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The n-loop expansion of the Reggeon calculus
Volume 59B, number 3
PHYSICS LETTERS
THE n-LOOP EXPANSION
OF THE REGGEON
10 November 1975
CALCULUS
J.W. DASH
Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA and S.J. HARRINGTON
Department of Physics, University of Washington, Seattle, Wash. 98195, USA Received 3 September 1975 We apply the technique known in solid state physics as the n-loop expansion to calculate the critical indices of the ~3 Gribov Reggeon calculus directly in two transverse dimensions. Infrared pathologies of the massless theory require the calculation to be done in the infinite momentum limit of the massive theory. For n = 1 the results are close to those of the e-expansion in O(e). For n = 2 the/3 function has no zero, analogously to the case in solid state physics. Use of a Pad~ approximant for/3 yields Otot ~ (In s) °'27 at infinity, close to the O(e 2) result.
Gribov's Reggeon calculus [1 ] has provided a sound theoretical base for an understanding o f the asymptotic properties o f diffraction scattering. The treatment o f the theory in strong coupling [2] is appealing because it provides the possibility o f a connection with methods utilized in solid state physics [3] for the calculation o f critical indices associated with the scaling behavior o f Green's functions in second order phase transitions. While grave problems associated with energy scales set b y Brookhaven-FNAL data render the probability o f the application o f the Gribov calculus to phenomenology remote (even in perturbation theory unless one is careful to include nondiffractive and diffractive effective thresholds explicitly [4] ), the fact that the calculus explicitly satisfies all constraints o f t-channel unitarity makes a calculation of its critical indices o f great interest. One would certainly like to know, for instance, whether or not the calculus is consistent with s-channel unitarity to the extent o f satisfying the Froissart bound, even if the exact values o f the critical exponents may not turn out to provide anything o f phenomenological interesl at present energies. There are several methods utilized in the calculation o f critical indices. The most reliable is the high temperature expansion when carried out to high order [3] ; a low order calculation has now been performed for the Reggeon-calculus [5]. The e-expansion and the loop expansion [3] are two reasonable approximation schemes. F o r example the 1 and 2 loop results [3] in the two dimensional Ising model are u = 0.75, 1.10 and ~ = 0, 0.42 which bracket the exact values v = 1, r / = 1/4 and are qualitatively as good as those o f the e-expansion. The e-expansion in the Reggeon calculus has been carried out to O(e 2) [2, 6, 7]. The loop expansion consists o f evaluating the critical exponents to some order o f perturbation theory directly in the dimension o f interest (here D = 2). This cannot, however, be carried out in the Reggeon calculus directly. The problem lies in the mass counter terms which can be ignored in the e-expansion but which necessarily enter into the theory evaluated directly at D = 2. Graphs with self-energy insertions develop poles at D = 2 which cannot be removed in the massless theory because the mass counter term 6A is nonanalytic in the bare coupling r o and cannot be expanded perturbatively in r o [3, 8, 9]. Instead one formally constructs the massive theory with a nonzero renormalized mass 2x 4 : 0 (i.e. the renormalized intercept is not equal to one). Scaling laws and critical exponents are defined using the Callan-Symanzik equation in the infinite m o m e n t u m limit. One then employs a theorem [3, 9] ~1 which assures us that these scaling laws and critical exponents are precisely those of the original massless theory in the infrared limit, and the latter are the ones we want. We thus perform renormalized perturba~1 We have checked the theorem in the e-expansion to O(e). 249
Volume 59B, number 3
PHYSICS LETTERS
10 November 1975
tion theory in the massive theory to some order o f r o for the unrenormalized Green's functions, subtracting appropriate counter terms, leaving us with a calculation which is finite and well-defined at D = 2 [10]. The critical exponents thus obtained are inserted into the scaling laws to obtain nonperturbative results. The connection with the massless theory is made explicit by introducing a normalization energy E N and taking the limit E N / A ~ oo with 2 ' ,N fixed [9, 10]. This is valid for the complete theory. In finite order perturbation theory the limit can be ro/t~o/~ shown to bring us back to the incorrect results obtained in the massless theory at D = 2 by picking out the most singular terms of graphs at D = 2. Explicitly one realizes the connection in this limit by replacing the quantity ln(EN/A) arising in the massive theory from bubbles and counter terms by (D/2 1) -1 in the massless theory. In fact, 3' is finite in this limit, but the value thus obtained is not that actually predicted by the theory since all higher order terms are then necessary for cancelling the A = 0 infrared divergences. Our results are as follows [10]. We shall use the notation of ref. [7]. Choosing the normalization point E N = 0 we obtain(g2o = ro/c%A 2 ' is the dimensionless bare coupling)
(1)
Z -1 = ~ iF(1,1)(E, k 2, F o, s o, = 1 +a2g2 +a4g 4 ' A) E=-eN =0 k 2 =0
~ '-Z 1-/ a o' - - -
(2)
1, 3 ip(1,1)(E, k 2 , r o , % ', A ) [ E = O = 1 + c2g 2 + c4g 4 c~o 3k2 k2=O
F(a'2)(Ei, ki, r o, O~o, A)[ Ei=ki.k]= 0
r°
(27r)3/2
(3)
(1 + d3g 2 + d5g 4)
where p(n,m) are the unrenormalized connected proper vertex functions. Here 87r a 2 = - 1 / 4 ,
8rr c 2 = - 1 / 8 ,
8n d 3 = --1.
