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539.171

22.383.5

86

. .

86 :

/ . .

. . : ,

2010. 298 . : ., .

ISBN 978-5-91304-154-8

539.171 K 22.383.5 . 15.12.2010. 6084 /16. .

. 40 . -279.

, , ӻ.

./ (495) 939-44-91;

www.kdu.ru;

e-mail: press@kdu.ru , 2010.

, 2010.

. ., 2010.

ISBN 978-5-91304-154-8 , , 2010.

x xF xF 1 3 m2 m 2 P m m P 2 = const = m P T p E = T + m E p P P 2 = E 2 p2 = m m P 2 = m2 E m n N n N = Pj, Pi = Pj, Pi i=1 j=1 i=1 j= n Pi i N Pi n Pj A0, A1, A2, A3 A = 0, 1, 2, 3 A = A0, A1, A2, A A = A,A A 2 2 2 A2 = A0 A1 A2 A3.

A A A2 A2 = A A A0 = A0, A1 = A1, A2 = A2, A3 = A A A A2 = A A = A0 A0 + A1 A1 + A2 A2 + A3 A3, = A A A A A B AB AB = A B = A B = A0 B 0 + A1 B 1 + A2 B 2 + A3 B 3 = A0 B 0 AB.

= A B g = g g 10 0 0 1 0 g = (g ) = (g ) = 0 0 1 0 = g.

00 0 A = g A, A = g A.

(AB) = A g B.

g P = (E, p) S p E S S p E S p E S = E p, E = p E, p = p, p = (p, 0, p ), p S S p p p Z Z p X X = =,.

2 1 S p E S p E p =, =, =.

E m m A= (A0, A) (A0, A1, A2, A3 ) (A0, Ax, Ay, Az ) S 0 A0 A3 = A0 A3, = A1, = = A A, 1 2 A2, A = A3 A0 = A3 A0.

= A 0 0 A A A 0 0 0 A1 = .

A 0 1 0 A2 A 0 A G = (, ) 2 G = = 0 1 = A G, A = A1, A = A2, A 1 3 A3 (A G) = A3 A = A.

1 S1 S 1 + 3 =, 3 = 1 2 (1 + 1 2 ) = 1 2 + 1 2, 1 + 1 G = (, ) 3 = 1 2 + 1 2, 3 = 1 2 + 1 2.

= tanh, = cosh, = sinh ;

3 = 1 + 2.

0 cosh 0 0 sinh A A A A1 0 1 0 = 2.

A A 0 0 1 A sinh 0 0 cosh A Ptarg Pproj Ptot Ptot = Pproj + Ptarg, (Etot, ptot ) = (Eproj + Etarg, pproj + ptarg ), (Etot, p ) Etarg, (p p Ptot = = + + = 0).

Eproj tot proj targ ptarg = 0 (Etot, p )2 = (Eproj + Etarg, pproj )2 = 2 Ptot = tot = Eproj + Etarg.

s s Ptot = m2 + m2 + 2mtarg Eproj = Eproj + Etarg 2.

proj targ (Eproj + Etarg, pproj ) E p = m2 2 m s a+ s bX X s a+b c+d+X Etot s pp Cross section (mb) total 10 elastic 1 2 3 4 5 6 7 8 10 1 10 10 10 10 10 10 10 10 10 3 10 1.9 2 10 Center of mass energy (GeV) _ pp Cross section (mb) total 10 elastic 1 2 3 4 5 6 7 8 10 1 10 10 10 10 10 10 10 10 Laboratory beam momentum (GeV/c) Cross section (mb) pd total pn total npelastic 1 2 10 1 10 10 1.9 2 3 4 5 6 7 8 910 20 30 pn pd 2.9 3 4 5 6 7 8 9 10 20 30 40 50 Center of mass energy (GeV) Cross section (mb) _ pd total _ p ntotal _ p nelastic 1 2 10 1 10 10 Laboratory beam momentum (GeV/c) Cross section (mb) + p total +pelastic 1 2 10 1 10 10 p 1.2 2 3 4 5 6 7 8 910 20 30 d 2.1 3 4 5 6 7 8 910 20 30 40 50 Center of mass energy (GeV) d total p total Cross section (mb) p elastic 1 2 10 1 10 10 Laboratory beam momentum (GeV/c) Cross section (mb) K ptotal K pelastic 1 2 10 1 10 10 1.6 2 3 4 5 6 7 8 9 10 20 30 KN 20 30 40 50 Kd 2.5 3 4 5 6 7 8 9 Center of mass energy (GeV) K dtotal Cross section (mb) K ntotal K n elastic 1 2 1 10 10 Laboratory beam energy (GeV/c) 22. K +ptotal 17. Cross section (mb) 12. 7. K +pelastic 2. 1 2 10 1 10 10 K +N 1.5 2 3 4 5 6 7 8 9 10 20 30 K +d 2.5 3 4 5 6 7 8 9 10 20 30 40 50 Center of mass energy (GeV) K +dtotal Cross section (mb) K +ntotal 1 2 10 1 10 10 Laboratory beam momentum (GeV/c) 2 ptotal pelastic 10 Cross section (mb) 10 -1 2 -1 10 1 10 10 10 1 10 Laboratory beam momentum (GeV/c) Laboratory beam momentum (GeV/c) 2.1 3 4 5 6 7 8 10 13 2.1 3 4 5 6 7 8 10 Center of mass energy (GeV) Center of mass energy (GeV) dtotal ptotal - Cross section (mb) - - total - 10 1 10 Center of mass energy (GeV) p 0.3 1 10 100 1000 d 0.1 1 10 100 1000 Laboratory beam momentum (GeV/c) + p p pp Kp tot, [mb] + Kp 0 1 2 3 4 -nuclear QCD transition region physics b12 = (u1 u2 )2 1 2 ui btp p = 1 + btp / btp K 0.8 c0 A b b+A A b (3 He, t) (d, d ) (, )

sN N = 4 9 B B B Pre-cooking by e.m. interaction??

Z1+Z2160137;

B 10151017 Gs 1970 time 1986 1996 2008 2015 a b c d a+bc+d+X, b(a, c)dX b(a, c).

X PX PX MX = PX PX m2 = E 2 p2 s t u s t u a+bc+d, b(a, c)d c d a b Pi 2 s = (Pa + Pb ) = (Pc + Pd ) t = (Pa Pc ) (b + d) a (a + c) b t = (Pa Pc )2 = (Pb Pd ) Pa Pb Pc Pd 2 u = (Pa Pd ) = (Pb Pc ) s + t + u = m2 + m2 + m2 + m2.

a b c d Pa + Pb = Pc + Pd.

s Pb Pc t Pa Pb Pc Pd u a+b 1 + 2 + 3 +... + n (n 1) Pi ) 2 2 n (2+3+...+n) X =( Mef f MX i= X t ma = mc mb = md t sin t = (Pa Pc )2 = (Ea Ec )2 (p p ) = a c 2 2 = 2 (p ) + 2 (| p |) cos = 2 (| p |) (1 cos ) = 2 2 = 4 (| p | sin /2) (| p | ) (p ).

a p p a |t| t | t | R R Rk f J1 (2kR sin /2) = J1 (2Rk sin /2).

