New effect in wave-packet scatterings of quantum fields: Saddle points, Lefschetz thimbles, and Stokes phenomenon
NNew effect in wave-packet scatterings of quantum fields:Saddle points, Lefschetz thimbles, and Stokes phenomenon
Kenzo Ishikawa, ∗ Kenji Nishiwaki, † and Kin-ya Oda ‡∗ Department of Physics, Hokkaido University, Hokkaido 060-0810, Japan ∗ Research and Education Center for Natural Sciences, Keio University Kanagawa 223-8521, Japan † Department of Physics, Shiv Nadar University, Gautam Buddha Nagar 201314, India ‡ Department of Physics, Osaka University, Osaka 560-0043, Japan ‡ Department of Mathematics, Tokyo Woman’s Christian University, Tokyo 167-8585, Japan
We find a new contribution in wave-packet scatterings, which has been overlooked in the standardformulation of S-matrix. As a concrete example, we consider a two-to-two scattering of light scalars φ by another intermediate heavy scalar Φ, in the Gaussian wave-packet formalism: φφ → Φ → φφ .This contribution can be interpreted as an “in-time-boundary effect” of Φ for the correspondingΦ → φφ decay, proposed by Ishikawa et al., with a newly found modification that would cure thepreviously observed ultraviolet divergence. We show that such an effect can be understood as aStokes phenomenon in an integral over complex energy plane: The number of relevant saddle pointsand Lefschetz thimbles (steepest descent paths) discretely changes depending on the configurationsof initial and final states in the scattering. INTRODUCTION
Particle scattering in quantum field theory requireswave packets in its very foundation, whereas the plane-wave formulation, which involves the square of energy-momentum delta function in S-matrix, is “more amnemonic than a derivation” [1]. However, it has longbeen believed that there are no new phenomena arisingfrom taking into account the wave-packet effects.On the other hand, Ishikawa et al. claim that indeed awave-packet effect—more specifically the time-boundaryeffect due to localization of wave-packet overlap in time—is responsible for diverse phenomena in science such asthe LSND neutrino anomaly [2, 3]; violation of selectionrules [4]; the solar coronal heating problem [5]; anoma-lous Thomson scattering and a speculative alternativeto dark matter, as well as modified Haag theorem [6];the anomalous excitation energy transfer in photosyn-thesis [7]; and anomalies in width in e + e − → γγ , in π lifetime, in Raman scattering, and in the water vaporcontinuum absorption [8, 9]. There is an ongoing exper-imental project for this effect [8, 10].However, the time-boundary effect has not been paidhigh attention, because so far it depends on whetherone accepts a priori the concept of a finite-time scatter-ing that yields the time-boundary effect (when combinedwith the localization of wave-packet overlap in time).Here, we fill the gap by proving existence of a new effect,which shares the same property as the time-boundary ef-fect of a sub-process, even in an S-matrix with an infinitetransition time.Now we turn to more concrete backgrounds of this Let- ter. The authors of Ref. [11] have proposed the Gaus-sian wave-packet formalism that fully takes advantage ofthe completeness of the Gaussian basis. Based on thisformalism, it has been argued that one-to-two processessuch as the scalar Φ → φφ decay have extra contribu-tions arising from the in and out time-boundaries thatare respectively the starting and end points of the tran-sition [12].In general, validity of the time-boundary effect de-pends [13, 14] on the assumption that the in and outwave-packet states can be well approximated by the freeGaussian states even when interactions are non-negligiblethere. Physically, the claimed in and out time-boundaryeffects would represent some yet unspecified productionand detection processes of Φ and φφ , respectively.In this Letter, we study a φφ → Φ → φφ scatteringwithout assuming any boundary effect of initial and final φφ states, and show existence of new contribution thatshare the same property as the in-boundary effect of Φ → φφ . This is how we take into account the productionprocess of Φ and resolve the in-boundary ambiguity forΦ → φφ . S-MATRIX IN GAUSSIAN FORMALISM
