Neutrino oscillations in matter of varying density

Transcription

1 Neutrio oscillatios i atter of varyig desity Paul M. Fishbae * Physics Dept. ad Istitute for Nuclear ad Particle Physics, Uiv. of Virgiia, Charlottesville, VA 904 Peter Kaus ** Aspe Ceter for Physics, Aspe, CO 86 We cosider two-faily eutrio oscillatios i a ediu of cotiuously-varyig desity as a liit of the process i a series of costat-desity layers. We costruct aalytic expressios for the coversio aplitude at high eergies withi a ediu with a desity profile that is piecewise liear. We copare soe cases to uderstad the type of effects that deped o the order of the aterial traversed by a eutrio bea. I. Trasitio aplitudes for structured atter. The proble of eutrio oscillatios i atter is of obvious iportace, ad it is iterestig to see to what extet ad i what aer aalytic solutio is possible. I this ote we discuss a geeral approach to the proble which allows us to solve i a ew way the case of eutrio passage through atter whose desity varies liearly with distace. Uder the assuptio that a two-chael approxiatio to eutrio ixig holds, the aplitude A for passage of a eutrio bea of eergy E through a ediu of costat electro desity, whose properties (desity N ad thicess x we label by, is a atrix whose idices label flavors. It is give by the expressio ( ( A = cosφ + isiφ cos θ σ isiφ si θ σ (. z x where the wor [] of Miheyev ad Sirov ad of Wolfestei [MSW] showed that the effect of the atter is suarized by * e-ail address pfr@virgiia.edu ** e-ail address paus@futureoe.co

2 ad ( = cosθ ξ + si θ ϕ ϕ = 4E x θ θ si θ = EV si ( θ cosθ ξ + si θ (. ξ with V = GFN. (. The ass paraeter = is positive. We recover the vacuu result, Cabibbo agle θ, for V = 0. Let us ow cosider the eleets of the aplitude A.. = A A...A for passage through a series of layers labeled sequetially fro to. These ca be foud either by trace techiques or by the followig direct techique: We extract a factor of cosφ fro each factor A. The the aplitude taes the for where ad A = cosφcosφ cosφ β = β = + σ B B = itaφ si θ,0,cosθ (.4 ( To fid the ordered product over the β, we require products of the for ( B ( B σ σ ( σ B, ad these ca be foud recursively usig the = result B B σ σ B = B + i σ B B. ( ( ( (.5 (.6 (I the recursive developet, it is helpful that the vector B has o y-copoet. The geeral for for our product is thereby foud to be ( σ B( σ B ( σ B = i taφ = (.7 σz cos( θ θ+ + θ σx si( θ θ + + θ odd cos( θ θ + θ + iσ y si( θ θ + + θ eve I tur the trasitio atrix eleets of iterest are

3 ad A A + i taφ cosθ + ( i taφ taφ cos( θ θ = < = cosφ + taφ taφ taφ cos θ θ + θ + (.8a ( i l ( l = < < l ( + ( i taφtaφ taφ cos θ θ + ( θ i taφ siθ + ( i taφ taφ si( θ θ = < = cosφ + taφ taφ taφ si θ θ + θ + (.8b ( i l ( l = < < l ( + ( i taφtaφ taφsi θ θ + ( θ Oe ca iediately chec that A + A =. Recovery of the vacuu result. A useful chec o Eqs. (.8 follows fro the assuptio that the paraeter ξ is sall copared to cosθ for all. We ca thus approxiate θ θ as well as. Assuig that the slabs have equal width, we i additio have taφ ta(φ /, where φ = X/(4E is the agle appropriate to passage of a total legth X through vacuu. Thus whe there are a odd uber of agles, θ i θ θ θ, ad whe there are a eve uber of agles θ i θ +... θ 0. Thus, with φ φ /, the eleet of the aplitude is ( ( + i taφ + A = cos φ ( ( cosθ i ( taφ ( itaφ + + +! iφ iφ iφ iφ = { e e cosθ e e } + + (.9a = cosφ + i cosθsi φ. We have here used Siilarly, ( ( ( (! ( ( (! + x x = x+ x + + x + x = + x +

