zi^o(0, y)i y)

In document The analytic properties of the triangle and box diagram amplitudes (Page 117-124)

SPG((-Ä0)2+(-Äi)5)Ä0 Qy)

2J J 2

~ r

and y^ < y Q (0, zy) if and only if h^( . y) > 0 . Similarly

A.^ < AQ (0, y ) , if and only if h ^ ( y ) > 0 . For subcase (i) we see that h (0, y, y ) strictly decreases as y increases from 0 to y CO, y ) and h (A, y(zy), zy) strictly decreases as A increases from 0 to A( y ) . Here 0 < y(zy) < y^ < y Q (0,

y)

so that

1 >

h [

o, y(y), y) >

g + iy)

,

where 9 + ( y ) is given by eq. (4-36). Also, from the properties of

H

(y,

y)

(subcase (la)) and

y)

(subcase (lb)) we see that min{A(A, y, y ) | A > 0, y > 0} = min{£1 + , £2+) >

and

max{^1 + , ^2+> < 0 + Q/) .

For subcase (ii) we see from the properties of Ä^(y, y ) (subcase (2a)) and zy) (subcase (3b)) that

g + (y) > ^1+ = Ä1 (0, zy) > ^ ( y ^ , zy) = h { 0, y^, zy) > £2+ .

Similarly for subcase (iii)

g + (y) - C2+ > Z1+ •

In appendix 4-A, some further results involving the quantities , ^2± and g ± (y) are given.

Finally we summarize our results for the subcases (i), (ii) and (iii) of the case where (a + b ) < 0 , (c+d) < 0 :

(i) 7z (zy) > 0 , h ^ ( y ) > 0 . For this subcase

0 < X(zy) < \ z < XQ (0 j zy) , 0 < y(zy) < y^ < yQ (0, zy) , max{51 + , ^2+} < g + (y) < 1 ,

min{/z(X, y, zy) | X > 0, y > 0} = min{£1+, £2+} ; (ii) /z (zy) < 0 , 7z (zy) > 0 . For this subcase

0 < XQ (0, zy) 5 Xz < X (zy) , y (zy) 5 0 , 0 < y^ < y Q (0, zy) , 52+ < S1+ s g + <.y) ,

min{/z(X, y , zy) | X > 0, y > 0} = £2 + ; (iii) /z (zy) > 0 , /z (zy) 5 0 . For this subcase

X(zy) < 0 , 0 < X z < XQ (0, zy) , 0 < yQ(0, zy) 5 y z 5 y(zy) ,

^1+ < ^2+ ~ 5

min{/z(X, y, zy) | X > 0, y > 0} = £ .

4-5. Solutions of [/(£, X, y, zy) = 0

In this section we study the behaviour of the zeros of

£/(£, X, y, zy) , first when 5, y and zy are held fixed and then when X and zy are fixed. The approach is very similar to that used in sect. 3-5. From eqs. (4-21), (4-22), (4-16) and (4-17) we have

where ^ ( S ) = ^ [ ( ^ ( y , z y)- q^( y, z / ) ) 2 - p i y ( 0 ] = 4 G2 [ ( E a - F b ) 2- v ( £ j } , ( 4 - 5 0 ) b ^ z , y , y ) = 2[ f o 1 ( y 9 z / ) - q [ ( y , zy)) ( z ^ C y ) - r [ ( y ) ) - ( ^ ( y , #)+<?[ ( y , y j ) v { Z ) \ = 2[267Z{(£a-PZ>)(£c-Pd)+zyy(£)}y +6 {(£a-Pb) (#2- F 2) - {Ea+Fb )v ( Z )}] , (4-51) c-L(^s y) = (r1 (y )-r|(y ))2 - 2y (£) (r^y )+r^(y)) +