We consider the scattering φφ → Φ → φφ withan interaction Lagrangian density L = − κ φ Φ, where φ and Φ are real scalar fields with masses m and M ( > m ), respectively, and κ is a coupling constant. Weonly take into account the tree-level s -channel scatter-ing as we are mostly interested in amplitudes near the a r X i v : . [ h e p - t h ] F e b resonance pole of Φ. Input parameters for each wavepacket a (with a = 1 , , σ a , spacetime position ofits center X a = (cid:0) X a , X a (cid:1) , and its central momentum P a = ( E a , P a ), where E a := ( m + P a ) / . We maytrade P a for V a := P a /E a as independent parameters.We write the plane-wave propagator of Φ with anoff-shell momentum p = (cid:0) p , p (cid:1) as − i − ( p ) + E p , where E p := ( E p − i(cid:15) ) / with E p := ( M + p ) / > (cid:15) > (cid:15) = M Γ, with Γ being the decaywidth of Φ [15]. In total, ( κ, m, M, (cid:15) ) and ( σ a , X a , P a )are all the independent parameters in this Letter: Theformer fixes the theory [16], while the latter parametrizeseach wave packet a .For the given Gaussian wave packets, f σ a ; X a , P a , thetree-level s -channel S-matrix S is given as Eq. (42) inRef. [14] with the limit T in → −∞ and T out → ∞ [17].We work at the leading order in the plane-wave expan-sion for large σ a [11, 13, 14]. Then the eight-dimensionalspacetime integral over the in- and out-interaction pointsbecomes Gaussian, and the result is S = (2 π ) ( − iκ ) (cid:32) (cid:89) a =1 √ E a ( πσ a ) / (cid:33) (cid:113) σ σ ς in ς out × (cid:90) d p (2 π ) − ip + M − i(cid:15) e F ( p ; p ) , (1)where σ in := σ σ σ + σ and σ out := σ σ σ + σ ( ς in = σ + σ ( V − V ) and ς out = σ + σ ( V − V ) ) are the spatial (time-like) width-squared for the in and out interaction regions, respec-tively, and the exponent becomes quadratic in p : F (cid:0) p ; p (cid:1) = F ∗ ( p ) − ς + (cid:0) p − p ∗ ( p ) (cid:1) , (2)in which ς + := ς in + ς out , p ∗ ( p ) = Ω( p ) − i δ T ς + , and F ∗ ( p )can be read off from Eq. (107) in Ref. [14] (its explicitform is mostly irrelevant for the following discussion);see Refs. [13, 14] for further notation; we have also pre-pared Supplemental Material. Physically, Ω( p ) is theoff-shell energy of Φ, and δ T is the time passed from thein-interaction to the out-interaction; see each Fig. 1 inRefs. [13, 14]. Both of them vary according to the wavepacket configurations ( σ a , X a , P a ). SADDLE POINT
The second line in Eq. (1), a “wave-packet Feynmanpropagator”, may be written as (cid:82) d p (2 π ) e F ∗ ( p ) I ( p ), where I ( p ) := (cid:90) ∞−∞ d p π − i − ( p ) + E p e − ς +2 ( p − p ∗ ( p ) ) . (3) E p (cid:45) E p Im p (cid:60) Im E p Im p (cid:62) Im (cid:72) (cid:45) E p (cid:76) Im E p (cid:60) Im p (cid:60) Im (cid:72) (cid:45) E p (cid:76) Re p Im p FIG. 1. Shaded region represents convergent directions (cid:60) F → −∞ for (cid:12)(cid:12) p (cid:12)(cid:12) → ∞ . The points ± E p denote the polesof the propagator of Φ. The orange, purple, and blue linesrepresent the integral path of I ∗ in Eq. (4) for (cid:61) p > (cid:61) ( − E p ), (cid:61) E p < (cid:61) p < (cid:61) ( − E