2 sin /2 2k sin / J1 2k sin / t Rk f J1 R t.

t R2 k d | f |2 J1 R t, t d d = dd cos = d2k 2 d cos /2k 2 = ddt/2k k 2 d R2 k d | f |2 = J1 R t.

t d dt t d/dt R d J1 R t.

t dt t Pinit = Pf inal, (Einit, pinit ) = (Ef inal, pf inal )), Pinit Pf inal E p 2 s (Pinit ) = (Pf inal ).

n n Mef f = P = Pi = i= 2 2 n n n = Ei pi mi, i=1 i=1 i= mi n s Mef f b(a, c)dX 2 s smin max (ma + mb ), (mc + md + MX ), smin s a Tthresh = Ethresh ma T E E =T +m pp pp + meson Tthresh t pp pp + meson 2 Tthresh s s m = 2m 1 + Tthresh, 4M = (Pa P1 ) = mM, tpp m = (Pa P2 ) = M 2 1 2 tpmeson.

M pA p + A + meson A b 3 a MX MX = MX + mproj. + thresh Tprojectile, Mtarg = M3 + m2 Mtarg, M1 = Ma.

MX MX pp pp p + + K + MY mi i = 1,...

Mp tpp = MY + Mp mi, Mp + MY + mi mi MY tpmesons = Mp 1 1+, Mp Mp MY mi tpY = Mp 1 1+, Mp Mp tpa a = p, Y, meson a 2 pp 2 (2M + MX ) spp (2M + MX + m ).

pA MX MX 4M 2 1 + 1+ spp (2M + MX ).

2M 2Mtarg MX M sthresh spp thresh = 2M MX 1 + 1.

2M pp Mtarg , ( ) , /c p DC 3500 d C 3000 COSY +3K +2K Nuclotron (, ) 0 2 4 6 8 10 12 14 16 ( ..), Tkin, p 12 C pp p d d pp m u = P/m P S S S u = (, ) S = (0, 0, ) = v/c = 1/ 1 2 c = Z S A = (A0, A) (A0, A1, A2, A3 ) (A0, Ax, Ay, Az ) v 0 = A0 A3 = A1, = A, A c 2 = A2, A = A3 A0, = A 1 = tanh, = cosh, = sinh ;

S S S S S Z S 1 S S 2 S 3 S 3 = 1 + 2, i 1 1 u1 u u u S u u = 1 u S u 1 u u0 = (u1 u2 ) u0 + u 1 = u1 u2 u12.

1 + u u0 = E1 /m1 = 1 u1 = p1 /m1 = 1 1 S S u = (u A) A 0 A +A = Au A.

1 + u s 2 s = (Pa + Pb ) = (Ea + Eb ), p = p m2 = Ei p2 a i i b Eb = s Ea, s + m2 m Ea 2 m2 + m2 = s + Ea 2 2Ea s Ea = a b.

2s a b p a b s + m2 m p2 = Ea 2 m2 = Eb 2 m2 = m2, a b 2s a b a 1/2 s, m2, m p = a b, 2s s, m2, m a b (a, b, c) = a2 + b2 + c2 2ab 2bc 2ac.

(a, b, c) = (a b c) 4bc.

p s ma + mb c d ma mb md mc a+b a +b p, p p p p p mc = ma mb = md p in p f p, p c Pd m2 = P 2 d c 2m2 s m2 + m pa s b cm = 1, cm = a b.

Ea + mb 2mb 2mb s s c c p in f p in f z c p c pc c s p c p (p, pz ) p p c p = p cm p + cm cm E = pz z cm cm p = cm E + E, z 2 p p + =1.

z p p p = pz cm E, z cm (p, pz ) 2 p pz H + =1, A B A = p, B = cm p, H = cm cm E.

cm cm cm E = p /E p = (p = 0, p (p 0, p = p ) p= = = p ) z z p1 = (0, cm E (cm )), p2 = (0, cm E (cm + )).

p1 p2 pmin pmax cm pmin cm = pmin = cm pmin Z max E = cm cm p cos + cm E, pz = p cos.

p 1/ E + cm cm p cos = cm p2 + m2.

p() p 1/ cm cos 2 2 cm cm sin 2 p = m 2 cos2 ) cm (1 cm cos g D p = p , cm (1 cm cos2 ) 2 2 cm cm sin 2 D = 1 + cm 1 g 2 tan2 =, 2 2 cos cm g =.

(p )2 + m E = = cm D=0 g = p+ = p g 1 p p+ p g 1 p+ sin max = =.

g cm cm cm sin tan =.

cm (cos + g ) NN dN p2 + p2 p= pL pT L T p + p p + p + 0 d+p d+p+ 0 NN p(p, )X p(d, )X p(p, )X p(d, )X X p(p, )X p(d, )X p d 2 +m +M m M | tmin | p pi F (x) P0 x x x F (x) x (x) p /M 1 M (t, x = 0, y = Z 0, z) t= (0, 0, 0, 0) (t, z) 45 Z t t (t1, z1 ) t = t1 (t2, z2 ) t = t (t1, z1 ) t = t2 t (t, z1 ) t = t1 (t + t, z2 ) t2 = t1 + t Z (, ) (t, z) 1 = (t + z) ;

= (t z) 2 (t, r) (, ) (x, y, z) r = (z, r ) = (, r ) p H= 2m p2 + m 2 ;

= (E + pz ).

H= 2 p /p N = 3.15254166(28) d = 0.85742 N n = 1.9130428 N p = 2.79284739 N d p + n d+d + d(d, ) (p + d + + t) R= 2, (p + d 0 +3 He) D d (p, d) p p + Espect =, pd + Ed p Espect pd Ed p p k k m2 + p m2 + p ;

k2 = m2, p p k = p ;

kz = 2 (1 ) 4 (1 ) p k p k 0 t 2 t k() k() = 1 4 (1 ) = 1 2bdp ;

bdp = 2 (1 ).

m2 mp mp p mn mp k q q qmax = mN mN 0 p = Md pd k m2 + p2 d3 p 2 2 |rel (k)| d3 k = |nrl (k)| p , 4 (1 ) (1 ) Ep nrl (k) rel (k) |nrl (k)| (kx, ky kz ) (px, py pz ) p d (sn, t )(n,targ) Ep dp m2 + p 2 p |nrl (k)| R (n, d), 4 (1 ) (1 ) (n, targ) 1/2 sn, Mtarg, m n R (n, d) = 1/2 sd, Mtarg, m d (n, targ) (n, targ) R (n, d) (N, targ) (N, targ) tot tot inel (N, targ) (N, targ) Mef f Mef f inel (N, targ) Mef f = p(d, p )d (N, targ) |nrl (k)| k

k 200

| k | k mn mp Pp = (Ep, q) m2 q2 m2 q Ep = + + q | q | mp p p Pn = (Md Ep, q) (Pp + Pn )2 = Md Md Md + m2 (Md Ep ) q p p = = Ep, 2Md Md + (Md Ep ) q 2 m p n = = Md Ep, 2Md 1/ qcm = 2 m2 )=q, p p up = (Ep /mp, q/mp) En + En rel En = (Pn up ), qrel = pn up rel.

1 + up n Md Md Pn = Ep mp, q rel.

mp mp P n = m2 rel n Pn Pn = mn qmax qmax Pn q qmax Pn q qmax = mN.

3/4mN q k k P M W X W Z q =kk Q2 xBj y E E qP = =EE 0.