( v ( Z ) ) 2

= 4

H 2 [_(Fc-Fd)2 - v ( Z ) J u 2

t 2n [ ( E c - F d ) (E2 - F 2 ) - ( E c + F d ) v ( Z ) ] v + 4£2P2 (£2-l) , (4-52) and

v C Z )

is given in eq. (4-18). By analogy with eq. (3-53) we see that

(2^(5, U, y ) ) 2 - a 1(5)c1 (5, U) > o for 5 > -(2EF)'1 (fi2+F2) if $ (y, zy^ty, y ) < 0 where ^(y, y ) and ^(y, y ) are given by (4-26). For cases where $_^(y, y)^l^(,\i, y ) - 0 ,

(2>1 (5, u, y))2 - a 1(5)c1 (5, P) = i6F2F2C2y(5)(5-F1(y, j/)} y)) ,

Note t h a t # . ( y , zy) i s j u s t t h e q u a n t i t y d e f i n e d i n e q. ( 4 - 2 9 ) f o r ( 4 - 5 3 ) where

= P 1 1| p 1 ( ^ i ( y ) +2, | ( y ) } -2q 1 ( y , y ) q [ ( v , y )

f i ^ u , y ) f

- M o , y , iy) - ( 2EFp ) 1 l / [q ( y, y Hv ' p Tr ( y ) ) ( M ( y, y ) +y£7r ! ( y ))

^ ( y , y ) i l l i l l l i

+ / f e i Cy, i / ) - / p 1r 1 ( y ) J ( q | ( y , 2 / ) - / p 1r ^ ( y " 5 } | a n d h e n c e

K ^ M , y ) < ^ ( y , zy) 5 M O , y , zy) . ( 4 - 5 5 ) The t wo e q u a l i t y s i g n s i n e q . ( 4 - 5 5 ) c a n n o t o c c u r t o g e t h e r . When 4>^(y, zy) > 0 , ¥ ( y , y ) - 0 , t h e r e i s a n o t h e r f orm f o r # ^ ( y , zy) , A l ( y , y ) , n a m e l y t f - ^ y , zy) ^ ^ _____________ = M O , y , zy) + ( 2 5 ^ ) X| / ( / p 1r 1 ( y ) + q 1 ( y , y )) ( V £ ^ ( y ) - q ^ ( y , zy)) y ) 1 ± / ( / p 1r 1 ( y ) - q 1 ( y , zy)) ( / p ^ T y T + q ^ ( y , zy)}j , ( 4 - 5 6 ) w h e r e M O , y , zy) i s g i v e n i n e q . ( 4 - 2 2 ) . T h u s , f o r t h i s c a s e - ( 2 £ ' F ) ‘ 1 (e2+F2) < u o , w, y ) s ^ ( y , y ) < f l - ^ y , y) s 0 , y , t/) . ( 4 - 5 7 ) Eq s . ( 4 - 5 7 ) a n d ( 4 - 5 3 ) show t h a t when (a + b) < 0 and 0 < y 5 y^ ,

( ^ 1 ( C S y , zy))2 - a 1 ( C ) ö 1 ( ^ , y ) > o f o r £ > ^ ( y , zy) , e q u a l i t y o c c u r i n g i f a n d o n l y i f t, = # ^ ( y , y ) . A l s o , f rom e q s . ( 4 - 5 5 ) a n d ( 4 - 5 3 ) i t f o l l o w s t h a t when (a + b) > 0 an d y > 0 o r when ( a+b) < 0 a n d y > y^ , ( , y , y ) ) 2 ~ a 1 (^)c?1 ( ^ , y ) > 0 f o r £ > M O , y , y ) . We h a v e now shown t h a t f o r e a c h £ > m i n ( M A , y , y ) | A > 0 , y f i x e d a n d > 0} , t h e q u a d r a t i c e q u a t i o n i n A , A, y, y ) = 0

has two real roots A+ (£, y, zy) given by

M $ > ^ 2/)

t x(5)]n -i

-M S , Vi, M ((M S , y, y))-a1W o

1 (€, u)j

(4-58) A.,

2

except for the one point at most where a^(£) = 0 .