p ), and (cid:61) p < (cid:61) E p , respectively. The exponential factor has a saddle point at p = p ∗ ( p ) with the steepest descent and ascent paths being (cid:61) (cid:0) p − p ∗ ( p ) (cid:1) = 0 and (cid:60) (cid:0) p − p ∗ ( p ) (cid:1) = 0, where (cid:60) and (cid:61) denote the real and imaginary parts, respectively. Thesaddle-point contribution is I ∗ ( p ) = (cid:90) ∞ + (cid:61) p ∗ −∞ + (cid:61) p ∗ d p π − i − ( p ) + E p e − ς +2 ( p − p ∗ ( p ) ) ≈ √ πς + − i − (Ω( p ) − i δ T ς + ) + E p , (4)which leads to Eq. (110) in Ref. [14]. The standard plane-wave result has been obtained by taking into account thiscontribution only, and no in-boundary effect for Φ → φφ has been found in the limit δ T → I ∗ over the steepestdescent path from −∞ + (cid:61) p ∗ to ∞ + (cid:61) p ∗ differs from I in Eq. (3) by the residue at the poles p = E p and − E p when (cid:61) p < (cid:61) E p and (cid:61) p > −(cid:61) E p , respectively: I ( p ) = I ∗ ( p ) + e − ς +2 (cid:16) Ω( p ) − E p − i δ T ς + (cid:17) E p θ (cid:18) δ T ς + + (cid:61) E p (cid:19) + e − ς +2 (cid:16) Ω( p )+ E p − i δ T ς + (cid:17) E p θ (cid:18) (cid:61) E p − δ T ς + (cid:19) , (5)where θ is the Heaviside step function. Eq. (5) is one ofour main results. When (cid:15) is small, we have (cid:61) E p (cid:39) − (cid:15) E p ,and the second and third terms are non-zero when δ T > (cid:15)ς + E p and δ T < − (cid:15)ς + E p , respectively; each of them cor-responds to a configuration of wave packets that givesforward and backward propagation of Φ in time, respec-tively; see Fig. 1 in Ref. [14]. Note that, droppoing theHeaviside function, the ratio of the second term to thethird is exp (cid:0) − ς + E p p ∗ (cid:1) , and hence the third term is ex-ponentially small when ς + E p Ω( p ) is large.The second term in Eq. (5) is the new contributionin addition to the ordinary I ∗ ( p ). This term sharethe same property with the in-time-boundary effect forΦ → φφ [14]: An exponential suppression exp[ − ( δ T ) ς + ]that comes from F ∗ ( p ) is cancelled by exp[ ( δ T ) ς + ] fromEq. (5) for large δ T , unlike the ordinary I ∗ ( p ). Namely,the propagation of Φ in time is not exponentially sup-pressed by the passed time δ T , which is a characteristic ofthe time-boundary effect: If it comes from the boundary,it does not matter how large is the bulk. Also, we newlyfind the extra suppression factor exp[ − ς + (Ω( p ) − E p ) ]in the second term in Eq. (5), which would cure the pre-viously found possible ultraviolet divergence in the time-boundary contribution, related to energy-non-conservingconfigurations that are non-vanishing for | Ω( p ) − E p | →∞ ; see Sec. 4.3 in Ref. [13]. STOKES PHENOMENON
Here we show that the result (5) can be understood asa Stokes phenomenon. We first rewrite the integral (3)into an exponential form: I ( p ) = (cid:82) ∞−∞ d p πi e F ( p ; p ), where F (cid:0) p (cid:1) = − ς + (cid:0) p − p ∗ (cid:1) − ln (cid:16) − (cid:0) p (cid:1) + E p (cid:17) . (6)Here and hereafter, we drop the p -dependence and write F (cid:0) p (cid:1) etc. for simplicity.By solving ∂ F ∂p = 0, we obtain the following three sad-dle points: p ∗ ) = p ∗ + 1 ς + p ∗ − ( p ∗ ) + E p + · · · , (7) p ± ) = ± E p + 1 ς + p ∗ ∓ E p + · · · , (8)with F ( ∗ ) = ln 1 − ( p ∗ ) + E p + 1 ς + (cid:0) p ∗ (cid:1) (cid:16) ( p ∗ ) − E p (cid:17) + · · · , (9) F ( ± ) = − ς + (cid:0) p ∗ ∓ E p (cid:1) + ln (cid:18) ς + (cid:18) ∓ p ∗ E p (cid:19)(cid:19) + 1 + · · · , (10)and ∂ F ( ∗ ) ∂p = − ς + + 1( p ∗ + E p ) + 1( p ∗ − E p ) + · · · , (11) ∂ F ( ± ) ∂p = ς (cid:0) p ∗ ∓ E p (cid:1) − (cid:18) ± p ∗ E p (cid:19) ς + + · · · , (12)where we have included