M Q2 = q 2 = 2 (EE kk ) m2 m2 0 ;

l l Q2 4EE sin2 (/2), Q2 t Q t Q x= ;

2M qP y= =.

kP E X W 2 = (P + q) = M 2 + 2M Q2.

Q + M 2 + m2 = M 2 + m2 + 2M E, s = (k + P) = l l xy h W Eh z=.

Eh z x xp (xp P + q) = x2 P 2 +q 2 +2xp P q = x2 M 2 +q 2 +2xp P q = m2, p p m q2 Q q 2 + 2xp P q = 0, xp = =, 2qP 2M xp x xBjorken xB x 2M = Q2 /x n Pi )2 m2 f = ( Pi i n ef n Pi ) m miss = (Pbeam + Ptarg Pbeam Ptarg n Pi Pi Pmiss = Pbeam + Ptarg n xF a b c X a+b c+X.

(Ec,pc ) c 1/ p min = Emax m2 p p p max = 1/ Emax m2 p =, 1/ 0 p p, m2 + p 2 E Emax = s + m2 m2, min X = ;

2s mX, min X p p p p p p p(p, )X p A B C p xF =, p max p(p, )X p(d, )X p(p, )X p(d, )X X p(p, )X p max B p max xF p xF = 2, s s p p 1 Ec + p c c = ln, 2 Ec p c Ec + p c c, long = ln.

2 Ec p c p = E + p, p = E p, + p+ long = ln.

2 p (p, xF ) (p, long ) E+p = ln ln tan long, m m2 p 2 + m2 = p p, = + c m m 1;

p |p | ;

p p m.

m 1, p 1 1 + cos long ln ln tan.

2 1 cos pseudo = ln tan.

xF c sh(c, long ) xF =.

sh(c,max ) long | xF | | c, long | 0 xF s s | xF | xF xF max long long Ec max pc max Ec + pc max long = ln = long 2 Ec max + pc max Ec pc Ec max pc max Ec + pc = ln max.

2 Ec pc + pc max Ec mc Ec Ec max pc max 2pc max long long ln max, 2 Ec pc 2pc m Ec max pc max pc max 1+ pc max = c max ) 2(pc m = c, max ) 2(pc m2 m Ec pc pc 1+ pc = c c, 2(pc )2 2(pc ) | xF | 1 pc pc pc max long long ln = ln (| xF |), pc max max exp long long, xF 0, xF max exp | | long, xF 0.

xF long a+b c+d 0 1+ (a + b) 0 M0 = s (a + b) p0 = pa + pb c1 d a+b c+d 01+ M s p0 T M T T 0 = M 0 m1 m2, M s M M 0 + m2 m 1 = E 2M (M0 m1 ) m = E1 m1 = = T 2M m2 = + O T0 /M T M m = E2 m2 T T2.

M T1 m =m.

T2 M 0 1+ 2m1 m p = T0.

M 0 P0 = (E0, p0 ) Z p0 P1 = (E1, p1 ) p1 Z (P0 P1 ) (P0 P1 ) = E0 E1 p0 p1 = E0 E1 p0 p1 cos 1 = E0 E1.

M0 E1 p0 cos 1 E0 D p1 =, E0 p2 cos2 D1 = M0 p 2 m2 p2 sin2 1.

1 M 0 p sin 1.

m1 p M0 p 180 m1 p M0 p m1 p M0 p = m1 p M0 p m1 p M 0 p 1 = arcsin = arcsin 1, m1 p 0 0 M0 m T0 p M p0 p , 0 0 1 1.

m p0 M0 m p0 m p p0 , M 0 1 + (E1 ) p1 p E (P1 + P2 ) = P0 ;

q 2 = M 0 m2 m.

1 E1 E2 q cos =.

p1 p E2 p2 E E1 (E0 E1 ) q cos =, E1 m 2 (E0 E1 ) m 1 E1 E1, E1, 1 0 E0 E1 p0 p E1, min/max =, E1, min E1 E1, max.

M M E1 E2 =, 2 (1 cos ) 2 E0 E0 M E1 = , 2 4 2 (1 cos ) 2 E0 E0 M E2 =, 2 4 2 (1 cos ) min M sin = = 2 E0 min cos = 0, 2 E0 M =.

4 2 (1 cos ) KS 0 0 0 KS V V V 0 0 0 0 0 KS V0 V2 V1 KS 0 KS V0 V 0 V0 V 0 V 0 pt p+ p = L L, p+ + p L L + pL p V0 V L (, p t ) V 90 V 0 V0 V V ( pt ) (, p t ) KS 0 K + p + (K, ) b+A c c + A + d b A d A c A c d c c = pc, p c = pc = (0, cm Ec (cm c )) = (0, 0) cm = c ;

c pc pc c c p + p pp + mc mp c c p 0., GeV/c 0. min q 0. 0. 0. 0. 0 2 4 6 8 p, GeV/c beam qmin pbeam , GeV/c, GeV/c 0. 1 0. min min 0. q q 0. 0. 0.6 0. 0. 0. 0. 0.2 0. 0 0 1 2 3 4 5 0 2 4 6 8 10 12 14 p, GeV/c p, GeV/c beam beam c A qmin c qmin | pc |= cm Ec | (cm c ) |.

qmin p (K, ) p (, )p p (p, )pp p (, )p p (p, )pp p (p, )pp p (p, )pp pp s , GeV/c 0. 0. min 0. q 0. 0. 0. 0. 0. 0. 0 2 4 6 8 10 12 p, GeV/c beam 1.25 mp mp, GeV/c 0., GeV/c 0. 0. 0. min min 0. q q 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 2 4 6 8 10 12 0 2 4 6 8 10 12 p, GeV/c pbeam, GeV/c beam 1.30 mp 1.35 mp b+t c+X c b t mtarg X M 1/2 s, m2, M s + m2 M, p = c Ec = c.

2s 2s c 2mtarg cm 1.

s c = p /Ec c m2 m2.

c cm targ c s 2mtarg (mtarg + mc ) mc c cm 1.

s mtarg 0 (MX t) (d, d ) 0 (Mmiss, t) Q

Q t MX Ed Q = E0 Ed E t = Q2 (p0 pd ) Q = E0 Ed MX = (Q + Mtarg )2 (p0 pd ).

t MX m2 2 bound = E p Pproj A Ptar Pdetect Pejected Precoil (A 1) PN A (A 1) A(p, n) A(3 He, t) A A (A 1) Precoil Ptar = (MA, 0) PN Pejected Ptransf = (, p) p PN = A Ptar Precoil PN = (EN, pf ) = (Mtar Erecoil, pf ) = = (Mtar Mrecoil Trecoil, pf ), pf A Mrecoil Trecoil (A 1) p f Trecoil.

2Mrecoil (A 1) s = Mtar (Mrecoil + mN ), mN = Mtar Mrecoil, s s mN (A1) s Mrecoil m N PN Mtar m 2 = PN = (mN p2, s) N f MtarmN + s Mtar PN m2 = 2mN p2 +, s N f s MtarmN + s s Mejected Pejected 2 2 Pejected F = (PN + Ptransf ) = ( + EN, p + pf ), = t + (mN s) + 2 (mN s ) F p f (Mtar + ) 2pf p.

Mtar mN + s t F R 0 12 R (1 + 2) R (R 1 + 2) .

