Aninda Sinha Centre for High Energy Physics, Indian Institute of Science, Bangalore, India. East Asian Strings Webinar, 2021
QFT Quantum Field Theory 1960- 1916- Geometric Function Theory EFT Effective Field Theory EFT-hedron 1980- 22
QFT Quantum Field Theory 1960- 1916- Geometric Function Theory 1980- Effective Field Theory EFT-hedron 
シン4 • Consider 2-2 scattering of identical massive scalars. •  In perturbative QFT we use Feynman diagrams. These manifest crossing symmetry. •  A non-perturbative representation of the amplitude follows from fixed-t dispersion relation. This loses crossing symmetry, which needs to be imposed as a constraint. 5 •  Where are the Feynman diagrams then from the dispersion relation point of view? •  There is an analogous question one can ask in conformal field theory—this was our motivation to look into this question: Polyakov in 1974 proposed a crossing symmetric bootstrap. This looked like...... 6 Gopakumar, Kaviraj, Sen, AS PRL '16; Gopakumar, AS '18,………………. 7 •  How do we constrain Effective Field Theories? What role does the inaccessible high energies (unknown) play? •  What are the correct mathematical ingredients?  Are there known mathematical theorems which can help us? 88 •  So imagine you were an EFT specialist and had access to 2-2 scattering. •  Schematically you have low energy information of the form
subject to s 1 + s2 + s3 = 0 9 •  Typically you have information about the first few Wilson coefficients at low energies. •  Can these take on arbitrary values? Knowing these are there anything useful we can say about the higher order ones which we don't know anything about? Arkani-hamed, Huang, Huang; Caron-huot, Mazac, Rastelli, Simmons-Duffin, van-Duong; Tolley, Wang, Zhou; Bern, Kosmopoulous, Zhiboedov.........
Table 2: W p,q for 1-loop φ4 amplitude

2103.12108
Table 1: W p,q for tree level type II superstring theory amplitude l l l
Table 3: W p,q for pion scattering from S-matrix bootstrap with s 0 = 0.35 Wo1 W 10 12 •  Based on 2012.04877 (PRL, 21) with Ahmadullah Zahed and 2101.09017 (PRL, 21) with Rajesh Gopakumar and Ahmadullah Zahed, 2103.12108 with Parthiv Haldar and Ahmadullah Zahed and work in progress with Prashanth Raman. l 3 PHYSICAL REVIEW D V O L U M E  6,  N U M B E R  1 0 15 NOVEMBER 1972 Rigorous Parametric Dispersion Representation with Three-Channel Symmetry* G. Auberson and N. N. Khuri Rockefeller University, New York, New York 10021 (Received 30 June 1972) Starting with an analyticity domain in the two Mandelstam variables which is contained in the domain obtained by Martin, we derive a parametric dispersion representation for scat- tering amplitudes in the equal-mass case.  For pion-pion scattering this representation is a rigorous consequence of the axioms of local field theory; it displays in a symmetric and explicit way the contributions of all three channels, and it has only “‘physical’’ absorptive parts.  This representation is useful for deriving sum rules involving only absorptive parts and relating all three channels.  Some of these sum rules are given in this paper, the most important of which form a set of independent physical relations that lead to necessary and sufficient conditions ensuring full crossing symmetry. A relatively unknown paper from 1972! 6 -
RIGOROUS PARAMETRIC DISPERSION REPRESENTATION...
