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f-pole corrections to the critical parameters of the pomeron
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
f-POLE CORRECTIONS TO THE CRITICAL PARAMETERS OF THE POMERON M. BAIG Departamento de Ftsica TeOrica, UniversidadAutOnoma de Barcelona, Bellaterra, Barcelona, Spain and C. PAJARES Departamento de Ftsica T~orica, Universidad de Santiago de Compostela, Santiago, La Corufta, Spain Received 23 February 1981
A reggeon field theory with two vacuum quantum numbers Regge poles, the pomeron and the f-reggeonis presented. In such a theory only one stable fLxedpoint appears which is sensibly different from that governing a RFT with the pomeron alone. This fact is responsible for the modifications of the critical parameters of the pomeron. In particular, the intercept of the bare pomeron increases and approaches the phenomenologieal value.
The pomeron has been extensively studied in the context of Reggeon Field Theory (RFT) [ 1 - 8 ] . In particular, its critical exponents as well as the intercept of the bare pomeron have been determined. Unfortunately, this last value seems to be in contradiction with some phenomenological analysis that generally requires a higher value for the intercept [9-11 ]. However, before rejecting this critical theory in favor of a supercritical one [ 12,13], it is interesting to analyze a correction that can change the values of the critical parameters and improve the agreement. The purpose of this note is to present the main resuits of a study of an extension of RFT that includes both pomeron and f-reggeon [14,15] fields. The interplay of the f-pole with the pomeron originates the above mentioned correction. The quantum numbers of the f-reggeon allow specific reggeon-pomeron couplings that have not been considered previously in standard works on secondary trajectories [16-18]. The fixed point governing the infrared behaviour of the theory is modified and, consequently, the values of critical parameters of the pomeron are changed. Our main result is that the intercept of the bare pomeron increases sensibly, due to the simple presence of the f-reggeon and that this value turns out to be very close to the phenomenological one.
We denote by ~ and X the pomeron and f-reggeon fields, respectively. The free lagrangian density is constructed in the standard form [2-18] 1
~+
~ 0 = ~ i ~ ( 8 / S T ) ~ 0 -- ¢ r b V ~ V ~ 0 1
+~"
+ ~ ix~(a/8O)XO - a'R o 7X~ V.X0 -
-
RoX
(1)
X0 •
Our interaction lagrangian includes the following terms 21
= _ - ir 0
+
- ixlo [ o;o o +
+
+
o o]t
-iX20 [X0 ~t0~0 + X g ~ 0 ~ + - ~ 2 X 0 + -~~02Xt0] - ~i)~Ro [X~2X0 + X2Xg] ,
(2)
where the subindex 0 stands for bare magnitudes and the subindex R means reggeon. One has to notice that we have used recently [19] a similar lagrangian to show the compatibility of the f.dominance of the pomeron hypothesis [20,21] with the reggeon field theory. Notice, also, that strictly speaking, the pole that couples to the pomeron is the SU(3) singlet combination fl of the f and f' poles, and
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
353
Volume 101B, number 5
PHYSICS LETTERS
not these poles themselves. However, along this work the definite value of the intercept of the non-critical reggeon is irrelevant, and, for this reason, we may consider the singlet combination as a simple pole with an effective intercept. The intercept of this fl'regge°n is defined in terms of the parameter a0 = ARo -- A 0 ,
(3)
or by means of its renormalized expression 8 = A R -- A -- Z ~ I Z R a 5 0 ,
(4)
In the following, we will consider the pomeron as a critical pole (A = 0) and the fl'regge°n as a noncritical one (A R > 0). We fix the renormalized theory imposing conditions on the renormalized truncated Green's functions r ( k ,l,m ,n )'(corresponding to the processes k pomerons + m reggeons ~ l pomerons + n reggeons) which are related to the unrenormalized ones by the renormalization constants Z 3 and Z R a p(k],m ,n) (Ei, k 2, a)
=Z~k+t)/ZZ~m+n)/2r(uk,t,m,n)(Ei, kZ, ao).
