Gluon scattering in N = 4 super Yang-Mills at finite temperature

1. Gluon scattering in N = 4 super Yang-Mills at finite temperature Koh Iwasaki Tokyo Institute of Technology Based on the work in collaboration with K.Ito and H.Nastase (Titech) ArXiv:0711.3532 [hep-th](To be published in Prog. Theor. Phys)

2. AdS/CFTCorrespondence Maldacena,Adv.Theor.Math.Phys.2,231(1998) r r DUALITY CONJECTURE In this talk this conjecture is tested using gluon scattering amplitude and gravity-side computations are extended to finite temperature

3. Contents • Introduction • AdS/CFT duality and gluon scattering amplitude • Gluon 4-point function at finite temperature • Conclusion and Outlook

4. Gluon Scattering Amplitude

5. Gluon scattering amplitude Gluon 4-point function In 4D N=4 Super Yang-Mills Disk amplitude In ⅡB sring theory Recently Alday and Maldacena show that both calculation match very nicely

6. Field Theory Side Calculation–BDS conjecture General form of L-loop, n-point gluon amplitude in 4D Maximally Supersymmetric Yang-Mills Perturbative computation color factor momentum ,helicity factor Up to three loop the amplitude is completely described in terms of one-loop amplitude CONJECTURE Bern,Dixon and Smirnov,Phys.Rev.D72:085001(2005)

7. Especially when we consider 4-point function the conjecture becomes Extrapolating to Strong Coupling Case Here and are related to cusp anomalous dimension and collinear cusp anomalous dimension and their forms in strong coupling region is already known

8. String Theory Side Calculation Alday and Maldacena,JHEP 0706 (2007) 064 Set up is the same as ordinary AdS/CFT set up Only the disk amplitude contributes Keeping field theory (boundary) momentum fixed and setting D3 brane near the throat of AdS5 space If we set D3-brane at the point We can set string proper momentum very large boundary of AdS5 In this high momentum setup. The amplitude is dominated by Saddle point of the action and the computation becomes easier Gross and Mende,Phys.Rev.Lett.D197,129(1987)

9. T-dual coordinates Computing minimum value of the action in curved space is difficult ? Neumann boundary We take a “T-dual” in the brane-direction and fix the boundary as light-like segments Dirichlet boundary

10. Gluon Scattering Amplitude / Wilson Loop Duality GLUON SCATTERING AMPLITUDE WILSON LOOP boundary In this T-dual coordinates the surface is Wilson loop. This is equal to minimal surface with boundary on the AdS boundary Maldacena,Phys.Rev.Lett.80,4859(1998) Finding classical solution in original coordinatesis equivalent tofinding minimal surface in T-dual coordinates

11. The result Using dimensionally regularized metric Itzhaki, Maldacena, Sonnenschein and Yankielowicz,Phys.Rev.D58,046004(1998) We can nicely regularize the IR divergence of the solution and obtain

12. Gluon Amplitude at Finite Temperature

13. AdS/CFT correspondence at finite temperature Witten,Adv.Theor.Math.Phys,2,505(1998) DUALITY Temperature: T-DUAL COORDINATE ORIGINAL COORDINATE DIFFERENT FORM Different interpretation is needed for string world sheet in each metric

14. Ansatz for the solution As in the case of T=0 its is difficult to find the classical solution, but in a specific case the solution is already known Liu, Rajagopal and Wiedemann,Phys.Rev.Lett.97,182301(2006) r We try to find similar solution r0 boundary With this ansatz the translational invariance along the long side of parallelogram This setup brings symmetry to solutions and makes computation easier

15. 1. Original coordinates String world sheet ending on the boundary of AdS = Usual Wilson loop Infinite boost Good! Liu et al. solution 2. T-dual coordinates String world sheet ending on the horizon (small r end) of AdS This configuration has a good interpretation As agluon scattering amplitude in the Alday-Maldacena’s context In this case the value of classical action is

16. Regularization and a result 1.Dimensional regularization in the same sense as T=0 case 2. Cut-off regularization ～ Using these regularization we finally obtain the gluon 4-point function a4 at finite temperature T

17. Conclusion We calculated finite temperature version of gluon scattering amplitude in string theory side originally proposed by Alday and Maldacena. The result is different from T=0 case , but ・We took translational invariant solution. In this solution main-divergent region (cusps) are gone away. Then it is natural not apperring divergentterms. ・We also took high temperature limit. In general finite temperature systems have phase transition then it is nontrivial that we can approach T=0 →　T=∞　continuously ・the configuration of the solution in our work is different from that in Alday and Maldacena’s work These facts indicate a possibility that our result is a new proposal and the test of the Alday and Maldacena’s Work in more general case, but to be more clear ・construction of solutions of the same configuration in AM or of more general configuration (including intermediate temperature solution) is needed

18. Outlook ・computing classical action in more general set up (over six-point amplitudes , Alday-Maldacena like solution at finite temperature, . . . ) Numerical calculation In a specific configuration We have succeeded in reproducing IR divergent property of BDS formula S.Dobashi, K.Ito and K.I, arXiv:0805.3594