(4-6)
The two loop coefficients (a4, c4, d5) are numerically equal to (8rr) -2 (0.232, 0.101, 1.56), respectively. They are given by (8rr)2a4 =½(1
ln2)+2
f dx
0
1
(87r)2c4 = ~ (1 - I n 2)+ 2 f dx 0 of +4
(9
ln6) + 10
~ (v2 +3) [(02 +3)
(2x 3 + 3x 2 g+ 18x + 9) ln(1 ~
(1
3--o)(1+o)
(7)
X~
(8) o)(o3+o2+11o+27)}
4(9-v2i
(1 do tn[ 2(3+v) 7
o iv2 + 3) L(3 -- o ) ( l + o)_]" 0 The/3 function AOg/OA at fixed r o, ~'o is then
/3(g) •
In
(9 - x2) 3
do { ( 5 - V 2 ) ln[ 2 ( 3 + 0 ) 0 ) ] (v2+3)2[(02+3) k(3~v)~l+
(8~)2a5 ~--1(1
+4
x2)2(3-x)
21 [1 + 2wg 2 + (4W4 -- 6w2)g 4 ] ,
(9)
(10)
where the dimensionless renormalized coupling g is defined by (2rr)3/2 Z 3/2 F(1,2)(0, 0, ro, t~'o, A) = go(1 + w g 2 + w434).
g = (~,a)1/2
The coefficients w and w 4 are equal to (-11/16)/87r and 1.04/(8702. They are 250
(11)
Volume 59B, number 3
w=d 3 -a 2 - c2/2 ,
PHYSICS LETTERS
w4 =d 5 -d3(a 2 +c2/2)+a2
+ 3c2/8 + a 2 c 2 / 2 - a 4 - c 4 / 2 .
10 November 1975
03)
The two critical exponents 3' and z = 1 - ~'/oe' are given by A 8Z
3' - Z ~ A - a 2 g 2 + (2a4 - 2 w a 2 - a 2 ) g 4
(14)
z = 1 -- ~ - ~ = 1 -- 3' + c 2 9 2 + (2c 4 - 2 w e 2 - c22)g4
(15)
where the derivatives in eqs. (14, 15) are taken at fixed ro2 and ~'o" To O(g 2) we obtain - 7 = 2/1 1 and z = 12/11. These values should be compared to the O(e) results o f 1/6 and 13/12, respectively [2]. The agreement is good. In the two loop calculation a problem arises. As in the case o f solid state physics,/3(g)/g has no real zero to this order [3]. Inclusion o f higher order terms can reinstate the zero. To estimate the higher order terms in/3(g) (and thus its asymptotic behavior ing), one may follow ref. [3] and employ a Pad~ approximant for/3(g), which is unique and has a zero at gc2 = w(2w 4 - 5w2) - 1 . This leads to the final results *2 o f the two loop calculation at E N = 0, - 7 = 0.27,
z = 1.05.