(51
The conditions, imposed in an arbitrary normalization point E N can be seen in refs. [19,22]. The dimensionless coupling constants and the intercept parameter are defined as
p =a/(-EN), 1
t
1
t
where we have defined
[3= EN(a/aEN) in glp , ~i=EN(a/aEN)lngilp
(i = 1,2, R)
(8)
and where t = In ~ is the scale parameter of the renormalization group transformations. Notice that we fix the parameter p = constant in the above definitions, according to the method developed by Frazer et al. [5]. The expressions of the/3-functions at one-loop level are complicated and they can be seen in ref. [22]. For the present purposes it is sufficient to present a table with all the fixed points and the corresponding degrees of stability (table 1), at p = 0. One has to notice the presence of a single fully stable fixed point among a total of 11 points. Although the values of the couplings g2 and gR in the stable point are zero, the value of the triple pomeron coupling g is modified by the presence of f-reggeon loops in the propagator. This fact is responsible for the modifications of the critical parameters of the pomeron. Although the f-dominance of the pomeron hypothesis is compatible with the renormalization structure of the RFT, the stable fixed point does not satisfy the factorization requirements [19].
ggl = g 2 g 2 ,
ggR = glg2,
(9)
imposed by this hypothesis. In consequence, the criti-
g= r(a'o)-l/2(EN)-l/2, Table 1 Fixed points of the theory for p = 0. The degree of stability and an indication of the fulfilment of the relations imposed by the f-dominance of the pomeron hypothesis are explicitly shown.
g l = •1 [~ (OtRo + t~;)]-1/2(EN)-1/2 g2 = X2 [~(~Ro + ~ 0 ) I - I / 2 ( E N ) - I / 2
gR = ~kR(Ot'Ro)-I/Z(EN) -1/2 "
(6)
Notice that we do not take into account the renormalization of the slopes. We will always restrict our discussion to the forward direction and, for this reason, the error introduced by such a simplification is very small [5]. The infrared behaviour of the theory is determined by the FLxed points of the renormalization group/3functions according to the equations
ag(t)/at = -~(g(t),gi(t)) , agi(t)/at = -[3i(g(t),gi(t)) 354
21 May 1981
(i = 1,2; R ) ,
(7)
Fixed point (gc, g 1c, g2 c, gRc
Degree of stability
f-dominance
(2.031, 1.665, 0 ,0 ) (2.492, 0 ,0 ,2.492) (0 ,0 , 1.665, 2.031) (2.492, 0 ,0 ,0 ) (0.992, 0.992, 0.992, 0.992) (0 ,0 ,0 ,2.492) (1.762, 1.762, 0 ,0 ) (0 ,0 , 1.762, 1.762) (0.190, 0.904, 0.583, 1.413) (1.413, 0.583, 0.904, 0.190) (0 ,0 ,0 ,0 )
4 3 3 2 2 2 2 2 1 1 0
no no no yes yes yes no no no no yes
Volume 101B, number 5
PHYSICS LETTERS
cal pomeron cannot be f-dominated. One can see in table 1 the fulfilment of such relations by the different f'Lxedpoints of the theory. The knowledge of the fixed points allows to construct scaling functions for the asymptotic behaviour of the theory by means of the renormalization group equation. Nevertheless, these scaling laws are not sufficient to determine the intercept of the bare pomeron and, for this reason, the method of integral representations of the propagators is introduced, following the work of Frazer et al. [5]. For each renormalization constant Z x we define
(lO)
Tx = EN (0/0EN) In Z x Ip ,
21 May 1981
and 0iF(,1,1,0, 0)
_ 0iP(u 1'1'0'0) i}E
180
iV(1,1,o,o) (E, p)
=f
-
d e t ( g ' R 2 ' R 3 ' R 4 ) ) ~xfge)/r*
,
(12)
where R a and r~ are, respectively, the eigenvectors and eigenvalues of the matrix Aii defined as
Aii= a[3i/agilg=ge .
