Constraints on multiparticle production in scalar field theory from classical simulations

Transcription

1 EPJ Web o Conerences 191, 221 (218) Constrants on multpartcle producton n scalar eld theory rom classcal smulatons S.V. Demdov 1,2, and B.R. Farkhtdnov 1,2, 1 Insttute or Nuclear Research o the Russan Academy o Scences, 6th October Annversary prospect 7a, Moscow , Russa 2 Moscow Insttute o Physcs and Technology, Insttutsky per. 9, Dolgoprudny 1417, Russa Abstract. We report numercal results on classcal scatterng o waves n a scalar eld theory and dscuss ther connecton to multpartcle producton. 1 Introducton We start wth some known results on multpartcle producton n standard φ 4 scalar eld theory n 3+1 dmensons. Tree level calculatons [1 4] show that the ampltudes o multpartcle producton, e.g. 1 N process, grow actorally at the threshold wth number N o nal partcles. Correspondng probablty also shows actoral growth and t can be wrtten as ollows ( 1 s assumed) P tree 1 N N!N e N(E) exp( 1 F(N, E)), E = E Nm N where E s total energy o colldng partcles and m s mass o eld quanta. Inspred by ths exponental orm, semclasscal methods to calculate the probablty were developed [5 7] whch among other thngs allowed to estmate ts energy dependence. In partcular, at ultrarelatvstc regme the probablty s energy ndependent and ts behavour as uncton o N s shown n Fgure 1. One can see that t s exponentally suppresses up to N o order 2. Let us note that correspondng calculatons are vald only at N 1 whle at larger N loop correctons are mportant. Also there are untary-base arguments [9, 1] whch tell that ths probablty should be exponentally suppressed even at large N and energes E but actual behavor o the multpartcle probablty n ths regme s stll unknown. Let us make several steps away rom the problem o calculaton o multpartcle ampltudes. Frstly, let us consder nstead o ew N processes the scatterng processes n whch both ntal and nal partcle numbers are large,.e. N N, N, N 1/. In ths case both ntal and nal states o collson process are semclasscal and t s natural to study ths system rom the semclasscal pont o vew. Next, these partcle numbers are large t s natural to study classcal solutons to the eld equatons, whch descrbe scatterng o waves. Namely, we consder solutons whch lnearze at nnte tmes correspondng ntal and nal states can be assocated wth coherent states wth some average partcle numbers N, e-mal: demdov@ms2.nr.ac.ru e-mal: arkhtdnov@phystech.edu, The Authors, publshed by EDP Scences. Ths s an open access artcle dstrbuted under the terms o the Creatve Commons Attrbuton Lcense 4. (

2 EPJ Web o Conerences 191, 221 (218) 5 log P tree 1 N N Fgure 1. Partcle number dependence o log P tree 1 N, see e.g. [8]. N and energy E. Common lore here s that such a classcal soluton wth some partcular partcle numbers N, N and energy E exsts then correspondng quantum scatterng process s classcally allowed and thus ts probablty s not exponentally suppressed. In the opposte case such a process belongs to classcally orbdden regon and ts probablty s expected to be suppressed exponentally. In ths talk we report numercal results [11] on classcal scatterng o wave packets n φ 4 theory. 2 The method For convenence we rescale coordnates and eld to dmensonless unts as x m 1 x and m φ 2 φ and obtan acton n the orm S [φ] = 1 [ 1 d 4 x 2 ( µφ) φ2 1 ] 4 φ4. (1) Ater that the equaton o moton becomes ndependent o couplng constant whch appears as a actor n ront o the acton and plays a role o semclasscal parameter. In what ollows we lmt ourselves to sphercally symmetrc eld conguratons and n ths case t s useul to substtute ollowng anzatz φ(t, r) = 1 r χ(t, r) and obtan the acton n the orm S = 4π dtdr 1 ( ) 2 χ 1 ( ) 2 χ χ2 2 t 2 r 2 χ4 4r 2. (2) In what ollows we solve classcal e.o.m. numercally. To do ths we restrct our solutons to a nte space nterval r [, R] where R s sucently large. At boundares we mpose condtons χ(t, r = ) = and r χ(t, r = R) =. The boundary condtons on the eld χ allow us to expand eld conguraton as ollows 2 χ(t, r) = c n (t) R sn k nr, (3) n= where k n are correspondng wave numbers. We study solutons whch lnearze at ntal and nal tmes. In the lnear regme the tme-dependent Fourer components c n (t) can be convenently wrtten va postve and negatve requency components as ollows ( 1 an c n (t) 2ωn e ωnt + a ne ) ω nt as t ( 1 bn 2ωn e ωnt + b ne ) ω nt as t +. (4) 2