(16)
These numbers are close to the O(e 2) values [6, 7] - 3 ' = 0.32 and z = 1.15. - 3 ' is somewhat smaller than results indicated b y the low order high temperature expansion [5] and somewhat larger than the "twisted fan" value - 3 ' =0.17. One caveat in interpreting our results should be mentioned. Although the exact critical exponents are independent Of EN, this is not true in perturbation theory. The two loop calculation redone with 0 ~ 0. We believe these considerations ameliorate the caveat. We wish to thank Prof. M. Baker, Prof. J. Bronzan, and especially Dr. A. White for very helpful discussions. One o f us (JD) enjoyed the hospitality o f the University of Washington during part o f the time this work was carried out. t2 Two loop results quoted in an earlier version of this paper were in error due to an error in ref. [6, 7]. All exponents are correctly quoted here.
References [1] V.N. Gribov, Sov. Phys. JETP 26 (1968)414; see H.D. Abarbanel and A.R. White, Lectures at the SLAC summer school (1974) for reviews. [2] A. Migdal, A. Polyakov and K. Ter Martirosyan, Phys. Lett. 48B (1974) 239; H.D. Abarbanel and J.D. Bronzan, Phys. Lett. 48B (1974) 345. [3] G. Parisi, Lectures at the Cargese summer school (1973). [4] J.W. Dash and J. Koplik, to be published in Phys. Rev.; N.F. Bali and J.W. Dash, Phys. Lett. 51B (1974) 496; [5] J.W. Dash, Phys. Rev. D9 (1974) 200. 251
Volume 59B, number 3 [5] [6] [7] [8] [9] [10] [11] [12] [13]
252
PHYSICS LETTERS
J. Ellis and R. Savit, TH 1974 (1975). M. Baker, Phys. Lett. 51B (1974) t58; Nucl. Phys. 80B (1974) 62. J.B. Bronzan and J.W. Dash, Phys. Lett. 51B (1974) 496, and Phys. Rev. D10 (1974) 4208. R. Sugar and A. White, Phys. Rev. D10 ~1974) 4063, 4074, J. Zinn-Justin, lectures at the Cargese summer school, 1973. Details of the calculation will be presented in J.W. Dash and S.J. Harrington, to appear. J.W. Dash and S.J. Harrington, Phys. Lett. 57B (1975) 78. J. Bronzan, private communication. J.W. Dash and S.J. Harrington, LBL 3885 (1975), submitted to Phys. Rev. B.
10 November 1975
PHYSICS LETTERS
THE n-LOOP EXPANSION
OF THE REGGEON
10 November 1975
CALCULUS
J.W. DASH
Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA and S.J. HARRINGTON
Department of Physics, University of Washington, Seattle, Wash. 98195, USA Received 3 September 1975 We apply the technique known in solid state physics as the n-loop expansion to calculate the critical indices of the ~3 Gribov Reggeon calculus directly in two transverse dimensions. Infrared pathologies of the massless theory require the calculation to be done in the infinite momentum limit of the massive theory. For n = 1 the results are close to those of the e-expansion in O(e). For n = 2 the/3 function has no zero, analogously to the case in solid state physics. Use of a Pad~ approximant for/3 yields Otot ~ (In s) °'27 at infinity, close to the O(e 2) result.