(13)
It can be shown that 0(~,x(gc, p)/rl)/O p = 0 ,
(18)
\Z~I p
0
Zx = (1
(1+alnz alnE
as it is expected in a critical exponent [23] and, therefore, the values of "},x/rl at p = 0 can be used in the exponent of eq. (12). The propagator of the pomeron is related to Z 3 through the normalization conditions. Inverting the equation we may write P(u1'1,0,0) as an integral of Z~ 1 from E' = 0 to E' = E. Unfortunately, at E' = 0, p runs to infinity and the scheme breaks down. In order to avoid this problem we define E (oir(1,l,O,O) ~ ir(ul,l,°,°)(E,p) = f dE' (15) 0 \ ~E; Ip]'
P
&l" (19)
This is the full expression for the pomeron propagator. Details of such a calculation can be seen in ref. [22]. The expression o f Z 3 can be obtained using eq. (12). The one loop level expression for the corresponding 7-function is
g2 7(g, P) -- - 16~
gl 16~(1-2p)
d 8~(1-p)"
(20)
Unfortunately, with this expression of Z 3 the integral (19) cannot be computed analytically. Nevertheless, one can extract explicitly its dominant contribution. First we define
Zg =g/go, (14)
(16)
(17)
8 o = ~/z~ = ~ p / z ~ ,
OIn Z x '¥x=/3 0 ~ - + / 3 1
Solving this equation for Z x one obtains a more convenient expression than the original perturbative one. The complexity of 3-functions does not allow us to obtain a full solution and it is necessary to linearize at the neighbourhood of a flexed point, ge = (ge, glc, g2c, gRc)" We finally obtain
38 0 E'~'- p.
using that
we finally obtain
(11)
~
If we define also 0ip(1,1,0,0) Z~p = - Z 3 ~8 o
and from eq. (8) we deduce the equation 0 In Z x 0 In Z x 01nZ x ~ - ~ 1 +~2 0g----~+/3R - 0gR
Oip(l'l'O'O)[ +
(21)
and using eq. (12) one obtains (1
det(g, R2, R3, R4) ~l/2r 1 (22)
Now, it is easy to see that Z3 = (if-0) 2"(ge)'~tgO]`ge'2"r(gc) = (E~)-'Y(ge)__ , (23) where E 0 -- ro/aog 2 , c2 •
(24)
This is the relevant approximated expression for the Z 3 function valid in the region E @ E o. Since at one-
355
Volume 101B, number 5
PHYSICS LETTERS
loop level (Zsp/Z~) ~ 0 and gc is almost independent of p, one finally obtains iP(ul,l,0,O)(E)
This expression differs from that obtained without the f-pole interaction only in the value of the critical exponent. Indeed, now the exponent is 1.1 36 in favor o f the value 1.124 obtained in the case o f pomerons alone. In order to evaluate the intercept of the bare pomeron a 0 = 1 - A 0, one has to compute - A 0 = 8A = f d E ' [ 1 - Z ~ I ( E ' ) ] . (26) 0 As is well known this kind of integral diverges. To remove the divergence one usually introduces an ultraviolet cut-off in the propagators 1
E-
ot~k 2 - A 0
e -bk2
-" E-
b > 0.
(27)
otbk 2 - A 0 '
In our case we will use a simplified method following the work o f Sugar and White [2], introducing a cutoff directly in the integral A 8A= f dE'[1 -z~a(E')] . (28) 0 To obtain a value for the cut-off we have computed the Z 3 function for the case o f pomerons alone and we have fixed the value of A imposing that the integral (28) reproduces the value 6 AFR = 0.038 obtained by Frazer et al. [5] using a cut-off in the propagators. Introducing the full expression for Z 3 into the integral (28) and assuming ~'10 ~- )'20 ~- r0 we finally obtain 8A = 0 . 0 5 6 . This value for the intercept of the bare critical pomeron is clearly higher than that obtained previously in a theory with critical pomerons alone (6AFR). Moreover, if we consider Xl0 > r0, as accepted in many reasonable models [24], then one obtains 6/X > 0.056, i.e. a value very close to the phenomenological one which is accepted to be in the range [ 9 - 1 1 ] 6 A ~ 0.06 - 0.08. Finally, one has to remark that the above results show an apparently surprising high influence o f the f-pole on the pomeron. Taken into account the in356
21 May 1981
