Lauricella弦散射振幅及其在各式極限下的行為

Abstract

在這一篇論文中，我們計算了三個迅子跟一個任意弦態跟四點玻色開弦振幅，並且將這一振幅表示成D-型Lauricella函數。D-型Lauricella函數可以用來研究在各種極限下的弦散射振幅。尤其是在Regge極限跟非相對論性極限底下，D-型Lauricella函數弦散射振幅可以被約化成Appell函數跟高斯超幾何函數。再者，在Regge極限底下存在著無限多個可以將無限多的獨立散射振幅約化成一個散射振幅的遞迴關係式。

另一方面，在高能固定角極限底下，我們可以從Lauricella弦散射振幅重現當年Gross在1988年提出的線性關係。而這些聯繫著不同自旋同質量的線性關係在不久之後在被修正且嚴格證明。此外，這些無限多的線性關係可以被用來約化獨立的高能固定角散射振幅的數量，從無限多到一。

我們也詳細證明了四點都是任意弦態的四點散射振幅在不同通道之間的弦BCJ關係。我們詳盡的證明方式比利用了單質性來證明弦BCJ關係的方法還要來的更具一般性。更重要的是，可以利用非相對論極限(取代文獻中的零斜率極限)得到質量階級相關的弦BCJ關係。

在高能固定角/Regge散射極限中，我們可以把線性/遞迴關係跟高能固定角/Regge關係結合成"推廣後的線性/遞迴關係"

最後我們展示了對任意能量尺度都成立Lauricella弦散射振幅有無限多個遞迴關係。我們也利用了這些遞迴關係解了這無限多種的Lauricella弦散射振幅，並且發現這些Lauricella弦散射振幅都可以被表示成一個四點迅子的散射振幅。這個結果把弦高能振幅的可解性延伸到了所有能量範圍。

In this thesis, we calculated the four-point bosonic open string scattering am-plitudes (SSA) of an arbitrary string state and three tachyons and express them interms of the D-type Lauricella functions. The D-type Lauricella functions can beused to study various scattering limits of SSA. In particular, the D-type LauricellaSSA can be reduced to the Appell functions and the Gauss hypergeometric func-tions in the Regge and non-relativistic scattering limits respectively. Moreover, inthe Regge scattering limit, there exist infinite recurrence relations which can beused to reduce the number of independent Regge SSA from∞down to 1.On the other hand, in the hard scattering limit, one reproduced from theLauricella SSA the linear relations conjectured by Gross in 1988and later corrected and explicitly proved in among SSA ofdifferent string states at the same mass level. Moreover, the infinite linear relationscan be used to reduce the number of independent hard SSA from∞down to 1.We also proved the exact string BCJ relations for the four-point SSA with fourarbitrarystring states. Our explicit proof of the BCJ relations is more general thanthe one demonstrated previously using monodromy approach. More importantly,one can calculate the non-relativistic limit (instead of zero slope limit taken in theliterature) of string BCJ relations to obtain the mass level dependent string BCJrelations, or the ”stringy BCJ relations”.In the Hard/Regge scattering limit, one can combine the linear/recurrence re-lations and string hard/Regge BCJ relation to obtain ”extended linear/recurrenceiii relations”.Finally, we showed that there exist infinite recurrence relations valid forallenergies among all the Lauricella SSA. Moreover, one can use these infinite recur-rence relations to solve all Lauricella string scattering amplitudes and express themin terms of one single four-tachyon amplitude. This result extend the solvabilityof high energy SSA to all kinematic regimes.