up to the sub-leading terms (aswell as the order-ln ς + term) for large ς + [18].For each saddle point ( i ) with i = ∗ and ± , the steep-est decent and ascent paths are obtained from the con-dition (cid:61) (cid:0) F (cid:0) p (cid:1) − F ( i ) (cid:1) = 0. The steepest descent path J ( i ) (ascent path K ( i ) ) from the saddle point ( i ) is calledthe Lefschetz thimble or the Stokes line (the anti-thimbleor the anti-Stokes line). Along J ( i ) , we may evaluatethe approximate Gaussian integral I ( i ) = (cid:82) J ( i ) d p πi e F ( p )without the oscillation of integrand: I ( ∗ ) (cid:39) √ πς + − i − ( p ∗ ) + E p , I ( ± ) (cid:39) e √ π e − ς +2 ( p ∗ ∓ E p ) E p . (13)The integral I can be decomposed into those on the Lef-schetz thimbles: I = (cid:88) i = ∗ , ± (cid:10) K ( i ) , R (cid:11) I ( i ) , (14)where (cid:10) K ( i ) , R (cid:11) is the intersection number between theanti-thimble K ( i ) and the original integration path R ; seee.g. Ref. [19] for a review. We see that the result (5) isreproduced under the saddle-point approximation up tothe extra factor e √ π (cid:39) . e (cid:60)F in the first panel, while in others,we show contours for (cid:61)F (mod 2 π ) on a density plot of (cid:60)F ; see the caption for more details.In each panel for (cid:61)F , through the saddle point p ∗ ) ,the horizontal and vertical contours (red-solid) are thethimble J ( ∗ ) and the anti-thimble K ( ∗ ) , respectively. Onthe other hand, through the saddle point p , the yellow-dashed contour that connects E p and a (cid:12)(cid:12) (cid:61) p (cid:12)(cid:12) → ∞ regionis the anti-thimble K (+) , while its perpendicular contouris the thimble J (+) . Similarly, through p − ) , the green-dotted contour connecting − E p and a (cid:12)(cid:12) (cid:61) p (cid:12)(cid:12) → ∞ regionis K ( − ) , and its perpendicular, J ( − ) . All the other con-tours that are disconnected from any of p i ) are irrelevantfor the integral.We comment on the branch cuts for the logarithmicfunction in Eq. (6). In the case (cid:61) p ∗ ) (cid:46) (cid:61) E p in thesecond panel, we may put a branch cut from − E p to −∞ + (cid:61) ( − E p ) and another from E p along a, say, red-solidor green-dotted contour. This way, we may continuouslydeform R into the thimbles J ( ∗ ) and J (+) without cross-ing the cuts. It is obvious that we may do similarly forother cases. In Fig. 3, we further explain artifact linesdue to the brach cuts in the numerical computation.There are three cases depending on the relative posi-tion of J ( ∗ ) and ± E p , represented by the last three pan-els in Fig. 2. When (cid:61) p ∗ ) (cid:46) (cid:61) E p , we see in the sec-ond panel (as magnified in Fig. 3) that the anti-thimbles K ( − ) (green-dotted) terminate in − E p , and hence do notintersect with the real axis: (cid:10) K ( − ) , R (cid:11) = 0. Similarlywhen (cid:61) p ∗ ) (cid:38) (cid:61) ( − E p ), in the fourth panel, K (+) (yellow-dashed) terminate in E p , and hence (cid:10) K (+) , R (cid:11) = 0. When (cid:61) E p (cid:46) (cid:61) p ∗ ) (cid:46) (cid:61) ( − E p ), in the third panels, both anti-thimbles terminate in ± E p , and hence (cid:10) K ( ± ) , R (cid:11) = 0.This is how the discrete change in the amplitude (5) isunderstood as a Stokes phenomenon in Eq. (14). (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) FIG. 2. In the first panel, we plot e (cid:60)F , whereas in the otherthree, we show contours for (cid:61)F (mod 2 π ), on a density plot of (cid:60)F in [ − , ς + = 10,Ω (= (cid:60) p ∗ ) = 5, and (cid:15) = 0 . E p = 1 units. Both the firstand second panels correspond to forward propagation of Φwith δ T ς + (= −(cid:61) p ∗ ) = 0 .
