VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

FIXED POLES IN VIRTUAL COMPTON AMPLITUDES

T. P . Cheng* and Wu-Ki Tung Institute for Advanced Study, Princeton, New Jersey 08540

(Received 5 February 1970)

It is suggested that the leading fixed-pole terms in virtual Compton amplitudes are polynomial functions of the photon mass variable q2. Several arguments are given in sup­port of this conjecture. Implications of this behavior in the calculation of the electromag­netic mass differences and in the q2 dependence of the e~p inelastic-scattering structure functions are discussed.

It has been suggested on theore t i ca l grounds that ce r t a in weak ampli tudes should have fixed poles in the complex angular momentum (J) plane at both r igh t - and wrong-s igna tu re nonsense p o i n t s . 1 ' 2 This r e su l t is supported by numer i ca l s tudies of var ious sum ru les 3 and, recen t ly , by a detai led ana lys is of the forward Compton ampli tude as cons t ruc ted from the observed total photoabsorpt ion c r o s s sec t ions via d i spe rs ion re la t ions . 4 In th is note, we study the dependence of f ixed-pole t e r m s in Compton ampl i ­tudes on the v i r tua l -photon m a s s var iab le (q2).

Consider the forward sp in -averaged v i r tua l Compton amplitude

T^=ifd\e~f«x(p\T*[j«(x)jM]\p), (D where a,j8 a r e i sospin indices . We shal l denote ampli tudes s y m m e t r i c and a n t i s y m m e t r i c in ot,fi by the s u p e r s c r i p t s + and - , respect ive ly . Each of these ampli tudes may be s epa ra t ed into two p a r t s :

where SJtlv(±) a r e the " s ingu la r " p a r t s in the low-energy l i m i t (#M-*0) and satisfy the Ward ident i t ies :

Q^ILV^ =Pv anc* ^MSMy(+) = 0 (for all values of q)\ whereas R^^ vanish in the l imit #M—0 and satisfy

q^Ruj}^ = 0. The b a r e Born t e r m sa t i s f ies the two r equ i r emen t s on S^v and thus explicitly

V H = t 4 ^ -P)PnPv-q2(<!vPv +PvVv) + fo -Ptetf J / (4wi V - t f 4 ) , (3)

V+) = 2fo^*-to-/0te^+^ (4) R{J

±)=t1{±\vyq

2){q2g^-q^qv)+t2^\v,q2)\^ (5)

where v = -q *p/m = {s-u)/4m. The invar iant functions £x(±) and £2

(±) a r e free of k inemat ic s ingu la r i t i e s and z e r o s . The regu la r i zed ^-channel double-hel ic i ty-f l ip ampl i tudes [essent ia l ly the coefficients of the pypv t e r m s in Eqs . (3)-(5)] a r e

f1 ^^mqn^^—^jr^y (6) 1 , 1 2 y2-(74/4ra2

7 1 ^ 1( + ) = m ^ 2

( + ) + 0 ,oq\/A 2 V (7) 1 , 1 2 2m(ir-q*/4m2)

We note that the assumption 5 that t2^ is Regge behaved (thus superconvergent in this case) impl ies a fixed pole at J= 1 in/ 1 >« 1

(" ) with the res idue equal to 1. Using the F r o i s s a r t - G r i b o v formula , one ob­ta ins

(2/ir)flmfu^(u9q2)dV=l. (8)

This is the t = 0 c u r r e n t - a l g e b r a sum rule of Adler , Dashen, Gel l -Mann, and Fubini . 1 A s i m i l a r a s ­sumption that t2

i+) is Regge behaved leads to a J = 0 fixed pole i n / ^ . j ^ with a r e s idue function given by the c l a s s i ca l Thomson ampli tude. Strong evidence for such a t e r m at q2 = 0 has been found by D a m a -shek and Gilman.4

These d i scuss ions suggest that fixed po les , being a spec ia l fea ture of local ized c u r r e n t - c u r r e n t in­t e r a c t i o n s , a r e closely assoc ia ted with the " s ingu la r " p a r t s of the ampli tude which a r e insens i t ive to the s t r u c t u r e a r i s ing from s t rong in te rac t ions . A pa r t i cu la r ly in te res t ing consequence of th is o b s e r v a ­tion is the following Ansatz : The res idue functions of these leading r igh t - s igna tu red fixed poles a r e in­dependent (or s imple polynomials) of the v i r tua l photon m a s s va r i ab le q2. In what follows we shal l give support ing a rguments to this conjecture and cons ider i t s impl icat ions for the e lec t romagne t i c m a s s

