April 20-24, 2006 DIS Soft Gluon Ressumation in Effective Field Theory Feng Yuan, RBRC, Brookhaven National Laboratory References: Ildibi, Ji, Yuan, PLB 625, 253 (2005); Ildibi, Ji, Ma, Yuan, PRD73, (2006).

April 20-24, DIS 2006 Two Large Scales Generate Large Double Logs in pQCD For example, a differential cross section depends on Q 1, where Q 2 À Q 1 2 À  QCD 2 For example, a differential cross section depends on Q 1, where Q 2 À Q 1 2 À  QCD 2 We have to resum these large logs to make reliable predictions We have to resum these large logs to make reliable predictions Q ? : Dokshitzer, Diakonov, Troian, 78; Parisi Petronzio, 79; Collins, Soper, Sterman, 85 Q ? : Dokshitzer, Diakonov, Troian, 78; Parisi Petronzio, 79; Collins, Soper, Sterman, 85 Threshold: Sterman 87; Catani and Trentadue 89 Threshold: Sterman 87; Catani and Trentadue 89

April 20-24, DIS 2006 Soft-Collinear Effective Theory Effective Theory approach Effective Theory approach Choose low-energy degree of freedom Choose low-energy degree of freedom Soft, Collinear fields (  n, A n, A s, …) Soft, Collinear fields (  n, A n, A s, …) Write down effective Lagrangian through gauge symmetry and power counting Write down effective Lagrangian through gauge symmetry and power counting Applications: Applications: Derive factorization theorems consistent with low energy degree of freedom Derive factorization theorems consistent with low energy degree of freedom Derive twist expansion for complicated QCD processes (B   decay, B  D decay, jet structure) Derive twist expansion for complicated QCD processes (B   decay, B  D decay, jet structure) Resumming large logarithms Resumming large logarithms Bauer, Fleming, Pirjol, Stewart

April 20-24, DIS 2006 Two Steps Matching At scale Q, match the quark (gluon) current between full QCD and SCET At scale Q, match the quark (gluon) current between full QCD and SCET Derive the matching coefficient and the anomalous dimension, which controls the running Derive the matching coefficient and the anomalous dimension, which controls the running At lower scale in SCET, match to the (product of) parton distribution, like a usual pQCD factorization for the cross section At lower scale in SCET, match to the (product of) parton distribution, like a usual pQCD factorization for the cross section

April 20-24, DIS 2006 Resummation in SCET DIS Structure function at large x, Manohar, 03 DIS Structure function at large x, Manohar, 03 Two scales, Q 2, (1-x) Q 2 Two scales, Q 2, (1-x) Q 2 Extended to Drell-Yan, Idilbi, Ji, 05 Extended to Drell-Yan, Idilbi, Ji, 05 Applications to Q ? resummation Applications to Q ? resummation Gao, Li, Liu, NLL, 05 Gao, Li, Liu, NLL, 05 Idilbi, Ji, Yuan, NLL, 05 Idilbi, Ji, Yuan, NLL, 05 All order equivalence for threshold resummation All order equivalence for threshold resummation Idilbi, Ji, Ma, Yuan, 05 Idilbi, Ji, Ma, Yuan, 05

April 20-24, DIS 2006 Matching at Q At scale Q, one can integrate out the fluctuations of order Q. Since all other scales are small, we can set them to zero, and the processes are very similar to elastic form factors. Therefore, the integration can be done by matching the full theory current to effective theory current, and only virtual diagrams contribute At scale Q, one can integrate out the fluctuations of order Q. Since all other scales are small, we can set them to zero, and the processes are very similar to elastic form factors. Therefore, the integration can be done by matching the full theory current to effective theory current, and only virtual diagrams contribute

April 20-24, DIS 2006 Form Factors High order corrections to the current in the full theory is represented as on-shell quark (gluon) form factors, High order corrections to the current in the full theory is represented as on-shell quark (gluon) form factors, F(Q 2,  )=1+  s F (1) +  s 2 F (2) +  Similarly, we can also calculate the form factors in the effective theory, which has the same IR structure as the full theory form factor. So their matching can be expressed as Similarly, we can also calculate the form factors in the effective theory, which has the same IR structure as the full theory form factor. So their matching can be expressed as F(Q 2,  )=C(Q 2 /  2 )F eff (Q 2 /  2,  )

April 20-24, DIS 2006 Matching coefficients at  =M H The matching C(M H ) can be expanded in terms of  s (M H ) The matching C(M H ) can be expanded in terms of  s (M H )

April 20-24, DIS 2006 Anomalous dimension The anomalous dimension controls the running of C(  ) The anomalous dimension controls the running of C(  ) Using the current known quark (gluon) form factors up to three-loop (MVV 05) Using the current known quark (gluon) form factors up to three-loop (MVV 05) A (i), the cusp anomalous dimension A (i), the cusp anomalous dimension B (i), the  (1-x) coefficient in the splitting function B (i), the  (1-x) coefficient in the splitting function f (i), a universal structure, similar to A (i) f (i), a universal structure, similar to A (i)

April 20-24, DIS 2006 Matching at  L Can be calculated from the cross section, Can be calculated from the cross section,  eff (  ) L )=M N (  ) L ) ­ f 1 (  L ) ­ f 2 (  L ) A detailed formulation in EFT is not necessary, rather we can use the result from the full QCD calculations in the soft-collinear limit, A detailed formulation in EFT is not necessary, rather we can use the result from the full QCD calculations in the soft-collinear limit, M N is universal, C A ->C F will give the quark one M N is universal, C A ->C F will give the quark one

April 20-24, DIS 2006 Final Result for the Threshold Resummation The cross section in the moment space, The cross section in the moment space, C(M H ) and M N (  L ) only depend on  s, the large logs are contained in the exponential factor C(M H ) and M N (  L ) only depend on  s, the large logs are contained in the exponential factor

April 20-24, DIS 2006 Exponential Form Factor  1 controls running from M H to  L =M H /N,  2 controls  L to  F  1 controls running from M H to  L =M H /N,  2 controls  L to  F A 1, A 2, B 1, B 2 are known and calculated up to 3- loop, and introduce the third integral, A 1, A 2, B 1, B 2 are known and calculated up to 3- loop, and introduce the third integral,

April 20-24, DIS 2006 Final result Final result N-dependent terms are entirely in I 1, I 2, I 3 N-dependent terms are entirely in I 1, I 2, I 3

April 20-24, DIS 2006 All Orders Equivalence In conventional resummation formalism In conventional resummation formalism Sterman 87, Catani and Trentadue 89 Sterman 87, Catani and Trentadue 89 All order relation between these two All order relation between these two I 1 +I 2 +I 3 =I M +ln C G with

April 20-24, DIS 2006 At three-loop We have calculated D (2) and D (3) from SCET formalism, agree with the results from full theory expansions (Vogt, et al., 05) We have calculated D (2) and D (3) from SCET formalism, agree with the results from full theory expansions (Vogt, et al., 05)

April 20-24, DIS 2006 Conclusion As a perfect tool, SCET has shown great ability to do resummation for the Q ? and threshold cases As a perfect tool, SCET has shown great ability to do resummation for the Q ? and threshold cases SCET provide an intuitive way to understand the resummation, and it is much simpler to perform the calculations SCET provide an intuitive way to understand the resummation, and it is much simpler to perform the calculations Further application of SCET to higher order corrections and other processes are desirable Further application of SCET to higher order corrections and other processes are desirable