5, while the third and fourth to equal-time and backward with δ T ς + = 0 and − .
5, respectively. Thered-solid, yellow-dashed, and green-dotted contours are (cid:61)F = (cid:61)F ( ∗ ) , (cid:61)F (+) , and (cid:61)F ( − ) respectively. The white dots denotethe poles at p = ± E p and the blue single circles nearby themare correspondingly the saddle points p ± ) . The blue doublecircle denote the saddle point p ∗ ) . FIG. 3. Magnified plot for (cid:61)F (mod 2 π ) of the second panelin Fig. 2 near the saddle point p − ) (blue circle) and the pole − E p (white dot). We see that three contours and an artifactline terminate in − E p : The three contours are (cid:61)F = (cid:61)F ( ∗ ) (red-solid), (cid:61)F = (cid:61)F (+) (yellow-dashed), and especially theanti-thimble K ( − ) (green-dotted) from p − ) . The extra arti-fact line, seemingly consisting of three degenerate contours,is from a branch cut of logarithmic function in the numeri-cal computation, along which contributions from all the otherRiemann surfaces appear. This way, in each pole in Fig. 2,there always terminate three contours and an artifact line. SUMMARY AND DISCUSSION
We have studied the wave-packet scattering φφ → Φ → φφ within the infinite time interval and hence withoutany time-boundary effect for φ . We have obtained thenew contribution that shares the same property as thein-time-boundary effect for Φ → φφ . The appearance ofthis effect is consistently understood as the Stokes phe-nomenon.This effect is intrinsically absent in the plane-wave S-matrix. Note that the Wick rotation from J ( ∗ ) to K ( ∗ ) makes the p -integral exponentially divergent.In this Letter, we have integrated over the times ofinteraction before p . Instead we may first integrate over p before them, in which all the exponents are linear in p and the Wick rotation is allowed, and confirm thatthe time-boundary effect emerges at the time-boundaries T in/out . This will be presented in detail in a separatepublication.It would be interesting to pursue the deviation e/ √ π (cid:39) .
08 between the residue computation and thesaddle-point approximation. This factor does not dependon any physical parameter, and there may be mathemat-ical account for it. In the case, say, (cid:61) p ∗ ) (cid:46) (cid:61) E p , in theplane-wave limit ς + → ∞ , the saddle-point p becomescloser and closer to the pole E p , and hence the thimble J (+) becomes more and more curved near the pole. Thismight be a cause of the deviation of the approximateGaussian integral along the straight line that is tangentto the curve. Acknowledgement:
We thank Osamu Jinnouchi foruseful discussion and Juntaro Wada for reading themanucript. The work of K.O. is in part supported byJSPS Kakenhi Grant No. 19H01899. [1] S. Weinberg,