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VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

differences and for the q2 dependence of the e-p i ne l a s t i c - sca t t e r ing s t r u c t u r e functions. F o r s impl ic i ty , we will f i r s t cons ider the J= 1 fixed pole i n / ^ - / " * . Without making use of cu r r en t

a lgebra , the res idue i s a genera l function of q2 [cf. Eq. (8) and footnote 5],

R{q2) = {2/n)Jlmfl^1{-\v,q2)dv,

We a s s u m e that R(q2) sa t i s f ies a d i spe rs ion relation6*7 in q2

(9)

N (a2\N C , lmR(q>2) (q'2)N{q>2-q2)

(10)

with a finite number (N) of sub t rac t ions . Bear ing in mind the fact that lmflt^.1 i s the ma t r i x e lement of a product of two c u r r e n t s [cf. Eq. (1)], we can graphical ly r e p r e s e n t the r ight-hand s ide of Eq. (10) as in Fig. 1. T h e r e , a c r o s s at the end of a photon line indicates poss ib le subt rac t ions in q2 (contact t e r m s ) . F igure 1(a) co r r e sponds to the subt rac t ion t e r m in Eq. (10) while F igs . l (b ) - l (d ) al l contr ibute to the d i spers ion in tegra l . The contributions from Figs . 1(b) and 1(c) a r e propor t ional to the res idue functions of a fixed pole at J=l for photoproduction ampl i tudes , and from Fig. 1(d) for a hadron-had-ron sca t t e r ing ampli tude. P u r e hadronic ampli tudes cannot have fixed poles at r igh t - s igna tured points . T h e r e is a lso evidence, both theore t ica l 8 and exper imenta l , 9 against the p r e s e n c e of fixed poles for photoproduction of hadrons . The absence of the J= 1 fixed pole in these ampli tudes (or equivalently, the validity of the cor responding superconvergence re la t ions) allows us to conclude that the only s u r ­viving contribution to the r e s idue function (9) is the polynomial t e r m in Eq. (10), i . e . ,

m*)=TA<i2)B~1Rn. (11)

F o r the case under cons idera t ion , c u r r e n t - a l g e b r a sum rule Eq. (8) ver i f ies th is behavior with N=l zndR(q2)^l.

The s a m e considera t ions can be applied t o / l g - x( + ) . However, he re the fixed-pole contribution is not

expected to be the leading t e r m in the h igh-energy limit. T h e r e a r e Regge poles lying above the point J = 0 at t = 0. The F r o i s s a r t - G r i b o v formula cannot be naively used down to J ^ 0 to give us a r e p r e s e n ­tation of the res idue corresponding to that of Eq. (9). This si tuation can be saved by consider ing the amplitude with i ts leading Regge contributions subt rac ted (in the sense of the f ini te-energy sum rule) . F o r the reduced ampli tude, the F r o i s s a r t - G r i b o v definition is valid down to J^0 and the a rguments p resen ted in the previous pa r ag raph may be c a r r i e d through. One r eaches the s a m e conclusion about the q2 dependence of R{q2) as before.

Next, cons ider the f ie ld- theory model of Bronzan et a l . 1 It cons is t s of self-coupled s c a l a r pa r t i c l e s Thus , except for the isospin indices , it is essent ia l ly the cp3 theory. Without going into deta i ls (which can be found in Ref. 1) we wr i te the ^-channel (y{y2-~(p(p) pa r t i a l -wave ampli tudes in the following form:

\ix/<f,9i 2 „ 2

<*2 > ,m2)=IXlX/{t,q2,q2,m2,m2)

Jdk2dk22IXiX/(t,q2,q2

2,k2,k: 2 )p-l(m2,k12,k2

2)TJ(t,kx 2 k2,m2,m2) (12)

(b) (c) (d)

FIG. 1. Unitary diagrams in the q2 variable for R (q2).

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P, r2

FIG. 2. Virtual Compton amplitudes in the field-the­ory model, Eq. (12).

VOLUME 24, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 13 APRIL 1970

where ( X ^ ) a r e hel ic i t ies of the photons; IXl x/ cons is t s of al l the i r r educ ib le g raphs ; TJis the ampli tude for (off-shell) (pep s ca t t e r ing ; and p _ 1 , as ide from some t r iv ia l normal iza t ion f ac to r s , is the product of two propaga tor functions of the in te rmedia te (p's. This equation and the re levant k inemat ics a r e i l lus t ra ted in Fig. 2. Since the r ight -hand side of Eq. (12) is l inear mIXl\2

J, any fixed s ingular i ty of I^x/ which is not sha r ed by TJ will show up in the full ampli tude F Xl x/. F u r ­t h e r m o r e , s ince I\1\2

J is the only q2-dependent factor on the r ight -hand s ide , the q2 dependence of any fixed-pole t e r m in the full ampli tude can be d i rec t ly in fer red from that of the i r r educ ib le pa r t I\1\2