terest of such a result, we will attempt a deeper understanding of this fact, using alternative techniques * l In conclusion, the presence of the f-pole changes the critical parameters of the pomeron in such a way that the intercept o f the bare critical pomeron approaches the phenomenological value. The attractive possibility of having a critical theory for the pomeron seems, therefore, not ruled out by the present phenomenology. We are grateful to Dr. A. Bramon for many discussions and to Instituto de Estudios Nucleares (Madrid) for partial financial support. ,1 We thank Dr. J.W. Dash for useful comments and suggestions on this point. [1] H.D.I. Abarbanel and J.B. Bronzan, Phys. Rev. D9 (1974) 2397. [2] R.L. Sugar and A.R. White, Phys. Rev. D10 (1974) 4074. [3] W.R. Frazer and M. Moshe, Phys. Rev. D12 (1975) 2370. [4] H.D.I. Abarbanel, J. Bartels, J.B. Bronzan and D. Sidhu, Phys. Rev. D12 (1975) 2798. [5] W.R. Frazer, H. Hoffman, J.L. Fulco and R.L. Sugar, Phys. Rev. D14 (1976) 2387. [6] J.B. Bronzan and J. Dash, Phys. Rev. D10 (1974) 4208; D12 (1975) 1850. [7] M. Baker, NucL Phys. B80 (1974) 62. [8] J. Dash and S. Harrington, Phys. Lett. 57B (1975) 78. [9] A. Capella and J. Kaplan, Phys. Lett. 52B (1974) 448. [10] A. CapeUa, J. Kaplan and Tran Thanh Van, Nucl. Phys. B97 (1975) 493. [11] C. Pajares and R. Pascual, Phys. Rev. D16 (1977) 1359. [12] D. Amati, M. Ciafaloni, M. Le Bellae and G. Marchesini, Nucl. Phys. l12B (1976) 107. [13] A.R. White, CERN preprint TH 2449 (1978). [14] J.C. Romao and P.G.O. Freund, NucL Phys. 121B (1977) 413. [15] A.C. Irving, Nucl. Phys. 121 B (1977) 176. [ 16] H.D.I. Abarbanel and R.L. Sugar, Phys. Rev. D10 (1974) 721. [17] R. Savit and J. Barrels, Phys. Rev. D l l (1975) 2300. [18] V.L.V. Baltar, Phys. Rev. D19 (1979) 370. [191 M. Baig and C. Pajares, Z. Phys. C7 (1980) 39. [20] R. Carlitz, M.B. Green and A. Zee, Phys. Rev. D4 (1971) 3439. [21] P.G.O. Freund, H.F. Jones and R.J. Rivers, Phys. Lett; 36B (1971) 89. [22] M. Baig, Ph.D. Thesis, Universidad Aut6noma de Barcelona (1980); M. Baig and C. Pajares, to be published. [23] J. Kogut and K.G. Wilson, Phys. Rep. 12C (1974) 75. [24] A.B. Kaidalov and K.A. Ter-Martirosyan, Nucl. Phys. B75 (1974) 471.
PHYSICS LETTERS
21 May 1981
f-POLE CORRECTIONS TO THE CRITICAL PARAMETERS OF THE POMERON M. BAIG Departamento de Ftsica TeOrica, UniversidadAutOnoma de Barcelona, Bellaterra, Barcelona, Spain and C. PAJARES Departamento de Ftsica T~orica, Universidad de Santiago de Compostela, Santiago, La Corufta, Spain Received 23 February 1981
A reggeon field theory with two vacuum quantum numbers Regge poles, the pomeron and the f-reggeonis presented. In such a theory only one stable fLxedpoint appears which is sensibly different from that governing a RFT with the pomeron alone. This fact is responsible for the modifications of the critical parameters of the pomeron. In particular, the intercept of the bare pomeron increases and approaches the phenomenologieal value.
The pomeron has been extensively studied in the context of Reggeon Field Theory (RFT) [ 1 - 8 ] . In particular, its critical exponents as well as the intercept of the bare pomeron have been determined. Unfortunately, this last value seems to be in contradiction with some phenomenological analysis that generally requires a higher value for the intercept [9-11 ]. However, before rejecting this critical theory in favor of a supercritical one [ 12,13], it is interesting to analyze a correction that can change the values of the critical parameters and improve the agreement. The purpose of this note is to present the main resuits of a study of an extension of RFT that includes both pomeron and f-reggeon [14,15] fields. The interplay of the f-pole with the pomeron originates the above mentioned correction. The quantum numbers of the f-reggeon allow specific reggeon-pomeron couplings that have not been considered previously in standard works on secondary trajectories [16-18]. The fixed point governing the infrared behaviour of the theory is modified and, consequently, the values of critical parameters of the pomeron are changed. Our main result is that the intercept of the bare pomeron increases sensibly, due to the simple presence of the f-reggeon and that this value turns out to be very close to the phenomenological one.