Gribov's Reggeon calculus [1 ] has provided a sound theoretical base for an understanding o f the asymptotic properties o f diffraction scattering. The treatment o f the theory in strong coupling [2] is appealing because it provides the possibility o f a connection with methods utilized in solid state physics [3] for the calculation o f critical indices associated with the scaling behavior o f Green's functions in second order phase transitions. While grave problems associated with energy scales set b y Brookhaven-FNAL data render the probability o f the application o f the Gribov calculus to phenomenology remote (even in perturbation theory unless one is careful to include nondiffractive and diffractive effective thresholds explicitly [4] ), the fact that the calculus explicitly satisfies all constraints o f t-channel unitarity makes a calculation of its critical indices o f great interest. One would certainly like to know, for instance, whether or not the calculus is consistent with s-channel unitarity to the extent o f satisfying the Froissart bound, even if the exact values o f the critical exponents may not turn out to provide anything o f phenomenological interesl at present energies. There are several methods utilized in the calculation o f critical indices. The most reliable is the high temperature expansion when carried out to high order [3] ; a low order calculation has now been performed for the Reggeon-calculus [5]. The e-expansion and the loop expansion [3] are two reasonable approximation schemes. F o r example the 1 and 2 loop results [3] in the two dimensional Ising model are u = 0.75, 1.10 and ~ = 0, 0.42 which bracket the exact values v = 1, r / = 1/4 and are qualitatively as good as those o f the e-expansion. The e-expansion in the Reggeon calculus has been carried out to O(e 2) [2, 6, 7]. The loop expansion consists o f evaluating the critical exponents to some order o f perturbation theory directly in the dimension o f interest (here D = 2). This cannot, however, be carried out in the Reggeon calculus directly. The problem lies in the mass counter terms which can be ignored in the e-expansion but which necessarily enter into the theory evaluated directly at D = 2. Graphs with self-energy insertions develop poles at D = 2 which cannot be removed in the massless theory because the mass counter term 6A is nonanalytic in the bare coupling r o and cannot be expanded perturbatively in r o [3, 8, 9]. Instead one formally constructs the massive theory with a nonzero renormalized mass 2x 4 : 0 (i.e. the renormalized intercept is not equal to one). Scaling laws and critical exponents are defined using the Callan-Symanzik equation in the infinite m o m e n t u m limit. One then employs a theorem [3, 9] ~1 which assures us that these scaling laws and critical exponents are precisely those of the original massless theory in the infrared limit, and the latter are the ones we want. We thus perform renormalized perturba~1 We have checked the theorem in the e-expansion to O(e). 249
Volume 59B, number 3
PHYSICS LETTERS
10 November 1975
tion theory in the massive theory to some order o f r o for the unrenormalized Green's functions, subtracting appropriate counter terms, leaving us with a calculation which is finite and well-defined at D = 2 [10]. The critical exponents thus obtained are inserted into the scaling laws to obtain nonperturbative results. The connection with the massless theory is made explicit by introducing a normalization energy E N and taking the limit E N / A ~ oo with 2 ' ,N fixed [9, 10]. This is valid for the complete theory. In finite order perturbation theory the limit can be ro/t~o/~ shown to bring us back to the incorrect results obtained in the massless theory at D = 2 by picking out the most singular terms of graphs at D = 2. Explicitly one realizes the connection in this limit by replacing the quantity ln(EN/A) arising in the massive theory from bubbles and counter terms by (D/2 1) -1 in the massless theory. In fact, 3' is finite in this limit, but the value thus obtained is not that actually predicted by the theory since all higher order terms are then necessary for cancelling the A = 0 infrared divergences. Our results are as follows [10]. We shall use the notation of ref. [7]. Choosing the normalization point E N = 0 we obtain(g2o = ro/c%A 2 ' is the dimensionless bare coupling)
(1)
Z -1 = ~ iF(1,1)(E, k 2, F o, s o, = 1 +a2g2 +a4g 4 ' A) E=-eN =0 k 2 =0
~ '-Z 1-/ a o' - - -
(2)
1, 3 ip(1,1)(E, k 2 , r o , % ', A ) [ E = O = 1 + c2g 2 + c4g 4 c~o 3k2 k2=O
F(a'2)(Ei, ki, r o, O~o, A)[ Ei=ki.k]= 0
r°
(27r)3/2
(3)
(1 + d3g 2 + d5g 4)
where p(n,m) are the unrenormalized connected proper vertex functions. Here 87r a 2 = - 1 / 4 ,
8rr c 2 = - 1 / 8 ,
8n d 3 = --1.
(4-6)
The two loop coefficients (a4, c4, d5) are numerically equal to (8rr) -2 (0.232, 0.101, 1.56), respectively. They are given by (8rr)2a4 =½(1
ln2)+2
f dx
0
1
(87r)2c4 = ~ (1 - I n 2)+ 2 f dx 0 of +4
(9
ln6) + 10
~ (v2 +3) [(02 +3)
(2x 3 + 3x 2 g+ 18x + 9) ln(1 ~
(1
3--o)(1+o)
(7)
X~
(8) o)(o3+o2+11o+27)}
4(9-v2i
(1 do tn[ 2(3+v) 7
o iv2 + 3) L(3 -- o ) ( l + o)_]" 0 The/3 function AOg/OA at fixed r o, ~'o is then
/3(g) •
In
(9 - x2) 3
do { ( 5 - V 2 ) ln[ 2 ( 3 + 0 ) 0 ) ] (v2+3)2[(02+3) k(3~v)~l+
(8~)2a5 ~--1(1
+4
x2)2(3-x)
21 [1 + 2wg 2 + (4W4 -- 6w2)g 4 ] ,
(9)
(10)
where the dimensionless renormalized coupling g is defined by (2rr)3/2 Z 3/2 F(1,2)(0, 0, ro, t~'o, A) = go(1 + w g 2 + w434).