(0 12 ) + 0 2 / 2 2 R (1 + 2) R (R 1 + 2) 2l+ (q/q0 ) 1 R (1 + 2) q 2 (1 + 2) 12 q (1 + 2) PN Ptransf 12 1 F 12 = F F q1 2 Ptransf Ptransf q1 F, m 2, t q1 2 = ;

N 4F s12 q s12 E1,2 = q1 2 + 12 = m2 m r N N E1, 2 r 2 t + m E1, + m2 m 2, =F N 4F r N N m N na nb va vb b vrel (b) a vrel d dV dt d = nb dV na vrel dt , na vrel dt vrel dt a nb b a b a t c ct (0) d = nb dV n(0) | uab | c dt a uab a b ((Pa Pb )) m2 m2 = ma mb | uab |.

InvF lux = a b Pi i mi sab a b 1 1/ sab, m2, m2 = p sab, InvF lux = 2 a b p a b 0 p b (0) nb dV = l dS NA /A l dS (0) c dt | uab | na dt dS T S NA | uab | c dS dt n = ltarg = a Atarg NA I = ltarg Ncycl, Atarg T Ncycl I L NA I = L, L = ltarg Ncycl.

Atarg x P0 (x) NA Ncenters (x) = x nmol, M M NA nmol tot Ncenters (x) tot I I(x) = I0 P0 (x) dx x P0 (x) dx Ncenters (dx) tot P0 (dx) = 1 Ncenters(dx) tot NA P0 (x + dx) = P0 (x) P0 (dx) = P0 (x)(1 dx nmol tot ).

M dP0 (x) NA = P0 (x) nmol tot.

M dx P0 (0) = NA P0 (x) = ex M nmol tot = extot ncenters, ncenters I(x) = I0 P0 (x) = I0 exp(x tot ncenters ).

R(x) R N (x) I(x) R(x) = = = exp(x tot ncenters ), I0 M N (x) = I(x)Tmeasur M = I0 Tmeasur Tmeasur i i d N (x, ) = N (x, 0) + N (x, 0) x ncenters d = d = M etot xncenters 1+ d + x ncenters d d N (x, ) = M etot xncenters f ( ) sin d + d +x ncenters f ( ) ( ) d d.

d 0 f () (, +d) f () 0 d/d 13. x x =2 z 1 + 0.038 ln moliere, cp X0 X p c z x X (3 4) moliere | t |2 t 103 2 c |t| | t | Tkin S |i |f S Pi Pf S f | |i f |S|i T f |S|i = f, i + i (2)4 (Pi Pf ) N f |T |i, N |i |f W = (2) (Pi Pf ) |N |2 V | f |T |i |2, t V Vt [ (P)] (P) dx eiPx = (P), 4 (2) (2) W/t V |N | = (2) m2 (En, f ) (Pi Pf ) | f |T |i |2, Pn, f n |f Pn, f n Pf = n Pn, f P n, f = m n n = v0 /V V2 |N | = (2) v Pn, f m2 (En, f ) (Pi Pf ) | f |T |i |2, n v Ea |pa | |pa | mb |uab | = a v0 = = = ma Ea ma mb Ea m2 m2 m = = ab b ma mb (Pa Pb )2 m2 m ab =, ma mb InvF lux Ea Eb v0 = ma mb |uab | = InvF lux a b Na =, V Ea 1 |N1 |2 Pn, f m2 (En, f ) = (2) n Ea Eb v (Pi Pf ) | f |T |i |2, N d4 Pn, f P n, f = m2 n 1 dq1 dqn |N1 |2 = (2)......

2E1 2En Ea Eb v Pa + Pb Pj | f |T |i |2, j=1, n d3 p d3 p V / (2) n 1 V dq1 dqn (2)4 |N1 |2 =......

3 2E1 2En (2) Ea Eb v Pa + Pb Pj | f |T |i |2, j=1, n (2)3 |N1 |2 1 1 dq1 dqn = (2) ......

3n 2E1 2En (2) Ea Eb v Pa + Pb Pj | f |T |i |2.

j=1, n n n dRn = d4 P1 d4 P2....d4 Pn n P 2 = Ei p2 = m2 i i n n d4 Pi Pi2 m2 dRn = Pi Pn, i 1 Pn n M2 dRn dW n n dW = M2 d4 Pi Pi2 m2 4 Pi Pn, i 1 E F (E1 )dE dRn F (E1 ) E1 = const E1

M2 Pi s Pi d3 pi d4 Pi Pi2 m2 = d3 pi dEi Ei p2 m2 =, 2Ei i i i n n n d3 pi (3) dRn = p E pi Ei.

2Ei i=1 i=1 i= n n n d3 pi (4) Rn (s) = P Pi.

2Ei i=1 i= d3 p1 d3 p2 (3) R2 = (p1 + p2 ) E1 + E2 s.

2E1 2E p2 d3 p1 p2 + m R2 = E1 + s.

1 2E1 2 p2 + m 1 d3 p1 = p2 dp1 dd cos p 0 2 cos 1 + p1 p2 dp1 = p1 E1 dE1, E1 m2 + m2, = E2 1 E1 m2 + m = E1 + s;

x 1 E1 dE1 E = dE1 + = dE1 1+ = dx 2 m2 m2 2 m2 + m + E1 E 1 2 1 s = dE1, E1 m2 + m 1 E1 m2 + m 1 = dx, dE s 1 p2 dp1 p2 + m E1 + = s 1 1 2E1 2 p2 + m 2 E1 m2 + m 1 1 = dx (x) = p1 E 2E1 2 2 m2 m + s E1 1 p1 dx (x), = 4s p s (m1 + m2 )2 s (m1 m2 ) pc.m. = p1, 2s s + m2 m 1 = E1.

2s dx 1/2 s, m2, m p 1 R2 (s) = 1 =, 2s s p (a, b, c) = a2 + b2 + c2 2ab 2ac 2bc s s R2 (s) s (m1 + m2 ) =, R2 (s) = ur.

( 2 sm1) ds2 1/ s2, s, m2 1/2 s2, m2, m2 ;

R3 (s) = 1 2 4s s (m2 +m3 ) mi 0 R3 (s) = ur s.

R3 (s) ur = s.

R2 (s) ur n n (/2) Rn (s) = sn2.

ur (n 1)! (n 2)!

sthresh n M s n s thresh s sthresh n (n1)/2 1/ 2 3 ( mi ) (3n5)/2.

Rn () nr 3 3/ 2 2 (n 1) ( mi ) 1/ 3 (m1 m2 m3 ) R3 () = s = nr mi 2 (m1 + m2 + m3 )3/2 i= 1/ 3 (m1 m2 m3 ) 2.

= 2 (m1 + m2 + m3 )3/ p+pp+p+V V 1/ mp mV 2.

R3 () = nr 2 (2mp + mV ) 2mp + mV V1 V 1/2 3/ R3 (;

V1 ) 2mp + mV nr mV = .

R3 (;

V2 ) 2mp + mV nr mV n s = 2433. pp pp dm21 dm Vps = 4s p p = pp S 1 P 1 + 2 + P1 P2 P3 P s1 s (1 + 2) (2 + 3) P 943 = P 1+2+3 s s1 = (P1 + P2 )2 = (P P3 ) s s2 = (P2 + P3 )2 = (P P1 ) s 2 s3 = (P3 + P1 ) = (P P2 ) s s 1 + s 2 + s 3 = s + m2 + m2 + m2.