3 EPJ Web o Conerences 191, 221 (218) Usng these representatons one can compute the energy o colldng wave packets as well as ntal and nal partcle numbers E = 4π ω n a n 2 = 4π ω n b n 2, N = 4π a n 2, N = 4π b n 2, (5) n n where ω n = (kn 2 + 1). For convenence we ntroduce the notatons Ẽ = 4π E, Ñ = 4π N, Ñ = 4π N. For numercal mplementaton o the eld equatons on c n (t) we truncate the Fourer expanson (3) at nte number o terms n = N r and solve evoluton equatons usng Bulrsch-Stoer method. We take ntal condtons whch correspond to a wave packet whch s localzed well away rom the nteracton regon and whch propagates towards r =. We choose an nterval I = [r 1, r 2 ] and construct ntal wave packet as ollowng superposton n sn ( k n (r r 1 ) ), r I χ(r) = (6), r I, wave numbers k n are chosen n such a way that ths wave packet nulles at the boundares o the nterval I. The rst dervatve o the eld at ntal tme t n s chosen n such a way that the wave packet starts propagatng to the let: n sn ( k n (r r 1 ) + ω n (t t n ) ), r [r 1, r 2 ] χ (t, r) = (7), r [r 1, r 2 ]. Schematc vew o the ntal wave packet s shown n Fg. 2. n n r 1 r 2 R Fgure 2. Schematc vew o ntal wave packet. Our am s to nd classcal solutons descrbng scatterng o wave packets n whch partcle number changes as much as possble at a gven energy. Ths corresponds to ndng absolute mnmum (or maxmum) o Ñ at xed energy and ntal partcle number wth respect to ntal condtons,.e. Fourer ampltudes n. To nd ts absolute mnmum (maxmum) we use stochastc samplng technque n combnaton wth smulated annealng method, see e.g. [12, 13]. Namely, we generate an ensemble o the classcal solutons wth xed ntal partcle number Ñ weghted by the ollowng probablty p e F, where F = β ( Ñ + ξ(ẽ Ẽ ) 2). At large postve β and ξ the ensemble wll be domnated by solutons havng small F thus they wll concentrate near the smallest possble values o Ñ at energy Ẽ. To reach the the upper boundary Ñ at energy E the sgns o β and ξ should be negatve. To generate such ensemble we use Metropols Monte Carlo 3

4 EPJ Web o Conerences 191, 221 (218) algorthm and vary values o β and ξ to reach the boundary o classcally allowed soluton n the parameter space (E, N ), see Re. [11] or detals. In the next Secton we present obtaned results. 3 Results Below we show regons o the classcal solutons n the parameter space (E, N ) or several derent values o ntal partcle number N. In Fg. 3 we plot the results o the stochastc 1.15 N max(e ) 1.1 N N = 1 1 N mn(e ) E Fgure 3. Classcally allowed regon or wave packet scatterng process n (E, N ) plane. samplng n the plane (E, N ) or N = 1. Here we x R = 2 and Nr = 4. Each dot n the Fg. 3 represents a classcal soluton obtaned on the way to the lower and upper boundares. The pcture s a combnaton o several runs wth several derent values o E chosen to populate representatve energy nterval. The very rst soluton o our procedure was randomly chosen wth the only condtons that t s lnear at ntal and nal tmes and N = 1. Such ntal solutons typcally have very small change n partcle number durng the tme evoluton and thus are stuated near the horzontal lne N = 1 on ths gure. Then we run our procedure as descrbed above. The obtaned regon o the classcal solutons n ths plane max has a smooth envelopes, N mn ( E ) and N ( E ) whch represent boundary o the classcally allowed regon n the parameter space. In Fg.4 we show typcal tme evoluton o an exemplary upper boundary soluton wth E 6.. Form o the ntal wave packet s the same or all boundary solutons whch we have ound or N = 1. Namely, t has relatvely sharp part near ts end where most o ts energy s concentrated. The wave packet reaches nteracton regon, relects and goes back to the lnear regme and the nal wave packet has smlar orm as the ntal one. We study the dependence o the obtaned results on the lattce, namely on the spacal cuto R and on number o Fourer modes Nr. We repeat the same numercal procedure to nd the lower and upper boundares, or the case R = 2, N x = 6 wth smaller lattce spacng and or the case R = 3, N x = 6 wth larger space nterval but the same lattce spacng. We have ound nce concdence n the poston o the boundares o classcally allowed regon and orm o correspondng soluton. A devaton has been observed or the case N x = 6, R = 2 or energes larger than 1 whch relects appearance o new hgh energy modes. 4

