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Evidence for energy dependence of photo-nucleon cross section from cosmic ray experiments
Volume 13, number 3
PHYSICS LETTERS
References 1 S Wemberg, Phys. Rev 128 (1962) 1457 2 J Bernstem, M Ruderman and G Femberg, Phys Rev 132 (1963) 1227, G Femberg, Int Conf cnCesmac Rays, Jakuur (1963) 3 D H Perkins, Int. Conf on Cosmac Rays,Ja~pur (1963) Vol 6 4. J.L m s el y , Intern Conf. on Cosmic Rays, Ja~pur (1963) Vol 4, 77 5 J Lmsely, Phys. Rev. Letters 9 (1962) 126), (Lmsely finds very small fluctuations m the muon ratio
1 December 1964
of air-showers produced by prtmarles of 108-109 GeV and concludes that the chemical composihon of these primaries must be pure. Presence of yrays amongst these prnnar~es will produce a~r showers w~th abnormally low number of muons. Hence, m view of h~s observations, ~ ~s a safe upper limit on the fraction of y-rays amongst the primarles m this energy range.) T D Lee and C N Yang, Phys Rev. Letters 4 (1960) 307 Ya B Zeltdowch and Ya A.Smorodmskn, Zh El~sperlm 1 Teor Flz 41 (1961)907 (Sowet Phys JETP, 14 (1962) 647)
EVIDENCE FOR ENERGY DEPENDENCE OF PHOTON-NUCLEON CROSS SECTION FROM COSMIC RAY EXPERIMENTS * H S MURDOCH and H D R A T H G E B E R
Falk~ner Nuclear Department Received 30 October 1964
The e n e r g y dependence of the photon-nucleon c r o s s s e c t i o n at high e n e r g i e s i s one of the p r o b l e m s of e l e m e n t a r y p a r t i c l e p h y s i c s which awaits solution both f r o m the t h e o r e t i c a l and e x p e r i m e n t a l points of v i e w The r e c e n t w o r k of K r a m e r and Stichel [1] and J a m e e l [2] shows that e s s e n t i a l l y only the 0 and w - m e s o n s c o n t r i b u t e to the c r o s s s e c t i o n and that at high e n e r g i e s it f a l l s off as E - ~ The photon-nucleon c r o s s s e c t io n ~ should r i s e s h a r p l y at e n e r g i e s a l i t t l e above the pion r e s t m a s s , go through a m a x i m u m and s t a r t to f a l l off t o w a r d s h i g h e r e n e r g i e s above a few 0 restmasses D i r e c t m e a s u r e m e n t s of ~ have been m a d e up to 1150 MeV [3,4] They show m f a c t that it r i s e s quickly to 270 ± 40 /lb at 300 MeV and r e m a i n s e s s e n t i a l l y constant to 1 GeV T h i s h a s been e x tended to 4 GeV but only f o r the d i f f e r e n t i a l c r o s s s e c t i o n s f o r s i n g l e pion production [5] at c.m. a n g l e s of 60 ° and 90 ° . T h e g e n e r a l p i c t u r e is one of r a p i d d e c r e a s e with e n e r g y and angle. T h e s e r e s u l t s e n c o u r a g e u s to p u b h s h conc l u s i o n s we have r e a c h e d f r o m a study of c o s m i c r a y muon e x p e r i m e n t s . Our i n t e r e s t a r o s e f r o m the d i s c r e p a n c y [6] m the c o m p a r i s o n of the m o m e n t u m s p e c t r u m of m u o n s a s m e a s u r e d by m a g n e t i c d e f l e c t i o n and t h e i r a b s o r p t i o n c u r v e m thick l a y e r s of r o c k In f a c t the only s o u r c e of * Thin work has been supported by the Nuclear Research Foundation w~thm the University of Sydney
i n f o r m a t i o n on a7 above a few GeV i s the n u c l e a r mteractxon of muons [7] whxch t a k e s p l a c e by the exchange of a v i r t u a l photon wxth a nucleon I m txally the muon m t e r a c t l o n c r o s s section a g was obtained f r o m a constant ay and f r o m a w r t u a l photon s p e c t r u m c a l c u l a t e d by the method of Wxlh a m s and W m z a c k e r [8]. M o r e r e c e n t quantum d y n a m i c a l c a l c u l a t i o n s [9-11] follow this b a s i c a p p r o a c h H o w e v e r they a r e not, h k e the method of W i l l i a m s and W e l z a c k e r , su b j ect to any h m l tatlon on the upper h m l t of the v i r t u a l photon energy In shower m e a s u r e m e n t s u n d e r g r o u n d [12] t r a n s f e r r e d