J- Thus we will concent ra te on the i r ­reducible g raphs . Jus t as in the case of the J= 1 pole in the i s o s p i n - a n t i s y m m e t r i c ampl i tude, 1 the Born d i a g r a m s genera te &J = 0 fixed pole in the i s o s p i n - s y m m e t r i c p a r t of I^-/. This t e r m is independent of q2 and q2

2, and it behaves as s~2

for l a rge s. But, unlike the an t i symmet r i c c a s e , a number of the s e c o n d - o r d e r d i a g r a m s in IXt-i can also behave1 0 asymptot ical ly as s~2 and s ~ 2 l n s , again ref lect ing the fact that t he r e a r e moving poles lying above the J = 0 point.1 1 How­eve r , we note that (i) the coefficients of these s~2

t e r m s a r e also independent of q2 for l a rge s and (ii) these t e r m s do not cancel the Born -d i ag ram contribution. We conclude, t he r e fo re , that the J = 0 fixed pole in Ih-x

J, and thus also in the full ampl i tudes Flr./f is independent of q2.

The p r e s e n c e of a J = 0 fixed pole in the i sosp in-s y m m e t r i c v i r tua l Compton ampli tude has d i rec t consequences for the calculat ion of the e l e c t r o ­magnet ic m a s s differences via the Cottingham formula . The evidence for the p r e s e n c e of this pole at q2 =0 , 4 together with the lack of s u c c e s s in recent a t tempted calculat ions of the n-p m a s s difference without fixed poles , 1 2 have led to the belief13 that such a fixed pole is indeed re levant , at leas t for the 1=1 m a s s differences. However, if the res idue function of this J=0 fixed pole is a polynomial function of q2, as suggested h e r e , the contribution of this t e r m to the m a s s difference must d iverge. 1 2 (The divergence is logar i thmic if the q2 dependence is taken to be that suggested by the ba re Born t e r m . ) It s e e m s , t he re fo re , b a r r i n g unforseen cancel la t ions , that the addition of a f ixed-pole t e r m is not the solution to the d i ­l e m m a confronting this approach to the e l e c t r o ­magnet ic mass -d i f fe rence prob lem.

Final ly , we d i scuss briefly the fixed poles at wrong-s igna tu red , nonsense points . A pole at J = 1 in the i s o s p i n - s y m m e t r i c ampli tude flt

of this type. In the cp3 model , s ince the Born t e r m alone is respons ib le for the J = l fixed pole in the i r r educ ib le pa r t 1^-/, the q2 dependence of th is wrong-s igna tured pole is again a t r i v i a l one.1 4 On the o ther hand, the poss ib le p r e s e n c e of this pole in the cor responding s t rong a m p l i ­tudes 1 5 r e n d e r s our a rguments leading to the van­ishing of the d i spe r s ion in tegra l in Eq. (1) inva l ­id. We note, however , that even in the absence of wrong-s igna tured fixed poles in s t rong a m p l i ­tudes , the v i r tua l Compton ampli tude can s t i l l have such poles . 1 6 This means that the s u b t r a c ­tion t e r m in Eq. (10) is in gene ra l p r e s e n t ; t h e r e ­fore the res idue function R(q2) behaves as a poly­nomial for la rge q2.

The p r e s e n c e of a wrong-s igna tu red pole at J = 1 for the 1 = 0 Compton ampli tude is n e c e s s a r y 2

to r e s t o r e the Pomeranchuk pole contr ibut ions to fit-i at £ = 0, thus maintaining a nonvanishing h igh-energy photo absorpt ion c r o s s sec t ion , as exper imenta l ly observed . The two poles of the p a r t i a l wave ampli tude appear in mult ipl icat ive form 2 :

lim P(g2,*) (J-l)[j-aP(t)]

(13) r = o

(+) i s

According to our a rgumen t s , j3 contains a t e r m which is polynomial in q2.17 If we again u s e the b a r e Born t e r m as a guide for th is q2 dependence, we can easi ly conclude that the e-p i n e l a s t i c -sca t t e r ing s t r u c t u r e function vW2 (the absorp t ive pa r t of vflt-i) must behave, at l a rge v and q2, as a constant in both va r i ab l e s . This a g r e e s with the exper imenta l resu l t . 1 8

In conclusion, we have p r e sen t ed s e v e r a l a r g u ­ments for the polynomial q2 dependence of fixed pole t e r m s in Compton ampl i tudes . None of these a rguments const i tu tes a proof of the s ta ted q2 d e ­pendence. However, taken as a whole they do suggest a reasonab le phys ica l p i c tu re : The fixed po les , being assoc ia ted with local ized i n t e r a c ­t ions of c u r r e n t s , a r e insens i t ive to the de ta i l s of s t rong in te rac t ions and consequently do not have any s t r u c t u r e in the i r q2 dependence. Appli­cations of th is idea to o ther p r o c e s s e s , as well as genera l iza t ion to o ther types of s ingu la r i t i e s (for ins tance , Kronecker de l tas ) , d e s e r v e fur ther investigation.

