One-loop CHY-integrand of bi-adjoint scalar theory
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Received: January 13, 2020 Accepted: February 11, 2020 Published: February 28, 2020
Bo Fenga,b and Chang Hua,1 a
Zhejiang Institute of Modern Physics, Zhejiang University, Zheda Road No. 38, Hangzhou, 310027, P.R. China b Center of Mathematical Science, Zhejiang University, Zheda Road No. 38, Hangzhou, 310027, P.R. China
E-mail: [email protected], [email protected] Abstract: In this paper, the one-loop CHY-integrands of bi-adjoint scalar theory has been reinvestigated. Differing from previous constructions, we have explicitly removed contributions from tadpole and massless bubbles when taking the forward limit of corresponding tree-level amplitudes. The way to remove those singular contributions is to exploit the idea of “picking poles”, which is to multiply a special cross ratio factor with the role of isolating terms having a particular pole structure. Keywords: Scattering Amplitudes, Field Theories in Higher Dimensions, Field Theories in Lower Dimensions ArXiv ePrint: 1912.12960
1
Corresponding author.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP02(2020)187
JHEP02(2020)187
One-loop CHY-integrand of bi-adjoint scalar theory
Contents 1 Introduction
1
2 Backgrounds 2.1 Integrate rules 2.2 Effective Feynman diagram 2.3 Picking out poles
3 3 6 8 10 12 16 20 25 26 27
4 Conclusion
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1
Introduction
In recent years, a novel formulism for tree-level amplitudes of various theories has been proposed by Cachazo, He and Yuan [CHY] in a series of papers [1–5]. The formula is given as an integral over the moduli space of Riemann spheres Q Z Y ( ni=1 dzi ) An = zij zjk zki δ(Ea ) · I(z1 , · · · , zn ) (1.1) vol(SL(2, C)) a6=i,j,k
where zi are puncture locations of (the) i-th external particles, and the denominator i +b vol(SL(2, C)) comes from the M¨ obius invariance. i.e., the transformation zi → az czi +d with ad − bc 6= 0. The E’s are the scattering equation defined as X sab Ea ≡ = 0, a = 1, 2, . . . , n (1.2) za − zb b6=a
with sab ≡ (ka +kb )2 being the Mandelstam invariants. As shown in (1.1), the CHY formalism includes two parts: the integration measure with δ functions of scattering equations, which is universal for all theories, and formulating different CHY-integrands I for different theories. For a theory with n particles, there are (n − 3)! solutions to these scattering equations. A proof of this construction for bi-adjoint scalar theory and Yang-Mills theory has been provided by Dolan and Goddard in [6]. Working out all (n − 3)! solutions of
–1–
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3 One loop 3.1 π = ρ 3.2 π = ρT 3.3 For general orderings π and ρ 3.4 Example 3.4.1 The first example 3.4.2 The second example
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A special case, i.e., Yang-Mills amplitudes, has been discussed in [20]. Although we will focus on one-loop case only in this paper, it is worth to mention that using the same idea, some two loop constructions are shown in [37–40]. 2
–2–
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scattering equation is very burdensome even at small value of n. In general there is no effective met
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