The Quantum theory of fields. Vol. 1: Foun-dations (Cambridge University Press, 2005).[2] K. Ishikawa and Y. Tobita, Resolving LSND Anomaly byNeutrino Diffraction, (2011), arXiv:1109.3105 [hep-ph].[3] K. Ishikawa and Y. Tobita, Matter-enhanced transitionprobabilities in quantum field theory, Annals Phys. ,118 (2014), arXiv:1206.2593 [hep-ph].[4] K. Ishikawa, T. Tajima, and Y. Tobita, Anoma-lous radiative transitions, PTEP , 013B02 (2015),arXiv:1409.4339 [hep-ph].[5] K. Ishikawa and Y. Tobita, Electroweak Hall Effect ofNeutrino and Coronal Heating, (2015), arXiv:1503.07285[hep-ph].[6] K. Ishikawa and Y. Tobita, Finite-size corrections toFermi’s Golden rule II: Quasi-stationary compositestates, (2016), arXiv:1607.08522 [hep-ph].[7] N. Maeda, T. Yabuki, Y. Tobita, and K. Ishikawa, Finite-size corrections to the excitation energy transfer in amassless scalar interaction model, PTEP , 053J01(2017), arXiv:1609.00160 [physics.chem-ph].[8] K. Ishikawa, O. Jinnouchi, A. Kubota, T. Sloan, T. H.Tatsuishi, and R. Ushioda, On experimental confirmationof the corrections to Fermi’s golden rule, PTEP ,033B02 (2019), arXiv:1901.03019 [hep-ph].[9] K. Ishikawa,
Implications of the correction to theFermi’s golden rule , in VIII · LA PARTE Y ELTODO: Workshop on advanced topics on high-energy physics and gravitation—Online via Zoom,Afunalhue, Villarrica, Chile, 4–8 January 2021, https://laparteyeltodo.files.wordpress.com/2021/01/kishikawa_chille.pdf , .[10] R. Ushioda, O. Jinnouchi, K. Ishikawa, and T. Sloan,Search for the correction term to the Fermi’s golden rulein positron annihilation, PTEP , 043C01 (2020),arXiv:1907.01264 [hep-ex].[11] K. Ishikawa and T. Shimomura, Generalized S-matrixin mixed representations, Prog. Theor. Phys. , 1201(2006), arXiv:hep-ph/0508303 [hep-ph].[12] K. Ishikawa and Y. Tobita, Finite-size corrections toFermi’s golden rule: I. Decay rates, PTEP , 073B02(2013), arXiv:1303.4568 [hep-ph].[13] K. Ishikawa and K.-Y. Oda, Particle decay in Gaussianwave-packet formalism revisited, PTEP , 123B01(2018), arXiv:1809.04285 [hep-ph].[14] K. Ishikawa, K. Nishiwaki, and K.-y. Oda, Scalar scat-tering amplitude in the Gaussian wave-packet formalism,PTEP , 103B04 (2020), arXiv:2006.14159 [hep-th].[15] See e.g. Appendix C and D in Ref. [20] and referencestherein, for subtleties when Γ becomes comparable to M .[16] Within this particular theory, (cid:15) is not an independentvariable as it can be computed for a given set of param-eters ( κ, m, M ). Here it is left independent for ease ofextension.[17] In this Letter, we first neglect the time-boundary effectsof φ ; focus on the “bulk effect” [14] for φφ → Φ → φφ ;and then show that the in-time-boundary effect of Φ for Φ → φφ still emerges .[18] The saddle-point equation is cubic in p , and we obtainexact solutions for p i ) . Here we show the large- ς + resultsas an illustration, though we use the exact ones in thefollowing numerical plots.[19] Y. Tanizaki, Study on sign problem via Lefschetz-thimblepath integral, https://ribf.riken.jp/~tanizaki/thesis/yuya_phd.pdf , Ph.D. thesis, Tokyo U. (2015).[20] K. Nishiwaki, K.-y. Oda, N. Okuda, and R. Watanabe,Heavy Higgs at Tevatron and LHC in Universal ExtraDimension Models, Phys. Rev. D , 035026 (2012),arXiv:1108.1765 [hep-ph]. SUPPLEMENTAL MATERIAL
Here we provide more detailed expressions that are omitted in the main text. These are not necessary to follow itbut may be useful.
