln M =lnh ln n ln n ln ln 3 ln S n = R n x n, s,l?, ln µ n = R n x n, s,l?, ln ln S = R s s,l?, ln µ Q µ Each collinear contribution depends on the relevant regulator

17 Splitting of the Soft Function (/3) The soft function can be split in two pieces. On one hand: M = H(Q /µ ) n ( ) n ( ) S(, ) Pure-collinear ln M =lnh ln n ln n ln ln 3 ln S n = R n x n, s,l?, ln µ n = R n x n, s,l?, ln ln S = R s s,l?, ln µ Q µ Each collinear contribution depends on the relevant regulator The soft function connects the two collinear sectors

18 Splitting of the Soft Function (/3) On the other: After OPE we can match onto the Integrated PDFs H C f M = H(Q /µ ) C n (x n ; L T,Q /µ ) C n (x n ; L T,Q /µ ) f n (x n ; /µ ) f n (x n ; /µ ) ln M =lnh ln C n ln C n lnf n lnf n f: The Integrated PDFs lnf n = R f (x n, s )R f (x n, s )ln µ lnf n = R f (x n, s )R f (x n, s )ln µ [Korchemsky-Radyushkin 87] The IR collinear divergence in the PDFs is encoded in a single log (or single IR pole in dimensional regularization) 4

19 Splitting of the Soft Function (3/3) M = H(Q /µ ) n ( ) n ( ) S(, ) M = H(Q /µ ) C n (x n ; L T,Q /µ ) C n (x n ; L T,Q /µ ) f n (x n ; /µ ) f n (x n ; /µ ) Then we have to all orders in perturbation theory: ln n = R n (x n, s,l? )R n (x n, s,l? )ln µ ln n = R n (x n, s,l? )R n (x n, s,l? )ln ln S = R s ( s,l? )R s ( s,l? )ln Q µ µ Q = p p (Log) Soft Function: Single Logarithm To all Orders in PT. Given Boost Invariance and Symmetry Arguments (between the two collinear sectors) S p, p = s S p, p s S p, p 5

20 Splitting of the Soft Function (3/3) M = H(Q /µ ) n ( ) n ( ) S(, ) M = H(Q /µ ) C n (x n ; L T,Q /µ ) C n (x n ; L T,Q /µ ) f n (x n ; /µ ) f n (x n ; /µ ) Then we have to all orders in perturbation theory: ln n = R n (x n, s,l? )R n (x n, s,l? )ln µ ln n = R n (x n, s,l? )R n (x n, s,l? )ln ln S = R s ( s,l? )R s ( s,l? )ln Q µ µ Q = p p (Log) Soft Function: Single Logarithm To all Orders in PT. Given Boost Invariance and Symmetry Arguments (between the two collinear sectors) n Sector S p, p = s S p, p s S p, p nbar Sector 5

21 Definition of TMDPDF Collecting all the positive & negative rapidity quanta we (re)define the TMDPDFs as: F n (x n,b; Q, µ) = n ( ) s S p, p F n (x n,b; Q, µ) = n ( ) s S p, p 6

22 Definition of TMDPDF Collecting all the positive & negative rapidity quanta we (re)define the TMDPDFs as: F n (x n,b; Q, µ) = n ( ) s S p, p F n (x n,b; Q, µ) = n ( ) s S p, p M = H(Q /µ ) F n (x n,b; Q,µ ) F n (x n,b; Q,µ ) No soft function in the factorization theorem!! 6

23 Results at One Loop The pure collinear is the same as before. apple n ( )= sc F ( x) " UV ln " UV µ 3 " UV 4 L T 3 L T L T ln µ ( x)ln( x) P q/q ln µ L T P q/q The soft function is split in two pieces: S p, = apple sc F p " ln UV " UV µ Q L T L T ln = apple S p, p S p, p µ Q 6 S p, p = sc F apple " UV ln " UV µ Q L T L T ln µ Q 6 7

without any mixed UV/nUV divergences (anomalous dimension).

24 F n (x n,b; Q, µ) = n ( ) s TMDPDF S Positive rapidity modes p, p C f F n = sc F ( x) apple " UV " UV ln Q µ 3 " UV L T 3 L T L T ln Q µ ( x) L T P q/q P q/q ln µ 4 ( x) ( x)[ ln( x)] First line: UV contribution without any mixed UV/nUV divergences (anomalous dimension). Second line: Matching coefficient of the TMDPDF onto integrated PDF. Does not depend on any nuv regulator!! Third line: integrated PDF [EIS, arxiv.947] 8

25 Evolution: Anomalous Dimension Applying the RGE to the hadronic tensor we get the AD of the TMDPDF: F n (x, b; Q,µ f )= F n (x, b; Q,µ i ) exp Z µf µ i dµ 0 µ 0 n s (µ), ln Q µ 9

Evolution: Anomalous Dimension Applying the RGE to the hadronic tensor we get the AD of the TMDPDF: F n (x, b; Q,µ f )= F n (x, b; Q,µ i ) exp Z µf µ i dµ 0 µ 0 n s (µ), ln Q µ A is the cusp AD A and

26 Evolution: Anomalous Dimension Applying the RGE to the hadronic tensor we get the AD of the TMDPDF: F n (x, b; Q,µ f )= F n (x, b; Q,µ i ) exp Z µf µ i dµ 0 µ 0 n s (µ), ln Q µ A is the cusp AD A and B are known up to three loops!! We have ``by free the AD of the TMDPDF at 3-loops!! [Moch,Vermaseren and Vogt 04-05] 9 n = H = sc F [Idilbi, Ji, Ma and Yuan 05] 3ln µ Q See Miguel s talk: phenomenology