4 ( ( ( A = cos φ siθ i ( taφ + itaφ +! = cos φ siθ ( + itaφ ( itaφ (.9b si iφ = θ e e + = isiθsi φ. iφ The expressios (.9 atch the vacuu result for a sigle width X, Eq. (. with the agles replaced by their vacuu values. A Give orderig ad its reverse. Oe of the iportat features of oscillatios withi atter is that the aplitude for trasitios depeds o the order of the desity of the layers through which a bea passes. The aplitude for passage through a sigle layer, Eq. (., is syetric, A = A T. Therefore A = ( A T, (.0 where we recall that the sequece of superscripts atches the layer order. That eas i particular that the diagoal (survival eleets are equal, e.g., A = A (. Uitarity i the two chael proble the gives us equal probabilities for the off-diagoal (coversio eleets for a give order ad its reverse, A = A (. This proof fails for the three-chael coversio proble []. To lear the relatio betwee the off-diagoal eleets of the aplitudes theselves, we ote that A has the siple for α β A =, β * α * (. To see this, oe ca for exaple use a recursive proof: Equatio (. shows it is true for oe layer, ad explicit calculatio shows that it is also true for two layers. The oe calculates α β γ δ αγ * * ( A A A βδ αδ + βγ + = + = =, β* α* δ * γ * β* γ α * δ * β * δ + α * γ * ad this has the requisite property. We ca ow put Eqs. (.0 ad (. together to show that α β* A = β α*. (.4 Thus i particular A = A. (.5 ( * These results have bee verified for the particular profiles we study below. 4

5 High eergy liit. Let us call ξ i = EV i / the iiu value tae o by the paraeter ξ as rus fro to. The we ca study a high eergy liit, ξ i >> cosθ, siθ (.6 I this liit siθ siθ θ ξ ξ (.7 ad, because ξ i this liit, we also have φ φ ξ (.8 For arbitrary this could be large or sall; however, because we are ultiately iterested i large ad because for ay situatios this quatity is sall i ay case, we shall also assue, as part of the defiitio of the high eergy liit, that φ is sall for all ad retai oly first order ters i φ. I particular the prefactor of the product over the cosφ i Eqs. (.9 is uity. The effect of our liit is ost easily see i the eleet of the trasitio aplitude A, Eq. (.8b. Let us refer to the -tuple su i the curly bracets o the right of Eq. (.8b as T, so that A is a su over these sus. Geerally T is a fuctio of the potetial V; however for =, which is the oe ter i A that is order idepedet, the potetial depedece cacels, T = i taφ siθ = iφ siθ. (.9 = This leadig ter essetially reproduces the vacuu result, idepedet of. Both the potetial ad the order depedece are preset i the = ter, T ( V = taφ taφ si θ θ < i ( E φ siθ V ( x V ( x < This expressio sets the patter for the geeral ter, ( ( T ( V = taφ taφ si θ θ + θ < < (.0 (. E i φ si θ S (, V where the ultiple su S is defied by S( V = V ( z V( z + (. (. ( ( ( < < V z V z V z I the followig sectio, we cosider these sus for specific potetials. The cotiuu liit of the sus of Eq. (. ca also be foud i the usual aer. With the scalig variables z /, the large liit of S (V is z z S( V dz dz dzv ( z V( z + ( V ( z V ( z V z ( 5

6 (. II. Liear desity profile Here we cosider a siple liear profile V give by V (x = V i + V X/. (. The liear case has i fact bee solved i other ways. I wor [, 4] o passage of eutrios through layers of costat desity atter, a liear desity profile was used to iterpolate the layers, ad i so doig it was oticed that a forally idetical proble had bee solved uch earlier i the cotext of atoic physics [5, 6]. This wor was ore thoroughly recalled ad refied by Petcov [7]. More recetly, aother approach has produced a solutio to the case of liear atter for a arbitrary uber of chaels [8]. I additio there is a body of wor based o various approxiatios[9]. What we preset here differs cosiderably i techique fro the exact wor cited ad the high eergy approxiatio that applies to the exaples below copleets the approxiate wor. For the profile of Eq. (. the double su ter, Eq. (., taes the for X X X ( + ( S( V = V ( = V ( = V (. < = = 6 This behaves as at large, ad sice there is a additioal factor i T, Eq. (.0, T itself has a fiite large liit, aely X ( + ( E T( V = V si 6 φ θ (. φ si θ. The patter is repeated for the geeral ter. We have S( V ( ( + (.4 < < This ultiple su is explicitly calculable for ay fiite but is ot very elighteig. The large behavior is sipler, ad we give here the first results for the first few values of : S V /! ( ( ( ( ( ( 7/5! S V ( 4 ( ( 7/7! ( 4 5 ( ( /9! ( 5 6 ( ( 65/! ( 6 ( 7 ( 7575/! S V S V S V S V (.5 We have igored the ters cotaiig V i because they are oleadig as becoes large. I other words, the expressio of Eq. (.4 is already a large- approxiatio. 6