S-MATRIX IN GAUSSIAN FORMALISM
We write the plane-wave propagator of Φ: − ip + M − i(cid:15) = − i − ( p ) + p + M − i(cid:15) = − i − ( p ) + E p − i(cid:15) = − i − ( p ) + E p , (15)where E p := (cid:112) M + p and E p := (cid:113) E p − i(cid:15) .For given Gaussian wave packets, f σ a ; X a , P a , the tree-level s -channel S-matrix reads [14] S = ( − iκ ) (cid:90) d p (2 π ) − ip + M − i(cid:15) (cid:90) d x f σ ; X , P ( x ) f σ ; X , P ( x ) e − ip · x (cid:90) d y f ∗ σ ; X , P ( y ) f ∗ σ ; X , P ( y ) e ip · y , (16)where the exponential factors come from the plane-wave expansion of Φ, and we integrate for the spacetime positionof “in-interaction” x and that of “out-interaction” y . The result does not differ if we expand Φ by the Gaussian wavesinstead of the plane waves [14].In principle, there can be a “time-boundary effect” of φ for each a at the time X a . As said in the main text, weneglect them. In order to get the bulk effect, we could have introduced time boundaries for the two-to-two scattering, T in and T out , at which interactions are negligible; have cut off the interaction-time integrals (cid:82) T out T in d x and (cid:82) T out T in d y ;have focused on the “bulk terms”; and have taken T in → −∞ and T out → ∞ .We work at the leading order in the plane-wave expansion for large σ a [11, 13, 14]: f σ a ; X a , P a ( z ) (cid:39) e iP a · ( z − X a ) ( πσ ) / √ E a e − σa [ z − X a − ( z − X a ) V a ] . (17)Then all the exponents in S become quadratic in x and y , and we may perform the eight-dimensional Gaussianintegral to get the form S = (2 π ) ( − iκ ) (cid:32) (cid:89) a =1 √ E a ( πσ a ) / (cid:33) (cid:113) σ σ ς in ς out (cid:90) d p (2 π ) − ip + M − i(cid:15) e F ( p ; p ) , (18)where σ in := σ σ σ + σ and σ out := σ σ σ + σ ( ς in = σ + σ ( V − V ) and ς out = σ + σ ( V − V ) ) are the spatial (time-like) width-squaredfor the in and out interaction regions, respectively; see Refs. [13, 14] for further details.We also show expressions for σ = σ = σ = σ (=: s ) after “ (cid:32) ” as an illustration: σ in (cid:32) s , σ out (cid:32) s , ς in (cid:32) s ( V − V ) , ς out (cid:32) s ( V − V ) , and S (cid:32) ( − iκ ) π s √ E E | V − V | √ E E | V − V | (cid:90) d p (2 π ) − ip + M − i(cid:15) e F ( p ; p ) . (19)Note that a head-on collision in the center-of-mass frame corresponds to V + V = 0 and, further in the masslesslimit m →
0, to | V − V | → p . After the Gaussian integral over x and y , the exponents inEq. (18) become quadratic in p : F (cid:0) p ; p (cid:1) = F ∗ ( p ) − ς + (cid:0) p − p ∗ ( p ) (cid:1) , (20)where ς + := ς in + ς out . The location of saddle point p ∗ ( p ) in the complex p plane becomes p ∗ ( p ) = Ω( p ) − i δ T ς + , (21)where we define a wave-average of (what-we-call) “wave-packet kinetic energies”, Ω( p ), and the time difference betweenthe in and out interactions, δ T ,Ω( p ) := ς in ω in ( p ) + ς out ω out ( p ) ς in + ς out (cid:32) ω in ( p )( V − V ) + ω out ( p )( V − V ) V − V ) + V − V ) , (22) δ T := T out-int − T in-int , (23)in which we use the following notations: • Ξ a := X a − V a X a denotes the spatial center of each wave packet a at a reference time t = 0. (In Refs. [13, 14], Ξ a has been written as X a .) • The wave-packet kinetic energies are ω in ( p ) := E in + V in · ( p − P in ) (cid:32) E in + V + V · ( p − P in ) , (24) ω out ( p ) := E out + V out · ( p − P out ) (cid:32) E out + V + V · ( p − P out ) , (25)where E in := E + E and E out := E + E ( P in := P + P and P out := P + P ) are the incoming