2965 Note added in proof.  One should note that Eq. (4.13) is tremendously simplified in the fully symmetric case, π⁰π⁰ → π⁰π⁰.  In that case one obtains A(s'; t +(s'; s, t , u)) H(s'; s, t , u) , where H(ミ';ミ, T, α) =[ 3(3' − 2)-1 +1(2' −1)-1 +1((2' −1)-1],  and Î„(š'; š, Î, ũ) = l,(š'; ā)  with ā= š Î ū(š ī + ľ ū + š ū)-1 and ï,(ã', ã) given in Eq. (5.20).  This representation holds for any point (s, t) for which τ≡ι,(ã'; š, t, ï)+ ័ lies in the Martin-Lehmann ellipses E(s') for A(s', τ) given in Eq. (A2).  The similarity of this representa- tion to the Cini-Fubini approximation4 is striking.  This representation follows most directly from Eq. (5.2) by transforming from the (z, a) variables to the s, t, u variables.,
ACKNOWLEDGMENTS
We are indebted to F. J. Dyson and G. Wanders for useful comments. The most useful/ encouraging formula was in a NOTE ADDED! See also Mahoux, Roy, Wanders 1974. 14
| X H ( S ′ ; S 1, S 2, S ′
S1 + S2 + S3 = 0   μ = 4m²
α0 ≡ M(s1 = 0,s2 = 0) NB: s 1 = s − 3 m², s 2 = t − 3 m², s 3 = u − 3 m² 15 Expand In terms of poles in S1, S2 &
S3 ＝一S1一S2 Answer
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Similar expansion exists for the 2d Ising Mellin Amplitude. Also similar expansion exists for the open string amplitude with only s,t symmetry (w P. Raman).
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• Good match for complex values
• Notice something nontrivial. If we consider just a fixed k. • Expand around a=0. •  You will get negative powers of x. •  But LHS has no such powers. This means that once we sum over k, these negative powers will cancel. Keep this in mind. •  Lesson: To have a crossing symmetric expansion, it seems we have to introduce "spurious", "non-local" singularities to the basis elements. LOCALITY CONSTRAINTS/NULL CONSTRAINTS A heuristic derivation of the crossing symmetric dispersion Auberson, Khuri 1972; AS, Zahed '20. ● The key steps • We can parametrize the solution using a new variable z
•  Idea now is to write a dispersion relation in z for fixed a. •  Conveniently one can show
Make a mental note of the forms of these. These are what are called Koebe functions in the context of univalent functio
•  So that an expansion in powers of x, y is an expansion in powers of a, z³.
s1 → a, | s2 | , | s3 | → ∞   when   z → z1 etc Low energy High energy
24 The key steps • Imposing s1 + s2 + s3 = 0 the equation
S1S2 + S1S3 + S2S3 • Becomes a quadratic equation giving s² in terms of s 1 . Call these sみ(s1, a)
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Dyson block expansion — “Regge bounded”
Feynman block expansion —not “Regge bounded” + eg.   polyշ = c1x + c2y 28
Expansion of the massless pole subtracted dilation amplitude in terms of crossing symmetric partial waves (locality constraints implicit). 29
Expansion of the massless pole subtracted dilation amplitude in terms of crossing symmetric partial waves (locality constraints imposed—operationally throw away negative powers of ×, partialwave*kernel wise). kmax = 6 30
■ S1=0.1 ■ S1=0.3 ■ S1=0.7 Feynman block seems to do 32
QFT CFT 33 ● 2103.12108 with P. Haldar and A. Zahed; but mainly work in progress with P. Raman •  Is dispersion (closing of contour, dropping of arcs) enough to give bounds? • No. What gives two-sided bounds is crossing symmetry and locality. On the math side we have additional analytic properties like univalence, typically realness etc on either a disc, punctured disc or annulus. ● 2103.12108 with P. Haldar and A. Zahed; but mainly work in progress with P. Raman 35
Definition Example in a disk Reverse triangle Triangle BOUND!! 36
Most general version proved by Louis de Branges in 1985 Interesting subclasses, more relevant to give stronger bounds ● 2103.12108 with P. Haldar and A. Zahed; but mainly work in progress with P. Raman 37
Im f(z) Im z, > 0 Kernel
Mobius transformation of Koebe function— univalent with real coefficients. Univalent functions with real coefficients are also typically Real** When absorptive part>0, we have a convex sum ** Convex sum of univalent functions are not necessarily univalent BUT convex sum of typically real functions are typically real!  With P. Raman, to appear Geometric function theory—Bieberbach, Rogosinski, Wigner, Nehari, Schwarz, Komatu, Goodman, papers dating over a 100 years or so. 38 27a²(a(aW 02 + W 11) + W 20) 40 W01 42
QFT Quantum Field Theory 1960- 1916- Geometric Function Theory 1980- EFT Effective Field Theory EFT-hedron
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