1 2 s1, s2 P 1+2+ s1 s si, sj Ei, Ej Ti, Tj i, j = 1, 2, 3 P 1+2+3 R3 (s) = d pi = (p p1 p2 p3 ) s E1 E2 E3.

2Ei i= p d3 p1 d3 p R3 (s) = s E1 E2 E3, 8E1 E2 E E2 =| p1 + p3 |2 +m2 = p2 + p2 + 2p1 p3 cos 13 + m2.

2 1 3 p1 p d3 p1 d3 p d3 p1 d3 p3 = p2 dp1 d1 p2 dp3 d3 = 1 = p1 E1 dE1 d1 p3 E3 dE3 d cos 13 d3, 3 = (cos 13, 3 ) p1 3 1 cos 13 dE2 /d cos 13 = p1 p3 /E2 dE1 dE3 d1 d3 1 cos2 13.

R3 (s) = cos cos 13 = (E1, E3 ) = E1 m2 + E3 m2 2 s E1 E3 1 1/ E1 m E3 m + m2, 2 1 3 =| p1 p3 |2 +m2.

s E1 E3 4 E1 m E3 m2 = 1 = s + 2E1 E3 2 s (E1 + E3 ) + m2 m2 + m2.

1 2 E1 E3 s1 s (E1, E2 ) / (s1, s2 ) 1/4s s1 s d1 2 ds1 ds2 G(s1, s2, s, m2, m2, m2 ), R3 (s) = 1 2 4s G(s1, s2, s, m2, m2, m2 ) 1 2 R3 (s) = 2 dE1 dE3 1 cos2 13.

s1, s2 d2 R3 d2 R3 = 2 ;

=, 4s dE1 dE3 ds1 ds s p2, p2, p2 = 0.

1 2 P 1+2+ i = Ei m T M0 s T1 + T2 + T3 = s 3m = Q, Q T1, T2, T3 Q (T1, T2 ) s t u 3 pp T1 = T2 = T3 = Q/3 (r, ) Q = (1 + r cos ), T Q = 1 + r cos + T2, 3 Q = 1 + r cos T3.

3 Q (1 + x) r2 + xr3 cos 3 = 1, x =, =.

(2 )2 s m2 m ij ij (mi + mj ) (i + j) 3 pp 3 pp s1 s (1+2) s1 (2 + 3) s2 p1 p2 p3 m2 = (P1 + P2 )2 = m2 2 m2 = (P1 + P3 )2 = (m1 + m2 ) = (P2 + P3 ) = (m2 + m3 ) 23 A1 A2 A (m1 + m3 ) p1 /p2 = m1 /m m2 = ( s m3 )2 B1 B2 B3 B1 B2 B3 m2 m2 m2 A1 A2 A 12 23 pp K KS pp (K KS ) 3, S ( p) p + 0 pp K + K pp K KS p p + ( p) p 2 t1 t2 a+b 1+2+ pa = pb 2 sab = (Pa + Pb ) = (P1 + P2 + P3 ), s 2 s12 = (P1 + P2 ) = (Pa + Pb P3 ), s 2 s23 = (P2 + P3 ) = (Pa + Pb P1 ), s 2 ta1 = (Pa P1 ) = (P2 + P3 Pb ), t 2 tb3 = (Pb P3 ) = (P1 + P2 Pa ).

t ta2 tb2 ta3 tb1 s13 Pi Pj i, j = a, b, 1, 2, 2n 2 (n 1) 1 2 22 1 a + b 1 + X X 2 + 3 X s2 t X s (t1, s2 ) t1 = (ma m 1 ) d3 p1 d3 p2 d3 p3 R3 (s) = (pa + pb p1 p2 p3 ).

2E1 2E2 2E d3 p23 1= (p23 p2 p3 ), ds 2E E23 = p2 + s2 d3 p1 d3 p23 = (pa + pb p1 p23 ) R3 ds 2E1 2E d3 p2 d3 p3 (p23 p2 p3 ), 2E2 2E ds2 R2 s;

m2, s2 R2 s2 ;

m2, m R3 (s) =.

1 2 1/2 s2, m2, m 2 R3 = dc.m..

d dt1 ds2 8 sPa 8s t1 = (ma mb = m2 = m1 = 1 m 1 ) mb m5 = 5 m3 = 2 s = ma + mb m1 + s s2 t m2 + m3 s 2 s m1 ;

| cos 1 | 1.

s s + m = s + m 2 m 2 m 2 + m 2 t 2m2 a b a a 1/2 s, m2, m2 1/2 t1, m2, m2.

2m2 a b a a s a t t1 s 2 1/2 s2, m2, m d3 R 2 = .

4s2 1/2 (s, m2, m2 ) ds2 dt1 a b (t1, s2 ) (s, s2, m2 ) 0, (s2, m2, m2 ) 0, 1 2 s s 2+ a+b 1+2+3 1+a+b t1 s a+b 2 + 2 + (1 ) n + nT nT f |T |nT | P haseV olume | =.

InvF lux(nT ) InvF lux(nT ) nT nT | f |T |nT |2 = P haseV olume = inel (nT ) InvF lux(ntransf T ).

InvF lux(dT ) 1/2 sn, Mtarg, m InvF lux(ntransf T ) n R (n, d) = = 1/2.

sd, Mtarg, m InvF lux(dT ) d a b 2 a b Xab s s 1 X 2 R2 a + b 1 + 2 + 3 +...

M M + 3 | T | M = M (pi, Ei ) (E1, E2, E3 ) M (pi, Ei ) = (1)PM M (pi, Ei ), PM Iin = | +, 0, + | 0,, + + | 3, 0 = + |, +, 0 | +,, | 0, +, |, 0, +.

3 | T | M M M (pi, Ei ) M M (pi, Ei ) E1 = E2 E3 = E2 E1 = E3 M (pi, Ei ) = (E1 E2 ) (E2 E3 ) (E3 E1 ) f (E1, E2, E3 ), f (E1, E2, E3 ) r (E1 E2 )2 (E2 E3 )2 (E3 E1 )2 = dc.m.

= dS = r6 sin2. n/6 n = 1, 3,..11 n/ n = 0, 2,.. M +1 M M = [p1 p2 + p2 p3 + p3 p1 ] f, p1 + p2 + p3 = 0, f M = 3f [p1 p2 ].

f p2, p2, p dc.m. 1 2 =, dS 2 dc.m.

r2 r3 cos(3) = 1 1+, 2 (1 ) (2 ) dS + M M = f [E1 (p2 p3 ) + E2 (p3 p1 ) + E3 (p1 p2 )].

M = f [p1 (M 3E2 ) p2 (M 3E1 )], E1 = E2 = E3 = M/3 p1 = p2 p2 = p3 p1 = p p1 = p2 p2 = p3 p1 = p dc.m.

r cos (3) r2.

= dS 0 1+ f M P I J (I J P ) (0, 1) 0 (0, 1) x [A, B] f (x) f (x) [A, B] N xi x0 = A ;

xN = B i x Wi f (xi ) (xi xi1 i xi N Wi = f (x)dx ;

Wi = 1.

i= xi Wi [0, 1] w j Wj1 w Wj xj (xj +xj1 )/ x x w(x) y x x B y= f (x )dx ;

f (x )dx = 1.

A A (0, 1) = g(x) g (x) = f (x) d = g (x)dx = f (x)dx [0, 1] g(x) = x exp (x/X0 ) X N0 Xmax x 1 x/X0 ex /X0 dx = 1 ex/X = dx ;

= d e X0 X x= X0 ln(1 ).