g = (~,a)1/2
The coefficients w and w 4 are equal to (-11/16)/87r and 1.04/(8702. They are 250
(11)
Volume 59B, number 3
w=d 3 -a 2 - c2/2 ,
PHYSICS LETTERS
w4 =d 5 -d3(a 2 +c2/2)+a2
+ 3c2/8 + a 2 c 2 / 2 - a 4 - c 4 / 2 .
10 November 1975
03)
The two critical exponents 3' and z = 1 - ~'/oe' are given by A 8Z
3' - Z ~ A - a 2 g 2 + (2a4 - 2 w a 2 - a 2 ) g 4
(14)
z = 1 -- ~ - ~ = 1 -- 3' + c 2 9 2 + (2c 4 - 2 w e 2 - c22)g4
(15)
where the derivatives in eqs. (14, 15) are taken at fixed ro2 and ~'o" To O(g 2) we obtain - 7 = 2/1 1 and z = 12/11. These values should be compared to the O(e) results o f 1/6 and 13/12, respectively [2]. The agreement is good. In the two loop calculation a problem arises. As in the case o f solid state physics,/3(g)/g has no real zero to this order [3]. Inclusion o f higher order terms can reinstate the zero. To estimate the higher order terms in/3(g) (and thus its asymptotic behavior ing), one may follow ref. [3] and employ a Pad~ approximant for/3(g), which is unique and has a zero at gc2 = w(2w 4 - 5w2) - 1 . This leads to the final results *2 o f the two loop calculation at E N = 0, - 7 = 0.27,
z = 1.05.
(16)
These numbers are close to the O(e 2) values [6, 7] - 3 ' = 0.32 and z = 1.15. - 3 ' is somewhat smaller than results indicated b y the low order high temperature expansion [5] and somewhat larger than the "twisted fan" value - 3 ' =0.17. One caveat in interpreting our results should be mentioned. Although the exact critical exponents are independent Of EN, this is not true in perturbation theory. The two loop calculation redone with 0 ~
References [1] V.N. Gribov, Sov. Phys. JETP 26 (1968)414; see H.D. Abarbanel and A.R. White, Lectures at the SLAC summer school (1974) for reviews. [2] A. Migdal, A. Polyakov and K. Ter Martirosyan, Phys. Lett. 48B (1974) 239; H.D. Abarbanel and J.D. Bronzan, Phys. Lett. 48B (1974) 345. [3] G. Parisi, Lectures at the Cargese summer school (1973). [4] J.W. Dash and J. Koplik, to be published in Phys. Rev.; N.F. Bali and J.W. Dash, Phys. Lett. 51B (1974) 496; [5] J.W. Dash, Phys. Rev. D9 (1974) 200. 251
Volume 59B, number 3 [5] [6] [7] [8] [9] [10] [11] [12] [13]
252
PHYSICS LETTERS
J. Ellis and R. Savit, TH 1974 (1975). M. Baker, Phys. Lett. 51B (1974) t58; Nucl. Phys. 80B (1974) 62. J.B. Bronzan and J.W. Dash, Phys. Lett. 51B (1974) 496, and Phys. Rev. D10 (1974) 4208. R. Sugar and A. White, Phys. Rev. D10 ~1974) 4063, 4074, J. Zinn-Justin, lectures at the Cargese summer school, 1973. Details of the calculation will be presented in J.W. Dash and S.J. Harrington, to appear. J.W. Dash and S.J. Harrington, Phys. Lett. 57B (1975) 78. J. Bronzan, private communication. J.W. Dash and S.J. Harrington, LBL 3885 (1975), submitted to Phys. Rev. B.
10 November 1975