It is a p l ea su re to acknowledge ve ry helpful d i s ­cuss ions with S. L. Adler and R. F . Dashen. We have also benefited from conversa t ions with C. C r o n s t r o m , Y. Dothan, R. Hwa, and J. Man-dula. We would like to thank Dr. C a r l Kaysen for hospital i ty at the Inst i tute for Advanced Study.

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VOLUME 24, NUMBER 15 PHYSICAL REVIEW LETTERS 13 APRIL 1970

F. E. Low, I. J . Muzinich, Schwarz, Phys. Rev. 160, 1329

Freedman, Phys. Rev. 182, 1628

* Re search sponsored by the National Science Found­ation, Grant No. GP-16147.

*J. B. Bronzan, I. S. Gerstein, B. W. Lee, and F. E. Low, Phys. Rev. 157, 1448 (1967), and references cited therein.

2H. D. I. Abarbanel, S. Nussinov, and J. H (1967).

3G. C. Fox and D. Z (1969).

4M. Damashek and F . J . Gilman, to be published; also M. J . Creutz, S. D. Drell, and E. A. Paschos, Phys. Rev. 178, 2300 (1969).

5It is interesting to note that without this assumption Eq. (6) already implies the existence of this fixed pole at q2 = 0 with residue R (q2 = 0) = 1, t2 being free of kine­matic singularities.

6The M l amplitude /+_ has three sets of singularities generated by unitarity in the channels corresponding to the s , u, and q2 variables. (These variables are relat­ed by s +u=2m2-2q2.) Because of s-u crossing symme­try, however, the function R receives equal contribu­tion from the s - and u-channel absorptive parts, con­sequently one can forget about the w-channel contribu­tion in (9). Since the only remaining singularities come from s and q2 channels, it is reasonable to assume that one only encounters normal thresholds in q2 when s is integrated over. Rigorous proof of the analyticity prop­erties in q2 is rather difficult. We have checked that the stated analyticity holds to each order in perturba­tion theory for ^-channel ladder diagrams.

7The possible usefulness of #2-dispersion relations in this context was first suggested to us by R. F. Dash-en.

8R. F. Dashen and S. Y. Lee, Phys. Rev. Letters 22, 366 (1969), and references cited therein. The argu­ment for absence of fixed poles is not valid if there is an "elementary" spin-1 boson that can couple to the photon. There is , of course, no experimental evidence

for such a particle. 9D. Tompkins et ah, Phys. Rev. Letters 23., 725

(1969). 10P. G. Federbush and M. T. Grisaru, Ann. Phys.

(N.Y.) 22, 263, 299 (1963); G. Tiktopoulos, Phys. Rev. 131, 480, 2373 (1963).

11 The precise J-plane nature of the s~2 terms from the second-order diagrams cannot be ascertained with­out the explicit evaluation of the Froissart-Gribov for­mula for the partial-wave amplitude at large Re J and continuing down to «/ = 0, the formula itself being diver­gent near J = 0. For our purposes, however, it is suf­ficient to know that any such terms are necessarily in­dependent of q2. (These terms, together with other diagrams, may sum up to give the Regge poles men­tioned in the text.)

12M. Elitzur and H. Harari, Ann. Phys. (N.Y.) 56, 81 (1970); R. Chanda, Phys. Rev. ^88, 1988 (1969); D . J . Gross and H. Pagels, Phys. Rev. 172, 1381 (1968).

13See, for example, F . J . Gilman, in Proceedings of the International Symposium on Electron and Photon Interactions at High Energies, Liverpool, England, September 1969 (unpublished).

14The difference between the symmetric and antisym­metric cases lies in the different Regge poles in the strong part T* in Eq. (12): P,P\ A2 in the former, p in the latter.

15S. Mandelstam and L. L. Wang, Phys. Rev. 1<30, 1490 (1967).

16That is, there are two independent mechanisms for generating wrong-signatured fixed poles in Compton amplitudes; see Ref. 2.

17This is to be contrasted to the contributions from "ordinary" Regge poles. The latter, being closely r e ­lated to resonance production, have form-factor—type rapid-falloff behavior in q2. See H. Harari, Phys. Rev. Letters 22, 1078 (1969).

18E. D. Bloom et al., Phys. Rev. Letters 23., 930, 935 (1969).

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