We denote by ~ and X the pomeron and f-reggeon fields, respectively. The free lagrangian density is constructed in the standard form [2-18] 1
~+
~ 0 = ~ i ~ ( 8 / S T ) ~ 0 -- ¢ r b V ~ V ~ 0 1
+~"
+ ~ ix~(a/8O)XO - a'R o 7X~ V.X0 -
-
RoX
(1)
X0 •
Our interaction lagrangian includes the following terms 21
= _ - ir 0
+
- ixlo [ o;o o +
+
+
o o]t
-iX20 [X0 ~t0~0 + X g ~ 0 ~ + - ~ 2 X 0 + -~~02Xt0] - ~i)~Ro [X~2X0 + X2Xg] ,
(2)
where the subindex 0 stands for bare magnitudes and the subindex R means reggeon. One has to notice that we have used recently [19] a similar lagrangian to show the compatibility of the f.dominance of the pomeron hypothesis [20,21] with the reggeon field theory. Notice, also, that strictly speaking, the pole that couples to the pomeron is the SU(3) singlet combination fl of the f and f' poles, and
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
353
Volume 101B, number 5
PHYSICS LETTERS
not these poles themselves. However, along this work the definite value of the intercept of the non-critical reggeon is irrelevant, and, for this reason, we may consider the singlet combination as a simple pole with an effective intercept. The intercept of this fl'regge°n is defined in terms of the parameter a0 = ARo -- A 0 ,
(3)
or by means of its renormalized expression 8 = A R -- A -- Z ~ I Z R a 5 0 ,
(4)
In the following, we will consider the pomeron as a critical pole (A = 0) and the fl'regge°n as a noncritical one (A R > 0). We fix the renormalized theory imposing conditions on the renormalized truncated Green's functions r ( k ,l,m ,n )'(corresponding to the processes k pomerons + m reggeons ~ l pomerons + n reggeons) which are related to the unrenormalized ones by the renormalization constants Z 3 and Z R a p(k],m ,n) (Ei, k 2, a)
=Z~k+t)/ZZ~m+n)/2r(uk,t,m,n)(Ei, kZ, ao).
(51
The conditions, imposed in an arbitrary normalization point E N can be seen in refs. [19,22]. The dimensionless coupling constants and the intercept parameter are defined as
p =a/(-EN), 1
t
1
t
where we have defined
[3= EN(a/aEN) in glp , ~i=EN(a/aEN)lngilp
(i = 1,2, R)
(8)
and where t = In ~ is the scale parameter of the renormalization group transformations. Notice that we fix the parameter p = constant in the above definitions, according to the method developed by Frazer et al. [5]. The expressions of the/3-functions at one-loop level are complicated and they can be seen in ref. [22]. For the present purposes it is sufficient to present a table with all the fixed points and the corresponding degrees of stability (table 1), at p = 0. One has to notice the presence of a single fully stable fixed point among a total of 11 points. Although the values of the couplings g2 and gR in the stable point are zero, the value of the triple pomeron coupling g is modified by the presence of f-reggeon loops in the propagator. This fact is responsible for the modifications of the critical parameters of the pomeron. Although the f-dominance of the pomeron hypothesis is compatible with the renormalization structure of the RFT, the stable fixed point does not satisfy the factorization requirements [19].
ggl = g 2 g 2 ,
ggR = glg2,
(9)
imposed by this hypothesis. In consequence, the criti-
g= r(a'o)-l/2(EN)-l/2, Table 1 Fixed points of the theory for p = 0. The degree of stability and an indication of the fulfilment of the relations imposed by the f-dominance of the pomeron hypothesis are explicitly shown.