e n e r g i e s m the r a n g e 5 to 50 C-eV w e r e r e c o r d e d . A s the e n e r g y of the incident m u ons was not m e a s u r e d it was n e c e s s a r y to roteg r a t e o v e r t h e i r e n e r g y s p e c t r u m . Since the steepn e s s of t h i s s p e c t r u m e n h a n c e s the influence of f r a c t i o n a l l y l a r g e e n e r g y t r a n s f e r s , the t h e o r y of W l l l m m s and W e l z a c k e r cannot l e g i t i m a t e l y be applied The r e s u l t s have been i n t e r p r e t e d a s f o l lows the s e v e r a l d e r i v e d p h o t o n - s p e c t r a with the a s s u m p t i o n of a constant a 7 o v e r the e n e r g y r a n g e ~(~b)
Theory
Interpretation
260
Wxlllams and Welzacker [8]
Hlgashi et al [11]
62
Dalyasu et al [8]
Dalyasu et al [8]
24
Kessler and Kessler [9]
Higashi et al [11]
Crossland and Fowler [10]
Crossland and Fowler [10]
9-15
267
Volume 13, number 3
PHYSICS LETTERS
C o m p a r i s o n of t h e s e r e s u l t s with the d n ' e c t l y m e a s u r e d ~r at 1 GeV i n d i c a t e s an e n e r g y d e p e n d e n c e ~ E - 1 above a f e w GeV. It i s a l s o w o r t h y of note that the quantum d y n a m i c a l t h e o r i e s p r e d i c t the c o r r e c t shape of the d i s t r i b u t i o n m four-vector momentum. F u r t h e r e v i d e n c e can be obtained f r o m a c o m p a r i s o n of the muon e n e r g y s p e c t r u m and u n d e r ground muon i n t e n s i t y T h i s c o m p a r i s o n g i v e s the e n e r g y l o s s of muons. A s e n e r g y l o s s p r o c e s s o t h e r than t h e n u c l e a r i n t e r a c t i o n component a r e f a i r l y well u n d e r s t o o d , the l a t t e r i s obtained as a residual term A c a r e f u l c o m p a r i s o n has been m a d e f o r m e s o n e n e r g i e s up to about 600 GeV [6] (where the muon s p e c t r u m has been d i r e c t l y m e a s u r e d by Hayman and Wolfendale [13]) It was found that a s s u m i n g ~r to be constant at 100 ~b and using the t h e o r y of K e s s l e r and K e s s l e r [9], the c a r e f u l l y m e a s u r e d i n t e n s i t y of B a r r e t t et al [14] was about 50% h ig h e r than p r e d i c t e d . The d i s c r e p a n c y w a s r e d u c e d but not e h m m a t e d by u s m g the t h e o r y of W i l l i a m s and W e l z a c k e r with the s a m e v a l u e of ~y (i e only 40% of the v al u e at 1 GeV). It now s e e m s p o s s i b l e to r e s o l v e t h i s d i s c r e pancy If we take the r e s u l t s of H lg a s h l as i n d i c a t ing ~y ~ E - 1 above 1 GeV then n u c l e a r e n e r g y l o s s of m u o n s i s l e s s than one p e r cent of the t o t a l e n e r g y l o s s at a l l muon e n e r g i e s (a n e g a t i v e exponent of s o m e w h a t l e s s than unity would s t i l l suffice to m a k e the contribution n e g h g l b l e f o r c a l c u l a t i n g u n d e r g r o u n d intensity) The e h m m a tlon of n u c l e a r e n e r g y l o s s b r i n g s the B a r r e t t m t e n m t y [14] into a g r e e m e n t with the p r e d m t e d va lue within e x p e r n n e n t a l e r r o r ( ~ 10%) The s m a l l d i s c r e p a n c y which r e m a i n s o v e r the whole r e g i o n between 50 and 500 GeV and which i s not r e l e v a n t to the m a i n a r g u m e n t m a y find its e x planation in the low r e l i a b i l i t y of the i n t e n s i t y m e a s u r e m e n t s m this r e g i o n and m the s m a l l e r c o n t r i b u t i o n s to e n e r g y l o s s of both p a i r p r o d u c tion [15] and l o n i s a t l o n [16] which have been a d vanced recently The muon i n t e n s i t y has r e c e n t l y been m e a s u r e d to depths of 840 kg cm -2 by Mlyake et al [17] T h i s c o r r e s p o n d s to muon e n e r g i e s of o r d e r 10 000 C-eV At t h e s e e n e r g i e s the muon e n e r g y s p e c t r u m h as not been d i r e c t l y m e a s u r e d but i s e s t i m a t e d f r o m the high altitude r - r a y s p e c t r u m of Duthle et al [18] Mlyake et al. [19] fred r e a sonable a g r e e m e n t by r a t h e r a r b i t r a r i l y r e j e c t i n g n u c l e a r e n e r g y l o s s , e x c e p t that the u n d e r g r o u n d i n t e n s i t y i s a little high and they i n t e r p r e t t h i s in t e r m s of kaon p a r e n t a g e c o m p e t i n g with plon p a r e n t a g e of the m u o n s at high e n e r g y T h i s d i s c r e pa nc y would be v e r y m u c h g r e a t e r ff n u c l e a r l o s s 268