5 EPJ Web o Conerences 191, 221 (218) t = t =6.5 t = 12.4 χ(t, r).4 t = 21.3 t = 25.7 t = r Fgure 4. Tme evoluton o an exemplary upper boundary soluton at Ẽ 6.. Now we turn to dscusson o numercal results or another values o ntal partcle number. In Fg. 5 the boundary o the classcally allowed regon or Ñ = 1 s shown wth thck 12. Ñ max(ẽ) 11. Ñ 1. two spkes three spkes Ñ = 1 Ñ mn(ẽ) Ẽ Fgure 5. Boundares o classcally allowed regon or Ñ = 1. sold (red) and thck dashed (blue) lnes. We nd that although the orm o the boundary s n the whole smlar to that o or smaller values o N, ts upper and lower parts actually consst o two derent branches o classcal solutons. At energes lower than about 7 the boundary solutons have two dstnct spky parts n ts ntal wave packet. Correspondng example s presented n the let panel o Fg. 6. We observe that the conguraton conssts o two smlar but space shted conguratons. At energes larger than about 7 the boundary solutons have three spkes n the ntal and nal wave packet. Example o such a soluton s presented n Fg. 6, rght panel. In ths branch o solutons the ntal and nal eld conguratons looks as a sum o three separated n space wavetrans wth smlar Fourer mage. Upper boundary solutons have very smlar propertes. The pcture becomes even more complcated at larger values o N. For Ñ = 3 we have ound that the correspondng boundary solutons contan already 4 7 spkes n the ntal (and 5

6 EPJ Web o Conerences 191, 221 (218) φ(,r), Ẽ = 65 φ(,r), Ẽ = r r Fgure 6. Intal wave packets o lower boundary solutons or Ẽ 65 (let panel) and Ẽ 8 (rght panel). nal) wavetrans. We note that the task o ndng the boundares becomes more complcated n ths case due to the presence o multple local extrema o N. They correspond to solutons wth larger dstances between spkes n the ntal wave packets. Our results ndcate that at larger N the ntal wave packets o boundary solutons tend to consst o more and more space separated wavetrans. It s easy to see that two classcal boundary solutons wth derent sets o parameters exst then there should exst also a classcal soluton descrbng scatterng whch has energy and partcle numbers equal to the sums o the correspondng energes and partcle numbers o these two solutons. One can construct ths soluton explctly by takng t as a sucently separated n space sum o those two solutons. In partcular, ths means that the wdth o the classcally allowed regon n parameter space (Ẽ, Ñ ) should grow wth ncrease o ntal partcle number aster than lnear uncton. Comparson o classcally allowed regons or derent values o Ñ s shown n Fg. 7 (let panel). On the rght panel o the Fgure we Ñ/Ñ Ñ = 3 Ñ = 1 Ñ =1 Ñ =.1 Ñ Ñ Ñ Ñ at = 1 ẼÑ αñ Ẽ/Ñ Ñ Fgure 7. Let panel: comparson o classcally allowed regons or derent values o Ñ. Rght panel: wdth o classcally allowed regon as a uncton o Ẽ/Ñ. present dependence o the derence o ntal and nal partcle number or partcular rato Ẽ/Ñ = 1 as a uncton o ntal partcle number Ñ. One can see that at large values o ntal partcle number the behavour o ths uncton tends to lnear. Ths can be consdered as an ndcaton on exstence o a lmtng boundary o classcally allowed regon at large N. Fnally let us make a connecton o our results wth the most nterestng case whch s 2 N partcle scatterng where N s semclasscally large,.e. 1/. Frst, let us consder 6

