PHYSICAL REVIEW D, VOLUME 61, 077301

Long distance effects inK¿\p¿nn

Gino Segre`Department of Physics, DRL 2N-1, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396

~Received 16 August 1999; published 6 March 2000!

A long distance contribution to the amplitude forK1→p1nn is calculated. Its magnitude, comparable tothat of other long distance effects, is three orders of magnitude smaller than the short distance effects, but thetechniques involved may have other applications.

PACS number~s!: 12.15.Ji, 13.20.Eb

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Rare K decay amplitudes such asK1→p1nn are pre-sumed to receive contributions from two separate sourusually referred to as short-distance~SD! and long-distance~LD! effects. The former is an effective four-fermion inteaction resulting from quark and lepton contributions to a bdiagram while the latter is typically due to hadronic intermdiate states.

Long-range effects have been considered by a serieauthors. For instance, Rein and Seghal@1#, Lu and Wise@2#,and Genget al. @3# have evaluated in a variety of ways theffective matrix element by modeling theK2p2Z vertexand then having theZ couple ton2 n. Others, such as Marciano and Passa@4# or Buchalla and Buras@5# have evaluatedgluonic contributions to penguin diagrams. Bigi and Gabani @6# have considered extensions of the standard moincluding those due to right-handed currents, nonminimHiggs sectors, additional families, and supersymmetry. Nof these effects significantly enhance or suppress the sdard model estimate. There is also an excellent compresive review of rareK decays in the encyclopedic article bBuchalla, Buras, and Lautenbacher@7#. At the same time,we should add that there is continued interest in explorthis rare decay mode as a test of the standard model. Frecent experimental review of the situation see, e.g., theper by Adler et al. @8#. Part of the interest in theK1

→p1nn decay is that the experiment, though very difficuis feasible and the results give us some clear informaabout standard model parameters since the the LD effewith their hadronic unknown quantities, appear to be cleasome three orders of magnitude smaller than the SD effeThe latter contribution to the branching ratio@9# is of order10210, while the LD effects are of order 10213. A secondreason for interest in the decay is the connection toCP vio-lation, which could be studied@10# in KL→pnn.

In this paper we plan to study a not as of yet calculaclass of long-range interactions and show how their conbution to the decay amplitude can be evaluated. The returns out to be surprisingly simple and potentially of interfor other applications as well. The relevant diagram, simin some ways to one estimated for doubleb decay, is dis-played in Fig. 1. To complete the calculation we must callate the amplitude forus→K1 and ud→p1. This is easilydone by using partial conservation of axial vector curredenoting by f p,K the pion and kaon decay constants, tmatrix element forK1→p1nn is evaluated with the resul@11#

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2f p f Kq1

aq2bTab sinuc cosuc .

In the aboveTab is the amplitude forW1W2→nn byexchange of an electron, the central part of Fig. 1. The aplitudeTab is reasonably complicated, but taking the doubdivergence reduces drastically the kinematic complexparticularly in the limit of vanishing electron mass. We finfor the diagram of Fig. 1 that

q1aq2

bTab52 n~p1!gsq1sn~p2!F11

me2

~mp2 22q1•p12me

2!G .In the limit me→0, the total result for the amplitude i

then

M54S GF

&D 2

f p f Kn~p1!gs~q1s2q2

s!n~p2!sinuc cosuc ,

and the comparison to the standard decay amplitudeN forK1→p0e1n is very simple. We find

M

N52 cosucf p f KS GF

&D ,

so that the branching ratio forK1→p1nn is approximately2.5310214. The muon contribution is also significant thougsmaller than the electron contribution since there is a pacancellation in the expression forq1

aq2bTab . Not surprisingly

the cancellation is essentially complete for thet lepton. Tosummarize, the LD effects for the diagrams of Fig. 1 a<5310214, comparable to other LD contributions, but cetainly not larger.

FIG. 1. Contribution to long distance effects inK1→p1nn.

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BRIEF REPORTS PHYSICAL REVIEW D 61 077301

We mentioned in the beginning that these diagrams bsome resemblance to those in neutrinoless doubleb decay@12#. There, of course, both outgoing particles are leptorather than one lepton and one antilepton, and the intermate propagator is lepton number violating. The double divgence still simplifies the calculation considerably so that ocould get a reasonably simple estimate of, e.g., the lo

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range pion exchange contribution to neutrinoless doublbdecay.

This research was supported by the Department of Eneunder Grant No. 3071-T. We would also like to thank Prfessor Paul Langacker and William Marciano for helpfremarks.

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@1# D. Rein and L. M. Seghal, Phys. Rev. D39, 3325~1989!.@2# D. Lu and M. Wise, Phys. Lett. B355, 569 ~1995!.@3# R. Genget al., Phys. Lett. B355, 569 ~1995!.@4# W. Marciano and Z. Parsa, Phys. Rev. D53, 1 ~1996!.@5# A. Buchalla and A. Buras, Nucl. Phys.B412, 106 ~1994!.@6# I. Bigi and F. Gabbiani, Nucl. Phys.B367, 3 ~1991!.@7# A. Buchalla, A. Buras, and G. Lautenbacher, Rev. Mod. Ph

68, 1125~1996!.

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@8# S. Adleret al., Phys. Rev. Lett.79, 2204~1997!.@9# G. Buchalla and A. J. Buras, Phys. Rev. D54, 6782~1996!.

@10# Y. Grossman, Y. Nir, and R. Rattazzi, ‘‘CP Violation Beyondthe Standard Model,’’ report SLAC-PUB-7379~1999!.

@11# T. P. Cheng and L. F. Li,Gauge Theory of Elementary Particles ~Clarendon, Oxford, 1984!.

@12# C. W. Kim and A. Pevsner,Neutrinos in Physics and Astrophysics~Gordon and Breach, London, 1994!.

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