From eqs. (4-51), (4-22), (4-16) and (4-17) we see that

M?2(0, V*»

y

)» U, 2/) = -4(/r»1(y)+/r^(y)J»/5*i (y)»/r^(y)Z1 (y, zy) , (4-59) where Z.(y, zy) is given in eq. (4-31). Consequently

X+ (fc(0, y, zy), y,

y) -

0 * X_(^(0, y , zy), y, zy) when Z-^y, zy) > 0 ,

X_(h(0,

y, zy), y, zy) = 0 ^

X + (h(09

y, zy), y, zy) when ^ ( y , zy) < 0 , and

X+ (M0, y, zy), y, zy) = Xj/i(0, y, zy), y, zy) = 0 when ^ ( y , zy) = 0 . Notice too that as £ -► +°°

X+ (5. u, y )

^

— ±

G

1 (BF)J 55 . When

(a+b)

< 0 and 0 5 y $ y^ , we see from eqs. (4-50) and (4-29) that

l l

*lv 1( M y , = ~4 (pi^1 (y)-fa1Cy, zy))2 | +[p

r i

M ( y ) - (y,

y))

r

-

< o and since a^(£) is linear in 4 ,

a1 (C) <

0

for ^ (y, zy) . (4-60) Also l 2 F2 p), u,

y)

= j/))2] 2+^p1r|(u)-(<7|(u, p))2| 1 1 • q x ( y , zy)[^p

1

r [ ( y ) - ( q | ( y , y )) 2| % M y , y ) ^ ^ ^ ) - [ ^ ( y , zy) )

2|2

, so that

A+ ^ i ( y ’

=

,

y ) ,

u,

y) = XQ( y , y ) ( 4 - 6 1 ) where X ( y , y ) i s g i v e n i n eq. ( 4 - 2 5 ) . Th us , when Z () i , y ) < 0 , X_(£, y , y ) i s t h e i n v e r s e o f t h e s t r i c t l y d e c r e a s i n g f u n c t i o n /z(X, y , y ) on j o , X ( y , y) \ ; t h e f u n c t i o n X_( £, y , y ) t h e r e f o r e s t r i c t l y d e c r e a s e s from X ( y , y ) t o 0 a s £ i n c r e a s e s from t f ^ y , y ) (= 7i(X ( y , y ) , y , y ) ) t o h i0, y , y ) S i m i l a r l y X+ ( £ , y , y ) i s t h e i n v e r s e o f t h e s t r i c t l y i n c r e a s i n g f u n c t i o n h ( \ , y , y ) on [XQ( y , y ) , +°°) and t h e r e f o r e s t r i c t l y i n c r e a s e s from XQ( y , y ) t o +°° a s £ i n c r e a s e s from tf ( y , y ) t o +«> . When Z ^ y , y ) > 0 we s e e t h a t X + ( £ , y , y ) i s a c o n t i n u o u s f u n c t i o n o f £ on [hi 0 , y , y ) , t°°) e x c e p t f o r t h e one p o i n t a t most where = 0 . Thus X+ ( £ , y , y ) i s t h e i n v e r s e o f t h e s t r i c t l y i n c r e a s i n g f u n c t i o n 7z(X, y , y ) on [ 0 , +00) and X+ ( £ , y , y ) s t r i c t l y i n c r e a s e s from 0 t o +°° a s £ i n c r e a s e s from h( Q, y , y ) t o +°° . We ded u ce f u r t h e r t h a t i f > h(Q, y , y ) , where ^ i s t h e s o l u t i o n o f a ^ ( £ ) = 0 , t h e n from e q s . ( 4 - 4 9 ) and ( 4 - 2 2 ) , U) - 0 , Z>1 ( ? a , y , y ) < 0 and