7 EPJ Web o Conerences 191, 221 (218) the ollowng nclusve probablty P(N, N, E) = ˆP N Ŝ ˆP N ˆP E 2, (8), where Ŝ s S -matrx, ˆP E, ˆP N and ˆP N are projectors onto subspaces wth xed energy, ntal and nal partcle numbers and sum goes over all ntal and nal states. Ths quantty can be vewed as a total probablty o transton rom states wth ntal partcle number N and nal partcle number N wth energy E. Suppose that the process N N scatterng at energy E conssts o two subprocesses wth some energes E (1), E (2) and partcle numbers N (1), N (2), N (1), N (2). I these subprocesses are separated by very large dstance then ther total probablty s gven by the product P(E (1), N (1), N (1) )P(E (2), N (2), N (2) ). As such subprocesses represent a subset among all possble N N scatterngs at energy E then the ollowng nequalty should be vald P(N (1) + N (2), N (1) + N (2), E (1) + E (2) ) P(N (1), N (1), E (1) )P(N (2), N (2), E (2) ). (9) Let us apply ths property to subprocesses 2 N at energy E and N N at energy E, where N s semclasscally large,.e. N > 1/. We obtan ollowng nequalty P(2 + N, 2N, E + E ) P(2, N, E)P(N, N, E ) (1) The process N N scatterng s classcally allowed at all energes above the threshold. At the same tme the ntal state wth N + 2 number o partcles s semclasscally ndstngushable rom the state wth partcle number equal to N. And our numercal results ndcate that processes N 2N are classcally orbdden at least or consdered range o energes. Then the above nequalty tells us that the processes 2 N should be exponentally suppressed. Another even smpler argument n avor o suppresson o 2 N processes ollows rom T nvarance whch n partcular ensures that P(E, N, N ) = P(E, N, N ). The processes N ew correspond to the lowest part o the (Ẽ, Ñ ) plane n the Fgures 3 and 5 whch mples that they le deep n the classcally orbdden regon. Ths means that the probablty o the process N ew and thus o the process ew N s expected to be exponentally suppressed. 4 Conclusons In summary, we numercally study classcally allowed regon n the parameter space Ẽ, Ñ, Ñ or processes descrbng scatterng o waves n unbroken scalar φ 4 theory and study propertes o boundary solutons at derent values o Ñ. Our results ndcate on exstence o lmtng (at Ñ ) boundary regon o classcally allowed transtons and they mply that the probablty o o 2 N processes at any N (not only at small N) s exponentally suppressed. O course, perormed classcal smulatons tell nothng about actual value o probablty o these processes. We plan to calculate suppresson exponent semclasscally n the classcally orbdden regon startng wth processes when both ntal and nal partcle numbers are semclasscally large. In ths case the problem s reduced to soluton o a correspondng semclasscal boundary value problem and at the end by takng lmt Ñ, see Res. [14 17]. When approachng the boundary o the classcally allowed regon the suppresson exponent should go to zero whch can be used as a check o our procedure. Acknowledgments The work was supported by the RSCF grant The numercal part o the work was perormed on Calculatonal Cluster o the Theory Dvson o INR RAS. 7

8 EPJ Web o Conerences 191, 221 (218) Reerences [1] J. M. Cornwall, Phys. Lett. B (199). [2] H. Goldberg, Phys. Lett. B (199). [3] L. S. Brown, Phys. Rev. D 46 R4125 (1992) [hep-ph/92923]. [4] M. B. Voloshn, Nucl. Phys. B (1992). [5] S. Y. Khlebnkov, V. A. Rubakov and P. G. Tnyakov, Nucl. Phys. B (1991). [6] V. A. Rubakov and P. G. Tnyakov, Phys. Lett. B (1992). [7] D. T. Son, Nucl. Phys. B (1996) [hep-ph/955338]. [8] F. L. Bezrukov, M. V. Lbanov and S. V. Trotsky, Mod. Phys. Lett. A (1995) [hep-ph/95822]. [9] V. I. Zakharov, Phys. Rev. Lett (1991). [1] V. A. Rubakov, hep-ph/ [11] S. V. Demdov and B. R. Farkhtdnov, arxv: [hep-ph]. [12] C. Rebb and R. L. Sngleton, Jr, Phys. Rev. D (1996) [hep-ph/96126]. [13] S. V. Demdov and D. G. Levkov, JHEP (211) [arxv: [hep-th]]. [14] F. L. Bezrukov, D. Levkov, C. Rebb, V. A. Rubakov and P. Tnyakov, Phys. Rev. D (23) [hep-ph/3418]. [15] D. G. Levkov and S. M. Sbryakov, Phys. Rev. D (25) [hep-th/41198]. [16] S. V. Demdov and D. G. Levkov, Phys. Rev. Lett (211) [arxv: [hep-th]]. [17] S. V. Demdov and D. G. Levkov, JHEP (215) [arxv: [hep-th]]. 8