1December 1964
w e r e a p p r e c i a b l e as it would be f o r c o n s t a n t a r It r e m a i n s to be shown why c o m p a r i s o n of ex p e r i m e n t a l r e s u l t s with c a l c u l a t i o n s b ased on a constant photon-nucleon c r o s s s e c t i o n and a W i l h a m s and W e l z a c k e r v i r t u a l photon s p e c t r u m do not v i o l e n t l y d i s a g r e e [12, 20]. Th e cut off m the v i r t u a l photon s p e c t r u m f o r s m a l l i m p a c t p a r a m e t e r s m the W i l l i a m s and W e l z a c k e r t h e o r y p r o v i d e s b r o a d l y that kind of d e c r e a s e of ap with e n e r g y which r e s u l t s f r o m a d e c r e a s m g a r Howe v e r the W l l l m m s and W m z a c k e r a p p r o a c h n e i t h e r f i t s al l r e s u l t s as w e l l a s the solution p r o p o s e d h e r e n o r is it a c c e p t a b l e f r o m a t h e o r e t i c a l pomt of v i e w A l l the a v a i l a b l e c o s m i c r a y data a r e not m e r e l y c o n s i s t e n t with but f a v o u r a t o t a l p h o t o n - n u cl eo n c r o s s sect i o n which d e c r e a s e s above a few GeV as E - ~ w h e r e a is c l o s e to unity This e n e r g y dependence is quite d i f f e r e n t f r o m that of n u c l e o n - n u cleon and p I o n - n u c l e o n i n t e r a c t i o n s which a r e e s s e n t i a l l y constant up to at l e a s t 1013 eV [21] T h i s d i f f e r e n c e i s u n d e r s t a n d a b l e si n ce the l a t t e r take p l a c e through P o m e r a n c h u k t r a ] e c t o r m s w h e r e a s the p h o t o n - n u c l e o n i n t e r a c t i o n i s e x c e p t i o n a l m that it is g o v e r n e d by l o w e r Reg g e t r a j e c t o r i e s We would h k e to acknowledge u s e f u l d i s c u s sions with D o c t o r M J a m e e l and P r o f e s s o r S T Butler on the t h e o r e t i c a l p o i n t s i n v o l v e d
References 1 G Kramer and P Stmhel, Zs f Phys 178 (1964) 519 2 M Jameel, private commumcatlon 3 C Castagnoh, M Muchmk, G Ghlgo and R Rmzlvlllo, Nuovo Clmento 16 (1960) 683 4 C E Roos and V Z Petersen, Phys Rcv 124 (1961) 1610 5 R Alvarcz, Z Bar-Yam, W Kern, D Lukey, L S Osborne and S Tazzar, Phys Rev Letters 12 (1964) 707 6 M J Campbell, H S Murdoch, K W Ogllvle and H D Rathgeber, Nuovo Clmento 28 (1963) 885 7 G N Fowler and A W Wolfcndale, Progr m Elementary Partmle and Cosmm Ray Physms, Vol 4 (North-Hol] Publ Comp ,Amsterdam 1958) 8 E P George and J Evans, Proc Phys Soc A63 (1950) 1248 9 D Kessler and P Kcsslcr, Compt Rcnd 244(1957) 1896 10 K Dalyasu, K Kobayakawa, T Murota and T Nakano, J Phys Soc Japan 17, suppl A-HI, 1962 Part III p 344 11 A D Crossland and G N Fowler, Nuclear Physms 53 (1964) 273 12 S Hlgashl, T Kltamura, Y Mlshlma, S Mltagl, S Mlyamoto, T Oshm, H Shlbata, K Wanatabe and Y Watase, J Phys Soc Japan, Suppl A-III (1962) Part III, 362 13 P J Hayrnan and A W Wolfendale, Proc Phys Soc 80 (1962} 710
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14. P.H Barrett, L.M.Bollinger, G.Coccom, Y Etsenberg and K Gretsen, Rev. Mod. Phys. 24 (1952) 133. 15. C.Castagnoh, A De Marco and R Scrimaglio, Nuovo Cimento 33 (1964) 722. 16. K . I . A l e k s e e v a , G B.Zhdanov, M.I.Tret'Yakova and M N.Sheherbakova, Soviet Phys JETP 17 (1963) 1254. 17 S.Miyake, V.S.Narastmham and P V Ramana Murthy, Nuovo Cimento 32 (1964) 1505.
