The Center for Nuclear Research Seminar, 2022

𝓝 = 𝟒 supersymmetric Yang-Mills thermodynamics to order 𝝀^𝟐

Authors: Jens O. Andersen^a, Qianqian Du^b,c, Michael Strickland^c and Ubaid Tantary^c

^aNorwegian University of Science and Technology, Norway

^bCentral China Normal University, China

^cKent State University, Kent Ohio

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Speaker: Ubaid Tantary

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• The SYM_4,4 theory can be obtained by dimensional reduction of SYM_1,D in D = D_max= 10 with all fields being in the adjoint representation of SU(N_c).

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• The action and Lagrangian that generates the perturbative expansion for SYM_4,4 in Minkowski-space can be expressed as

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• The action of N = 1 supersymmetric Yang-Mills in D dimensions (SYM_1,D) can be written in Minkowski space as

N = 4 supersymmetric Yang-Mills theory in 4-dimensions (SYM_4,4)

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Background and Motivation

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Background and Motivation

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The dashed lines indicate a scaler field and dotted lines indicate a ghost field. The crosses are the thermal counter terms.

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• The resumed one-loop free energy

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• The resumed two-loop free energy

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• The resumed three-loop free energy

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• By combining eqns. (6) (9) and (14), the result of the resumed free energy up to 3 loop level for SYM_4,4 under the RDR scheme is

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(15)

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• This result holds for all N_c .
• The result (15) is manifestly finite due to an explicit cancellation between three-loop infrared singularities and the three-loop counter term diagrams.
• These cancellations remove all infrared divergent contributions.
• In addition, there are no remaining poles due to ultraviolet divergences, since the coupling does not run in SYM_4,4 and, hence, no coupling constant renormalization counter term is required.

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PART II

Dimensional Reduction and EFT Technique

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Construction of SUSY-EFT

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Parameters of the effective theory

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A. Coefficient of the unit operator

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A.2 Two loop contribution

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A.3 Three loop contribution

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B. Mass parameters

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One-loop A₀ self-energy graphs in the SYM4;4 dimensionally-reduced EFT. Spiral lines represent three-dimensional gluons, sinusoidal lines represent the adjoint scalar A₀, dashed lines represent scalars, and dotted lines (not appearing in this particular figure) represent the three-dimensional ghost field

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Two-loop contribution:

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The divergence is cancelled by the counterterm:

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• With new perturbative coefficients in hand one can produce an updated Padé approximant. ^{J.P. Blaizot, E.Iancu, U.Kraemmer and A.Rebhan, hep-ph/0611393}

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• Based on the large-N_c structure of the strong-coupling expansion, we find that the following form can reconstruct all known coefficients in both the weak- and strong-coupling limits

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• We computed the thermodynamic function of SYM_4,4 to 𝒪(𝜆²) under RDR Scheme .

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• Having computed new coefficients, we then constructed a large-N_c Padé approximant that interpolates between the weak- and strong-coupling limits.

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• We are also computing the coefficient of λ^5/2 in the SYM_4,4 free energy.

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• We also plan to pursue a three-loop HTLpt calculation of SYM_4,4 thermodynamics to extend our previous two-loop HTLpt calculation.

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