7 These results ca be foud either with the large behavior of the ultiple sus of Eq. (. or, ore siply, with the ultiple itegrals of the large for Eq. (.. We first rear that the factors of are those ecessary to ae the result fiite, sice T cotais a additioal factor of. The sequece i the deoiator ca be idetified [0] as follows: a( a( T = ( i φ si θ T i, = (! (.6 (! where a( obeys the two-ter recurrece relatio a( + = a( + ( + a(, with a(0 = a( =. (.7 The quatity a( is a expasio coefficiet i several eleetary fuctios, icludig soe cobiatios of iverse trigoetric fuctios ad algebraic fuctios. Perhaps the ost iterestig relatio is y a ( exp / u y = y e du ( +! ( = 0 0 y π = exp / erf y ( y ( y + y exp /, ( y ( = y! = 0 a relatio that ca be applied to our case with the substitutio The result is ( + y= exp ( iπ /4. (.8 π A( V = T exp ( y / erf ( y (.9 y III. Copariso to other desity profiles It is istructive to copare the results of the previous sectio with two other desity profiles of the sae total thicess X, each represetig a rearrageet of the atter that coposes the profile V, i. e., each havig the sae itegral of V over x fro 0 to X. Specifically, we cosider a profile peaed at the ceter, X V =, V( x = Vi + X V (. = +, ad a costat profile, V (x = V i + V X/. (. We agai wor at our high eergy regie. 7

8 The profile V ca be treated by ay of the sae techiques that we used for V, eve if the algebra is rather ore coplicated. We fid for the double su ter, Eq. (. with =, S( V = V( x V( x =. (. < This result should be copared to the correspodig oe for S (V, Eq, (., which is proportioal to at large. The ter T (V vaishes i the large- liit. The cacellatio is a cosequece of the syetry of the atter distributio about X/ (as will be cofired below for the V case; ideed it is geerally true that S (V = 0 for eve values of i the large- liit. For odd values of, however, there is a ozero liit. We have wored through the first few odd- expressios for S (V for fiite. These are ot siple, eve i the large- liit, uless we set V i to 0, which we do to ae the expressios clear. The the large- results for all are 0 eve S ( V ( (.4 odd (!! This gives for the coversio aplitude i A( V = T =, (.5 (!! where T is the aplitude for passage through a sigle layer of vacuu, Eq. (.9. The su o w the right is a Loel fuctio [] U ( w,0 = cos w( t dt π, ad 0 π A( V = T U,0. (.6 The Loel fuctio is associated with diffractio fro edges. The aplitude for V is siply give by the MSW result of Eq. (., i.e. A( V = isiθv si φ V. I the high eergy liit we have φv φ ξv = ad the coversio aplitude becoes siθ φ siθ A( V i si i si ξ = φ ξ V V si si = T = T. φ ξv (.7 Strictly speaig, this violates our high eergy coditio Eq. (.6; however, because V is oly o-zero for a arbitrarily sall rage of x this should ot be troublesoe. 8

9 The coplete high eergy expressios of Eqs. (.6 ad (.7 are ot very useful as grouds for copariso with the result for V, Eq. (.9. It is ore trasparet to expad each result for sall values of V X (as well as for V i = 0. I that case, i 7 A( V T +! 5! (.8a A( V T + 5!! (.8b A( V T + 4 (.8c Fro these the coversio probabilities are, to leadig order i V X, 4 P( V T + 45 (.9a P( V T + 5 (.9b P( V T + (.9c The ratios of these probabilities is i priciple subect to experietal test, although it is clear that such tests would be very difficult. Acowledgeets We would lie to tha the Aspe Ceter for Physics, where uch of this wor was doe. PMF would also lie to tha Doiique Schiff ad the ebers of the LPTHE at Uiversité de Paris-Sud for their hospitality. This wor is supported i part by the U.S. Departet of Eergy uder grat uber DE FG0-97ER407. Refereces. L. Wolfestei, Phys. Rev. D7, 69 (978; S. P. Miheyev ad A. Yu. Sirov, Sov. J. Nucl. Phys. 4, 9 (985.. P. M. Fishbae ad P. Kaus, hep-ph/ W. C. Haxto, Phys. Rev. Lett. 57 ( S. J. Pare, Phys. Rev. Lett. 57 (

10 5. L. D. Ladau, Phys. Z. USSR ( C. Zeer, Proc. R. Soc. A 7 ( S. T. Petcov, Phys. Lett. B 9 ( H. Leha, P. Oslad, ad T. T. Wu, hep-ph/ See for exaple T. K. Kuo ad J. Pataleoe, Phys. Rev. Lett. 57 ( ; S. P. Rose ad J. M. Gelb, Phys. Rev. D 4 ( ; V. Barger, R. J. N. Phillips, ad K. Whisat, Phys. Rev. D 4 ( P. S. Bruca, Fib. Quarterly 0, 69 (97. This idetificatio was ade usig The Ecyclopedia of Iteger Sequeces ( G. N. Watso, A Treatise o the Theory of Bessel Fuctios, d ed., Cabridge [Eg.] The Uiversity Press; New Yor, The Macilla Copay, 944; K. W. Kocehauer, A. der Physi ud Cheie ( XLI, 04 (87. 0