ON THE
SUPERPOSITION
PRINCIPLE
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18 J Duthie, P .H .F o w l er , A.Kaddoura, D.H.Perkms and P.Pinkau, Nuovo Cnnento 24 (1962) 122. 19 S.Miyake, V . S . N a r a s i m h a n a n d R . V Ramana Murthy, Nuovo Cimento 32 (1964) 1524. 20 P.J.Hayman, N.S Palmer and A.W Wolfendale, l>roc. Roy Soc. 275A (1963) 391. 21. G.Day, C.F.Gauld, C.B.A.MeCusker and L.S. Peak, Nuovo Ctmento 27 (1963) 977.
AND
CP
INVARIANCE
IN K ° DECAY
B. LAURE NT NORDITA, Copenhagen, Denmark, and Instztute f o r Theoretwal Physws, Umverszty of Stockholm, Sweden and M. ROOS
NORDITA , Copenhagen, Denmark Received 5 November 1964
Since 1957 when Wu made the d m c o v e r y that p a r i t y is not c o n s e r v e d in weak l n t e r a c h o n s one has g e n e r a l l y b e l i e v e d that C P m the p a r t i c l e t r a n s f o r m a t x o n c o r r e s p o n d i n g to s p a c e lnversxon The r e c e n t e x p e r i m e n t by C h r i s t e n s o n et al [1] shows that the t h e o r y of KO d e c a y xs b a m c a l l y w r o n g in s o m e r e s p e c t The i n t e r p r e t a t i o n c l o s e s t at hand is that C P m v a r m n c e m v~olated. It s e e m s to be v e r y d f f h c u l t to r e p e a t the t r i c k used in c o n nectxon with P v l o l a h o n and fred a new p a r t i c l e o p e r a t o r c o r r e s p o n d i n g to s p a c e m v e r s l o n . Hence, we would probably have to abandon l n v a r l a n c e u n d e r s p a c e r e v e r s i o n c o m p l e t e l y if C P l n v a r m n c e did not hold T h i s would be such a s e r i o u s change m our m o s t profound c o n c e p t io n s that one should t r y a l t e r n a t i v e ch an ge s m the f u n d a m e n ta l p r i n c i p l e s of physxcs to s e e if they could a l s o explain the e x p e r i m e n t In this p a p e r , we would like to s u g g e s t that the quantum m e c h a m c a l s u p e r p o s l h o n p r l n m p l e be c a r e f u l l y e x a m i n e d in the c a s e of KO de c a y F o r s o m e t i m e we have c o n s i d e r e d g e n e r a l m e t h o d s f o r t e s t i n g the s u p e r p o s l t l o n p r i n c i p l e in d i f f e r e n t c a s e s . (Some a s p e c t s have a l r e a d y been publi sh ed by one of us [2] ) Postponing t h e s e c o n s l d e r a t i o n s , we s h a l l h e r e deal e x c l u s i v e l y with K° d e c a y and u s e a v e r y p a r h c u l a r m o d e l only to show q u a h t a h v e l y the kind of e f f e c t s which could be e x p e c t e d f r o m n o n - l m e a m t i e s in an m t e r f e r -
ence-type experiment Nobody can deny the p o s s l b l h t y that the s u p e r p o s i t i o n p r i n c i p l e m a h n e a r a p p r o x i m a t i o n of a m o r e g e n e r a l , n o n - h n e a r law of n at u r e. The obvious w e a k n e s s of such an a s s u m p t i o n is the ab s e n c e of any hint w h a t s o e v e r as to how the s u p e r p o s i t i o n p r i n c i p l e could be g e n e r a h z e d A w a r e of t h i s, we want to s t r e s s the ad hoc c h a r a c t e r of the n o n - l i n e a r m o d el f o r K ° decay that we now a r e golng to u se Thin m o d el has as its only p u r p o s e to show that a C P mvarzant but n o n - l i n e a r theory can g~ve r e s u l t s m a g r e e m e n t u'zth the e x p e r z m e n t s . Let KO and KO stand f o r a m p h t u d e s of KO and ~ o m e s o n s , as is c u s t o m a r y A s s u m e , a l s o as usual, that the s p a c e m v e r s x o n c o r r e s p o n d s to the t r a n s f o r m a t i o n K ° ~ ~ o m the c e n t r e of m a s s s y s t e m (The c e n t r e of m a s s s y s t e m is going to be used throughout ) We w i l l now i n t r o d u c e a n o n - h n a r t e r m m the w e l l - k n o w n e q u a h o n s of motion f o r K ° and ~ o Not to u p set the e q u a h o n s too much we shall l e a v e t h e i r h o m o g e n e o u s c h a r a c t e r unchanged, writing dK o Ko _ ~ o - d---~ = a K ° + b K O + E K ° . KO+~o , (1) K o _ ~o