X= = x exp (x2 /2).

(x, y) y x (x, x + dx;

y, y + dy) x y exp (x2 + y 2 )/2 dxdy = = dWxy exp (2 /2)dd = dW.

= dW = exp (2 /2)d W = exp (2 /2) ;

(0, 1) W W x y = 2 ln(W ) = cos() x = sin() y (0, 2) x y

L m c m=p 1, L c 1/ (x, y) (x, y) p

sN N = 4 U U Au Au E s s 4E 2 s Ebeam E s Ebeam =2, 2mtarget mtarget s 2E /mtarget I I n = l 6.022 1023 nmol =, L = Nnucl A I Nnucl T = /T l 3 A nmol n

= A mA 1 A = 2 6 H 1028 CH 1028 1028 6.251012 21012 0.16 A A

+ 33 2 + 34 2 31 2

9 Tmax Tmax

Z/A = 1/2 4 106 107 1010 d p 4 1012 1011 p 1010 5 108 n 1012 5 1010 d 2 1010 2 1010 (1 5) dpol 2 106 108 ppol 106 108 npol 2 He 5 1010 5 109 2 He 2 109 2 1010 5 Li 109 7 109 2 C 5 O 104 2 108 5 Ne 5 106 3 108 5 Mg 3 Si 3 107 2 Ar Fe 5 Zn 2 107 5 Kr Mo 2 Sn 107 2 Xe Ta 3 106 U , ( ) , /c p DC 3500 d C 3000 COSY +3K +2K Nuclotron (, ) 0 2 4 6 8 10 12 14 16 ( ..), Tkin, p 12 C pp p d d pp A Tkin T /T

6 p 101 p p/p

+ 3 + 3 106 1.6 3 105 + 3 + 3 9 Tkin 102 3

3. pmax 3.65

actual pmax 0.5 p/p 2

5

2 1028 1034

3 1011

5 4 1013 2 1 109

U 4 Research Communities at FAIR SIS 100/ Nuclear Matter Physics with CBM 35-45 GeV/u HI beams, x HADES HESR Rare Isotope Production Target Hadron Physics Super with antiprotons Antiproton FRS Production of 0 - 15 GeV Target Plasma Physics: x600 Nuclear Structure & Astrophysics FLAIR CR higher target energy with rare isotope beams, x10 RESR density 600kJ/g and excellent cooling Special Features: NESR 50ns Bunched beams High EM Field (HI) _ Electron cooling of secondary beams Fundamental Studies (HI & p) SC magnets fast ramping Applications (HI) 100 m Parallel operation 4 1010 C U

U 28+ 5 U 92+ cos p 240 p/p = p 3 p

e p p p 9.4 1015

He

Q2 1 He 10 5 B

max Tkin 2 104 2 p/p Li Z/A 0. Tkin /A Tkin /A

d n 7 1011 2 p 5 1011 3 d 4 He 4 He C N 2 O 2 Ne Ar 2 106 30+ Kr 8 106 26+ Kr 2.5 Xe 340 2. pA p A d A

Tkin 3.5 1 1027 2 11030 2 O r rz () OZ OZ Ar A Ar Ar i Ai A Ar = Rij Aj i Rij 33 R r Ai Ar = R A.

Y r = ry () cos() 0 sin() R [ry ()] = 0 1 sin() 0 cos() e(i) e(j) er e(j) (j) er (j) e(i) er = Rij e(i) = RT e(i) (j) ji R RT RT R = RRT = 1 R R A Aj S O A= Aj e(j).

j Sr O S r er (l) A= (Al )S r er.

(l) l (Ai )S r = Rij r1 Aj, = R r [R(r)].

r R(r1 ) Sr S s (2s + 1) (m) 1 0 0 1. , (s1) =.,..., (s) =., = (s)..

.

0 0 0 s = m (m) m m (m )S r m (i )S r = Dij (r1 ) r1 j, s Ds (r1 ) (2s+1) D D = 1 r (s) (m) = Dm m (m ) r (m) Or O S S (m ) r O S Or |sm s Sr S r |sm S r (s) |sm S r = Dm m (r1 ) |sm.

|sm |sm r |sm r (s) |sm r = Dm m (r) |sm, U (r) r |sm r = U (r)|sm.

(s) Dm m (r) = sm |U (r)|sm |sm S r = U r1 |sm.

1/ j sj 22 s (2s + 1) Si cos (/2) e1/ =.

sin (/2) e1/ P P , P = (sincos, sinsin, cos), P = 1.

P s P =, s P [2(2s + 1) 2] (2s + 1) Si si s D(s) (r)Si D+(s) (r) = Rij r1 Si.

1/ Sr S = Rij r i j.

Sr s s Ji sz s |ssz l r lj j = x, y, z S y S S v Z S S S S S = lz (v) ry () S.

l(v) l (v) r1 (v) lz (v) r (v).

r (v) e(z) v r1 (v) Z v A Al l Al A Al = (l)A, 44 l 10 0 0 (r) =, 0 R R l v z cosh u 0 0 sinh u 0 1 0 [lz (v)] =, 0 0 1 sinh u 0 0 cosh u cosh u = 1/ 1 v 2 tanh u = v SA A m |s, sz O SA p2 + m v v = p/ p SA O A p |p...

p O S SA v S S1 S p p A O O S SA l(v) |p, sz (, ) p O S p2 + m v = |p|/ AA Z S p = (p,, ) A () OY () Z OZ SA S = rz ()ry ()lz (v)SA r(,, ),, r(,, ) = rz ()ry ()rz (), S = r1 (,, 0) lz (v)SA.

= |;

s, sz = A SA |;

s, sz = S (m, 000) O S |p;

|p;

|p;

|;

s, sz = S, S S A |p;

SA S SA = lz (v)r (,, 0) S, sz = A h(p) h(p) lz (v)r (,, 0).

h(p) h(p) = r (,, 0) lz (v).

h(p) p S = h1 (p)SA, |p;

|;

s, sz = S = U [h(p)]|;

s, sz =, U h(p) |p, S O A p |p, Sl l S Ol A Ol |p, S l = U l1 |p,, U (l) l p Ol A p Ol p p (pmu )S l = l1 p.

|p, S l = |p, p U l1 |p, = U l1 U [h(p)] |,.

U [h(p )] U 1 [h(p )] = h(p ) |p, |p, U [h(p )] |,, U l1 |p, = U [h(p )] R|,, R R = U 1 [h(p )] U l1 U [h(p)] U h1 (p )l1 h(p).

U l = (m, 0, 0, 0, ) h1 (p )l1 h(p) h(p) : p l1 : p p h1 (p ) : p R l r(l, p) r(l, p) h1 (p )l1 h(p).

l pp (s) R|, = D [r (l, p)] |, D (s) |p, S l = D [r (l, p)] U [h (p )] |, = (s) = D [r (l, p)] |p,.

Sl S S Sl |p, S l |p, Sl S Y l r(l, p) l = ry () p = (p,, 0) p = (p,, 0) |p, |p,.


p |p, exp iJ p/|p| |p, = ei |p,.