g l = •1 [~ (OtRo + t~;)]-1/2(EN)-1/2 g2 = X2 [~(~Ro + ~ 0 ) I - I / 2 ( E N ) - I / 2
gR = ~kR(Ot'Ro)-I/Z(EN) -1/2 "
(6)
Notice that we do not take into account the renormalization of the slopes. We will always restrict our discussion to the forward direction and, for this reason, the error introduced by such a simplification is very small [5]. The infrared behaviour of the theory is determined by the FLxed points of the renormalization group/3functions according to the equations
ag(t)/at = -~(g(t),gi(t)) , agi(t)/at = -[3i(g(t),gi(t)) 354
21 May 1981
(i = 1,2; R ) ,
(7)
Fixed point (gc, g 1c, g2 c, gRc
Degree of stability
f-dominance
(2.031, 1.665, 0 ,0 ) (2.492, 0 ,0 ,2.492) (0 ,0 , 1.665, 2.031) (2.492, 0 ,0 ,0 ) (0.992, 0.992, 0.992, 0.992) (0 ,0 ,0 ,2.492) (1.762, 1.762, 0 ,0 ) (0 ,0 , 1.762, 1.762) (0.190, 0.904, 0.583, 1.413) (1.413, 0.583, 0.904, 0.190) (0 ,0 ,0 ,0 )
4 3 3 2 2 2 2 2 1 1 0
no no no yes yes yes no no no no yes
Volume 101B, number 5
PHYSICS LETTERS
cal pomeron cannot be f-dominated. One can see in table 1 the fulfilment of such relations by the different f'Lxedpoints of the theory. The knowledge of the fixed points allows to construct scaling functions for the asymptotic behaviour of the theory by means of the renormalization group equation. Nevertheless, these scaling laws are not sufficient to determine the intercept of the bare pomeron and, for this reason, the method of integral representations of the propagators is introduced, following the work of Frazer et al. [5]. For each renormalization constant Z x we define
(lO)
Tx = EN (0/0EN) In Z x Ip ,
21 May 1981
and 0iF(,1,1,0, 0)
_ 0iP(u 1'1'0'0) i}E
180
iV(1,1,o,o) (E, p)
=f
-
d e t ( g ' R 2 ' R 3 ' R 4 ) ) ~xfge)/r*
,
(12)
where R a and r~ are, respectively, the eigenvectors and eigenvalues of the matrix Aii defined as
Aii= a[3i/agilg=ge .
(13)
It can be shown that 0(~,x(gc, p)/rl)/O p = 0 ,
(18)
\Z~I p
0
Zx = (1
(1+alnz alnE
as it is expected in a critical exponent [23] and, therefore, the values of "},x/rl at p = 0 can be used in the exponent of eq. (12). The propagator of the pomeron is related to Z 3 through the normalization conditions. Inverting the equation we may write P(u1'1,0,0) as an integral of Z~ 1 from E' = 0 to E' = E. Unfortunately, at E' = 0, p runs to infinity and the scheme breaks down. In order to avoid this problem we define E (oir(1,l,O,O) ~ ir(ul,l,°,°)(E,p) = f dE' (15) 0 \ ~E; Ip]'
P
&l" (19)
This is the full expression for the pomeron propagator. Details of such a calculation can be seen in ref. [22]. The expression o f Z 3 can be obtained using eq. (12). The one loop level expression for the corresponding 7-function is
g2 7(g, P) -- - 16~
gl 16~(1-2p)
d 8~(1-p)"
(20)
Unfortunately, with this expression of Z 3 the integral (19) cannot be computed analytically. Nevertheless, one can extract explicitly its dominant contribution. First we define
Zg =g/go, (14)
(16)
(17)
8 o = ~/z~ = ~ p / z ~ ,
OIn Z x '¥x=/3 0 ~ - + / 3 1
Solving this equation for Z x one obtains a more convenient expression than the original perturbative one. The complexity of 3-functions does not allow us to obtain a full solution and it is necessary to linearize at the neighbourhood of a flexed point, ge = (ge, glc, g2c, gRc)" We finally obtain
38 0 E'~'- p.
using that
we finally obtain
(11)
~
If we define also 0ip(1,1,0,0) Z~p = - Z 3 ~8 o
and from eq. (8) we deduce the equation 0 In Z x 0 In Z x 01nZ x ~ - ~ 1 +~2 0g----~+/3R - 0gR
Oip(l'l'O'O)[ +
(21)
and using eq. (12) one obtains (1
det(g, R2, R3, R4) ~l/2r 1 (22)
Now, it is easy to see that Z3 = (if-0) 2"(ge)'~tgO]`ge'2"r(gc) = (E~)-'Y(ge)__ , (23) where E 0 -- ro/aog 2 , c2 •
(24)
This is the relevant approximated expression for the Z 3 function valid in the region E @ E o. Since at one-
355
Volume 101B, number 5
PHYSICS LETTERS
loop level (Zsp/Z~) ~ 0 and gc is almost independent of p, one finally obtains iP(ul,l,0,O)(E)
This expression differs from that obtained without the f-pole interaction only in the value of the critical exponent. Indeed, now the exponent is 1.1 36 in favor o f the value 1.124 obtained in the case o f pomerons alone. In order to evaluate the intercept of the bare pomeron a 0 = 1 - A 0, one has to compute - A 0 = 8A = f d E ' [ 1 - Z ~ I ( E ' ) ] . (26) 0 As is well known this kind of integral diverges. To remove the divergence one usually introduces an ultraviolet cut-off in the propagators 1
E-
ot~k 2 - A 0
e -bk2
-" E-
b > 0.