dK°dt = a K ° + b K ° - EK° KO+KO
(2)
Notice that this s y s t e m of e q u a h o n s is m v a m ant u n d er the C P t r a n s f o r m a t i o n K o ~ ~ o Adding 269
PHYSICS LETTERS
References 1 S Wemberg, Phys. Rev 128 (1962) 1457 2 J Bernstem, M Ruderman and G Femberg, Phys Rev 132 (1963) 1227, G Femberg, Int Conf cnCesmac Rays, Jakuur (1963) 3 D H Perkins, Int. Conf on Cosmac Rays,Ja~pur (1963) Vol 6 4. J.L m s el y , Intern Conf. on Cosmic Rays, Ja~pur (1963) Vol 4, 77 5 J Lmsely, Phys. Rev. Letters 9 (1962) 126), (Lmsely finds very small fluctuations m the muon ratio
1 December 1964
of air-showers produced by prtmarles of 108-109 GeV and concludes that the chemical composihon of these primaries must be pure. Presence of yrays amongst these prnnar~es will produce a~r showers w~th abnormally low number of muons. Hence, m view of h~s observations, ~ ~s a safe upper limit on the fraction of y-rays amongst the primarles m this energy range.) T D Lee and C N Yang, Phys Rev. Letters 4 (1960) 307 Ya B Zeltdowch and Ya A.Smorodmskn, Zh El~sperlm 1 Teor Flz 41 (1961)907 (Sowet Phys JETP, 14 (1962) 647)
EVIDENCE FOR ENERGY DEPENDENCE OF PHOTON-NUCLEON CROSS SECTION FROM COSMIC RAY EXPERIMENTS * H S MURDOCH and H D R A T H G E B E R
Falk~ner Nuclear Department Received 30 October 1964
The e n e r g y dependence of the photon-nucleon c r o s s s e c t i o n at high e n e r g i e s i s one of the p r o b l e m s of e l e m e n t a r y p a r t i c l e p h y s i c s which awaits solution both f r o m the t h e o r e t i c a l and e x p e r i m e n t a l points of v i e w The r e c e n t w o r k of K r a m e r and Stichel [1] and J a m e e l [2] shows that e s s e n t i a l l y only the 0 and w - m e s o n s c o n t r i b u t e to the c r o s s s e c t i o n and that at high e n e r g i e s it f a l l s off as E - ~ The photon-nucleon c r o s s s e c t io n ~ should r i s e s h a r p l y at e n e r g i e s a l i t t l e above the pion r e s t m a s s , go through a m a x i m u m and s t a r t to f a l l off t o w a r d s h i g h e r e n e r g i e s above a few 0 restmasses D i r e c t m e a s u r e m e n t s of ~ have been m a d e up to 1150 MeV [3,4] They show m f a c t that it r i s e s quickly to 270 ± 40 /lb at 300 MeV and r e m a i n s e s s e n t i a l l y constant to 1 GeV T h i s h a s been e x tended to 4 GeV but only f o r the d i f f e r e n t i a l c r o s s s e c t i o n s f o r s i n g l e pion production [5] at c.m. a n g l e s of 60 ° and 90 ° . T h e g e n e r a l p i c t u r e is one of r a p i d d e c r e a s e with e n e r g y and angle. T h e s e r e s u l t s e n c o u r a g e u s to p u b h s h conc l u s i o n s we have r e a c h e d f r o m a study of c o s m i c r a y muon e x p e r i m e n t s . Our i n t e r e s t a r o s e f r o m the d i s c r e p a n c y [6] m the c o m p a r i s o n of the m o m e n t u m s p e c t r u m of m u o n s a s m e a s u r e d by m a g n e t i c d e f l e c t i o n and t h e i r a b s o r p t i o n c u r v e m thick l a y e r s of r o c k In f a c t the only s o u r c e of * Thin work has been supported by the Nuclear Research Foundation w~thm the University of Sydney
i n f o r m a t i o n on a7 above a few GeV i s the n u c l e a r mteractxon of muons [7] whxch t a k e s p l a c e by the exchange of a v i r t u a l photon wxth a nucleon I m txally the muon m t e r a c t l o n c r o s s section a g was obtained f r o m a constant ay and f r o m a w r t u a l photon s p e c t r u m c a l c u l a t e d by the method of Wxlh a m s and W m z a c k e r [8]. M o r e r e c e n t quantum d y n a m i c a l c a l c u l a t i o n s [9-11] follow this b a s i c a p p r o a c h H o w e v e r they a r e not, h k e the method of W i l l i a m s and W e l z a c k e r , su b j ect to any h m l tatlon on the upper h m l t of the v i r t u a l photon energy In shower m e a s u r e m e n t s u n d e r g r o u n d [12] t r a n s f e r r e d e n e r g i e s m the r a n g e 5 to 50 C-eV w e r e r e c o r d e d . A s the e n e r g y of the incident m u ons was not m e a s u r e d it was n e c e s s a r y to roteg r a t e o v e r t h e i r e n e r g y s p e c t r u m . Since the steepn e s s of t h i s s p e c t r u m e n h a n c e s the influence of f r a c t i o n a l l y l a r g e e n e r g y t r a n s f e r s , the t h e o r y of W l l l m m s and W e l z a c k e r cannot l e g i t i m a t e l y be applied The r e s u l t s have been i n t e r p r e t e d a s f o l lows the s e v e r a l d e r i v e d p h o t o n - s p e c t r a with the a s s u m p t i o n of a constant a 7 o v e r the e n e r g y r a n g e ~(~b)
Theory
Interpretation
260
Wxlllams and Welzacker [8]
Hlgashi et al [11]
62
Dalyasu et al [8]
Dalyasu et al [8]
24
Kessler and Kessler [9]
Higashi et al [11]
Crossland and Fowler [10]
Crossland and Fowler [10]
9-15
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PHYSICS LETTERS
C o m p a r i s o n of t h e s e r e s u l t s with the d n ' e c t l y m e a s u r e d ~r at 1 GeV i n d i c a t e s an e n e r g y d e p e n d e n c e ~ E - 1 above a f e w GeV. It i s a l s o w o r t h y of note that the quantum d y n a m i c a l t h e o r i e s p r e d i c t the c o r r e c t shape of the d i s t r i b u t i o n m four-vector momentum. F u r t h e r e v i d e n c e can be obtained f r o m a c o m p a r i s o n of the muon e n e r g y s p e c t r u m and u n d e r ground muon i n t e n s i t y T h i s c o m p a r i s o n g i v e s the e n e r g y l o s s of muons. A s e n e r g y l o s s p r o c e s s o t h e r than t h e n u c l e a r i n t e r a c t i o n component a r e f a i r l y well u n d e r s t o o d , the l a t t e r i s obtained as a residual term A c a r e f u l c o m p a r i s o n has been m a d e f o r m e s o n e n e r g i e s up to about 600 GeV [6] (where the muon s p e c t r u m has been d i r e c t l y m e a s u r e d by Hayman and Wolfendale [13]) It was found that a s s u m i n g ~r to be constant at 100 ~b and using the t h e o r y of K e s s l e r and K e s s l e r [9], the c a r e f u l l y m e a s u r e d i n t e n s i t y of B a r r e t t et al [14] was about 50% h ig h e r than p r e d i c t e d . The d i s c r e p a n c y w a s r e d u c e d but not e h m m a t e d by u s m g the t h e o r y of W i l l i a m s and W e l z a c k e r with the s a m e v a l u e of ~y (i e only 40% of the v al u e at 1 GeV). It now s e e m s p o s s i b l e to r e s o l v e t h i s d i s c r e pancy If we take the r e s u l t s of H lg a s h l as i n d i c a t ing ~y ~ E - 1 above 1 GeV then n u c l e a r e n e r g y l o s s of m u o n s i s l e s s than one p e r cent of the t o t a l e n e r g y l o s s at a l l muon e n e r g i e s (a n e g a t i v e exponent of s o m e w h a t l e s s than unity would s t i l l suffice to m a k e the contribution n e g h g l b l e f o r c a l c u l a t i n g u n d e r g r o u n d intensity) The e h m m a tlon of n u c l e a r e n e r g y l o s s b r i n g s the B a r r e t t m t e n m t y [14] into a g r e e m e n t with the p r e d m t e d va lue within e x p e r n n e n t a l e r r o r ( ~ 10%) The s m a l l d i s c r e p a n c y which r e m a i n s o v e r the whole r e g i o n between 50 and 500 GeV and which i s not r e l e v a n t to the m a i n a r g u m e n t m a y find its e x planation in the low r e l i a b i l i t y of the i n t e n s i t y m e a s u r e m e n t s m this r e g i o n and m the s m a l l e r c o n t r i b u t i o n s to e n e r g y l o s s of both p a i r p r o d u c tion [15] and l o n i s a t l o n [16] which have been a d vanced recently The muon i n t e n s i t y has r e c e n t l y been m e a s u r e d to depths of 840 kg cm -2 by Mlyake et al [17] T h i s c o r r e s p o n d s to muon e n e r g i e s of o r d e r 10 000 C-eV At t h e s e e n e r g i e s the muon e n e r g y s p e c t r u m h as not been d i r e c t l