天球上的交叉對稱

Crossing Symmetry on the Celestial Sphere

黃子勲(Zi-Xun Huang)

指導教授 ： 黃宇廷

國立臺灣大學/理學院/物理學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202202248

摘要

在本篇論文，我們研究了各種物理條件對共形基底下四點無質量散射振幅（下稱為天球振幅）的意涵。我們先回顧了$D$維勞倫茲群與$D-2$維歐式共形群的同構，接著建立了$D=1+3$維無質量共形主族波函數，這牽涉了對於平面波進行梅林變換。這就給了$S$矩陣藉由梅林變換從平面波基底轉到共形基底，而四點天球振幅此時為一個共形維度總和$\beta=\sum_{i=1}^4\Delta_i-4$、二維自旋$\ell_i$，以及共形交叉比$z$的函數。而不同的動力學設定對應到$z$的不同定義域。然而，梅林變換疊加了$S$矩陣在動量基底下的所有能量尺度，使得交叉對稱變得不顯然。因此本篇論文的目標之一就是研究天球振幅上的交叉對稱，這涉及到解析延拓$z$至非物理的定義域。 除了交叉對稱，我們首先研究了體的局域性在天球上的意涵並且發現了，作為正則性、局域性，以及$S$矩陣之龐加萊不變性的結果，天球振幅的虛數部分可以被正的展在龐加萊偏波上。至於交叉對稱，我們找到了一條對於$z$的解析延拓路徑並且推導了天球上的交叉對稱。基於以上交叉關係，我們推導了天球分散關係，這聯繫了有效場論中的耦合強度到天球振幅的虛數部分，並通過第一與第二型超弦振幅數值驗證了天球分散關係。

關鍵字

散射振幅 ； 天球振幅 ； 全像對偶 ； 共形場論 ； 交叉對稱 ； 局域性 ； 正則性 ； 龐加萊對稱 ； 分散關係 ； 解析延拓

並列摘要

In this thesis, we study the implications of various physical constraints on the 4-pt massless scattering amplitude in the conformal basis (termed the $celestial$ $amplitude$). We start from reviewing the basics of group isomorphism between $D$ dimensional Lorentz group and the $D-2$ dimensional Euclidean conformal group, and we construct the massless conformal primary wavefuntions in $D=1+3$, which involves a Mellin transform of the plane-waves. This gives the transformation of the $\mathcal{S}$-matrix in plane-wave basis to the conformal basis via the Mellin transform, and the 4-pt celestial amplitude is a function of sum of conformal dimensions $\beta=\sum_{i=1}^4\Delta_i-4$, 2D spin $\ell_i$ and conformal cross-ratio $z$. And different kinematic setups correspond to different support region of $z$. However, the Mellin transform superposes all the energy scales of the $\mathcal{S}$-matrix in momentum space, which obscures the crossing symmetry. Therefore one of the main goals in this thesis is to study the crossing symmetry on the celestial sphere, which involves analytic continuation of $z$ into unphysical regions. Besides the crossing symmetry, we first study the implications of bulk locality on the celestial sphere and we find that the imaginary part of the celestial amplitude can be positively expanded on the Poincaré partial waves as a result of unitarity, locality, and Poincaré invariance of the $\mathcal{S}$-matrix. Moving on to the crossing symmetry, we find a path of analytic continuation in $z$ and derive the crossing symmetry on the celestial sphere. Based on the crossing relation, we derive the celestial dispersion relation that relates the EFT couplings to the imaginary part of the celestial amplitude and numerically check via the type-I and type-II string amplitudes.

並列關鍵字

Scattering Amplitude ； Celestial Amplitude ； Holographic Dual ； Conformal Field Theory ； Crossing Symmetry ； Locality ； Unitarity ； Poincaré Symmetry ； Dispersion Relation ； Analytic Continuation

參考文獻

[1] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, 1703.05448.

[2] T. McLoughlin, A. Puhm and A.-M. Raclariu, The SAGEX Review on Scattering Amplitudes, Chapter 11: Soft Theorems and Celestial Amplitudes, 2203.13022.

[3] S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062, [2108.04801].

[4] S. Pasterski, M. Pate and A.-M. Raclariu, Celestial Holography, in 2022 Snowmass Summer Study, 11, 2021. 2111.11392.

[5] A.-M. Raclariu, Lectures on Celestial Holography, 2107.02075.