andoutgoing energies (momenta), respectively, and V in := σ σ σ + σ (cid:18) V σ + V σ (cid:19) (cid:32) V + V , (26) V out := σ σ σ + σ (cid:18) V σ + V σ (cid:19) (cid:32) V + V • The in- and out-interaction times are given by T in-int = − ( V − V ) · ( Ξ − Ξ )( V − V ) , (28) T out-int = − ( V − V ) · ( Ξ − Ξ )( V − V ) . (29)The initial φφ wave packets intersect each other at T in-int , and the final φφ at T out-int .The exponent at saddle point F ∗ ( p ) is F ∗ ( p ) = (cid:18) − R in − σ in p − P in ) − i Ξ in · ( p − P in ) (cid:19) + (cid:18) − R out − σ out p − P out ) + i Ξ out · ( p − P out ) (cid:19) − ( δ T ) ς + + iς (cid:18) T in ς in + T out ς out (cid:19) δω ( p ) − ς δω ( p )) , (30)where ς := ς in ς out ς in + ς out is a wave-average of the time-like width-squared; δω ( p ) := ω out ( p ) − ω in ( p ) is the difference of thewave-packet kinetic energies; the wave-averaged position for the initial φφ (at the reference time t = 0) and that forthe final φφ are Ξ in = σ σ σ + σ (cid:18) Ξ σ + Ξ σ (cid:19) (cid:32) Ξ + Ξ , (31) Ξ out = σ σ σ + σ (cid:18) Ξ σ + Ξ σ (cid:19) (cid:32) Ξ + Ξ φφ wave packets and for the final φφ are, respectively, R in = ( Ξ − Ξ ) + (cid:104) (cid:98) V · ( Ξ − Ξ ) (cid:105) σ + σ , (33) R out = ( Ξ − Ξ ) + (cid:104) (cid:98) V · ( Ξ − Ξ ) (cid:105) σ + σ , (34)in which (cid:98) V := V − V | V − V | and (cid:98) V := V − V | V − V | . SADDLE POINT
We compute a “wave-packet Feynman propagator”: (cid:90) d p (2 π ) − ip + M − i(cid:15) e F ( p ; p ) = (cid:90) d p (2 π ) e F ∗ ( p ) I ( p ) , (35)where I ( p ) := (cid:90) ∞−∞ d p π − i − ( p ) + E p e − ς +2 ( p − p ∗ ( p ) ) . (36)The key observation is that the saddle-point integral I ∗ over the steepest descent path from −∞ + (cid:61) p ∗ ( p ) to ∞ + (cid:61) p ∗ ( p ) differs from the integral I over R by the residues: I ( p ) = I ∗ ( p ) − (cid:32) Res p = E p e − ς +2 ( p − p ∗ ( p ) ) − ( p ) + E p (cid:33) θ (cid:0) (cid:61) E p − (cid:61) p ∗ (cid:1) + (cid:32) Res p = − E p e − ς +2 ( p − p ∗ ( p ) ) − ( p ) + E p (cid:33) θ (cid:0) (cid:61) p ∗ − (cid:61) ( − E p ) (cid:1) = I ∗ ( p ) + e − ς +2 (cid:16) Ω( p ) − E p − i δ T ς + (cid:17) E p θ (cid:18) δ T ς + + (cid:61) E p (cid:19) + e − ς +2 (cid:16) Ω( p )+ E p − i δ T ς + (cid:17) E p θ (cid:18) (cid:61) E p − δ T ς + (cid:19) . (37) STOKES PHENOMENON
We first rewrite the integral (36) into an exponential form: I ( p ) = (cid:90) ∞−∞ d p πi e F ( p ; p ) , (38)where F (cid:0) p ; p (cid:1) = − ς + (cid:0) p − p ∗ ( p ) (cid:1) − ln (cid:16) − (cid:0) p (cid:1) + E p (cid:17) . (39)We take the direction of J ( ± ) such that it can be deformed to R without crossing the poles, namely, J (+) and J ( − ) circulate around E p and − E p clockwise and counterclockwise, respectively. Along J ( i ) , we may evaluate theapproximate Gaussian integral I ( i ) ( p ) = (cid:90) J ( i ) d p πi e F ( p ; p ) (40)without the oscillation of integrand: I ( ∗ ) ( p ) (cid:39) √ πς + − i − ( p ∗ ( p )) + E p , (41) I ( ± ) ( p ) (cid:39) ∓ − i √ π (cid:113) − ς ( p ∗ ( p ) ∓ E p ) e − ς +2 ( p ∗ ( p ) ∓ E p ) +1 ς + (cid:18) ∓ p ∗ ( p ) E p (cid:19) = e √ π e − ς +2 ( p ∗ ( p ) ∓ E p ) E p , (42)where we have taken √ re iθ = √ re iθ/ for − π < θ < π , namely, (cid:113) ( p ∗ ( p ) ∓ E p ) = E p ∓ p ∗ ( p ). SUMMARY AND DISCUSSION
In the main text, we have integrated over the times of interaction x and y before the Φ-energy p . Instead wemay first integrate over p before x and y , and see how the time-boundary effect emerges at x , y = T in/outin/out