y m e a s u r e d but i s e s t i m a t e d f r o m the high altitude r - r a y s p e c t r u m of Duthle et al [18] Mlyake et al. [19] fred r e a sonable a g r e e m e n t by r a t h e r a r b i t r a r i l y r e j e c t i n g n u c l e a r e n e r g y l o s s , e x c e p t that the u n d e r g r o u n d i n t e n s i t y i s a little high and they i n t e r p r e t t h i s in t e r m s of kaon p a r e n t a g e c o m p e t i n g with plon p a r e n t a g e of the m u o n s at high e n e r g y T h i s d i s c r e pa nc y would be v e r y m u c h g r e a t e r ff n u c l e a r l o s s 268
1December 1964
w e r e a p p r e c i a b l e as it would be f o r c o n s t a n t a r It r e m a i n s to be shown why c o m p a r i s o n of ex p e r i m e n t a l r e s u l t s with c a l c u l a t i o n s b ased on a constant photon-nucleon c r o s s s e c t i o n and a W i l h a m s and W e l z a c k e r v i r t u a l photon s p e c t r u m do not v i o l e n t l y d i s a g r e e [12, 20]. Th e cut off m the v i r t u a l photon s p e c t r u m f o r s m a l l i m p a c t p a r a m e t e r s m the W i l l i a m s and W e l z a c k e r t h e o r y p r o v i d e s b r o a d l y that kind of d e c r e a s e of ap with e n e r g y which r e s u l t s f r o m a d e c r e a s m g a r Howe v e r the W l l l m m s and W m z a c k e r a p p r o a c h n e i t h e r f i t s al l r e s u l t s as w e l l a s the solution p r o p o s e d h e r e n o r is it a c c e p t a b l e f r o m a t h e o r e t i c a l pomt of v i e w A l l the a v a i l a b l e c o s m i c r a y data a r e not m e r e l y c o n s i s t e n t with but f a v o u r a t o t a l p h o t o n - n u cl eo n c r o s s sect i o n which d e c r e a s e s above a few GeV as E - ~ w h e r e a is c l o s e to unity This e n e r g y dependence is quite d i f f e r e n t f r o m that of n u c l e o n - n u cleon and p I o n - n u c l e o n i n t e r a c t i o n s which a r e e s s e n t i a l l y constant up to at l e a s t 1013 eV [21] T h i s d i f f e r e n c e i s u n d e r s t a n d a b l e si n ce the l a t t e r take p l a c e through P o m e r a n c h u k t r a ] e c t o r m s w h e r e a s the p h o t o n - n u c l e o n i n t e r a c t i o n i s e x c e p t i o n a l m that it is g o v e r n e d by l o w e r Reg g e t r a j e c t o r i e s We would h k e to acknowledge u s e f u l d i s c u s sions with D o c t o r M J a m e e l and P r o f e s s o r S T Butler on the t h e o r e t i c a l p o i n t s i n v o l v e d
References 1 G Kramer and P Stmhel, Zs f Phys 178 (1964) 519 2 M Jameel, private commumcatlon 3 C Castagnoh, M Muchmk, G Ghlgo and R Rmzlvlllo, Nuovo Clmento 16 (1960) 683 4 C E Roos and V Z Petersen, Phys Rcv 124 (1961) 1610 5 R Alvarcz, Z Bar-Yam, W Kern, D Lukey, L S Osborne and S Tazzar, Phys Rev Letters 12 (1964) 707 6 M J Campbell, H S Murdoch, K W Ogllvle and H D Rathgeber, Nuovo Clmento 28 (1963) 885 7 G N Fowler and A W Wolfcndale, Progr m Elementary Partmle and Cosmm Ray Physms, Vol 4 (North-Hol] Publ Comp ,Amsterdam 1958) 8 E P George and J Evans, Proc Phys Soc A63 (1950) 1248 9 D Kessler and P Kcsslcr, Compt Rcnd 244(1957) 1896 10 K Dalyasu, K Kobayakawa, T Murota and T Nakano, J Phys Soc Japan 17, suppl A-HI, 1962 Part III p 344 11 A D Crossland and G N Fowler, Nuclear Physms 53 (1964) 273 12 S Hlgashl, T Kltamura, Y Mlshlma, S Mltagl, S Mlyamoto, T Oshm, H Shlbata, K Wanatabe and Y Watase, J Phys Soc Japan, Suppl A-III (1962) Part III, 362 13 P J Hayrnan and A W Wolfendale, Proc Phys Soc 80 (1962} 710
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14. P.H Barrett, L.M.Bollinger, G.Coccom, Y Etsenberg and K Gretsen, Rev. Mod. Phys. 24 (1952) 133. 15. C.Castagnoh, A De Marco and R Scrimaglio, Nuovo Cimento 33 (1964) 722. 16. K . I . A l e k s e e v a , G B.Zhdanov, M.I.Tret'Yakova and M N.Sheherbakova, Soviet Phys JETP 17 (1963) 1254. 17 S.Miyake, V.S.Narastmham and P V Ramana Murthy, Nuovo Cimento 32 (1964) 1505.