A + B... + K +... B m = 0 XZ p = (p,, 0), E ;

pL = (pL, L, 0), EL.

h (p) = ry ()lz (v) ;

h (pL ) = ry (L )lz (vL ), v vL Z S S SL CM Z CM SL lz (CM ) S l l = lz (CM ) p pL r(l, p) r lz (CM ), p r() = h1 (pL ) lz (CM )h (p).

Y e(y) = (0010) e(x) = (0100) Y X e(y) r() = ry () e(x) = (0100) e(x) = {0, cos cos L + CM sin sin L, e(x) m 0, (sin cos L CM cos sin L ).

E e(x) Y cos = cos cos L + CM sin sin L m sin = (sin cos L CM cos sin L ) E = CM.

1 CM (s) |p, D [ry ()] |pL, = ds ()|pL,.

p E p pL tanh(u) = ;

tanh(uL ) = ;

tanh(uCM ) = CM.

E EL sin sin L sin = =, sinh uCM sinh u sinh uL pL sin L = p sin cosh uCM = cosh u cosh uL sinh u sinh uL cos.

( L ) = ( L ) K p ry () A + B C + D C R L D cos C = cos cos L + CM sin sin L = m2 E 2 m2 E pB EC = cos L + B C L C B pC EB mB pC pC EB mC sin C = {sin cos L CM cos sin L } = EC mC CM CM mC pB = sin = sin L L mB pC pC cos D = cos cos R + CM sin sin R = m2 E 2 m2 E pB ED = cos R + B D L D B pD EB mB pD pD EB mD CM CM mD pB sin D = sin = sin R L mB pD pD.

C mA = mC mB = mD D = R, pp mA = mB C = L.

C 0 s I= | + | I = 1, Iz = 1 = | | I = 1, Iz = 0 = | | I = 1, Iz = 1 = I = 0, 1, | + 0 | 0 + | + | 1,1 = | + | + | | 1,0 = | 0 | 0 |, | 1, 1 = | 0, 0 = | + + | + | 0 0.

| | | 0, 0 = | + + | + | 0 0.

0, 1, 0 3 1 2 A B C D /2 4 2 /2 = /4 = L() = + 2 (1 tan ).

cos 4 sin L = 2 =0;

4 sin 2 = 0 ;

=, (cos ) = / = / P = (E, p) P P 2 = m a+bc+d s + t + u = m2 + m2 + m2 + m2.

a b c d b s R R d J1 R t, t dt |t| d/dt d/dt | t | d/dt d/dt R | t | d/d |t| d/dt u t a+b 1+2+ mproj = M a Mtarg = M3 + m2 Mtarg b MX 1 2 pp pp p + + K + 3 MY mi +p +p mp m M

M 750.

+p M+p M M m E p(, )X K K +K 1 1 u0 = (u1 u2 ) u0 + u 1 = u1 u2 u12.

1 + u u0 = E1 /m1 = 1 u1 = p1 /m1 = 1 1 s 2 s = (Pa + Pb ) = (Ea + Eb ), Ea Eb s a b a b a b (x, y, z) (x, y, z) = x2 + y 2 + z 2 2xy 2xz 2yz = (x y z)2 4yz s ma = a+b a +b mb = m cm b b a a+b c+d (px, py = 0, pz ) c Z c c p1 = (0, 0, cm E (cm )), p2 = (0, 0, cm E (cm + )).

n(K, ) K pK pK p = p cm p + cm cm E = pz z cm E + cm cm p, = E z c sin cm, g =.

tan = cm (cos + g ) c p() 1/ E + cm cm p cos = cm p2 + m2.

+ m + M m M tmin mi m s i l m n kf + ki kf ki ki kf l=, m=, n=, kf + ki kf ki ki kf ki kf l m n l=mn, m= nl, n=lm.

kf + ki ki kf ki kf l=, m=, n=.

kf + ki ki kf ki kf l = nm, m= nl, n=ml.

D + En qn = m 2 n Pn q mN.

1+2 1 + n(K, ) K K M + N M + N E M 0 1+2 0 1 M m1 m 1 P0 = (E0, p0 ) 0 Z p0 1 P1 = (E1, p1 ) p Z (P0 P1 ) (P0 P1 ) = E0 E1 p0 p1 = E0 E1 p0 p1 cos 1 = E0 E1.

M0 E1 p0 cos 1 E0 D p1 =, E0 p2 cos2 D1 = M0 p 2 m2 p2 sin2 1.

1 1 2+ 23 m = sin 2 2 E2 E min 2 = (E1 ) 0 1+ (E0, p0 ) p E p E1 E2 q cos =, p1 p E1 (E0 E1 ) q cos =, E1 m 2 (E0 E1 ) m 1 E1 E1, E1, 1 0 E0 E1 p0 p E1, min/max =, E1, min E1 E1, max.

M 0 1+2 2 E0 E0 M E1 = , 2 4 2 (1 cos ) 2 E0 E0 M E2 =.

2 4 2 (1 cos ) 1+2 1 + a + T c + X a T c X M c d3 pi d4 Pi Pi2 m2 = d3 pi dEi Ei p2 m2 =.

2Ei i i i s sthresh s R2 (s) s (m1 + m2 ) =, s sthresh R2 (s) = ur.

n+p M +d n+p M + n+p M R2 R3 a b R2 /R a b p + n+ p1 = p V0 pp P 1+2+ 3 = (cos 13, 3 ) p 3 d3 p1 d3 p3 = p2 dp1 d1 p2 dp3 d3 = p1 E1 dE1 d1 p3 E3 dE3 d cos 13 d3, 1 dE1 dE3 d1 d3 1 cos2 13.

R3 (s) = cos cos 13 = (E1, E3 ) 1/ = E1 m2 +E3 m2 E1 m 2 2 E3 m +m2, s E1 E3 1 3 1 3 =| p1 p3 |2 +m2.

s E1 E3 p2, p2, p2 = 0.

1 2 a + b 1 + 2 + ta2 tb2 ta3 tb1 s Pi Pj p2, p2, p dc.m. 1 2 =, dS 2 dc.m.

r2 r3 cos(3) = 1 1+, 2 (1 ) (2 ) dS dc.m.

r cos (3) r2.

= dS N (0, 1) N (X0, ) 0 (x, y) x Nx (X0, ) Ny (Y0, ) X0 = 0 Y0 = y (X, Y ) (x, y) x y r r r r R x y 2 / w (z), (z0 z) + 2 / z=x z=y x0 = y0 = 0 W (x, y) W (x, y) = i i =.

W (x, y)max u d x Pk (x) k Pk (x) Pscatt (x) 13.6 z x x moliere = 2 1 + 0.038 ln, cp X0 X p c z x X Z1 Z2 A2 A Au Au Cu Cu d Au = (Pa + Pb ) = m2 + m2 + 2Pa Pb, s a b = (Pa Pc ) = m2 + m2 2Pa Pc, t a c = (Pa Pd )2 = m2 + m2 2Pa Pd, u a d c m2 + 2P (Pb Pc Pd ) + 2m2 ;

s+t+u = I a i=a Pa + Pb = Pc + Pa = Pb + Pc + Pd P a = m Pd a R R d J1 R t, t dt t = 0 t R t = Zi J1 (x) Zi i ti = (Zi /R) |t| d/dt(0) R d (ln (d/dt)) /dt(t = 0) R d/dt |t| R (1)k 2k m z z Jm (z) = = 2 k!(m + k + 1) k= m (z 2 /4)k z = .