27 OPE of TMDPDF onto PDFs The TMDPDF has perturbative content when q T is perturbative. We can do an OPE of the TMDPDF onto the PDFs in impact parameter space, integrating out the intermediate scale q T. F f/p (x, b; Q,µ )= X j=q,g Z x dx 0 x x 0 Cf/j x 0,b; Q,µ f j/p (x 0 ; µ ) O (( QCD b) a ) 0

28 OPE of TMDPDF onto PDFs The TMDPDF has perturbative content when q T is perturbative. We can do an OPE of the TMDPDF onto the PDFs in impact parameter space, integrating out the intermediate scale q T. F f/p (x, b; Q,µ )= X j=q,g Z x dx 0 x 0 Cf/j x x 0,b; Q,µ f j/p (x 0 ; µ ) O (( QCD b) a ) L T =ln µ b 4e E It is supposed to live at q_t, but it has a subtle Q -dependence Logs cannot be combined in one log like at threshold: Resummation is needed!! (see Echvarria`s talk!) 0

29 Properties J v,dis n = J v,dy n sc F ( x) () ( ~ k nt ) J r,dis n = J r,dy n s C F ( x) () ( ~ k nt ) So the TMDPDF is UNIVERSAL!! (UN-polarized TMDPDF) Also that when integrated over the transverse momentum we recover the PDF (prior to renormalization) Z d ~ knt F n (x, ~ k nt ; Q,µ )=f n (x; µ)

30 Collins (JCC) Definition of TMDPDF F JCC(naive) n (x n,b; p n,µ)= lim y n! n (y n ) S(y c,y n ) n (y n )=A B n B n n =( e y n,,~0? ) n c =(, e y c,~0? ) S(y c,y n )=B n F n = A B n If we take into account the Wilson lines self-energy for finite rapidity y c, then: F n JCC (x n,b; p s S(y n,y c ) n,µ)= lim n (y n ) y n! S(y y n! c,y n ) S(y n,y n ) n =(p ) e y c

31 JCC and EIS Definitions By symmetry in the center of mass frame, and since the TMDPDF is boost-invariant: F n (x n,b; Q, µ) = n ( ) s S p, p F JCC n (x n,b; p n = Q, µ) = lim y n! y n! n (y n ) s S(y n,y c =0) S(y c =0,y n ) S(y n,y n ) In terms of rapidity analysis (rather than soft/collinear) both definitions (JCC and EIS) agree. Implementation in PT is rather different. JCC is bound to its construction (mainly to resum logs using Collins-Soper equation). Ours is ``more liberal while also the relevant logs are resummed (via differential equation w.r.t to rapidity divergences). 3

32 Conclusions & Outlook We have defined a renormalizable and regularization-scheme free TMDPDF. Properties: free from mixed UV/nUV divergences, universal, gauge invariant, boost invariant. We know its AD at 3-loops based on the factorization theorem. Together with Q^- resummation to perform phenomenology up to the highest possible accuracy! (NNLL) (polarized and un-polarized TMDs) The definition of quark-tmdpdfs can be extended to gluon TMDPDFs and quark/gluon TMDFFs (SIDIS). Soft and Collinear limits of QCD are ill-defined!! Different limits might exist(?) Important Consequences for the theory of TMD factorization (Possibly we talk about it Next Year). 4

33 F.ThP (x,x? )[x, 0] P i 5

34 We already showed that: EIS Definition Revisited ln S = R s ( s,l? )R s ( s,l? )ln Q µ ln S p, p = ln S p, p ln S p, p It can be generalized: ln S p, p = ln S p, p ln S p, p nu transforms like (p ) under boosts

35 Splitting of the Soft Revisited ln S p, p = ln S p, p ln S p, p nu transforms like (p ) under boosts y n =ln µp y n =ln µ p y 0 n =ln µp y 0 n =ln µ p lim y n! y n! S(y n,y n )= lim y n! y n! y 0 n! y 0 n! q S(y n,y 0 n) q S(y 0 n,y n ) y c = lim y n! y 0 n! (y n yn)= 0 lim 0 n! y y n! (y0 n y n )= ln 7

36 Prelude: Integrated PDF (/) f n (0,y,~0? )= X hp, n W n (0,y,~0? ) n/ W n n P, i W n (x) = P exp apple ig Z 0 ds n A n (x s n) 8

37 Prelude: Integrated PDF (/) f n (0,y,~0? )= X hp, n W n (0,y,~0? ) n/ W n n P, i W n (x) = P exp apple ig Z 0 ds n A n (x s n) Regulator: i(p/ k/ ) (p k) i! k i, = p i( p/ k/ ) ( p k) i! k i, = p 8

38 Integrated PDF (/) f n = ( x) sc F P q/q " UV ln µ 4 ( x) ( x)[ ln( x)] The UV pole is cancelled by renormalization The IR pole (logarithm) is washed out by confinement 9

TMD Fragmentation Function at NNLO

TMD Fragmentation Function at NNLO Institut für Theoretische Physik, Universität Regensburg, D-9040 Regensburg, Germany E-mail: vladimirov.aleksey@gmail.com The calculation of the unpolarized non-singlet