ON THE
SUPERPOSITION
PRINCIPLE
1 December 1964
18 J Duthie, P .H .F o w l er , A.Kaddoura, D.H.Perkms and P.Pinkau, Nuovo Cnnento 24 (1962) 122. 19 S.Miyake, V . S . N a r a s i m h a n a n d R . V Ramana Murthy, Nuovo Cimento 32 (1964) 1524. 20 P.J.Hayman, N.S Palmer and A.W Wolfendale, l>roc. Roy Soc. 275A (1963) 391. 21. G.Day, C.F.Gauld, C.B.A.MeCusker and L.S. Peak, Nuovo Ctmento 27 (1963) 977.
AND
CP
INVARIANCE
IN K ° DECAY
B. LAURE NT NORDITA, Copenhagen, Denmark, and Instztute f o r Theoretwal Physws, Umverszty of Stockholm, Sweden and M. ROOS
NORDITA , Copenhagen, Denmark Received 5 November 1964
Since 1957 when Wu made the d m c o v e r y that p a r i t y is not c o n s e r v e d in weak l n t e r a c h o n s one has g e n e r a l l y b e l i e v e d that C P m the p a r t i c l e t r a n s f o r m a t x o n c o r r e s p o n d i n g to s p a c e lnversxon The r e c e n t e x p e r i m e n t by C h r i s t e n s o n et al [1] shows that the t h e o r y of KO d e c a y xs b a m c a l l y w r o n g in s o m e r e s p e c t The i n t e r p r e t a t i o n c l o s e s t at hand is that C P m v a r m n c e m v~olated. It s e e m s to be v e r y d f f h c u l t to r e p e a t the t r i c k used in c o n nectxon with P v l o l a h o n and fred a new p a r t i c l e o p e r a t o r c o r r e s p o n d i n g to s p a c e m v e r s l o n . Hence, we would probably have to abandon l n v a r l a n c e u n d e r s p a c e r e v e r s i o n c o m p l e t e l y if C P l n v a r m n c e did not hold T h i s would be such a s e r i o u s change m our m o s t profound c o n c e p t io n s that one should t r y a l t e r n a t i v e ch an ge s m the f u n d a m e n ta l p r i n c i p l e s of physxcs to s e e if they could a l s o explain the e x p e r i m e n t In this p a p e r , we would like to s u g g e s t that the quantum m e c h a m c a l s u p e r p o s l h o n p r l n m p l e be c a r e f u l l y e x a m i n e d in the c a s e of KO de c a y F o r s o m e t i m e we have c o n s i d e r e d g e n e r a l m e t h o d s f o r t e s t i n g the s u p e r p o s l t l o n p r i n c i p l e in d i f f e r e n t c a s e s . (Some a s p e c t s have a l r e a d y been publi sh ed by one of us [2] ) Postponing t h e s e c o n s l d e r a t i o n s , we s h a l l h e r e deal e x c l u s i v e l y with K° d e c a y and u s e a v e r y p a r h c u l a r m o d e l only to show q u a h t a h v e l y the kind of e f f e c t s which could be e x p e c t e d f r o m n o n - l m e a m t i e s in an m t e r f e r -
ence-type experiment Nobody can deny the p o s s l b l h t y that the s u p e r p o s i t i o n p r i n c i p l e m a h n e a r a p p r o x i m a t i o n of a m o r e g e n e r a l , n o n - h n e a r law of n at u r e. The obvious w e a k n e s s of such an a s s u m p t i o n is the ab s e n c e of any hint w h a t s o e v e r as to how the s u p e r p o s i t i o n p r i n c i p l e could be g e n e r a h z e d A w a r e of t h i s, we want to s t r e s s the ad hoc c h a r a c t e r of the n o n - l i n e a r m o d el f o r K ° decay that we now a r e golng to u se Thin m o d el has as its only p u r p o s e to show that a C P mvarzant but n o n - l i n e a r theory can g~ve r e s u l t s m a g r e e m e n t u'zth the e x p e r z m e n t s . Let KO and KO stand f o r a m p h t u d e s of KO and ~ o m e s o n s , as is c u s t o m a r y A s s u m e , a l s o as usual, that the s p a c e m v e r s x o n c o r r e s p o n d s to the t r a n s f o r m a t i o n K ° ~ ~ o m the c e n t r e of m a s s s y s t e m (The c e n t r e of m a s s s y s t e m is going to be used throughout ) We w i l l now i n t r o d u c e a n o n - h n a r t e r m m the w e l l - k n o w n e q u a h o n s of motion f o r K ° and ~ o Not to u p set the e q u a h o n s too much we shall l e a v e t h e i r h o m o g e n e o u s c h a r a c t e r unchanged, writing dK o Ko _ ~ o - d---~ = a K ° + b K O + E K ° . KO+~o , (1) K o _ ~o
dK°dt = a K ° + b K ° - EK° KO+KO
(2)
Notice that this s y s t e m of e q u a h o n s is m v a m ant u n d er the C P t r a n s f o r m a t i o n K o ~ ~ o Adding 269