X+ ( £a , y , y) = - 0 l (5a , y ) / 2 ^ 1 (Ca , y , y ) • We can a l s o w r i t e

4 E2F 2U ( Z , X , U, y) = a 2 ( 5 ) u 2 + 2b2 ^ , X , y ) M + o 2 ( 5 , X) , ( 4 - 6 2 ) where

a 2 (£) = 4 H 2 [ ( E c - F d ) 2-v(^i\ , (4-63) b2(£, A, z/) = 2[>^{(Sa-Ffe-)(£’c-JP(i)+^(e)}y

(£2-F2}-(&M*tf)i>(£)Q , (4-64) <?2 (£, A) = 4G2 [(£a-Fb)2-i;(5[[y2

+ 2 G [ ( E a - F b ) [e2 - F2) - (Ea+Fb)v + 4£2F 2 (^2-l) . (4-65) For each £ Ü min{b(A, y, y ) j y > 0, A fixed and - 0} , the

quadratic equation in y ,

U(£,, A, y, 2/) = 0 ,

has two real roots y+ (£, A, y ) given by

y+C?, A, y ) - [a2 (£)] 1 •

-Jb2 (5, A, y ) ± { ( b 2 ( Z, A., y))2-a2 (^)c2 (C, A)}2 (4-66)

We can also write statements about the behaviour of y+ (£, A, y )

analogous to those for A+ (£, y, y ) . These results are sufficient to reverse the order of integration in eq. (4-20) and perform the required

integrations for all cases except the case where (a+b) < 0 ,

i a + d ) < 0 .

Now we look at the case (a+b) < 0 and consider the function i/(£, AQ(y, 2/)> y, y ) f o r 0 5 y 5 y^ . From eqs. (4-21), (4-22),

(4-25), (4-29) and (4-26), we have 4

E 2F 2U{Z,

AQ (y,

y ) ,

y,

y)

- 4F 2F 2 [5-^(A0 (y, y ) , y, j/J] [£-k(A0 (y, y ) , y, y)] ( / $ , ( y , y ) + / ' l '1( y , y ) J2- u ( $ ) + ( E a - F i>)2

(/^(y, y ) - A x (y, y)}2-u(5)t(£h-Fi):

where

4-, (5, y ) = 4[($1- ^ ) 2-P1{(£'a-Fi)2-y(S)}] ,

B1(S, y ) = 2

(

ä x

-

ä

[)- Oq+q) {(Fa-F2>)2-y(5)}

q ( S ) = (q-ÄlJ}2 - 2(i?1+Ä^)[(Ea-F&t-y(5)] + [(Ea-Fi)z-y(£)] (4-68) (4-69) 4F2F 2

(5-5. ) (5-5,) ,

'1+ (4-70)

and P 1#

R^,

are given by eq. (4-27) and £1± by eq. (4-32).

The discriminant of the quadratic function of y in eq. (4-67) is [B1 (5, J/)]2 - 4,(5, y)C,(5) = 4[(Fa-FZ>)2-y(5)] P 1{(Fa-FB)2-u(5)}2

+2{2ei$^-P1(P1+B|)H(Fa-FB)2-y(5)}+P1(B1-Bh+4( V e^

and the second factor vanishes when

2EFK + E2 + F2 - (Ea-Fb) 2

= P-1 2«1fli-P1 (B1« i ) ±2 [ ^ - P 1H 1 2 («i2-PiJ?i _ , 1 r

giving £ =

g + (y)

, with

g + (y)

defined in eq. (4-36). Thus

p)]2 - ^ ( 5 ,

y)c^K)

-

16

E2F2H2\{Ea-Fb)2-v^)\

(y2-i)

[Z~g+ ( y )) ( Z ~ g _ ( y )}

= 4

E2F2G~2H2a1(Z)F(£iy)

, (4-71)

using eqs. (4-50) and (4-A3).

In document The analytic properties of the triangle and box diagram amplitudes (Page 117-124)