(27)
otbk 2 - A 0 '
In our case we will use a simplified method following the work o f Sugar and White [2], introducing a cutoff directly in the integral A 8A= f dE'[1 -z~a(E')] . (28) 0 To obtain a value for the cut-off we have computed the Z 3 function for the case o f pomerons alone and we have fixed the value of A imposing that the integral (28) reproduces the value 6 AFR = 0.038 obtained by Frazer et al. [5] using a cut-off in the propagators. Introducing the full expression for Z 3 into the integral (28) and assuming ~'10 ~- )'20 ~- r0 we finally obtain 8A = 0 . 0 5 6 . This value for the intercept of the bare critical pomeron is clearly higher than that obtained previously in a theory with critical pomerons alone (6AFR). Moreover, if we consider Xl0 > r0, as accepted in many reasonable models [24], then one obtains 6/X > 0.056, i.e. a value very close to the phenomenological one which is accepted to be in the range [ 9 - 1 1 ] 6 A ~ 0.06 - 0.08. Finally, one has to remark that the above results show an apparently surprising high influence o f the f-pole on the pomeron. Taken into account the in356
21 May 1981
terest of such a result, we will attempt a deeper understanding of this fact, using alternative techniques * l In conclusion, the presence of the f-pole changes the critical parameters of the pomeron in such a way that the intercept o f the bare critical pomeron approaches the phenomenological value. The attractive possibility of having a critical theory for the pomeron seems, therefore, not ruled out by the present phenomenology. We are grateful to Dr. A. Bramon for many discussions and to Instituto de Estudios Nucleares (Madrid) for partial financial support. ,1 We thank Dr. J.W. Dash for useful comments and suggestions on this point. [1] H.D.I. Abarbanel and J.B. Bronzan, Phys. Rev. D9 (1974) 2397. [2] R.L. Sugar and A.R. White, Phys. Rev. D10 (1974) 4074. [3] W.R. Frazer and M. Moshe, Phys. Rev. D12 (1975) 2370. [4] H.D.I. Abarbanel, J. Bartels, J.B. Bronzan and D. Sidhu, Phys. Rev. D12 (1975) 2798. [5] W.R. Frazer, H. Hoffman, J.L. Fulco and R.L. Sugar, Phys. Rev. D14 (1976) 2387. [6] J.B. Bronzan and J. Dash, Phys. Rev. D10 (1974) 4208; D12 (1975) 1850. [7] M. Baker, NucL Phys. B80 (1974) 62. [8] J. Dash and S. Harrington, Phys. Lett. 57B (1975) 78. [9] A. Capella and J. Kaplan, Phys. Lett. 52B (1974) 448. [10] A. CapeUa, J. Kaplan and Tran Thanh Van, Nucl. Phys. B97 (1975) 493. [11] C. Pajares and R. Pascual, Phys. Rev. D16 (1977) 1359. [12] D. Amati, M. Ciafaloni, M. Le Bellae and G. Marchesini, Nucl. Phys. l12B (1976) 107. [13] A.R. White, CERN preprint TH 2449 (1978). [14] J.C. Romao and P.G.O. Freund, NucL Phys. 121B (1977) 413. [15] A.C. Irving, Nucl. Phys. 121 B (1977) 176. [ 16] H.D.I. Abarbanel and R.L. Sugar, Phys. Rev. D10 (1974) 721. [17] R. Savit and J. Barrels, Phys. Rev. D l l (1975) 2300. [18] V.L.V. Baltar, Phys. Rev. D19 (1979) 370. [191 M. Baig and C. Pajares, Z. Phys. C7 (1980) 39. [20] R. Carlitz, M.B. Green and A. Zee, Phys. Rev. D4 (1971) 3439. [21] P.G.O. Freund, H.F. Jones and R.J. Rivers, Phys. Lett; 36B (1971) 89. [22] M. Baig, Ph.D. Thesis, Universidad Aut6noma de Barcelona (1980); M. Baig and C. Pajares, to be published. [23] J. Kogut and K.G. Wilson, Phys. Rep. 12C (1974) 75. [24] A.B. Kaidalov and K.A. Ter-Martirosyan, Nucl. Phys. B75 (1974) 471.