2 (m + k + 1) k!

k= ex = xk /k!

k= (x2 )k 1 J1 (2x) ex /2 = J1 (2x) = x ;

(k + 1)!

k! x k= (x2 )k 1 = k ;

(k + 1)! k!

k= x J1 (2x) = ex /2 + O x6.

x J0 J 1 0.050661 0. xn + O [4n 1]5, J0 (xn ) = 0 = n + (4n 1) 4 4n 1 0.151982 0. Zn O [4n + 1]5.

J1 (Zn ) = 0 = n+ + (4n + 1) 4 4n + J1 (x) Z0 = 0 ;

Z1 2.233 ;

Z2 3.238 ;

Z3 4. x kR = R t 4.986 2 2.233 ;

(k1 ) x1 = k1 R ;

R2 tot (k1 ) | t | tot | t | k 2 R4 k2 R2 2 / d =| f |2 = e d R4 R2 |t|/4 tot R2 |t|/ d d = 2 = e = e 4 dt k d b R tot b= = 4 MX MX Tprojectile = MX + mproj. +, MX = Mproj +m2 Mtarg.

thresh Mtarg 2 sthresh = (Pproj + Ptarg ) = (Mproj + Mtarg + MX ), MX sthresh 2 sthresh = Mproj + Mtarg + 2 (Tthresh + Mproj ) Mtarg, Pproj = ((T + Mproj ), pproj ) Ptarg = (Mtarg, ptarg = 0) sthresh pp tpmeson p+Y + tpmeson = (Pa Pmeson )2 a Mp Y MY m Pmeson = (m, 0) = (Pa Pmeson ) = Ea 2 m, p = tpmeson a 2 2 = +m 2Ea m, Mp pa Ea 2Ea Ea = s/ s s 1 Ea = s = (Mp + MY + m).

2 m MY tpmeson = Mp 1 1+.

Mp Mp Y thresh E thresh cm E M = m M E = M 1 + thr.

2m p tot Etot cm =, cm =, Etot Mtot Etot = E + m Mtot p tot = p + ptarg = p s 1 + 2m M |p M, thr | = M m.

1+ M lab M=m E 2 1 u0 = (u1 u2 ) u0 + u 1 = u1 u2 u12.

1 + u u0 1 u0 1 u0 = (u1, u2 ) = u0 u0 (u1 u2 ) = 12 1 ( 1 2 ) u12 = u1 u2.

s s = (Pa + Pb ) = m2 + m2 + 2Ea mb ;

a b s m2 m Ea = a b.

2mb p 2 = E 2 m s 1/2 s, m2, m a pa = b, 2mb 2m2 s m2 + m pa s b cm = 1, cm = a b.

Ea + mb 2mb s 2mb s Ea + mb cm =, s Ea + mb s Ea s m2 m s 1 as b s m2 + m b= = a cm, 2mb s 2mb s m2 s m s a b s m2 + m2 s = a b cm.

2mb s 2mb 1, 2 = =, 1 mb / s cm 2m pa cm = 1 b.

Ea + mb s Ea + mb pa s 1/2 s, m2, m a cm = b, s m2 + m a b m2 s m s a b 1/ m2 m m2 + m 1/ m2, m2 a =s 12 a + b b s,, s a b s 1/s m2 + m 1/2 s, m2, m2 s 1 a b.

a b s m2 m 1 1 = 1+ a b.

s m2 + m2 m2 m2 s s s 1 as b a b 2m pa cm = 1 b.

Ea + mb s cm 1/2 s, m2, m2 s m2 m2 s m2 + m a pa =, Ea =, Ea + mb = b a a b b.

2mb 2mb 2mb ma = mb = m 1/2 s, m2, m pa cm = =.

Ea + mb s a p a =, E ma = mb = m 1/2 s, m2, m p a = =, E s cm cm = a ga = =1.

a cm pz = cm (cm E + p ), p = 0.

z | pz | | p z |=| | E pz = cm E (cm + ), p = 0, pz = cm E (cm ), p = 0, | p| p tan =.

pz pz pz = p cos p = p = p sin pz = cm E (cm + cos ), p = E.

sin cm, g =.

tan = + cos ) cm (g 1/ E + cm cm p cos = cm p2 + m2.

m E p2 1 cm cos2 2p mcm cos + m2 1 2 =0.

cm cm 4D 2 4D = 4m2 cm cos2 4 1 cm cos2 m2 1, cm cm D = m2 cm cos2 + 2.

cm 1/cm cos2 + sin2 = 1, 2 1 = 2 2.

m2 2 cm cm sin2.

2 4D = 4 cm 1/ m cm cos 2 2 cm cm sin 2 p = .

1 cm cos cm +m +M m M m M P P t 2 = (Pm PM ) = 22 2P P t = (P P ), t = 22 2 E E p p cos.

cos t 22 + 2E E cos = , 2p p | cos | cos +1 t cos = + | tmin | | tmin |= 0 cos = +1 tmin tmin = 22 2E E + 2p p.

t tmin 22 s + 2 m 2 s + 2 M = tmin 2s 1/2 s, 2, m2 1/2 s, 2, M.

m2 s i m M m = =, M = =.

2 m2 s 2 M 2E s 2E 1/2 s, 2, m 1/ 1/2 s, 2, m2 s 2 m 2 4m2 = = 1/ s 2 m 2 1 42 m = = s 2 m 2 1 22 m 24 m + O 6 m = ;

1/2 s, 2, M tmin 1 (1 + x)1/2 = 1 + x x2 +....

2 s + 2 M 2 M 2 m = = E E, 2s 2s M 2 m = E 2 E E E.

2s 2 =E p p p E 1/ 1 M 2 m E M 2 m = 1 2 + p p.

4s p p s p p 1/ 1 M 2 m E M 2 m p = 1 2 + p p .

4s p s p M 2 m 1 E M 2 m2 p 2 = + 2 p p 2 p 2p 4s s 1 p 4 + 2 M 2 m + 8 p 2 s M 2 m 1 E + ....

4 p 4s s p p E E p = 1/2 s, 2, m2 /2 s 1 2 M 2 m 2 + p p = +... = E E spm2 u 1 2 M 2 m = + +....

2 (s, 2, m2 ) p p E E m s i 2 M 2 m tmin +....

s kf + ki kf ki ki kf l=, m=, n=, kf + ki kf ki ki kf ki kf l m n l=mn, m=nl, n=lm.

l = m n kf ki ki kf kf [ki kf ] ki [ki kf ] mn= =.

kf ki ki kf kf ki ki kf a [b c] = b (ac) c (ab) kf [ki kf ] = ki (kf kf ) kf (kf ki ) = ki kf cos, ki [ki kf ] = ki (ki kf ) kf (ki ki ) = ki cos kf, kf [ki kf ] ki [ki kf ] = (ki + kf ) (1 cos ), ki kf kf ki ki kf = sin 2 (1 cos ) = 2 cos (1 cos ), ki + kf mn=.

2 cos / kf + ki = 2 (1 + cos ) = 2 cos / ki + kf mn= =l.

kf + ki Md Md Pn = Ep mp, q rel.

mp mp P n = m rel n Pn = q Pd = (Pp + Pn )2 = Md 2 K m2 m2 m mn EK = + K.

2(m mn ) m (2M m) E =.

2(M m) m M (P1 + P2 ) = P0, q2 = M 0 m2 m.

1 E1 E2 p

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